Vıbratıon energy harvestıng from a raılway vehıcle usıng commercıal pıezoelectrıc transducers

MARMARA UNIVERSITY INSTITUTE FOR GRADUATE STUDIES IN PURE AND APPLIED SCIENCES VIBRATION ENERGY HARVESTING FROM A RAILWAY VEHICLE USING COMMERCIAL PIEZOELECTRIC TRANSDUCERS NAZENİN GÜRE MASTER THESIS Department of Mechanical Engineering Thesis Supervisor Prof. Dr. Erturul TACGIN Thesis Co-Supervisor Assist. Prof. Dr. Alper ŞİŞMAN ISTANBUL, 2017 MARMARA UNIVERSITY INSTITUTE FOR GRADUATE STUDIES IN PURE AND APPLIED SCIENCES VIBRATION ENERGY HARVESTING FROM A RAILWAY VEHICLE USING COMMERCIAL PIEZOELECTRIC TRANSDUCERS NAZENİN GÜRE 524611017 MASTER THESIS Department of Mechanical Engineering Thesis Supervisor Prof. Dr. Erturul TACGIN Thesis Co-Supervisor Assist. Prof. Dr. Alper ŞİŞMAN ISTANBUL, 2017 ACKNOWLEDGEMENT From the beginning of my thesis subject and advisor selection to the final step, I have been through many good, hard, challenging and interesting layers of life, in which I have been wounded; yet, with the help of many; managed to reborn countless times. Thus, this whole procedure extended my acknowledgements to fill a full page. Throughout the other face of the world, there had been times that I was so jealoused as much as to be targeted against in order to block my productivity and freedom, undoubtedly, all made me feel so lonely and isolated. At those very in-need times, there had been many special people and corporations such as: A. Lengyel, M.C. van Schoor, B. Durant, S. Hanly, M. Schuller, A.Rousson and related Midé Technology Corporation, which is the greatest inspiration of this study and my start-up firm Enhas Energy Systems R&D, and their web-site, products, blogs, shared materials, advices are the main contributor of this thesis; Roy Freeland who is kind enough to deliver a selected sample of vibration energy harvester to me and supportive enough to offer scientific advice, and related Perpetuum Ltd.; especially Prof. Dr. Abdulkerim Kar and Prof. Dr. Erturul Tacgın who always protected, supported and stood for me, my achievements and each step of the research I ever attempt; Rahman Altın who helped me to deliver my test equipments from CA, A.A.Doner, my passed away budgie Bulut Kartopu, my relatives: Ali O. Gure and Sevin Osmay, my treasured: father Ataman Güre, intellectual mother Nakiye Güre, and wise brother Zeynel Abidin Yürür. Having a chance to be in contact with Them makes me so lucky that -though it will not be the case- even if there exists no one, I would always conclude as having these bonds with these essential building blocks of the world is certainly a blessing and far enough for me. Therefore, I conclude with thanks for every experience that ever enlightened me. Finally, I acknowledge financial supports of Scientific and Technological Research Council of Turkey (TUBITAK) 2210-C and Technological Entrepreneurship Industry Support (TGSD) by T.C. Ministiry of Science, Industry and Technology (MoSIT). i TABLE OF CONTENTS ACKNOWLEDGEMENT .............................................................................................. i TABLE OF CONTENTS ............................................................................................... ii ABSTRACT ................................................................................................................... vi ÖZET ............................................................................................................................. vii ABBREVIATIONS...................................................................................................... viii SYMBOLS........................................................................................................................ i LIST OF FIGURES ........................................................................................................ ii LIST OF TABLES ......................................................................................................... iv 1. INTRODUCTION ...................................................................................................... 1 1.1. Problem of the Thesis ............................................................................................ 1 1.2. Literature Review .................................................................................................. 1 1.2.1. Comparison of Piezoelectric and Electromagnetic Generators ................ 2 1.2.2. Classic HEH Systems ............................................................................... 3 (a) Fixed-Frequency Classic HEHs [not: 1.4.1.=(a)].................................................... 3 (b) Broadband Classic HEHs ........................................................................................ 6 (c) Overall Classic HEHs Comparision ..................................................................... 9 1.2.3. Novel HEH Systems ............................................................................... 13 (a) Fixed-Frequency Single-Source Powered HEHs ............................................... 13 (b) Broadband Single-Source Powered HEHs ......................................................... 14 1.3. Introduction to Piezoelectricity and Classic Methodologies for Output Power Generation............................................................................................. 16 1.4. Analytical Formulation Methods for the Power Generation-a Review ........ 23 1.4.1. Hehn and Manoli’s proposed expressions for power generation [1, 4– 6, 9]: ................................................................................................................... 26 1.4.2. Khalatkar et al.’s proposed expression for power generation [15]: ........... 29 1.4.3. du Toit’s proposed expression for power generation [3, 7, 11, 16] ............ 30 1.4.4. Erturk’s suggestion on mass correction on lumped parameter model and optimum load resistance for the maximum output power ii generation [11] : ................................................................................................ 35 1.5. Objective and Scope of the Thesis ...................................................................... 37 CHAPTER 2: DERIVATION OF EQUATIONS OF CLASSICAL AND NOVEL VIBRATION ENERGY HARVESTERS.................................................... 38 2.1. Introduction ......................................................................................................... 38 2.2. Lumped Parameter Modelling of Stacked PiezoelectricEnergy Harvester ......... 40 2.3. Lumped Parameter Modelling of Cantilever Piezoelectric Energy Harvester .... 42 2.4. Derivations of Equations for a Novel Energy Harvester ..................................... 44 CHAPTER 3: PRELIMINARY EVALUATION OF TRAIN VIBRATION DATA AND THE SELECTED VEHs ........................................................................ 47 3.1. Introduction ......................................................................................................... 47 3.2. Evaluation of Train Vibration Acceleration Data................................................ 47 3.2. Evaluation of Selected VEHs to Validate the Mathematical Model ................... 52 CHAPTER 4: PERFORMANCE EVALUATIONS OF CLASSICAL AND NOVEL HARVESTERS .............................................................................................. 54 4.1. Introduction ......................................................................................................... 54 4.2. Performance Evaluation of Tuned Single DOF Energy Harvesters .................... 55 4.2.1. Tuned Performance of Single DOF Energy Harvester at 7 Hz ........................ 58 4.2.2. Tuned Performance of Single DOF Energy Harvester at 21 Hz ...................... 58 4.2.3. Tuned Performance of Single DOF Energy Harvester at 80 Hz ...................... 59 4.3. Performance Evaluation of Tuned Novel Two DOF Energy Harvester Array .................................................................................................................. 60 CHAPTER 5: RESULTS AND DISCUSSION .......................................................... 62 5.1. Introduction ......................................................................................................... 62 5.2. Proposed approach Validation, Sensitivity Analysis and Correction .................. 62 CHAPTER 6: CONCLUSION .................................................................................... 73 6.1. Conclusion ........................................................................................................... 73 6.2. Future Research Recommendations .................................................................... 74 7. REFERENCES ......................................................................................................... 77 ÖZGEÇMİŞ .................................................................................................................. 84 APPENDIX 3A: CHAPTER 3: Preliminary Evaluation of Train Vibration Data and the Selected VEHs ANALYSIS OF TRAIN VIBRATION SIGNAL iii DATA AND DETERMINATION OF THE TUNING FREQUENCIES [63– 68, 70–75, 84, 85] ........................................................................................................... 86 3A.1. For 2 DOF VEH, Selected Train Vibration Signal Characteristics and Analysis: Lateral acceleration data at the middle train body ............................. 87 FFTs of acceleration and displacement signals .......................................................... 92 PSDs of acceleration FFT and displacement FFTs: ................................................... 97 Welch Power Spectral Estimator for Acceleration and displacement signals: ......... 102 RMS Values and Time-varying Plots of raw Acceleration signal, filtered Acceleration and Disp: ..................................................................................... 105 3A.2. For 3 DOF VEH, Selected Train Vibration Signal Characteristics and Analysis Vertical acceleration data at the middle train body (Interpolated, fs=600Hz) ......................................................................................................... 109 This program high pass (4,5 Hz) and low pass (100 Hz) filters and integrates acceleration signal to displacement signal. Plots time varying signals, FFT, Welch PSD and RMS of Acceleration and Displacement signals. ......... 110 FFTs of acceleration and displacement signals ........................................................ 114 FFT of Position signal: ............................................................................................. 116 ZOOMED FFT of position plot:............................................................................... 117 PSDs of acceleration FFT and displacement FFTs: ................................................. 119 PSD of position signal calculation for position: ....................................................... 121 ZOOMED PSD of FFT of the position signal Plot ................................................. 122 Welch Power Spectral Estimator for Acceleration and displacement signals: ......... 124 PSD of Welch spectral of Low Pass filtered Acceleration Signal Plot: ................... 125 Welch PSD of Displacement : .................................................................................. 126 RMS Values and Time-varying Plots of raw Acc signal, filtered Acc and Disp: .... 127 3A.3. Previously Investigated Train Vibration Signal Characteristics and Analysis Vertical train vibration at the third train bogie .................................. 132 3A.4. Previously Investigated Train Vibration Signal Characteristics and Analysis Lateral acceleration data at the middle train body ............................ 143 Appendix 3B Datasheet Details of the Evaluated Midé Volture Piezoelectric VEHs ............................................................................................................................ 147 APPENDIX 4 MATLAB Programs for Evaluation of Mathematical Model of iv Energy Harvesters with Various Natural Frequencies ........................................... 155 PPA-2011, Clamped at 0 and Input Vibration is at 60 Hz and 0.25g ....................... 155 Called Function for PPA-2011, Clamped at 0 and Input Vibration is at 60 Hz and 0.25g .......................................................................................................... 156 APPENDIX 5A Chapter 5: Results and Discussion MODAL ANALYSIS of PPA 1001 & 1021 in SAP2000 and Ansys, and VIBRATION BEHAVIOUR of PPA 1021 ..................................................................................................................... 158 5A.1. ANSYS SIMULATIONS ............................................................................... 158 5A.1.1. RESPONSE TO THE HARMONIC BASE INPUT ................................... 158 The first mode:.......................................................................................................... 158 The 2nd mode: ........................................................................................................... 159 The 3rd mode: ............................................................................................................ 160 The 4th mode: ............................................................................................................ 161 The 5th mode: ............................................................................................................ 163 The 6th mode: ............................................................................................................ 164 Note on 5th and 6th modes: ........................................................................................ 166 5A.1.2. RESPONSE TO THE RANDOM VIBRATION INPUT ............................ 166 5A.2. SAP2000 SIMULATIONS ............................................................................. 169 APPENDIX 5B Chapter 5: Results and Discussion VALIDATION & SENSITIVITY ANALYSIS ....................................................................................... 177 5B.1. VALIDATING MATLAB CODE OUTPUT WITH EXPERIMENTS ......... 177 5B.1.1. Determination of the Optimum Load ........................................................... 200 v ABSTRACT VIBRATION ENERGY HARVESTING FROM A RAILWAY VEHICLE USING COMMERCIAL PIEZOELECTRIC TRANSDUCERS This thesis study covers classic vibration spectral signal analysis, piezoelectricity and energy harvesting methodologies in literature, and offers a new perspective in all those field as well as presenting an accurate novel theoretical estimation of power output methodology for Midé’s PPA-1011 and 2011 commercial piezoelectric vibration energy harvester (VEH) models, in addition to proposed alternatives of novel 2 DOF and 3 DOF configurations that can be adjusted relative 2 and 3 dominant frequencies of input excitation. Investigations begin with the selection and analysis of train vibration as an input dissipated energy source. Among 16 vibration measurements of train test runs, spectrum analysis of 4 vertical and lateral acceleration data having maximum vibration amplitudes are analyzed, and dominant frequencies of acceleration and displacement excitations are detected for each. As a distinct consideration, it is seen that input displacement dominant frequency is suitable for efficient energy harvesting and thus, it is also taken as tuning frequency of Midé piezoelectric VEHs (PPA-1011 and 2011). Last of all, existing estimation procedures in literature on piezoelectric energy generation, single DOF VEH modelling and optimum load are covered in detail and new methodologies with high precision that is inline with Midé’s experiment data are expressed in this research. Additionally, to better visualize and investigate the vibration behavior, ANSYS (PPA-1021) and SAP2000 (PPA-1011) simulations are analyzed. Finally, further models and research complementary ideas that arose from the conducted examinations are represented. vi ÖZET TİCARİ PİEZOELEKTRİK GÜÇ DÖNÜŞTÜRÜCÜLERİ KULLANILARAK TREN GİRDİSİNDEN TİTREŞİM ENERJİ HASADININ İNCELEMESİ Bu tez çalışması kapsamında klasik titreşim spectral analizi, piezoelektik ve enerji hasadı metodolojileri literatürde yer aldığı üzere incelenmiş olup, tüm bu alanlarda yeni perspektifler sunulmuş olmanın yanı sıra Midé’nin PPA-1011 ve 2011 ticari piezoelektrik güç dönüştürücü, güç çıktısının yüksek doğrulukta hesaplanmasına yönelik özgün teorik hesap yöntemleri aktarılmış olup, tüm bunlara ek olarak özgün 2 ve 3 serbestlik dereceli, ilgili 2 ve 3 dominant frekanslara ayarlanabilen hasat sistemleri önerilmiştir. Tez kapsamındaki incelemeler, hali hazırda yayılmakta olan hasat edilecek titreşim kaynağının tren titreşimi olarak belirlenmesi ile başlamıştır. Test sürüşleri esnasında alınan 16 farklı noktadaki titreşim verilerinden genliği en yüksek olan 4 adet yatay ve dikey titreşim verisi seçilmiş ve spectral analizleri tamamlanılarak, ivme ve pozisyon titreşim girdilerinin dominant frekansları bulunmuştur. Literatürde yer alan yöntemlerden ayırd edici olarak, yer değiştirme dominant frekanslarındaki enerjinin de enerji hasadına uygun ve verimli düzeyde olduğu gözlemlenmiş olup, Midé piezoelectric enerji hasat sistemlerinde (PPA-1011 and 2011) ayarlanılacak olan doğal frekans değerlerine eklenilmiştir. Son bölümde, piezoelektrik enerji üretimi, tek serbestlik dereceli titreşim enerji hasat sistemlerinin modellenmesi ve optimum rezistans bulunmasına yönelik literatürdeki mevcut hesaplama prosedürleri uygulanmış ve detaylı bir şekilde incelenmiştir. Bunun devamında, sunulan yeni methodolojinin neticelerinin Midé’nin yayınladığı deney bulguları ile uyumunun, tez çalışmasında yürütülen hesaplama methodolojisinin doğruluğunu desteklemiştir. Ek olarak, titreşim davranışını daha iyi gözlemleyebilmek ve inceleyebilmek için ANSYS (PPA-1021) ve SAP2000 (PPA-1011) simulasyonları analiz edilmiştir. Son olarak, gelecekte incelenmesi planlanan modeler ve bu tez çalışmasından doğan yeni inceleme konuları sunulmuştur. vii ABBREVIATIONS a-Si : Amorphous Silicon AGS : Automatic Generating System DOF : Degree of Freedom DPE : Direct Piezoelectric Effect EHer : Energy Harvester EH : Energy Harvesting EM : Electromagnetic EMHs : Electromagnetic Energy Harvesters FFT : Fast Forier Transform HAWT : Horizontal Axis Wind Turbine HEHs : Hybrid Energy Harvesters HRTHs : Hybrid Rotary-Translational Harvesters MEMs : Microelectromechanical Systems PAGV : Power Augmentation Guide Vane PE : Piezoelectric PEHs : Piezoelectric Energy Harvesters PET : Polyethylene Terephthalate PV : Photo-voltaic PPA : Piezo Protection Advantage (Midé Volture Products) RF : Radio Frequency RMS : Root Mean Square PSD : Power Spectrum Density SME : Shape Memory Effect viii TENG : Triboelectric Nanogenerator VAWT : Lateral Axis Wind Turbine VEH : Vibration Energy Harvester ix SYMBOLS ekp : Piezoelectric Coefficient  ikS : Clamped Permittivity ( i,p,q are directions ) n : Natural Frequency  : Efficiency of VEH power generation  : Electromechanical Coupling Term -31 mode  : Electromechanical Coupling Term -33 mode µamplitude : Amplitude Correction Factor µmass : Mass Correction Factor µ: Correction Factor  2 is constant parameter  FF : Force Factor, also called Proportionality Constant (Gp ) i LIST OF FIGURES Figure 2.1. Classic energy HEH configurations are categorized and illustrated ........ 5 Figure 2.2. Classic HEH prototypes in test setup [25, 36, 37]. ................................... 6 Figure 2.3. Two (left) and four (right) pole magnet arrangements on classic HEH designs [20, 35]. ............................................................................................................... 6 Figure 2.4. Broadband HEH by Shan et al. (left) [15] and Ping Li et al. (right) [4, 29]. On Shan et al.’s prototype NdFe35 magnet and PZT-5H ceramics are used as EM and PE transducer components. ........................................................................................ 9 Figure 2.5. Halim et al.’s novel HEH design schematically represented in (A), fabricated components are listed in (B) and the assembled prototype is seen in (C) [14]. 14 Figure 2.6. The technical drawing of the Castagnetti’s HEH prototypes (A), Belleville spring scheme (B) and HEH scheme with two magnets (C) [39]. ................. 15 Figure 2.7. Karami and Inman’s design illustration and prototype [46, 47]. ............ 16 Figure 2.1.1. Schematic illustration of stacked piezoelectric energy harvester (a), cantilever beam energy harvester in 33 (b) and 31 (c) operation mode, and their common lumped parameter model (d)............................................................................ 39 Figure 2.1.2. (a) Piezoelectric material operated in (a) 33 and (b) 31 mode.[54] ......... 39 Figure 2.2.1. Model of 1DOF harvester. Total mass, effective spring coefficient and mechanical damping as well as input base excitation and relative displacement of the proof mass are represented. ............................................................................................ 40 Figure 2.3.1. Beam of piezomaterial loaded in 31 mode of the piezoelectric material. 43 Figure 2.4.1. Physical model (a) and mathematical model (b) of a novel vibration energy harvester. ............................................................................................................. 45 Figure 3.2.1. Time-varying low pass filtered train acceleration signal (m/s2, sec). ...... 49 Figure 3.2.2. Single-sided low-pass filtered acceleration amplitude spectrum (m/s2, Hz). ........................................................................................................................................ 49 Figure 3.2.3. Welch Power Spectrum of Raw Train Acceleration Signal (dB/Hz, Hz). Having half-power bandwidth of 28 Hz between 72 Hz and 100 Hz. .......................... 50 Figure 3.2.4. Time-varying train displacement signal (mm). ........................................ 50 ii Figure 3.2.5. Single-sided displacement amplitude spectrum (mm, Hz)....................... 51 Figure 3.2.6. Welch power spectrum of train displacement (dB/Hz, Hz). Having halfpower bandwidth of 7 Hz between 2.3 Hz and 9.3 Hz. .................................................. 51 Figure 4.2.1. Mide VEH possible increased length arrangement to tune low frequencies via clamp bar [80]. .......................................................................................................... 57 Figure 4.3.1. The scheme of novel two dof energy harvester........................................ 61 Figure 5.2.1. Vibration response of PPA 1021-tuned to 22 Hz- at resonance under random vibration acceleration amplitude of 1g, units are in mm. .................................. 63 Figure 5.2.2. Sensitivity analysis for output power at the input frequency of 21 Hz, tip mass of 25.3 g and the effective mass of 0.614 g. .......................................................... 66 Figure 5.2.3. Sensitivity analysis for output power at the input frequency of 60 Hz, tip mass of 2.7 g and the effective mass of 0.614 g. ............................................................ 67 Figure 5.2.4. Sensitivity analysis for output power at the input frequency of 146 Hz in average, no tip mass and the effective mass of 0.614 g.................................................. 68 Figure. 6.2.1. W. Mason’s mechanical filter and its electric equivalent. ...................... 76 Figure. 6.2.2. Modified 3 DOF VEH inspired from W. Mason’s mechanical filter...... 76 Figure. 6.2.3. Lumped Model of the 3 DOF VEH for future examinations. ................. 76 iii LIST OF TABLES Table 1.1. The overall comparison of the reviewed classic HEH systems in parts 3.1 and 3.2. .................................................................................... 11 Table 4.2.1. The piezoelectric, unique and common piezo patch and transducer properties for the chosen PPA-1011 at -6.0 and 0 clamp locations ....... 56 Table 4.2.2. The RMS output powers, voltages and efficiencies as a response to input excitation at tuned frequencies of 7 Hz, 21 Hz and 77 Hz. .......... 60 Table 5.2.1. Confidence interval table of all investigated cases for PPA-1011 power output estimation. Selected damping and correction factors are listed along with the optimum load investigation in comparision with the Midé’s experimental loads....................................................... 71 Table 5.2.2. Confidence interval table of all investigated cases for PPA-2011 power output estimation. Selected damping and correction factors are listed along with the optimum load investigation in comparision with the Midé’s experimental loads....................................................... 72 iv 1. INTRODUCTION 1.1. Problem of the Thesis In this study, it is aimed to find a solution to a one common problem in literature that is being limited with the resonance frequency of the vibration harvester devices. In this perspective, device model configurations, allowing multiple frequencies due to multiple degrees of freedom, are researched. In early thesis research process, investigation of the possible low-pass, high-pass and band-pass passive electric filters and their mechanical equivalent mechanical filter models derived from force voltage and force current analogies are studied. However, inconsistent and complex transfer functions lead already existing mechanical filter design adaptation that is used as electrical wave filters in telephone systems in 1940s. Proposed mechanical filter model also resembles mechanical suspension models and these simplified mechanical filter configurations with single, two and three degree(s) of freedom (DOF) are investigated in separate and coupled arrangements for each DOF model. Moreover, it is also aimed to adopt Midé Volture piezoelectric vibration energy harvesters into the studied configurations due to their advanced power generation potentials and easy tuning options. It is seen that multiple mode frequencies allow to vibrate multi-modes thus, suitable for the vibratory sources having multiple dominant vibration frequency. In the light of all these findings, proposed models integration to train is further researched. For this reason, train dominant frequencies are evaluated and the models are selected and designed according to fit mode frequencies with the dominant input excitation frequencies. 1.2. Literature Review Ever since the beginning of the industrial age, being independent from man and animal power sources, especially at greater energy levels, was the greatest innovation. As the time passes by, the more technological improvements occur along with wireless networks, the more devices are in our lives thus, elevating the quality of life, production and work. In spite of the efforts to decrease the energy input of the electronic devices, 1 this ever increasing demand on energy surprisingly takes us to seek using existing power sources like human movement, known as kinesiology, similar to the energy source before industrial age [1–3]. In this millennium, the methodologies to harvest existing dissipated powers not only supply input energy to our sophisticated devices, but also contribute the current technological researches and developments. Among energy harvesting systems, one of the innovative research trend is on hybrid energy harvesters [4, 5]. Obeying the first law of thermodynamics, conservation of energy implies that the existing and dissipated power sources can be scavenged and transduce into usable electrical energy [1]. Up until recently, energy harvester (EH) need is arose by the dominant use of electronic devices, biosensors, human, structural and machine health monitoring, and wireless sensor nodes [6–9]. Single harvester generator or harvesting single power source, also known as stand-alone EHs, may produce low output powers to supply energy to the system. For sufficient energy feed to these vast varieties of applications, hybridization of EHs takes place to increase the limited energy generation of stand-alone EHs [10–23]. Harnessing multiple power sources or combining multiple generators for energy extraction in a single unit is called “hybrid energy harvesting or multimodal energy harvesting” [24–27]. 1.2.1. Comparison of Piezoelectric and Electromagnetic Generators EM energy harvesters (EMHs) generate power based on faraday’s law of induction, which equates the time derivative of flux to the electromotive force. As the scale goes down to microscopic level, the decreased coil area results smaller magnetic flux. On the other hand, quasi-static (ultra-low-frequency) movements increase the time intervals. Thus, both factors lead electromotive force to approach to zero. Apart from the fabrication boundaries of coil diameters and turns, theoretically EM harvesters are bound to be limited at low speeds [28]. Thus, EM harvesters perform better at high frequencies and PE harvesters outperform at low frequencies [8]. Additionally, at microscale level, EMH output voltage generally stays lower than the need to power 2 devices [17]. As a result, piezoelectric and electrostatic harvesters are more suitable for microscale applications, while electrostatic systems hold greater advantage due to the ease of integration to microelectromechanical systems (MEMs) [27]. Similar to EMHs, PE harvesters (PEHs) do not require voltage source while electrostatic generators require separate voltage source and more difficult in practice, and in contrast to EMHs, PEHs produce sufficient output voltage but at low current level [8, 19, 27]. Among these three types of transducers, piezoelectric generators are the simplest ones in terms of required components, transducer geometries and directly converting mechanical energy to voltage output [9]. In addition to PEHs, at macroscopic level, EMHs also provide simplicity in geometry, design and production [19]. In conclusion, PEHs are applicable for micro-, meso- and large-scales, while EMHs are easily manufactured and although they perform better at mesoscale, they are integrable to MEMs. As a result, abundant PE and EM HEH are reviewed and compared in the following sections. 1.2.2. Classic HEH Systems While tremendous amounts of multimode energy harvesters (EHs) are possible, there exist such PE and EM combination that takes the greatest research and development interest and turns out to be classic. As listed in Figure 1, these are composed of piezoelectric plate or patch attachment on Euler-Bernoulli beam and either magnetic or coil tip mass is surrounded by respective coil or magnets to achieve faraday law of induction [29]. In this chapter, PE unimorph and bimorph structures; comparison of rectangular and trapezoidal beams; four and two poles magnet configurations and comparisons; serial, parallel and isolated connections of PE and EM transducers in HEHs; fixed-frequency applications; and rarely studied broadband HEH designs are reviewed. (a) Fixed-Frequency Classic HEHs [not: 1.4.1.=(a)] Figure 2.1 and 2.2 shows the classic HEH designs and fabrications, respectively. As a brief summary, in 2008, Wischke and Woias researched PE layers on unimorph and bimorph cantilevers with rectangular and trapezoidal layouts in HEH. It is seen that 3 trapezoidal shape is not superior in terms of power generation and its fabrication is more complicated. Since unimorph HEH has greater tip velocity, EM transducer generates greater power. In contrast to unimorph design, bimorph PE part produces greater output than EM part. Upon this contrary output, authors suggest using greater tip mass (magnet) to reduce EM coupling [11]. Becker et al. begin to test HEH prototype in Figure 2.1. (C) and research further for its adaptation into synchronized switch harvesting interface [30]. Xu et al., both theoretically and experimentally analyzed PE and EM HEH. Theoretical optimum output power is 1.02 mW at 77.8 Hz and experimental value is 0.845 mW at the resonance frequency of 66 Hz under the vibration acceleration of 9.8 m/s2. Respective to single PE and EM transducer, output powers are 667 mW and 0.32 mW at 9.8 m/s2 and 66 Hz. Xu et al. proves that presented HEH generates greater power than single EHs [31]. Ali et al., investigated total power outputs of PE and EM harvesters in serial, parallel connection and separately. As seen in Figure 1 (C), the HEH has approximately 1000 turns and 4 magnets (25 x 10 x 5 mm3) with opposite polarization at cantilever tip. At the fixed input frequency of 76.2 Hz, total generated power is the highest when PE and EM transducers are isolated. While PEH generates 27.56 mW, EMH generates the lowest power output. Besides, single PEH output power is 3 times greater than serial connected HEH and parallel connected HEH generates 3 times more than single EM transducer [32]. As a recent study, Xia et al. not only investigates the classic HEH but also compared the performances of HEH and EMH (Figure 1. (C)). Throughout the experiments best HEH case generated the output power of 2.26 mW with 41% efficiency at 23.3 Hz and 0.4 g input excitation. and thus, greater performance compared to EMH alone. HEH not only owns greater output power and efficiency, but also enables broadband operation [33, 34]. In contrast to these findings, Sang and Shan et al.’s experiments resulted that HEH has the almost the same resonance frequency with the PEH. Sang et al., considered the valuable Classic HEH configurations and yet, similar to Figure 1 (C) with only difference of having lateral coil placement on both sides of magnet is researched for four different cantilever lengths. HEH with cantilever and PE layer respective sizes of 4 50 x 15 x 1 mm3 and 30 x 15 x 0.5 mm3, generated 10.7 mW while EM alone was 5.9 mW at 50 Hz with the acceleration of 0.4 g [21]. Supportively, Shan et al. also reported that HEH produced greater output power of 4.25 mW than single PEH of 3.75 mW at 40.5 Hz and optimum loads. Their design is slightly modified version of Figure 1 (C) and (D). The U-shaped magnet cage is fixed at the beam end and coil was fixed filling the gap in U-shape magnets during vibration [16]. (B) (A) (C) (D) (A) [20, 25, 26, 33–36] and (C) [30, 32, 37] are the most common ones and the tip mass can either be two magnets as in (A), or single magnet surrounded by coils as in (B) [4, 15] and (C), or lateral coil surrounded by magnets as in (D) [20, 35, 38] as well as modified horizontal arrangement [21]. Along with varying PE length on beam, PE unimorph (D) and bimorph (A, B and C) cantilever configurations are also possible [11]. Interface circuit is schematically indicated on (D). Figure 2.1. Classic energy HEH configurations are categorized and illustrated Ab Rahman et al. studied two and four pole magnet arrangements on classic HEH in Figure 1 (A) and (D), respectively (see Figure 3). It is experimentally proved that each HEH transducer with four-pole magnets produce greater output voltage than two pole HEH. When the input excitation is 1 g, generated output powers of four pole type PE and EM parts were 2.3 mW and 3.5 mW at 15 Hz, whereas those outputs were 0.5 mW and 1 mW at 49 Hz for two pole HEH [20, 35]. More detailed comparison of four-, twoand single magnet novel HEH performances are studied by Castagnetti and covered in part 4.2. [39]. 5 Figure 2.2. Classic HEH prototypes in test setup [25, 36, 37]. Figure 2.3. Two (left) and four (right) pole magnet arrangements on classic HEH designs [20, 35]. (b) Broadband Classic HEHs The ever-demanded ideal EH efficiently performs in wide bandwidth. In order to satisfy this demand, many researches have been conducted and still being investigated. These 6 include passive tuning either manually or with linear, non-linear, multi-stable and bandpass harvester structures; and active tuning which may result in negative power outputs due to consumed energy by active parts and advanced electronic networks [26, 40, 41]. Among these efforts, there are some classic HEH design approaches exist. Operation of an EH within wider frequency band is a very significant advantage in EHs. A few studied novel micro scale HEHs are also mentioned in this section for being broadband. Linear Classic HEH: In 2013, Ping Li et al. designed similar to Figure 4 (right) without fixed magnets. They analyzed their linear HEH performance under white noise excitement to model random vibration. Their sensitivity analysis indicated that the power generation of HEH dominantly affected by vibration frequency, damping ratio, coupling coefficients, which widens the bandwidth and increase power output as increases, and load resistances to achieve HEH impedance matching and maximize power generation. While PE load directly proportional to resonance frequency of harvester, EM load has almost no effect. The most efficient performances have the mean power levels of 0.44, 1.93, 4.2 mW at 77.5 Hz [17, 42]. Tunable Classic HEH: Wischke Masur et al. focused on frequency tuning method for HEH exactly shown in Figure 1 (A) and the picture of the protptype is on the left in Figure 2. In this technique, voltage is introduced to the PE layer. Applied voltage changes the stiffness of the generator and relatedly, resonance frequency. By matching input frequency with tuned generator’s resonans frequency, broadband operation is achieved. Electrode length’s effect on tunability is also investigated and found that greater than 10 mm PE beam length, tunable range almost saturates between ~50 to 60 Hz. To invertigate widest tunable range, fabricated HEH cantilever length was 20 mm with the width of 5 mm. Extractable output powers from EM part is 60 W, for PE part with parallel connection is around 200 W and with serial connection it is 215 W. EM transducer generated minimum of 50 W at 56 Hz wide operation band width around the range of 267 Hz to 323 Hz [25, 36]. Non-linear Classic HEHs: One method to achieve broadband operation is that harvester to be nonlinear so that the frequency response range between half power outputs can be widen [39]. Li et al. summarized nonlinear broadband mechanism such 7 that as nonlinearity increases, resonance frequency decreases and as the acceleration increases, half power bandwidth broadens, whereas resonance frequency decreases [4]. As inspired from classic HEHs in Figure 1 (B) and (C), Shan et al.’s design and materials are illustrated in Figure 4 (left). In magnet configuration, poles are oppositely aligned so that indirectly exerted force on the suspended magnet can yield nonlinear mono-stable HEH [15]. Two peak powers and modes of HEH are 11.4 mW at 8.373 Hz by EM and 21.6 mW at 14.83 Hz by PE transducers. At half peak power, the device band is around 7-17 Hz [15]. In Addition to HEH in Figure 1 (C), Xu et al. used same pole magnet aligned in front of the tip magnet (see Figure 2 on left) so that HEH can be nonlinear and operate at larger frequency band. Their nonlinear HEH prototype achieved 5.66 mW power output at 1g, this result is 247% greater than at 0.5g and at half power level (3 dB), the frequency band is 83.3% wider than PE transducer alone [37]. Recently, HEH illustrated in Figure 4 (right) is researched by Mahmoudi et al. and Ping Li et al. Their device mirrors the configuration in Figure 1 (B); magnets are opposite pole magnet arrangement is used and the moving magnet is shared with both symmetrically placed beams. For Mahmoudi et al.’s HEH, EM and PE parts respectively produce 39% and 61% of the power output. This EM transducer can increase power density by 60% up to 1035 mW/cm3 and bandwidth by 29% (155 to 220 Hz) at 0.9g with respect to single EMHs [29]. Ping Li et al. deeply studied modeling, tests, effects of nonlinear factors, loads, input frequency and acceleration on amplitude, and found that their HEH design both enhance as wider band with low resonance frequency and greater power output compared to linear HEH designs. In contrast to linear EHs, optimal loads differ with excitation acceleration. Apart from Ping Li et al.’s statement, their theoretical and experimental frequency responses show no significant band widening other than shifting the linear resonance frequency from 119 Hz down to 113.5 Hz. Experimental analysis optimum results with respect to input accelerations of 0.2 and 0.45g are 0.14 and 1.19 mW for EM generator and 0.085 mW and 0.5 mW for PE generator. The HEH peak power output is 3.6 mW at 0.6g and ~110 Hz having half-power frequency range of ~107.5 to 112.5 Hz [4]. 8 Figure 2.4. Broadband HEH by Shan et al. (left) [15] and Ping Li et al. (right) [4, 29]. On Shan et al.’s prototype NdFe35 magnet and PZT-5H ceramics are used as EM and PE transducer components. As a different vibration source and application on airflow harvesting system example, hybrid aeroelastic vibration EH is modeled by Dias et al. Their system includes an airfoil that is connected to fixed spring and damper at around mid-plane and starting that point, cantilever beam as seen in Figure 1 (C) is connected. Dias et al. propose 2 and 3 degree of freedom system dynamic modeling [43, 44]. Relatedly, novel aeroelastic HEH harvesting incident sunlight is proposed by Chatterjee and Bryant (Figure 16), and their research is covered in part 4.2.4., under ‘two-multi source powered HEHs’ title. (c) Overall Classic HEHs Comparision Table 1.1 is prepared to list the reviewed classic HEH systems peak power generations, HEH volumes, magnet masses, input excitations, input frequencies and half power band width ranges in parts 3.1 and 3.2. 9 Among classic HEHs, Table 1.1 shows that Shan et al.’s HEH holds the greatest power output of 33 mW along with broadband performance in the range of 7 to 17 Hz [16]. By considering the device volume, Ali et al. holds greater power density than Shan et al [32]. Once PE and EM power generations are compared, generally, EM power generations are lower than PE parts with an only exception of Ab Rahman et al.’s HEH with four pole magnet arrangement [20, 35]. Apart from Wischke et al.’s microscale HEH, the lowest energy generation belongs to Xu et al.’s device [31] and the lightest device is Xu et al.’s HEH with 9.8 g of mass [37]. 10 Table 1.1. Type The overall comparison of the reviewed classic HEH systems in parts 3.1 and 3.2. Input Acceler ation References Input frequency/ Band width range Volume (mm3) Peak power generation Mass (g) FIXED-FREQUENCY CLASSIC HEHs [30] - 130 Hz - - - [31] 1g 66 Hz 187.2 0.845 mW - [32] 1g 76.2 Hz 5992 27.56 mW - [33, 34] 0.4g 23.3 Hz 2.26 mW 11 [21] 0.4g 50 Hz 2975 10.7 mW ~15 [16] - 40.5 Hz 760 4.25 mW ~21.5 Four-pole [35] 1g 15 Hz 2280 PEH: 2.2 mW EMH: 3.5 mW - Four-pole [20] 1g 15 Hz 2280 PEH: 2.3 mW EMH: 3.5 mW - Two-pole [20] 1g 49 Hz 1181 11 PEH: 1 mW EMH: 0.5 mW - Type Input Acceler ation References Input frequency/ Band width range Volume (mm3) Peak power generation Mass (g) BROADBAND CLASSIC HEHs Linear [17, 42] A* 77.5 Hz / ~70-80 Hz 29810 4.2 mW - Tunable [25, 36] 1g 299 Hz / ~267323 Hz 31201 215 μW - 8.373 Hz for EM Nonlinear [15] 0.5g Nonlinear [37] 1g ~45.5 Hz / ~4347 Hz 2257 5.66 mW 9.8 Nonlinear [29] 0.9g 93 Hz / 155-220 Hz 40000 B** - Nonlinear [4] 0.6g 110 Hz / ~107.5112.5 Hz 18437 3.6 mW - 14.83 Hz for PE / 7-17 Hz 20726 PEH: 21.6 mW EMH: 11.4 mW 100 (*) A: random acceleration with (0.1g)2/Hz spectral density of acceleration (**) B: Peak power density of 1035 mW/cm3. Note: Device volumes represent the minimum volume occupied by the harvester components and do not include and remaining device parts and the air gaps in HEHs, and Mass generally stands for the only stated magnet mass in references. 12 1.2.3. Novel HEH Systems While stand-alone systems are generally bound to be limited by one power source selection, multimode designs offer never-ending possibilities. Especially, when multisource powered energy harvesting concept is included as well as multimode HEHs, designs turn out to be novel. In this section, vibrational HEH novel designs are covered along with multiple vibration source harvesting, two and three multi-source powered harvesting systems in meso, micro and large scales. (a) Fixed-Frequency Single-Source Powered HEHs As a preliminary study, Reuschel et al. proposed axial flux and radial flux arrangements, where set of opposite pole magnets aligned radially in a radial coil house, designing EMH and PE cantilever modeling for the proposed arrangements. They announce to combine both transducers and analyze HEH as a whole system [45]. Harvesting from human motion is a demanded research subject specially to power personal electronics. Wei and Ramasamy studied harvesting kinesiology and it is shown that the HEH is suitable to feed personal electronics and charged mobile phone in experiments. The mechanical harvesting part composed of flywheel and in each foot fall, it runs the shaft connected to one-third of diameter of the actual wheel. Two piezoelectric configurations researched for shoe insole and it is seen that rolled piezoelectric plate is placed in shoe sole. It is seen that though this HEH is slow to charge mobile phone for being able to charge about 10% in 30 minutes, it is also found that starting from half- fully charged phone. the user can end up with 70% of charge with HEH, whereas without any harvester it would be 16%. Authors assume the potential over one million personal usage of their harvester. In this case, they foreseen the total power generation of 60,000 kW h [1]. Halim et al.’s unique components turns classic HEH into novel one. The main harvester body is almost same with the illustration in Figure 4 on the right. Novel pars are the parabolic top of the tip mass, which is intended to be moved laterally by the nonmagnetic ball action during horizontal input excitations. This mechanism also leads lateral PE bimorph displacement at center and EM induction with magnet attachment (see Figure 5). This design is aimed to harvest human motion, thus shaken manually by 13 hand at around 5 Hz during experiments. Resulted frequency responses of EM and PE transducers show that the first mode is at 816 Hz for both parts. Optimum power generation performances are 0.64 mW for EM part and 0.98 mW for PE part of HEH system [14]. (B)c (A) (C)c Figure 2.5. Halim et al.’s novel HEH design schematically represented in (A), fabricated components are listed in (B) and the assembled prototype is seen in (C) [14]. (b) Broadband Single-Source Powered HEHs Linear Novel HEHs: Castagnetti’s novel HEH is one of the most innovative one as well possessing 60 Hz-bandwidth. The design concept is composed of Belleville springs (B1 and B2 in Figure 6 (B) and (C)) and three different case of EM part for having a single magnet as in Figure 6 (A), 2 magnets (Figure 6 (C)) and four magnets configurations. The lateral frame in (A) is shown horizontally in (C), denoted by “F”. Experiments conducted for three cases of HEHs at 1g and 19.62 m/s2. Input acceleration of 19.62 m/s2 yield greater power outputs at resonance frequency. Among HEHs, the generated power of four-magnet HEH is 2 times of two-pole magnet HEH and 8 times of single magnet HEH. The four-magnet HEH configuration produced the greater power of 15.31 mW and enables broadband operation from 120 Hz to 180 Hz at 2g. Castagnetti also 14 reported their HEH is superior than commercial products like Perpetuum in terms of power generation and broadband width [39]. Bi Ci Ai Figure 2.6. The technical drawing of the Castagnetti’s HEH prototypes (A), Belleville spring scheme (B) and HEH scheme with two magnets (C) [39]. Non-linear Novel HEHs: Karami and Inman presented mono and bistable nonlinear thus, broadband novel HEH, as seen in Figure 7. As the horizontal input excitation is applied, then, the tip magnet moves harmonically. This oscillation yields EM and PE energy generation. HEH magnets are aligned with opposite polarization and these magnets’ distance is arranged such that the HEH can perform as mono or bistable but nonlinear unless the gap is set to 50 mm in order to see linear system performance. Additionally, the system behaves as linear at low input base excitation and nonlinear at greater acceleration inputs. The best power output results are close to 35 W for EM transducer and 1.5 mW for PE part at 1.7 m/s2. Worth to mentioned that linear dynamics of these HEH systems at low excitations [46, 47], is overcome by Leadenham and Erturk’s M-shaped PEH design [48]. 15 Figure 2.7. Karami and Inman’s design illustration and prototype [46, 47]. 1.3. Introduction to Piezoelectricity and Classic Methodologies for Output Power Generation Basic equations of piezoelectricity relate the stress (T) and electrical induction (D) with strain (S) and the electric field (E). These constitutive equations, summarized mono dimensional simplified and dimensional detailed sub-equations of the parameters are listed in Eqs (1.3.1), (1.3.1a-b) and (1.3.2), respectively[1–6]. 16 E E Sq  s pq Tq  d kp Ek Tp  c pq S q  ekp Ek Di  diqTq   ikS Ek Di  eiq Sq   ikS Ek D D Sq  s pq Tq  g kp Dk Tp  c pq S q  hkp Dk Ei   giqTq  Dk  T ik Ei  hiq S q  Dk       d .E     Y    D   .E  d .  simplified expression (1.3.1)  ikS  T   S  d .e (1.3.1a) c D  c E  h.e e   S h  c E d  [C/ m 2 ]  [N/ (m.V)] d   T g  s E e  [C/ N]  [m/ V] 1 g  S d  s D h  [m 2 / C]  [(m.V) / N] (1.3.1b)  h 1 S e  c D g  [N/ C]  [V/ m] where; S : Mechanical Strain ( ) [m/m] T : Mechanical Stress ( ) [N/m 2 ] D : Dielectric Displacement [C/m 2 ] E : Electric Field [V/m] x s pq : Elastic Complience [m 2 /N] at constant 'x' x x c pq or 1/ s pq  : Elastic Stiffness or Young's Modulus (Y) [N/m2 ] at constant 'x'  ikS : Dielectric Permittivity [F/m] at constant 'x' d , e ,g , h : Piezoelectric Coefficients i, k, p, q are directions. As expressed in Eqs (1.3.1, 2.1.1a-b), constant electric field -short circuit condition(E=0), constant strain (S=0), constant stress (T=0) and the constant dielectric displacement (D=0) are denoted with the respective superscripts of E, S, T and D [4–6]. In order to guide Eq (1.3.1) better for the following future derivations, the simplified coupled constitutive equation sets are given in Eqs (1.3.1c-d), and for being the fundamental of the common 6-crystal PZT representation, detailed matrix representation of Eqs (1.3.1 and 2.1.1d) is shown in Eqs (1.3.1e-f) [1]. The most common usage and requirement of Eqs (1.3.1 and (1.3.1e-f) is that the simulation software like Ansys. 17 Simplified 1-D Eq(2.1) : E T  c    D  e  e  S     S   E  E S  s    D   d (2.1.1c) d T  T     T   E  (1.3.1d) Detailed Matrix Notation of Eq(2.1) for 6-crystal PZT : E  S1   s11   sE  S 2   12E  S3   s13    S4   0 S   0  5   S6   0  D1   0 D    0  2   D3   d31 s12E s11E s13E 0 0 s13E s13E s33E 0 0 0 0 0 E s44 0 0 0 0 0 E s44 0 0 0 0 0 0 d31 0 0 d33 0 d15 0 d15 0 0  T1   0     T2   0  T   0  3  T4   0  T     5   d15 E E 2  s11  s12   T6   0 0 0 0 0 0 0 0 0 d15 0 0 d31   d31   E1  d33     E2 0    E3  0  0  T1  T  2 0    11T 0 0   E1    T3    0      0 11T 0   E2  T T  0   4   0 0  33   E3  T   5 T6  (1.3.1e) (1.3.1f) Note:  2  s11E  s12E   is also denoted as s66E . Mono dimensional piezoelectric coefficients summarized in Eq (1.3.1b) and they are expressed in the form of both main equations and derived versions with sub-parameters regarding dimensions (Eq (1.3.2)) [5]. 18   Di    Tp dip   Di    Sp eip E    Sp         Ei   E  i S  p hip E    Tp         Ei   E  i  T  p gip T D at constant 'x' and 'y': dip   ikx gip  siky eip S x y eip  c pq diq = pq hiq     Sp        Di   D T    Tp         Di   S gip  d kp  x ik  siky hkp hip  cqpx giq = (1.3.2) eiq  qpy One other important equation is for the conversion between mechanical and electrical energies and related electromechanical coupling factor (k, kiq), and the efficiency ( h ) at resonance are listed below [4–7]. k2  k  2 iq Stored Energy electrical Supplied Energy mech e2 d2 d2 d2   T E  T E  T cE  T Y  Supplied Energy mechanical Stored Energy elec  c  s  eiq2 eiq2 Wi ( electrical )   Wq( mechanical )  ikT c Epq  ikT c Epq   iq2 (1.3.3) k2 2 1  k 2  (1.3.4) k2 1  Q 2 1  k 2  where;  : Efficiency at Resonance [%] k 2 : Electromechanical Coupling Coefficient  kiq2  Q: Quality Factor In order to illustrate piezoelectricity modelling in equivalent-circuit representation (Figure (1.3.1 and (1.3.2), related parameters are expressed in the following equations. By assuming voltage coefficient gip as constant with the stress, the open circuit voltage is formulated in Eq (1.3.5). Further approximations in Eq (1.3.6-9) are set to relate 19 electrical voltage (V) and current (I) with mechanical force (F) and displacement (X, w) as shown in Eq (1.3.10, respectively [4–6]. Figure (1.3.1. The relation between mechanical and electrical properties of piezoelectricity in Nye- Heckmann diagram [2, 8]. (b) (a) Figure (1.3.2. (a) Electrical and mechanical parameters -voltage (V) and current (I), force (F) and displacement (w)- illustration on the piezoelectric element, having the thickness l, (b) equivalent circuit representation of electromechanical modelling [4]. As shown in Figure (1.3.2, the voltage across the piezoelectric electrodes at constant voltage coefficient (gip) is represented, and illustrated electro-mechanical piezoelectricity modelling equations are listed further in mono-dimensional and detailed representation forms. 20 V  Tp .gip (1.3.5) where; V : Voltage across the Piezoelectric Electrodes [V] : Gap between the electrodes, piezoelectric layer thickness [m] F A . p   A. Fp   AT  Q  D dD =A.J p since Q p  A.D  Ip  A A dt  X  S X  w  S.   .  V  E.  V E T (1.3.6)  If Fp   ATp  Note:    then V   E.  k PE  kSC  Cp   ikE A E c pq A 1      FF  G p    2 eiq A (1.3.7)  ikE A  and  2  d 2  c d  A  AY d Y  iq (1.3.8) (1.3.9) where; Fp : Restoring Force of the Piezoelectric Material [N] I p : Outgoing Current across the Electrodes [Amp] Q p : Outgoing Charge on the Piezoelectric Capacitance [C] J p : C urrent Density [C/(m 2 .s)] X : Displacement (w) [m] A : Surface Area of the Piezoelectric Layer [m 2 ] k PE : Short-circuited Stiffness (kSC ) [N/m] C p : Clamp Parasitic Capacitance [F]  FF : Force Factor, also called Proportionality Constant (G p ) or Electromechanical Coupling Term ( ) ( 2 is constant parameter) 21 As noted for Eq (1.3.6), sign is only a subject of matter for force and voltage and as long as their signs are opposite, minus sign for force or voltage is interchangeable. Regarding Figure (1.3.2-3, and substitution of Eq (1.3.6) into simplified 1-D Eq (1.3.1) gives the electro-mechanical equation sets in Eq (1.3.8). In these equations, it is aimed to express in two common forms, referring the literature usage. As a result, the same electro-mechanical equations are represented in terms of (i) short-circuited stiffness, force factor and clamp capacitance, and (ii) proportionality constant Gp (Γ or θ) for electromechanical energy terms (Figure (1.3.3)) listed in Eqs (1.3.8-9) [1, 4–6]. (a) (b) (c) Figure (1.3.3. Circuit equivalent of the electro-mechanical in terms of (a-b) force factor [4], and (c) proportionality constant [6]. (a) and (c) illustrates electrical and mechanical equivalent modellings, and (b) combines two separate circuit equivalent with transformer for a complete representation of the piezoelectric element. Representative electromechanical coupled equations illustrated in Figure (1.3.2-3): 22  cE A   eA  YA  YA  F  F X   d V  X   V       S  A  eA   YA   A 2 Q    X  or Q   d  X   V  1   V          YA  dX   A  2 dV  eA  I  d  I ( j )    j X  jC pSV  1    dt   dt     Fp  k PE X  V F  kX  G pV dV dV or I  C p  Gp X dt dt  since Q p   X  C pV   since Q  G p X  C pV  I p   FF X  C p f  G pV i  Gp X (1.3.10) or for 31-mode: Fe  V I e  X and 33-mode: (1.3.11) Fe  V Ie   X (1.3.12) where; f , i are electromechanical energy conversion terms 1.4. Analytical Formulation Methods for the Power Generation-a Review So far, piezoelectric constitutive equations, property constants, main parameters and electromechanical modelling are expressed. Further on, in literature there exists many approaches for the estimation of the piezoelectric current, voltage, power outputs. Here in this sub-section, some of these numerous simplified, derived, approximated and exact equations for the vibration input at resonance frequency and for any excitation frequency are tried to demonstrated separately as follows: To begin with, it is essential to model piezoelectric energy harvester dynamic equations to derive output power. Therefore, as seen in Figure 1.4.1, lumped element mode of piezoelectric EH is shown in Figure 1.4.4 and force balance equation is given in Eq (1.4.1) and descriptive base excitation formulas in terms of displacement, velocity and acceleration are given in Eq (1.4.1a) [1]. Derivation of output power is also carried out in this study, see Section 2.3 for details. 23 Figure 1.4.1. Lumped-element model of piezoelectric EH and its connection to harvester circuitry scheme [1]. mwB  mw  cw  kw  Fe  meff wB  meff w   cm ,eff  ce  w  keff w (2.2.1) wB  WB sin(t )  WB cos(t )   2WB sin(t ) (2.2.1a) Alternatively, wB  WB e it  iWB e it   WB e 2 it 33 mbeam  mtip 140 where; wB : Base (Frame) Displacement Excitation Input [m] meff  (2.2.1b) wB : Acceleration Excitation Input acting on the Harvester Base (Frame) [m/s 2 ] w : Relative Displacement of the Seismic Mass (or Dynamic Mass or Proof Mass) [m] meff : Effective Mass [kg] mtip : Tip Proof Mass [kg] mbeam : Cantilever Beam Mass [kg] cm,eff : Effective Mechanical Damping Coefficient [N.s/m] ce : Electrical Damping Coefficient [N.s/m] keff : Effective Stiffness [N/m] In addition to the basic expression is given in Eq (1.4.1), in order to gain the relative displacement on seismic mass, its laplace transform is taken in Eq (1.4.1b) and to simplify the representation, well-known damping ratios and dimensionless frequency are given in Eq (1.4.1c) and then, the transfer function of relative displacement to the frame displacement is derived in Eq (1.4.1d) and relative displacement amplitude of the seismic mass is represented in Eq (1.4.2) [1]. Finally, Hehn and Manoli’s generated 24 power estimation is covered in sub-section (a) in Eqs (1.4.3-5). meff s 2 wB  meff s 2 w   cm,eff  ce  sw  keff w m  cm,eff , e  2meff n (1.3.b) keff ce 3EI  and   ; n  and keff = 3 2meff n L meff n (1.4.1c) meff s 2 wB ( s )  meff s 2 w( s )   cm,eff  ce  sw( s )  keff w( s ) m s w (s)  eff 2 s 2   cm ,eff  ce  s  keff B meff w( s ) s2  2 wB ( s ) s  2n  m   e  s  n 2  w(s) and w( s )  2W sin(t ) W  = wB ( s )  2WB sin(t ) WB (1.4.1d) 1/ n 2  2  2 W W    . WB  2  2 jn  m   e   n 2 1/ n 2 WB 1  2   2 j  m   e  W WB 1      2  m   e    2 2 (1.4.2) 2 reminder : w(t )  W sin(t )   2W sin(t ) wB (t )  WB sin(t )   2WB sin(t ) where; W : Seismic Mass Displacement Amplitude [m] WB : Base Input Displacement Excitation Amplitude [m] n : Natural (Resonance) Frequency [rad/s]  : Input Excitation Frequency [rad/s]  : Dimensionless Frequency : Electrical (Transducer) Damping Ratio e  m : Mechanical Damping Ratio 25 1.4.1. Hehn and Manoli’s proposed expressions for power generation [1, 4–6, 9]: Hehn and Manoli derive power output from the dissipated energy. Since the only dissipative element is damping equivalent, in this thesis study, output power Eq (1.4.1.2) is calculated from the damping energy as in Eq (1.4.1.1). P ( )  2 1 1 2 W   e  W   e  2 2 (1.4.1.1) By substituting Eq (1.4.2) into Eq (1.4.1.1) output power derivation is carried out as such: 2  WB 1 2 P( )    2 2  1   2    2      2 e     m   2 2 2   WB  e 1 = 2 2 1    2   2       2 m e        e   substitution of: WB     2WB    4WB 2 2 2   2WB 2 W   B 2  2 WB   n 2  2 e 2 into P( ) : 2  P( )= P( )  1 2 1   2  2   2       2 m e     1 W  B 2 e 2n 2 1   2  2   2       2 m e     meff WB   e 1  P( )  2keff 1   2  2   2       2 m e     2 meff WB   e 2 at resonance:   n  , P (n )  8keff  m   e  for max extractable power:  m   e  , P (n )  26 2 or W  B 2 e 8n 2  m   e  meff WB  32keff  e 2 or (1.4.1.2) 2 W  2 B 32n 2 e Eq (1.4.1.2) is gained as a result of derivations. However, as seen in Eq (1.4.1.3), Hehn and Manoli’s power expression is different then the purely derived Eq (1.4.1.2). It is seen that for Hehn and Manoli’s output power expression to be equal to the exact derivation, keff has to be equal to “0.5”. In this thesis study, since effective stiffness can be calculated, Hehn and Manoli’s assumption for keff is not considered. In addition to Hehn and Manoli, many researchers followed the same derivations and keff assumption by neglecting the term “1/(2keff)” [1, 3, 5, 7, 10, 11]. meff 3 3 WB   e 2 P( )  1   2    2  m   e    2 2 meff n3 WB   e 2 at resonance: P(  n )  4  m   e  2 for maximum extractable power:  m   e  , Plim (n )  c m ,eff  ce  2meff n e  2meff n m  , Plim m ( )  eff n meff n WB  3 (1.4.1.3) 2 16 e WB  2 8ce For the studies conducted over the Mide Volture Piezoelectric EHs, in order to gain the damping ratio, given actual quality factor (Q) values are used as described in Eq (1.4.1.4) [1] for each selected product [12]. Qe  mn 1  2 e ce and Qm  1 2 m  mn cm,eff  1 1 1   QT Qe Qm (1.4.1.4) where; Qe : Electrical Quality Factor Qm : Mechanical Quality Factor Qm : Total Quality Factor Additionally, all of these transducers have wide-variety of multimorph structures [12]. For this reason, by using given piezoelectric and substructure properties, new elasticity of modulus needs to be calculated for the whole cantilevered beam. In literature, Andosca et al. suggestion for the calculation of the elastic modulus of a multimorph beam and definition for the required neutral axes of layers and cantilevered beam are 27 demonstrated in Eqs (1.4.1.5-6) and used for continuous beam modelling [13, 14] I 1  N 3 b  hi 12  i 1  (1.4.1.5) 2   hs 2  hP 2  2 EI  E p Ap  z pN  z N      Es1 As  zsN  z N   12  12      hi 2   2    Ei Ai  zi  z N     12   i 1    N 1 1 z pN  hp and zsN  hp  hs 2 2  zN  E p1hp z pN  Es1hs zsN E p1hp  Es1hs (1.4.1.6) (1.4.1.7) where; E : Modulus of Elasticity of the Piezoelectric Cantilever Beam [N/m 2 ] E p1 : Modulus of Elasticity of the Piezoelectric Layer [N/m 2 ] Es1 : Modulus of Elasticity of the Substrate Layer [N/m 2 ] Ei : Modulus of Elasticity of the ith Layer [N/m 2 ] Ap : Surface Area of the Piezoelectric Layer [m 2 ] As : Surface Area of the Substrate Layer [m 2 ] Ai : Surface Area of the ith Layer [m 2 ] hp : Piezoelectric Layer Thickness ( ) [m] hs : Substrate Layer Thickness [m] hi : ith Layer Thickness [m] I : Area Moment of Inertia [m 4 ] zi : Neutral Axis of the ith Layer [m] zsN : Neutral Axis of the Silicon Substrate [m] z pN : Neutral Axis of the Piezoelectric Layer [m] z N : Neutral Axis of the Total Piezoelectric Cantilever Beam [m] Priya et al.’s addition to Hehn and Manoli’s output power formulation [9]: Simlar to Hehn and Manoli’s proposed power output in Eq (1.4.1.3), Priya et al. also proposed total damping ratio in dominator for the total power in the system. Consequently, according to Priya et al., Eq (1.4.1.3) is electrical power generation alone and to find the total power in the system, mechanical power dissipation that is converted 28 to electrical power eventually, needs to be calculated. These steps are demonstrated in Eqs (1.4.1.8-1.9) [9]. meff 3 3 WB   T 2 PT ( )  1   2    2 T   2 2 meff 3 3 WB   e meff 3 3 WB   m 2 Pe ( )  2 1   2    2 T   2 2 and Pm ( )  1   2    2 T   2 (1.4.1.8) 2 where;  T   m   e  at resonance: meff n 3 WB   e 2 Pe (n )  4 T 2 meff n 3 WB   m 2 and Pm (n )  4 T 2 meff n 3 WB   e 2 PT (n )  Pe (n )  Pm (n )  4 T 2 meff n 3 WB   m 2  (1.4.1.9) 4 T 2 for maximum extractable power:  m   e : Pe,lim (n )  meff n 3 WB  16 e 2 equals to Pm ,lim (n )  meff n 3 WB  2 16 m 1.4.2. Khalatkar et al.’s proposed expression for power generation [15]: Khalatkar et al. derives power generation in Eq (1.4.20) by using coupling electromechanical equation sets stated in Eq (1.4.8) [15]. PRMS =  2 b 2 hbeam 2 e31 ARMS RL 1  Abeam 2 33 RL  4  h piezo   where; (1.4.2.1) 2 ARMS : Vibration Amplitude at Free-end of the Cantilevered Beam For maximum power output, the denominator needs to be as small as possible. This 29 means the squared part is required to be small yet it cannot be lower than 1 that means somehow right-side terms needs to cancel out. At this point, the only selected variable is load resistance; thus, it can be arranged to cancel out the terms as in Eq (1.4.2.1). As obvious, load resistance plays important role on the maximum extractable output and highly preferred subject in literature. Caliò et al. mentioned the result of the same derivation in Eq (1.4.2.2) [5]. However, during studies by sweeping the load resisting in future evaluations showed that optimum load value sometimes holds and sometimes differs than the calculated one. This is also observed from the given experimental values in Mide manual where usually, the optimum load resistance increases with added tip mass and decreased input excitation frequency [12]. 1  Abeam 2 33 RL   A  R   1  A. 33 L   A 33 Ropt  hp   h h piezo  p    hp 1 R opt = = A ii C p (1.4.2.2) 1.4.3. du Toit’s proposed expression for power generation [3, 7, 11, 16] du Toit follows the similar steps; however, does not use force terms representing electrical output (Eq (1.4.3.1)), then derives the displacement response instead and calls the common term as seen in the denominator in Eq (1.4.2) as the standard amplification factor and denotes as | G(iω) |, see Eq (1.4.3.2).  mwB  mz   c    z  kz  -wB  z   m   e  n z  n 2 z  meff WB 1  Z  nWB G (i ) 2 k 2 2 eff 1      2  m   e    (1.4.3.1) (1.4.3.2) where; z (t ) : Displacement of the System in Response to Input Excitation Input [m] Z : Displacement Amplitude of the System in Response to Input Excitation Input [m] G (i ) : Standard Amplification Factor 30 At resonance, standard amplification factor reduces to quality factor as seen in Eq (1.4.3.3). Eq (1.4.3.4) helps to enlighten about the half power bandwidth over the mechanical damping ratio. Half power bandwidth definition states that the frequency limits where the maximum power is reduced by a factor of 0.707 (1/√2). As the quality factor reduces, half-power bandwidth broadens. Z  meff WB 1   static .Q keff 2 m (1.4.3.3) where;  static : Static Deflection [m] G (i ) : Standard Amplification Factor m   a  b 2  a   and b  b   a  n n   (1.4.3.4) where; a : Lower Frequency Limit of Half Power Bandwidth (rad/s) b : Highest Frequency Limit of Half Power Bandwidth (rad/s) For further analysis, by using constitutive equations in Eq (1.4.1, 1.4.1c, and 1.4.8) for the 33-mode stress model in Fig 1.4.3.1, electromechanical coupling factor (θ, in Eq (1.4.7 and 9)) and mass per cross-sectional area are defined, and by taking the first equation among electromechanical coupling equation set in Eq (1.4.3.5), redefining by piezoelectric parameters in Eq (1.4.3.6), and multiplying each side by piezo surface area, du Toit rewrites dynamic equation of the model illustrated in Figure 1.4.3.1 as shown in Eq (1.4.3.7). Figure 1.4.3.1-a. Scheme of 33-stress mode piezoelectric EH [3, 7, 11, 16]. 31 Figure 1.4.3.1-b. Scheme of 33-stress mode piezoelectric EH [3, 7, 11, 16]. T3  c33E S3  e33 E3 (1.4.3.5) D3  e33E S3   33 E3 me wB  me z  c33E z V  e33 hp hp Q z V  e33E   33 Ap hp hp  meff wB  meff z  n 2 z  V Q   z  CPV  meff meff  z  wB  e33 Ap where;  me  , T  , z  w  wB and     Ap Ap hp  z : Relative Displacement [m] (1.4.3.6)    for maximum extractable power  m   e : -wB  t   z  t   2 mn z  t   n 2 z  t    meff V t  (1.4.3.7) As a result of the following Laplace transforms, mentioned electromechanical coupling factor in Eq (1.4.3) is used for 33-mode and for the rest of the same group of parameters, dimensionless time constant r (ωn.τ) regarding well known time constant (τ) is defined in Eq (1.4.3.9) and further used in Eq (1.4.3.10-13). Some initial steps of the derivation are given in Eq (1.4.3.10); yet the final derivation is different than the exact formulation that du Toit proposed in Eqs (1.4.3.11-13) therefore, derived ones are not mentioned. [7]. 32 dQ  RL  V ( s )  sQ( s ) RL  dt  Z ( s)  Q( s)  sQ( s ) RL   z Q  Z ( s)  Q( s)  CP  V (s)  V CP CP  V   Z ( s )  sQ( s ) RL CP  Q ( s )  Q ( s )  V (s)  s  Z (s) 1  sRL CP RL , V (i  )  r  n RL CP ,  =RL CP  r n   Z (s) 1  sRL CP i  Z (i  ) RL i  RL Z  V (i  )  1  i  RL CP 1  i  r (1.4.3.8) (1.4.3.9)  meff s 2 wB ( s )  meff s 2 Z ( s )  2s mn Z ( s )  n 2 Z ( s )  V  s  meff s 2 wB ( s )   meff s 2  2s mn  n 2  Z ( s )  V  s  meff  2 wB (i  )   meff  2  2i mn  n 2  Z (i  )   V (i  ) 2      meff wB (i  )   meff  2i m n   n   Z (i  )  V (i  )        2i m 1  i  2 RL   2  meff wB (i  )   meff   Z (i  )    1  i r    2i m 1   2  1  i r   meff     meff wB (i  )   1  i r      2   i  RL    Z (i  )    (1.4.3.10) 1 z (i  )  meff wB (i  ) keff 1 V (i  )  meff wB (i  )  m Pout (i  ) eff wB (i  ) 2  1   r  2   1  1  2 m r   2  2   2 m  1  ke 2  r   r  3  2   rke 2    1  1  2 m r      2 m  1  ke 2  r   r  3  2   2 N keff 2 rke 2  2 1  1  2 m r   2  2   2 m  1  ke 2  r   r  3  2     (1.4.3.11) (1.4.3.12) (1.4.3.13) Later on, du Toit modifies Eq (1.4.3.6) with Eq (1.3.2) and with the resonance 33 frequency depending on 33-mode operation parameters and and Eq (1.3.11) with load resistance as seen in Eq (1.4.3.14 and 15). Among latter derived FRFs, displacement frequency response function (FRF) in Eq (1.4.3.16) is exactly the same with Eq (1.4.3.11). However, not only output voltage FRF in Eq (1.4.3.17) and output power FRF in Eq (1.4.3.18) is different than Eq (1.4.3.12-13), but also output power FRF does not hold, still quite close to the power derivation from voltage FRF and load resistance. me wB  me z  c33E z V  e33  z  t   2 mn z  t   n 2 z  t   n 2 d 33 V  t  =-wB  t  hp hp (1.4.3.14)  IReq  V  Req C pV  Req X   V  Req C pV  Req X  0 (1.4.3.15)  2 c33E Ap V2 and P   n  meff h RL  (1.4.3.15a) z (i  ) 1  2 wB (i  ) n    1   r  2   1  1  2 m r   2  2   2 m  1  ke 2  r   r  3  2   meff Req d 33n  V (i  )  wB (i  ) 1  1  2 m r   2  2   2 m  1  ke 2  r   r  3  2    Pout (i  ) wB (i  ) 2   meff Req rke 2  2 1   RLn 1  1  2 m r   2  2   2 m  1  ke 2  r   r  3  2     (1.4.3.16) (1.4.3.17) (1.4.3.18) In addition to Calio et al.’s proposed optimum load in Eq (1.4.2) [5], du Toit also suggests a more complex one that is used for their derived output power (see Eq (1.4.3.19)). Ropt   4   4 m 2  2   2 1   2  6   4 m 2  2 1   keff C p     4  2    1       keff C p 34 2  2    (1.4.3.19) 1.4.4. Erturk’s suggestion on mass correction on lumped parameter model and optimum load resistance for the maximum output power generation [11] : Erturk studies transverse and longitudinal vibrations of cantilever beam and observes a correction need for the validation of the analytical results the experimental outputs. Correction factor for the Eqs (1.4.3.7 and 14) is suggested so that for all selected proof masses, the power generation estimation can hold. The only safe region where uncorrected lumped parameter validates experiments is that when tip mass is sufficiently larger than the beam mass. Derived correction factors for transverse and longitudinal vibrations are expressed in Eq (1.4.4.1) and Eq (1.4.4.2), respectively. Corrected lumped-model is given in Eq (1.4.4.3) and derived FRFs are listed in Eq (1.4.4.4-6). For FRFs, regarding the operation mode; in otherwords, transverse and longitudinal vibrations, related mass correction factor and respective piezoelectric charge (displacement) coefficient (d31 and d33) are used but expressed in main µ and diq terms.  transverse m / m  m / m m / m  m / m tip tip longitudinal beam tip tip   0.603 m / m   0.08955   0.4637  m / m   0.05718   0.7664  m / m   0.2049   0.6005  m / m   0.161 2 beam tip beam 2 tip beam 2 beam beam (1.4.4.1) tip beam 2 (1.4.4.2) beam tip where;  transverse : Transverse Vibration Mass Correction Factor longitudinal : Longitudinal Vibration Mass Correction Factor -longitudinal wB  t   z  t   2 mn z  t   n 2 z  t   n 2 d 33 V  t  (1.4.4.3) or - transverse wB  t   z  t   2 mn z  t   n z  t   n d 31V  t  2 2 35 z (i  )   2 wB (i  ) n 1   r  2  (1.4.4.4)  1  1  2 m r      2 m  1  ke 2  r   r  3  2   meff Req d iqn  2 2 V (i  )  wB (i  ) 1  1  2 m r   2  2   2 m  1  ke 2  r   r  3  2    Pout (i  ) wB (i  )  meff Req rke   RLn 1  1  2 m r   2  2   2 m  1  ke 2  r   r  3  2   2 2  (1.4.4.5)  2   (1.4.4.6)  Erturk also derives optimum load and defines dimensionless parameter, γ for unimorph beam as seen in Eq (1.4.4.7). This dimensional parameter evaluation differs according to serial and parallel connectionof piezoelectric electrodes in bimorph configuration [11]. Ropt  1 n C P 2 1   m 2      2 m  1     m2 1    2 m2  (1.4.4.7) where:  2 C p n 2 So far, beginning from the fundamental equations of piezoelectricity, classic 31-strain and 33-stress mode displacement response and power generation formulations in literature and their derivations are covered. It is seen that proposed power generation formulations basics depend on either calculation of the power output over dissipated energy or solving displacement response and voltage from constitutive and electromechanical coupling equations. As a result, concluded power output formulations differs in literature and due to inconsistencies in terms of power and expressions among sources, a novel derivation methods depending on the same basis are covered in the following Chapters 2 and 3 for single and two DOF VEHs. 36 1.5. Objective and Scope of the Thesis The objective of this thesis study is to investigate the most efficient VEH methodology for EHing from train vibrations. In the light of this objective, piezoelectric energy harvester modelling is covered in Chapter 2 along with the proposed novel 2 and 3 DOF configurations. 2 and 3 DOF structures are inspired from mechanical filter configurations [49–53]. In the goal of achieving dominant frequencies of input train acceleration and displacement excitations, vibration signal analysis covered in Chapter 3 and Appendix 3A in detail for 4 train vibration data. As the next step of tuning VEHs, commertial Midé Volture Piezo Protection Advantage (PPA) 1011 and 2011 along with the PPA-1021 and 1001 (Appendix 4) used in Ansys and SAP2000 simulations, as explained in Appendix 5A, are listed in Appendix 3B. Performance evaluation is briefly covered in Chapter 4 and corrected methodology results and discussion are explained in Chapter 5 and detailed in Appendix 5B. Lastly, thesis is finalized with the conclusion and future recommendations in Chapter 5. 37 CHAPTER 2: CLASSICAL DERIVATION AND NOVEL OF EQUATIONS VIBRATION OF ENERGY HARVESTERS 2.1. Introduction Piezoelectric effect is well-suited to convert vibrations to usable electric energy. Physically, there are two approaches: Cantilever beams operated at resonance and piezoelectric material in stack configuration. Modelling approaches concentrate on lumped parameter modelling and distributed parameter modelling. In this chapter, lumped parameter modelling will be used with mass correction obtained from distributed modelling. As shown in Figure 2.1.1, we will use concentrated lumped parameter models of cantilever beam and stack piezoelectric models. During vibration, piezoelectric material generates electritiy in 33 mode in stacked configuration and in 31 mode in cantilever configuration, as shown in Figure 2.1.2. 38 Figure 2.1.1. Schematic illustration of stacked piezoelectric energy harvester (a), cantilever beam energy harvester in 33 (b) and 31 (c) operation mode, and their common lumped parameter model (d). Figure 2.1.2. (a) Piezoelectric material operated in (a) 33 and (b) 31 mode.[54] 39 2.2. Lumped Parameter Modelling of Stacked PiezoelectricEnergy Harvester Model illustrated in Figure 2.2.1, where base is excited by acceleration input 𝑤̈𝐵 . Relative motion between mass and base is 𝑤 = 𝑧 − 𝑤𝐵 . Figure 2.2.1. Model of 1DOF harvester. Total mass, effective spring coefficient and mechanical damping as well as input base excitation and relative displacement of the proof mass are represented. For 1 DOF model, constitutive equations can be written as [55, 56] E T3  c33    D3  e33  e 33   S3     33S   E3  (2.1.1c) where D, E, S and T are electric displacement, electric field, strain and stress, respectively. E is the piezoelectric constant relating charge to strain and ε is permittivity of piezoelectric material. Superscripts E and S indicates constant electric field and strain, respectively [55–58]. Approximate mass of the system includes proof mass plus effective mass of piezoelectric material move to proof mass point. Therefore 40 M w V 1 M T  m p  m , me  T , S3  , E3   (2.2.2) Ap h h 3 where h is the height and Ap is the area of piezoelectric material. From these, it can be written as: T3   M T  w  wB  Ap  me  w  wB   c33E S3  e33 E3 (2.2.3) w V  e33 (2.2.4) h h w V me wB  me w  c33E  e33 (2.2.5) h h c33E AP e A  M T wB  M T w  w  33 P V (2.2.6) h h Here we can identify K as stiffness of piezoelectric element (Eq (2.2.7)) and Γ as me  w  wB   c33E electromechanical coupling term. c33E AP h e A = 33 P h K (2.2.7) (2.2.8) Damping of the piezoelectric material (Cp) is added and dynamic equation becomes: MT wB  MT w  C p w  Kw  V (2.2.9) The last term in the Eq (2.2.9) is the force 𝐹𝑝 = 𝛤𝑉 on mass due to electromechanical coupling. Furthermore, it can be identified that [54, 56–58]. e33 AP d cE A   33 33 P   d33 K h h e cE A where; d33  33E , K  33 P  M T  2 n c33 h  (2.2.10) Then, dynamic equation becomes  M T wB  M T w  C p w  Kw  d33 KV (2.2.11) or wB  w  2n w  n2 w  d33n2V (2.2.12) where;  =  d33n2 and    M T d33n2 41 Utilizing Eq (2.2.1) and (2.2.2) electrical charge equation becomes D3  e33 w q   33S V  h Ap (2.2.13) where q is the charge produced and i is the current produced. Since i  q dq , V  RL i dt S A c33E AP w  33 P V h h (2.2.14) c33E AP S A w  33 P V h h E S A dq c33 AP  i w  33 P V dt h h or i   w  C pV q or (2.2.14) (2.2.15) (2.2.16) V  RL i  RL w  RL C pV (2.2.17) From Eq (2.2.17) we can write V  RL C pV  RL w  0 where;  =  e33 AP e33 c33E AP cE A K  E   d33 K   d33 M T n2 and n2  33 P  h c33 h MT h MT  V  RL C pV  RL d 33 M T n2 w  0 (2.2.18) Equations (2.2.12) and (2.2.18) are the electromechanical equation of the stacked 1DOF piezoelectric energy harvester. 2.3. Lumped Parameter Modelling of Cantilever Piezoelectric Energy Harvester In this case, piezoelectric material is under axial tensile force F, as shown in Figure 2.3.2 [54] where L is the length, b is the width, h is the height of the cantilevered piezoelectric material. 42 Figure 2.3.1. Beam of piezomaterial loaded in 31 mode of the piezoelectric material. Constitutive equations are in the [54]: E  S1   s11    D3   d31 d31  T1     33T   E3  (2.3.1) These equations can be written in macroscopic variables F, ε, V, I instead of local varibles S, E, D, I [54]. Fp  k p   V (2.3.1a) i    C pV  V F dq where; E   , q  DhL, T  , S= , I  , h bh L dt  T d31  bL d b bh  E  ,   31E and k p E , C p    33 Ls11 s11  h s11  (2.3.2) In Eqs (2.3.1-2), kp is the stiffness and Cp is the capacitance of the piezoelectric material. Γ is the generalized electromechanical coupling factor. It shows that large deflection in 3 direction cause small elongation ε in 1 direction. Fp is the restoring force acting on the seismic mass, in Figure 2.1.1. Therefore, dynamic equation of motion can be written as: 43  M T wB  M T w  Cw  Kw  V (2.3.3) i    C pV where K is the stiffness of piezoelectric and mechanical components, C denotes equivalent damping of the mass. If we utilize 𝑖 = 𝜃= 𝛤 𝑀𝑇 𝑉 𝑅𝐿 , 𝜔𝑛2 = 𝐾 𝑀𝑇 , 𝐶 𝑀𝑇 = 2𝜁𝜔𝑛 and , we obtain similar equations as Eq (2.2.12) and Eq (2.2.18). However; K, Cp and θ must be calculated (or determined experimentally) for 31 operation mode. In this configuration [54–58]: AP 33   E AP 2   =  e31 h   d31c31 h  d31M T n     M T  mtip  140 mbeam   bL C p   3S1 (2.3.5) h (2.3.4) However, Cp is generally calculated experimentally due to nonlinear response of piezoelectric element to the applied strain. In this research it is taken from the catalog of MidéVolture piezoelectric transducers [59]. Therefore, equation in the form: w  2n w  n 2 w - V  - wB (2.3.6) V  RL C pV  RL w  0 are utilized, where µ is a correction factor for beam mass [57] in the form of m  m tip tip / mbeam   0.603  mtip / mbeam   0.08955 2 / mbeam   0.4637  mtip / mbeam   0.05718 2 (2.3.7) 2.4. Derivations of Equations for a Novel Energy Harvester Energy harvesters are designed (or modified) to harvest energy from harmonic vibration. Generally, excitation is in the form of acceleration. Acceleration values amplifies vibration at higher frequencies. However, displacement values concentrate on lower frequencies. In this thesis, a novel energy harvester of 2 DOF is developed to design the higher 44 natural frequency to the maximum value of acceleration input; and the lower frequency to the maximum value of displacement (obtained from acceleration signal by dual integration). Physical model and lumped parameter model are shown in Figure 2.4.1. Figure 2.4.1. Physical model (a) and mathematical model (b) of a novel vibration energy harvester. Assuming (𝑧2 > 𝑧1 > 𝑤𝐵 ) and defining 𝑧2 = 𝑤𝐵 + 𝑤1 and using the addition force 𝐹𝑘 = 𝐾(𝑧2 − 𝑧1 ) = 𝐾(𝑤2 − 𝑤1 ) on masses, equation can be derived based on Eq (2.3.3) and (2.3.6): Equations for the first mass: m1 w1  c1 w1  k1 w1  1V1  kcoupling  w2  w1   - m1wB ( 2.4.1) V1  RL C p1V1  1 RL w1  0 Equations for the second mass: m2 w2  c2 w2  k2 w2   2V2  kcoupling  w2  w1   - m2 wB V2  RL C p 2V2   2 RL w2  0 45 (2.4.2) Defining 1  m11 ,  2  m2 2 and 1  K K , 2  K1 K2 we obtain w1  2 m ,1n1 w1  n12 w1 1  1   1 2 n1 w2  1 V1  - wB V1  V1  m11 RL w1  0 RL C p1 w2  2 m,2n 2 w2  n 2 2 w2 1   2    2 2 n 2 w2   2 V2  - wB V2  (2.4.3) V2  m2 2 RL w2  0 RL C p 2 In this system, 𝛼1 = 𝛼2 = 0, (𝐾 = 0) decouples both harvester. Any value of 𝛼1 ≠ 0, 𝛼2 ≠ 0, (𝐾 ≠ 0) creates a coupled energy harvester. Effectiveness of coupling spring will be investigated in detail in the next chapter. 46 CHAPTER 3: PRELIMINARY EVALUATION OF TRAIN VIBRATION DATA AND THE SELECTED VEHs In this Chapter, random characteristics of train vibration data is investigated in detail. In order to harvest the maximum input power, dominant frequencies, where the input vibration has power peaks, are determined by computing Fast Forier Transform (FFT) and Power Spectrum Density (PSD) of acceleration and displacement. 3.1. Introduction Recently, Turkish Government and Metro Istanbul A.S. are paying great efforts to set and run Turkish suburban trains on new metro lines. During train test runnings, vibration measurements from 16 different points on the train body and bogies are collected. Among these points, the data set having the greatest displacements belongs to lateral vibrations at the middle train body is selected as the actual train vibration input data and further used in this research [60]. The acceleration frequency generally varies in between 0 to 100 Hz. Among this frequency bandwidth, there exist frequencies preserving maximum energies and tuning harvester to these dominant frequencies enables efficient VEH by utilizing the maximum input power. Therefore, harvesters are tuned for these selected dominant frequencies. For the evaluation, commertial Midé - Volture cantilever VEHs are analyzed and selected products [61, 62] are tested according to derived modellings in Chapter 2 and these findings are compared with the conducted experimental results that are presented in Volture Piezo Protection Advantage (PPA) [59]. 3.2. Evaluation of Train Vibration Acceleration Data In this research, actual suburban metro train acceleration data is used as input excitation [60]. In this section, acceleration data that is obtained from the middle body lateral motions during actual tests is selected and investigated intensely. To better highlight the reason of selection, comperative evaluation of the vertical acceleration measurement sample at the at the third train bogie is presented in Appendix 3A. 47 To begin with, the effect of the difference in the harvester resonance frequency when tuned to acceleration and displacement dominant frequencies on the output power is researched while subjecting to the same input random acceleration. Furthermore, the methodology is as follows: First of all, among 16 set of train vibration aceleration measurements, the one having the greatest magnitudes are selected (see comperative evaluations in Appendix 3A). Selected raw acceleration signal belonging to the lateral vibrations at the middle train body is filtered through 8th order Butterworth low-pass filter with a cut-off frequency of 100 Hz and then integrated twice to gain the train displacement. After that, FFTs and PSDs of acceleration and displacement data are evaluated with a sampling rate of 300 Hz and dominant frequencies of each are determined. Last step of selection and further evaluation of the commertial cantilever VEHers that is suitable for tuning to the gained dominant frequencies [59, 61–67] is presented in the next section. Sample of filtered acceleration and integrated displacement data of 200 sec-duration are shown in Figure 3.2.1 and 3.2.4, respectively. The difference of the dominant frequencies of acceleration and displacement are clearly seen in respective FFTs in Figure 3.2.2, 5 and Welch PSDs in Figure 3.2.3,6. In Appendix 3A, FFTs, PSDs, Welch PSDs, root mean squares (RMSs) and MovingRMSs evaluation results and related MATLAB codes are presented for acceleration and displacement signals[63, 65, 66, 68]. Decision on the dominant frequency is gathered from the Welch PSDs since unexpected rare shock frequencies can mislead as in Figure Y.10 in Appendix 3A. Figure 3.2.3 and 6 reveals that the acceleration and displacement respective peaks occur around 87.9 Hz and 5.9 Hz with half-power bandwidth –at which the magnitude drops by 3dB- of 28 Hz and 9.3 Hz. These dominant frequencies are further taken as the possible resonance frequencies of the energy harvester [68–79]. 48 Figure 3.2.1. Time-varying low pass filtered train acceleration signal (m/s2, sec). Figure 3.2.2. Single-sided low-pass filtered acceleration amplitude spectrum (m/s2, Hz). 49 Figure 3.2.3. Welch Power Spectrum of Raw Train Acceleration Signal (dB/Hz, Hz). Having half-power bandwidth of 28 Hz between 72 Hz and 100 Hz. Figure 3.2.4. Time-varying train displacement signal (mm). 50 Figure 3.2.5. Single-sided displacement amplitude spectrum (mm, Hz). Figure 3.2.6. Welch power spectrum of train displacement (dB/Hz, Hz). Having halfpower bandwidth of 7 Hz between 2.3 Hz and 9.3 Hz. 51 3.2. Evaluation of Selected VEHs to Validate the Mathematical Model In order totest the mathematical model derived in Chapter 2, an actual cantilever beam type piezoelectricVEH is chosen from the Volture PPA catalog [59]. This manual informs users about the RMS power, RMS voltage and RMS current outputs, optimum loads and peak to peak tip displacements only at the middle clamp location when subjected to harmonic excitation with varying acceleration amplitude and frequencies of the PPA models when tuned to input excitation frequency. Here, validation test conducted for PPA 1001 when sinusoidal acceleration input -having 1g amplitude at 60 - is applied. The expected RMS power and voltage outputs are 1.8 mW and 7.1 V, respectively when load resistance of 28.6 kΩ is used. The used parameters to run the derived mathematical model are taken from the datasheet and as follows: Cp = 100 nF, 1.7g of proof-mass, 2.8 g of beam mass, harvester length (L), width (b) and thickness (h) of 36 mm, 20.8 mm and 0.15 mm, respectively. For the calculation of 𝜃= 𝑑31 .𝑐11 .𝑏.ℎ 𝐿 as in Eq (2.3.5) piezoelectric parameters are taken as d31=-220 pm/V and c11=67 GPa. Damping ratio isobtained from the given electrical and mechanical quality factors [59]. Recalling the constitutive mathematical model equation in Eq (2.3.6): w  2n w  n 2 w - V  - wB (2.3.6) V  RL C pV  RL w  0 is converted to state space form by defining the state variables as relative displacement (w), relative acceleration (𝑤̇ ) and output voltage (V): y1  w , y2  w , y3  V (3.3.1) Then state equations of the energy harvester become: y1  w  y 2 y 2  w    wB  2n y 2  n 2 y1   y3 y3  V  (3.3.2) 1  y2  y3 Cp RL C p Matlab function and code to solve this equation sets are given in Appendix 3A. 52 Instantaneous power input due to acceleration input can be calculated as Pinst (t)  F (t ).z(t )  meff wB .z(t ) where;  F (t )  m eff (3.2.3) wB , z=w  wB  z=y2  wB  wB is the relative acceleration excitation to the base of energy harvester. z is the absolute velocity of the moving mass, and it can be obtained by z=y2  wB . wB is the speed obtained from base excitation acceleration due to integration with MATLAB program in Appendix 3A. However, acceleration data must be high passfiltered (with approximately 10% of Nyquist Butterworth high-pass) and then “cumsum()*dt” must be used to btain velocity data, as shown in Appendix 3A [9]. Instantaneous power output can be obtained from power relation as Pout ,inst (t )  V 2 (t ) RL (3.2.4) From Eqs (3.2.3-4), efficiency of conversion can be obtained from the output to input power ratio. Likewise, overall value for efficiency can be evaluated from the RMS power ratios as notedin Eq (3.2.5).  Pout Pin  PRMS ,out  Alternatively,   PRMS ,in  (3.2.5)    With these investigations, RMS power output of 1.8 mW, RMS voltage output of 6.37 V is obtained for PPA-1001. Compared to the experimental measurement outputs of 1.8 mW for RMS power and 7.1 V for RMS voltage, it is seen that there is some discrepancy of order about 10%. The reason of the difference with mathematical and experimental results is most probably due to the θ parameter used in the mathematical model but changes with geometry and frequency between -97 to -320 pC/N [7-8]. Detailed data values with geometric dimensions about PPA 1001, and PPA1011 is given in Appendix 3B. 53 CHAPTER 4: PERFORMANCE EVALUATIONS OF CLASSICAL AND NOVEL HARVESTERS First three harvesters are setup corresponding to peak frequencies obtained from train vibration acceleration and displacement FFTs and Welch PSDs. From catalog [59], three EHs are selected and parameters are setup so that the devices are tuned to these peak frequencies of 80 Hz, 21 Hz and 7 Hz. These tuned devices performance analyses are conducted independently and also for decoupled and coupled 2 DOF array configurations. 4.1. Introduction In this chapter, explained single and two DOF uncoupled and coupled cantilever beam piezoelectric EHs performance evaluations for output power and efficiency are conducted by following the derived equations in Chapter 2 with real train acceleration and displacement excitation input as characterized in Chapter 3. Through out the analysis in Chapter 2, the peak frequencies in welch PSDs of acceleration and displacement are found as 77 Hz and 21 Hz; and 7 Hz and 21 Hz, respectively. As a start, these three dominant frequencies: 77 Hz, 21 Hz and 7 Hz are taken as the resonance frequencies of harvester devices and selected all two types of Volture-PPA1011 and 2011 VEH models are tuned to these frequencies by adding the respective proof masses of 77 Hz, 21 Hz and 7 Hz. Parameters used in derivations are also set for the corresponding resonance frequencies and RMS power outputs and RMS voltage outputs are evaluated in the scope of single DOF individually. Two DOF designs include configurations of the decoupled array and the novel coupled array. For the decoupled array, the combination of two individual cantilever piezoelectric VEHs is considered by only connecting to the same base frame while having free end and the best combination according to power generation is determined. As described in Section 2.4, the final configuration is the developed novel design 54 consists of two piezoelectric transducer beams and additional connection at the tips with a spring while also sharing the same base frame. Since the array is connected at the end, they become coupled. The key point of both two DOF device models is to harvest vibration input mode than one frequency range and thus, electrical equivalent frequency response behaviour would be the same with band pass filter. This novel harvester is evaluated while having natural frequency combinations of 7 Hz and 21 Hz; 7 Hz and 77 Hz; and 20 Hz and 77 Hz so that those would respond at most to displacement and acceleration inputs. The last component affecting the natural frequency is the coupler in other words, the spring. Thus, spring stiffness K. Relating to the indicated key point, these coupling enables broadband energy harvesting and regarding this aim, mentioned components are explored for the maximum total power and voltage outputs, and efficiency in the following sections. 4.2. Performance Evaluation of Tuned Single DOF Energy Harvesters Midé Volture PPA-1011 piezoelectric VEH model is selected among other models for having low equivalent stiffness value [59]. Tuning and performance evaluation are conducted for two clap locations: -6.0 and 0. Claping at -6.0 enables to use the full range of piezoelectric patch. On the other hand, since stiffness increases as the length decreases, at clamp 0 location, the natural frequency is greater. This requires greater proof mass, which eventually increases the output power. PPA-1001 specifications at clamp -6.0 and 0 are listed in Table 4.2.1: 55 Table 4.2.1. The piezoelectric, unique and common piezo patch and transducer properties for the chosen PPA-1011 at -6.0 and 0 clamp locations [59]. at Clamp -6.0 267.45 N/m at Clamp 0 446.28 N/m at Clamp -6.0 0.645 g at Clamp 0 0.614 g at Clamp -6.0 46 mm at Clamp 0 40 mm Piezo Width, bp 20.8 mm Piezo Thickness, hp 0.15 mm Equivalent Stifness, Km Effective Mass, m Piezo Length , Lp Piezoelectric Charge (Displacement) Coefficient, d31 𝐸 Piezoelectric Elastic Modulus, 𝑐11 Capacitance, Cp -170 pm/V 67 GPa 100 nF In consideration of vertical acceleration dominant frequencies, the evaluation of the required proof masses in order to tune 7 Hz, 21 Hz and 77 Hz are represented in Eq 4.2.1 to 4.2.1c for clamp -6.0 location, and in Eq 4.2.1d to 4.2.1e for clamp 0 location. For the lowest frequency of 7 Hz, required tip mass for tuning is extremely high. As a result, tuning mass for 7 Hz is considered theoretically as in Eq 4.2.1a and 4.2.1d. On the other hand, referring 𝑘𝑒𝑓𝑓 = 3𝐸𝐼⁄𝑙 3 , it is also possible to tune low frequencies as seen in Figure 4.2.1. Used clamp bar and all other components in Figure 4.2.1. are provided with PPA-9011 Clamp Kit [80]. Throughout the calculations, tip mass attachment is chosen so that the effective stiffness value would be as stated in Mide Volture datasheet (Table 4.2.1) [59]. mtip at tuning frequency  Km  2 f tuning  2  meff (4.2.1) 56 Fixed at Clamp -6.0 : mtip at 7 Hz  267.45 N/m mtip at 21 Hz  267.45 N/m mtip at 77 Hz  267.45 N/m  2 .7Hz  2  2 .21Hz  2  2 .77Hz  0.645 kg  0.1376 kg=137.612 g 1000 (4.2.1a)  0.645 kg  0.0147 kg=14.717g 1000 (4.2.1b)  0.645 kg  0.4976 103 kg = 0.4976 g 1000 (4.2.1c)  2 Fixed at Clamp 0 : mtip at 7 Hz  446.28 N/m mtip at 21 Hz  446.28 N/m mtip at 77 Hz  446.28 N/m  2 .7Hz  2  2 .21Hz  2  2 .77Hz  2 0.614 kg  0.2301 kg=230.088g 1000 (4.2.1d)  0.614 kg  0.0250 kg=25.020g 1000 (4.2.1e)  0.614 kg  0.0013 kg = 1.293g 1000 (4.2.1f)  Figure 4.2.1. Mide VEH possible increased length arrangement to tune low frequencies via clamp bar [80]. 57 The optimum load resistance used is as follows [81, 82]: R opt = 1 2 f .C p (4.2.2) Approach :   2 b 2 hbeam 2 e31 ARMS RL 1  Abeam 2 33 RL   P = to maximize  1 P     RMS RMS   2 hpiezo   2    A R 1      beam 33 L 4  h   piezo     hp 1  A 33 RL     A 33 Ropt  hp  R opt = =   A.  hp  A ii C p   1  227360  227.4 k 2  7 Hz 100 109 F 1  75788  75.8 k R opt at 21Hz = 2  21Hz 100 109 F 1  20669  20.7 k R opt at 77 Hz = 2  77 Hz 100 109 F R opt at 7 Hz = (4.2.2a) (4.2.2b) (4.2.2c) As seen in Eq (4.2.2), this optimum load resistance is derived from assumed power output formula. Therefore, numerous load power approaches have been tested along with the manual load sweep. Evaluations have shown that the reasonable and optimum output power and voltage values are gathered when the Eq (4.2.2) is used. Some trials with different load resistances are listed in Table 4.2.2. 4.2.1. Tuned Performance of Single DOF Energy Harvester at 7 Hz Details of PPA-1011 are given in Appendix 4. Matlab code in Appendix 4 is used with train vibration data to evaluate the designed harvester. As seen from Table 4.2.1, Eq (4.2.1a) and (2.2.2a), fixing the base of VEH at clamp -6.0 and by using relative parameters performance evaluation is conducted. Hence, respective RMS power and voltage outputs are resulted as 5.16 mW and 20.7 V with 0.20% efficiency according to Eq (3.3.6). 4.2.2. Tuned Performance of Single DOF Energy Harvester at 21 Hz Similar to Part 4.2.1, PPA-1011 in Appendix 4 and Matlab code in Appendix 4 are used to evaluate the tuned performance at 21 Hz. Unlike previous part, this evaluation is 58 conducted for the base at clamp 0 location. As seen from Table 4.2.1, Eq (4.2.1e) and (2.2.2b), fixing the base of VEH at clamp 0 and by using relative parameters as well as the mass correctrion factor of1 (µ=1), performance evaluation is conducted. Hence, respective RMS power and voltage outputs are resulted as 0.326 mW and 3.252 V with 0.097% efficiency. However, here it is seen that the change in the optimal load has vast effecton the output power generation. As a result, frequency sweep resulted the optimum load as 40 kΩ and leading respective RMS power and voltage outputs of 3.279 mW and 7.335 V with an increased efficiency of 0.64 %. This massive change indicates that power generation is highly dependent on the load resistance. 4.2.3. Tuned Performance of Single DOF Energy Harvester at 80 Hz Similar to Part 4.2.2, PPA-1011 in Appendix 4 and Matlab code in Appendix 4 are used to evaluate the tuned performance at 21 Hz. As seen from Table 4.2.1, Eq (4.2.1f) and (2.2.2c), fixing the base of VEH at clamp 0 and by using relative parameters as well as the mass correctrion factor of 1.25 (µ=1.25) [REF4], performance evaluation is conducted. Hence, respective RMS power and voltage outputs are resulted as 0.005 mW and 0.169 V with 0.020% efficiency. The overall output powers, voltages and efficiencies summarized in Parts 4.2.1, 4.2.2 and 4.2.3 as a response to input excitation, and comparision of load resistances are listed in Table 4.2.2 regarding the selected tuning frequencies of 7 Hz, 21 Hz and 77 Hz. 59 Table 4.2.2. The RMS output powers, voltages and efficiencies as a response to input excitation at tuned frequencies of 7 Hz, 21 Hz and 77 Hz. Tuning Req RMS Power RMS Voltage Frequency (Ω) Output (mW) Output (V) 7 Hz 21 Hz 77 Hz Efficiency (%) 227,480 23.103 46.371 0.515 228,296 23.020 46.371 0.513 75,788 0.326 3.25 0.097 0.64 20,669 0.005 0.169 0.020 Since displacement FFT has peaks around 7 Hz and another one around 20 Hz, higher output power and voltage are obtained from the enrgy harvesters with natural frequencies of7 Hz and 20Hz. 4.3. Performance Evaluation of Tuned Novel Two DOF Energy Harvester Array A novel energy harvester is designed combining two piezoelectric VEHs having two different resonance frequency combinations among 7 Hz, 21 Hz and 77 Hz while having a coupling spring combining the tips of PPA VEHs as in Figure 2.4.1 and Figure 4.3.1. It is aimed to investigate the effect of the spring with a stiffness K on first modes and resulted output power and voltage outputs. This evaluation follows the derivation expressed in Section 2.4 and its Matlab code is given in Appendix E. 60 Figure 4.3.1. The scheme of novel two dof energy harvester. The expected behaviours of the novel VEH array are to response just as two seperate cantilevered beam when the spring stiffness is 0 and to response as if the VEHs and the spring are rigid body when K goes to infinity or in practice, when K value is very large. Therefore, for K value to be 0, the response will be the same as in Section 4.2. which is outlined so far. Recalling the maximum amplitudes of respective displacement and aceleration are 7 Hz and 77 Hz while the common secondary peak around 21 Hz, the selected two resonance frequency combinations are: 7 Hz and 20 Hz, 7 Hz and 77 Hz, and 20 Hz and 77 Hz. The effect of the spring stiffness on power generation for these selected resonance frequency combinations of each beam is researched for the K values of 100, 250, 450 and 1000 N/m, and the results are listed in Table 4.3.1. 61 CHAPTER 5: RESULTS AND DISCUSSION 5.1. Introduction This chapter covers the validation of the proposed approach on solving the constitutive set of equations, detection of the diverged results of the proposed approach from the experimental data via sensitivity analysis and set conditions for the correction to reach more precise evaluation. Simulations indicate the higher mode frequencies are higher than the dominant frequencies of the input train vibration. These investigations are detailed in Appendix 5A. Regarding this fact, only first modes, namely natural frequencies are tuned to the dominant frequencies of the input train vibration signal. 5.2. Proposed approach Validation, Sensitivity Analysis and Correction In this chapter, one major check is conducted over the optimum load resistance. the first step of validation started by chacking the optimum load resistance value. If our approach is correct, then the experimentally found optimum load should give peak on the output power. These validations are deeply covered in Appendix 5B along with the the sensitivity analysis. After correcting the estimation accuracy, it is again returned to optimum load investigation to complete the research. At this point, it is also seen that the optimum load formulas presented in literature [54–58, 82, 83] result sometimes slightly and sometimes widely from the values used in the experiments. As the difference extends, evaluated output power via our proposed approach decreases. Returning back to the second step of correction: Throughout the evaluations -starting the first raw form as in the previously represented constitutive equation sets to the corrected final form- piezoelectric power generation requires correction regarding the input excitation characteristics. Though some harvester structures may have non-linear behaviors for acting as a plate model and sometimes between plate and beam model, this is not the case for this research. Nonetheless, Erturk proposed the first correction factor empirically for the pure piezo structures in 31 and 33 modes [83]. As the correction on the damping coefficient and the correction factor are integrated into the code, it is seen that not only the power and voltage outputs but also the tip-to-tip 62 displacement is converged with the test results. As a reminder, Erturk’s correction factor value that is calculated from his proposed formula does not hold when it is integrated into our evaluation. However, the correction factor is still needed and for the accurately selected value in the range that Erturk suggested, accuracy of our evaluation increases for the researched Midé’s composite piezoelectric beam energy harvesters. All these indicate that not only for power generation of piezoelectric energy harvesters but also the beam vibration response needs correction. This can be seen by tracking the Figure 5.2.1 and the corresponding tip to tip deflection value from Midé PPA datasheet [59]. When PPA-1021 is tuned to 22 Hz with a tip mass of 12.7 g and for 1g input acceleration at resonance, the tip deflection is half of 16.1 mm, that is 8.05 mm. Whereas, Ansys simulation resulted 37% less than this value (Figure 5.2.1, detailed in Appendix 5A). Thus, along with piezoelectric power generation, vibration response of the beam needs correction. Changing the damping factor regarding the characteristics of input vibration eased the great portion of the correction. Figure 5.2.1. Vibration response of PPA 1021-tuned to 22 Hz- at resonance under random vibration acceleration amplitude of 1g, units are in mm. Theoretical evaluations are completed in Matlab for the same inputs as in Midé’s experiments and the results are compared. In comparision with the other proposed analitical formulas in literature [54–58, 82, 83], our approach on solving set of differential constitutive equations results vastly better in terms of the minimum error of the previously explained comparision with experiment data. However, even in our 63 proposed approach, there exists only very few cases come precisely close with the experimental results. Thus, the appraoch needs to be corrected due to the nonlinearities in application. The same issue is also covered by Erturk, and he stated the input amplitude is one of the major factor and derives amplitude correction factor if there is no tip mass and for added tip mass, he suggests mass correction factor. As mentioned in Section 2.3, mass term in the constitutive set of equations are multiplied with the correction factor and for no added tip mass, amplitude correction factor is multiplied with base amplitude value [83]. In constitutive equations, the only and obvious one place is shown as in Eqs (2.3.6, 2.4.1-3) and substituted as(µ𝑚𝑎𝑠𝑠 + µ𝑎𝑚𝑝𝑙𝑖𝑡𝑢𝑑𝑒 ) for no tip mass case asin Eq (2.3.6) and for added tip mass µ𝑚𝑎𝑠𝑠 as seen in Eq (2.3.7). w  2n w  n 2 w -  V  -   mass  amplitude  wB V  RL C pV  RL w  0 w  2n w  n 2 w -  V  - mass wB V  RL C pV  RL w  0 (2.3.6) (2.3.7) Moreover, even when all these corrections are made, amplitude and tip mass are not the only factors affecting the change in the output power of the piezoelectric energy harvesters. In this study, sensitivity analysis indicates that in addition to input amplitude and tip mass, damping coefficient and correction factor are also the terms needs to be arranged regarding these two corrected major components. The common sense in the correction depends basically on the importance of tip mass and the input amplitude for directly affecting the output voltage and power. When the input mass does not exist, total damping or transducer damping alone is too much and effective mass is already very low due to the nature of the harvester structure, which leads very high mass correction factor to be effective. On the other hand, when the tip mass is high, mechanical damping alone is very low and as the amplitude increases, the total damping and low mass correction factor gives better results. For having many terms affecting the output power and voltage, instead of the derivation of correction factor formula including tip mass, acceleration amplitude and damping coefficient variables, via trial and error, the optimum values of all these mentioned parameters are tried to be found 64 for the fixed tuned frequencies, namely tip mass values and acceleration amplitudes. While the details of this investigation are given in Appendix 5B, graphically summarized 3 tip mass case for the relative amplitude inputs of 0.25g, 0.5g, 1g and 2g are expressed in Figure 5.2.2-5.2.4. As discussed in Appendix 5B, confidence intervals based on the corrections on the correction factor and the damping coefficients are clarified in Table 5.2.1. Referring the table, it is seen that for no tip mass and low tip mass, mechanical damping coefficient is dominant with a small transducer damping coefficient transition up until the 25.3 g of tip mass at 0.5g acceleration amplitude. After this amplitude range, damping coefficient is stabilized for the sum of mechanical and transducer damping. In Table 5.2.1, it is also seen that not only as the amplitude increases, but also as tip mass increases, correction factor decreases to its ineffective value of 1. In other words, after around 0.5g amplitude at 25.3 g of tip mass, the model does not need any correction to gain the accurate power output. For no tip mass case, amplitude correction factor is also considered along with the mass correction factor in the model of constitutive set of equations both takes place in the same and only place with 𝑚𝑤𝑏̈ term. Though in this study the chosen method to find the correction factor is trial and error since the empirically driven formula does not hold, the correction factor of maximum 3 also gives clue that it is very close to the summation of the amplitude and mass correction factors. Thus, for improved derivations, this observation is expected to be used in the following future studies. 65 Figure 5.2.2. Sensitivity analysis for output power at the input frequency of 21 Hz, tip mass of 25.3 g and the effective mass of 0.614 g. 66 Figure 5.2.3. Sensitivity analysis for output power at the input frequency of 60 Hz, tip mass of 2.7 g and the effective mass of 0.614 g. 67 Figure 5.2.4. Sensitivity analysis for output power at the input frequency of 146 Hz in average, no tip mass and the effective mass of 0.614 g. 68 One of the most investigated parameter in literature concerns on the optimum load resistance. This subject is mentioned by by Calio et al. and widely covered by du Toit and Erturk. Previously covered determination of the optimum load in this chapter is also estimated via the proposed formulations by Calio et al, du Toit and Erturk as in Eqs (5.1.1-4), respectively. As seen from Table 5.1.1, Calio et al.’s proposed optimum load formula is steady among the same input frequency though the input excitation amplitude varies. Whereas, both the equivalent load sweep results of the runned codes and the estimated equivalent loads by use of duToit and Erturk formulas have tendency to response as the excitation amplitude increases. The referenced du Toit’s formulation is corrected by the multiplication of Calio et al.’s fundamental 𝜔 1 𝑛 𝐶𝑝 term in Eq (5.1.3) and re-calculated. Otherwise, du Toit’s equation results in very low values in between 0.5-2 (see Table 5.1.1).[82][55–57, 83] 1  Abeam 2 33 RL   A  R   1  A. 33 L   A 33 Ropt  hp   h h piezo  p    hp 1 R opt .Calio = = A ii C p Ropt ,duToit   4   4 m 2  2   2 1   2    4 m 2  2 1   k C  eff p   6 Ropt ,duToit ,corrected Ropt 1  n C P 1  n C p  4  2     1     keff C p (5.1.2) 2  2     4   4 m 2  2   2 1   2  6   4 m 2  2 1   keff C p    2 1   m 2      2 m  1     m2 1    2 m2   4  2    1       keff C p 2  2    (5.1.3) (5.1.4) where:  (5.1.1) 2 C p n 2 69 As Table 5.2.1 and 2 reveals both literature evaluation and the closer optimum load approach. For the estimation of optimum load, Calio and Erturk formulas seems close enough and it seems like the calculation load of Erturk is not necessary. Since du Toit takes account of the input frequency, its modification seems much closer results. Although Midé datasheet optimum load values varies, it is notified by Steve Hanly that some can be close to optimum but not the optimum load that are used in experiments and thus calculated as equivalent load. Regarding this fact, to gain the maximum power output of our code, optimum load gained by load resistance sweep is decided to be used in further evaluations. 70 Table 5.2.1. Confidence interval table of all investigated cases for PPA-1011 power output estimation. Selected damping and correction factors are listed along with the optimum load investigation in comparision with the Midé’s experimental loads. 146 Hz, no tip mass 60 Hz, 2.7 g tip mass 21 Hz, 25.3 g tip mass Amp 0.25g 0.5g 1g 2g 0.25g 0.5g 1g 2g Damping Zmech Zmech Zmech Zmech Zmech Zmech Zmech coefficient 0.0167 0.0167 0.0167 0.0167 0.0167 0.0167 0.0167 0.0284 0.25g 0.5g 1g 2g Ztotal Ztotal Ztotal* 0.0167 0.0451 0.0451 0.0451* Ztransducer Ztransducer Correction 3.0 factor Req, exp -Midé 12,100 2.95 2.55 2.28 1.54 1.35 1.16 1.44 1.25 1.57 1.28 12,600 10,200 10,100 25,000 15,800 19,500 17,300 76,600 41,200 33,700 Ropt-code sweep 4,000 5,000 5,000 10,100 16,000 15,000 15,000 20,000 60,000 60,000 60,000 Ropt Calio 11,904 11,904 11,904 11,904 27,250 26,836 26,836 26,836 76,201 76,201 76,201 Ropt Calio,w 10,827 10,901 10,901 10,976 26,526 26,526 26,526 26,526 76,517 75,788 75,788 Ropt duToit Ropt duToit 0.867 0.859 1.097 1.124 1.552 1.159 1.159 1.053 0.819 0.973 0.973 10,322 10,220 13,064 13,378 42,288 31,111 31,111 28,271 62,431 74,150 74,150 9,388 9,359 11,963 12,336 41,163 30,751 30,751 27,945 62,689 73,748 73,748 Ropt Erturk 6,264 6,264 6,264 6,264 14,339 14,121 14,121 14,125 40,109 40,156 40,156 Ropt Erturk,w 5,697 5,736 5,736 5,776 13,958 13,958 13,958 13,962 40,275 39,938 39,938 corrected Ropt duToit corrected,w (*) After 1g value, it is assumed no correction regarding as its previous behavior. 71 1.00* 𝟏 𝝎𝒏 ⋅ 𝑪𝑷 𝟏 𝝎 ⋅ 𝑪𝑷 - 𝟏 𝝎𝒏 ⋅ 𝑪𝑷 𝟏 𝝎 ⋅ 𝑪𝑷 𝟏 𝝎𝒏 ⋅ 𝑪𝑷 𝟏 𝝎 ⋅ 𝑪𝑷 Table 5.2.2. Confidence interval table of all investigated cases for PPA-2011 power output estimation. Selected damping and correction factors are listed along with the optimum load investigation in comparision with the Midé’s experimental loads. 154-147 Hz, no tip mass Amp 0.25g 0.5g 1g 2g 60 Hz, 3.5 g tip mass 0.25g 0.5g 1g 24 Hz, 25.3 g tip mass 2g 0.25g 0.5g 1g 2g Damping Zmech Zmech Zmech Zmech Zmech Zmech Zmech Ztransd. Zmech Ztransd. Ztransd. Ztotal* coefficient 0.0167 0.0167 0.0167 0.0167 0.0167 0.0167 0.0167 0.0284 0.0167 0.0284 0.0284 0.0451* 1.44 1.25 1.57 1.28 1.00* Correction 3.0 factor Req, exp -Midé 7,000 Ropt-code sweep 4,000 2.95 2.55 2.28 4,000 3,300 5,100 10,500 9,000 14,700 18,200 24,000 39,400 30,900 17,200 4,000 4,000 5,100 10,500 11,000 11,000 12,500 32,000 32,000 32,000 35,000 Ropt Calio 5,558 5,558 5,558 5,558 14,082 14,082 14,082 14,082 35,369 35,369 35,369 35,369 Ropt Calio,w 5,439 5,511 5,622 5,698 13,961 13,961 13,961 13,961 34,902 34,902 35,196 36,420 Ropt duToit Ropt duToit 1.525 1.010 0.624 0.681 1.013 1.000 1.241 1.083 0.935 0.856 8,477 5,611 3,470 3,787 14,266 14,266 14,266 14,081 43,898 38,308 33,083 30,284 8,296 5,563 3,509 3,882 14,143 14,143 14,143 13,959 43,319 37,802 32,920 31,184 Ropt Erturk 5,557 5,557 5,557 5,557 14,079 14,079 14,079 14,084 35,360 35,372 35,372 35,413 Ropt Erturk,w 5,438 5,510 5,620 5,697 13,958 13,958 13,958 13,962 34,894 34,905 35,198 36,465 1.54 1.013 1.35 1.013 1.16 corrected Ropt duToit corrected,w (*) At 2g amplitude input, no correction is made and the results are overestimated by 11% for output power. 72 Multiplied term 𝟏 𝝎𝒏 ⋅ 𝑪𝑷 𝟏 𝝎 ⋅ 𝑪𝑷 - 𝟏 𝝎𝒏 ⋅ 𝑪𝑷 𝟏 𝝎 ⋅ 𝑪𝑷 𝟏 𝝎𝒏 ⋅ 𝑪𝑷 𝟏 𝝎 ⋅ 𝑪𝑷 CHAPTER 6: CONCLUSION 6.1. Conclusion In the scope of this thesis, novel designs and methodologies on the estimation of the output power, voltage and tip deflection are introduced. It is also shown that usually discarded dominant frequency of displacement is also suitable as tuning frequency of the energy harvesters. As another contribution to literature, it is observed that the damping of the harvester changes according to the input vibration characteristics and regarding this change in evaluations hıghly contributes to increase the accuracy of the results. It is also shown that using Erturk’s improved version of the constitutive set of piezoelectric equations with acceleration amplitude (for no tip mass case) and mass correction factors also played the final polishing step to reach the accurate result especially in terms of focused RMS power output. Both of these corrections are found by iterative methods and set as boundary conditions for the respective input conditions and formed a new if condition block of our code and all the corrections are finally represented in the same Matlab code. Analysis of simulations are conducted in İTU and it is seen that Ansys modelling results better than SAP2000 simulations for allowing to introduce thin layers of Midé’s composite VEHs. Thus for future simulation studies, simulations will be carried out in Ansys. Completed modal analysis both in Ansys and SAP2000 indicated that higher mode frequencies are higher than the dominant frequencies and maximum band limit of our considered train vibration input. However, when the tip mass is very high or in other words, when the harvester is tuned to very low frequencies of 5 Hz, second mode frequency takes place in the input vibration frequency range. As stated in the following section 6.2, power generation at higer modes and multi DOF coupled designs needs to be investigated further. In conclusion, this thesis study brings new insights to the examination of output rms power and voltage, and tip displacement asn well as indicating EHing from the ignored dissipiated energy at the dominant frequency of the displacement input and finally, 73 suggesting a new coupled multi DOF energy harvester structures. Conclusions on Simulations Prior to theoretical analysis, simulation need is arosed from the aim of tuning higher modes of VEH. If the higher modes were in the range of input dominand frequency bands, thesis research aimed to focus on continuous vibrations. However, both regarding initial theorethical continuous vibration results without tip mass and simulation results with very high tip mass of 88.7 gr posesses greater second mode frequency than the train vibration range limit of 100 Hz. As well as regarding output power generation methodologies in literature, initial estimations without corrections lead to check any possible non-linear behaviour of Midé VEH at resonance. However, it is seen that non-linearity can only be considered at higher modes and certainly not the case for the resonance at first mode as supported by both Ansys and SAP2000. Following analysis also showed that regardless of the power generation, beam deflection alone is simulated less than experimental tip deflection values by 38 % for PPA 1021 at Clamp 0 position with 12.7 g tip mass and when vibration acceleration input is at 22 Hz and 1g. Combination of these outcomes supports the need of corrections in kinetic equations derived from the set of constitutive equations. Finally, SAP2000 is lack of accuracy for lumped mass modelling, yet very easy and fast for users. On the other hand, Ansys simulates much better for instance, for the fiirst mode fequency, natural frequency, is found by %5 error whereas the error of SAP2000 for the first mode frequency is 20% less. 6.2. Future Research Recommendations In spite of many researched subjects in scope of this thesis, there are still many to discover and investigate further. In terms of missing elements and inspired research issues, future recommendations are listed as follows: 74 1. Deep analysis of the measured train vibration data highlight inadequate low sampling frequency as well as noticing some peaks after 100 Hz. To eliminate this problem existing data is doubled by interpolation and increased the sampling frequency up to 600Hz. Thus, this manuel arrangement and consequent result indicated that train vibration measurements need to be repeated with higher sampling frequency. In this manner, new measurements with Slam-StickX of Midé Technology Corporation not only give more accurate data measurement but also provide easy to use Slam Stick Lab Software so that customers would be able to run vibration analysis- FFT, PSD and Spectrogram [70–75], 2. In addition to conducted investigations on train vibration analysis, signal spectrogram analysis could have been better on the selection of the tuning frequencies of VEH since spectrogram allows to detect the most frequently occurring frequencies [63], 3. Analyzing train vibration data along the whole railway line by Dynamic Time Warping (DTW) so that the analysis of the whole duration can be completed in less time, 4. Empirical formulation of amplitude and mass correction factor for the tested Midé Volture PPA products and considering mass correction factor value as less than 1.0 for high amplitudes when tuned to low frequencies, 5. Emprical and/or derived formulation of the damping coefficient instead of 3 boundaries of mechanical, transducer and total damping coefficients, 6. Derivation of the optimum resistance load formula by running more cases, 7. Considering secong mode shape of 88.7 g tip mass added harvester, researching energy harvesting from higher modes in the existence of the since the input vibration range collapses with the second mode frequency of PPA-1021 –tuned to 8.44 Hz (see Appendix 5A), 8. After investigating the coupled three VEHs (3DOF tuned to 3 dominant frequencies) and their power outputs (see Figures 6.2.1-3), analyzing the higher mode effects on modal shapes and output power and voltage when harvesters are coupled by a spring 75 9. Finally, these theorethical investigations should have been also validated and completed with the laboratory tests followed by applications on train and other transportation vehicles. Figure. 6.2.1. W. Mason’s mechanical filter and its electric equivalent. Figure. 6.2.2. Modified 3 DOF VEH inspired from W. Mason’s mechanical filter. Figure. 6.2.3. Lumped Model of the 3 DOF VEH for future examinations. 76 7. REFERENCES 1. Wei, C. K., & Ramasamy, G. (2011). A hybrid energy harvesting system for small battery powered applications. 2011 IEEE Conference on Sustainable Utilization and Development in Engineering and Technology (STUDENT), (October), 165–170. doi:10.1109/STUDENT.2011.6089346 2. Saini, P. K., Biswas, A., & Bhanja, D. (2015). Performance Evaluation and Simulation of Solar Panel, Wind Mill, Fuel Cell Hybrid System for Small Scale Energy Harvesting. Journal of Clean Energy Technologies, 3(6), 417–421. doi:10.7763/JOCET.2015.V3.234 3. Weddell, A. S., Magno, M., Merrett, G. V., Brunelli, D., Al-Hashimi, B. M., & Benini, L. (2013). A Survy of Multi-Source Energy Harvesting Systems. In Design, Automation & Test in Europe Conference & Exhibition (DATE), 2013 (pp. 905–908). New Jersey: IEEE Conference Publications. doi:10.7873/DATE.2013.190 4. Li, P., Gao, S., Cai, H., & Wu, L. (2015). Theoretical analysis and experimental study for nonlinear hybrid piezoelectric and electromagnetic energy harvester. Microsystem Technologies, 10(8). doi:10.1007/s00542-015-2440-8 5. Wu, X., Khaligh, A., & Xu, Y. (2008). Modeling , Design and Optimization of Hybrid Electromagnetic and Piezoelectric MEMS Energy Scavengers. In Custom Integrated Circuits Conference (CICC) (pp. 177–180). San Jose, CA, USA: IEEE. 6. Energy Research Center. (2015). Hybrid Adaptive Ambient Vibration Energy Harvesting. University of Maryland. Retrieved October 10, 2015, from http://www.umerc.umd.edu/projects/harvest01 7. IDTechEx. (2015). Multi­mode energy harvesting. energy harvesting journal. Retrieved October 10, 2015, from http://www.energyharvestingjournal.com/articles/8294/multi-mode-energyharvesting 8. Yang, B. (2010). Hybrid energy harvester based on piezoelectric and electromagnetic mechanisms. Journal of Micro/Nanolithography, MEMS, and MOEMS, 9(2), 023002. doi:10.1117/1.3373516 9. Cook-Chennault, K. a, Thambi, N., & Sastry, a M. (2008). Powering MEMS portable devices—a review of non-regenerative and regenerative power supply systems with special emphasis on piezoelectric energy harvesting systems. Smart Materials and Structures, 17(4), 043001. doi:10.1088/0964-1726/17/4/043001 10. Challa, V. R., Prasad, M. G., & Fisher, F. T. (2009). A coupled piezoelectric– electromagnetic energy harvesting technique for achieving increased power output through damping matching. Smart Materials and Structures, 18(9), 095029. doi:10.1088/0964-1726/18/9/095029 11. Wischke, M., & Woias, P. (2008). A multi-functional cantilever for energy 77 scavenging from vibrations. In Proc. of PowerMEMS’08 (pp. 73–76). Sendai, Japan: IEEE. Retrieved from http://cap.ee.ic.ac.uk/~pdm97/powermems/2008/pdfs/073-76 Wischke, M.pdf 12. Khbeis, M. T. (2010). Development of a Simplified, Mass Producible Hybridized Ambient, Low Frequency, Low Intensity Vibration Energy Scavenger (HalfLives). University of Maryland, College Park. Retrieved from http://drum.lib.umd.edu/handle/1903/10778 13. Eun, Y., Kwon, D., Kim, M., Yoo, I., Sim, J., Ko, H.-J., … Kim, J. (2015). A flexible hybrid strain energy harvester using piezoelectric and electrostatic conversion. Smart Materials and Structures, 23(4), 045040. doi:10.1088/09641726/23/4/045040 14. Halim, M. A., Cho, H. O., & Park, J. Y. (2014). A handy-motion driven, frequency up-converted hybrid vibration energy harvester using PZT bimorph and nonmagnetic ball. Journal of Physics: Conference Series, 557(1), 012042. doi:10.1088/1742-6596/557/1/012042 15. Shan, X., Guan, S., Liu, Z., Xu, Z., Xie, T., & The, I. (2013). A new energy harvester using a piezoelectric and suspension electromagnetic mechanism. Appl Phys & Eng, 14(12), 890–897. doi:10.1631/jzus.A1300210 16. Shan, X., Xu, Z., Song, R., & Xie, T. (2013). A New Mathematical Model for a Piezoelectric-Electromagnetic Hybrid Energy Harvester. Ferroelectrics, 450(1), 57–65. doi:10.1080/00150193.2013.838490 17. Li, P., Gao, S., & Cai, H. (2013). Modeling and analysis of hybrid piezoelectric and electromagnetic energy harvesting from random vibrations. Microsystem Technologies, 21(2), 401–414. doi:10.1007/s00542-013-2030-6 18. Chen, S., Sun, J., & Hu, J. (2013). A Vibration Energy Harvester with Internal Impact and Hybrid Transduction Mechanisms. In 13th International Conference on Fracture (pp. 1–10). Beijing, China: IEEE. 19. Karthik, K. S., Ali, S. F., Adhikari, S., & Friswell, M. I. (2013). Base excited hybrid energy harvesting. In 2013 IEEE International Conference on Control Applications (CCA) (pp. 978–982). Hyderabad, India: IEEE. doi:10.1109/CCA.2013.6662878 20. Ab Rahman, M. F., Kok, S. L., Ruslan, E., Dahalan, A. H., & Salam, S. (2013). Comparison Study between Four Poles and Two Poles Magnets Structure in the Hybrid Vibration Energy Harvester. In IEEE (Ed.), 2013 IEEE Student Conference on Research and Development (SCOReD) (pp. 16–17). Putrajaya, Malaysia Comparison. 21. Sang, Y., Huang, X., Liu, H., & Jin, P. (2012). A Vibration-Based Hybrid Energy Harvester for Wireless Sensor Systems. IEEE Transactions on Magnetics, 48(11), 4495–4498. 22. Tadesse, Y., & Priya, S. (2008). Multimodal Energy Harvesting System: Piezoelectric and Electromagnetic. Journal of Intelligent Material Systems and Structures, 20(5), 625–632. doi:10.1177/1045389X08099965 78 23. Matiko, J. W., Grabham, N. J., Beeby, S. P., & Tudor, M. J. (2014). Review of the application of energy harvesting in buildings. Measurement Science and Technology, 25(1), 012002. doi:10.1088/0957-0233/25/1/012002 24. Larkin, M., & Tadesse, Y. (2014). HM-EH-RT : hybrid multimodal energy harvesting from rotational and translational motions. International Journal of Smart and Nano Materials, 4(4), 257–285. 25. Wischke, M., Masur, M., Goldschmidtboeing, F., & Woias, P. (2010). Electromagnetic vibration harvester with piezoelectrically tunable resonance frequency. Journal of Micromechanics and Microengineering, 20(3), 035025. doi:10.1088/0960-1317/20/3/035025 26. Tang, L., Yang, Y., & Soh, C. K. (2010). Toward Broadband Vibration-based Energy Harvesting. Journal of Intelligent Material Systems and Structures, 21(18), 1867–1897. doi:10.1177/1045389X10390249 27. Khameneifar, F. (2011). Vibration-Based Piezoelectric Energy Harvesting System For Rotary Motion Applications. Simon Fraser University. Retrieved from http://summit.sfu.ca/item/11906 28. Lafont, T., Gimeno, L., Delamare, J., Lebedev, G. A., Zakharov, D. I., Viala, B., … Geoffroy, O. (2015). Magnetostrictive – piezoelectric composite structures for energy harvesting. Journal of Micromechanics and Microengineering, 22(9), 094009. doi:10.1088/0960-1317/22/9/094009 29. Mahmoudi, S., Kacem, N., & Bouhaddi, N. (2014). Enhancement of the performance of a hybrid nonlinear vibration energy harvester based on piezoelectric and electromagnetic transductions. Smart Materials and Structures, 23(7), 075024. doi:10.1088/0964-1726/23/7/075024 30. Becker, P., Folkmer, B., & Manoli, Y. (2009). The Hybrid Vibration Generator , a New Approach for a High Efficiency Energy Scavenger. In IEEE (Ed.), International Workshop on Micro and Nanotechnology for Power Generation and Energy Conversion Applications (pp. 439–442). Washington, DC, USA: PowerMEMS. 31. Xu, Z. L., Wang, X. X., Shan, X. B., & Xie, T. (2012). Modeling and Experimental Verification of a Hybrid Energy Harvester Using Piezoelectric and Electromagnetic Technologies. Advanced Materials Research, 569, 529–532. doi:10.4028/www.scientific.net/AMR.569.529 32. Ali, N. M., Mustapha, A. A., & Leong, K. S. (2013). Investigation of Hybrid Energy Harvesting Circuits Using Piezoelectric and Electromagnetic Mechanisms. IEEE Student Conference on Research and Developement, (December), 16–17. 33. Xia, H., Chen, R., & Ren, L. (2015). Analysis of piezoelectric–electromagnetic hybrid vibration energy harvester under different electrical boundary conditions. Sensors and Actuators A: Physical, 234, 87–98. doi:10.1016/j.sna.2015.08.014 34. Xia, H., Chen, R., & Ren, L. (2015). Sensors and Actuators A : Physical Analysis of piezoelectric – electromagnetic hybrid vibration energy harvester under different electrical boundary conditions. Sensors & Actuators: A. Physical, 234, 79 87–98. doi:10.1016/j.sna.2015.08.014 35. Ab Rahman, M. F., Kok, S. L., Ali, N. M., Hamzah, R. A., & Aziz, K. A. a. (2013). Hybrid vibration energy harvester based on piezoelectric and electromagnetic transduction mechanism. In 2013 IEEE Conference on Clean Energy and Technology (CEAT) (pp. 243–247). Langkawi TBD, Malaysia: Ieee. doi:10.1109/CEAT.2013.6775634 36. Wischke, M., Masur, M., Goldschmidtboeing, F., & Woias, P. (2010). Piezoelectrically tunable electromagnetic vibration harvester. In 2010 IEEE 23rd International Conference on Micro Electro Mechanical Systems (MEMS) (pp. 1199–1202). IEEE. doi:10.1109/MEMSYS.2010.5442427 37. Xu, Z. L., Shan, X. B., Song, R. J., & Xie, T. (2014). Electromechanical modeling and experimental verification of nonlinear hybrid vibration energy harvester. In Applications of Ferroelectrics (Ed.), 2014 Joint IEEE International Symposium on the Applications of Ferroelectric, International Workshop on Acoustic Transduction Materials and Devices & Workshop on Piezoresponse Force Microscopy (pp. 1–4). State College, PA, USA: IEEE. doi:10.1109/ISAF.2014.6923018 38. Ishikuro, H. (2011). Energy Harvesting Technology, System LSI Design. Tokyo, Japan. 39. Castagnetti, D. (2015). A Belleville-spring-based electromagnetic energy harvester. Smart Materials and Structures, 24(9), 94009. doi:10.1088/09641726/24/9/094009 40. Twiefel, J., & Westermann, H. (2013). Survey on broadband techniques for vibration energy harvesting. Journal of Intelligent Material Systems and Structures, 24(11), 1291–1302. doi:10.1177/1045389X13476149 41. Zhou, S., Cao, J., Inman, D. J., Lin, J., Liu, S., & Wang, Z. (2014). Broadband tristable energy harvester : Modeling and experiment verification. Applied Energy, 133, 33–39. doi:10.1016/j.apenergy.2014.07.077 42. Li, P., Gao, S., Niu, S., Liu, H., & Cai, H. (2014). An analysis of the coupling effect for a hybrid piezoelectric and electromagnetic energy harvester. Smart Materials and Structures, 23(6), 065016. doi:10.1088/0964-1726/23/6/065016 43. Dias, J. A. C., Marqui, C. De, & Erturk, A. (2013). Hybrid piezoelectricinductive flow energy harvesting and dimensionless electroaeroelastic analysis for scaling. American Institute of Physics, 044101(102), 1–6. 44. Dias, J. a. C., De Marqui, C., & Erturk, a. (2015). Three-Degree-of-Freedom Hybrid Piezoelectric-Inductive Aeroelastic Energy Harvester Exploiting a Control Surface. AIAA Journal, 53(2), 394–404. doi:10.2514/1.J053108 45. Reuschel, T., Salehian, A., Kotsireas, I., Melnik, R., & West, B. (2011). Analysis and Modelling towards Hybrid Piezo-Electromagnetic Vibrating Energy Harvesting Devices. doi:10.1063/1.3663465 46. Karami, M. A., & Inman, D. J. (2011). Equivalent damping and frequency change for linear and nonlinear hybrid vibrational energy harvesting systems. 80 Journal of Sound and Vibration, 330(23), 5583–5597. doi:10.1016/j.jsv.2011.06.021 47. Karami, M. a., & Inman, D. J. (2010). Nonlinear Hybrid Energy Harvesting utilizing a Piezo-magneto-elastic spring. In M. N. Ghasemi-Nejhad (Ed.), SPIE Smart Structures and Materials + Nondestructive Evaluation and Health Monitoring (Vol. 7643, p. 76430U–76430U–11). doi:10.1117/12.847566 48. Leadenham, S., & Erturk, a. (2015). Nonlinear M-shaped broadband piezoelectric energy harvester for very low base accelerations: primary and secondary resonances. Smart Materials and Structures, 24(5), 055021. doi:10.1088/0964-1726/24/5/055021 49. Gure, N. (2014). TUBITAK 2210-C Master Scholarship Program PROJECT. Ankara. 50. Gure, N. (2015). Technological Entrepreneurship Industry Support Application Project. Ankara. 51. Gure, N. (2015). Technological Entrepreneurship Industry Support Project Work Plan Document. Ankara. 52. Gure, N. (2015). Technological Entrepreneurship Industry Support Presentation Project. Ankara. 53. Meruane, V., & Pichara, K. (2016). A broadband vibration-based energy harvester using an array of piezoelectric beams connected by springs. Shock and Vibration, 2016. doi:10.1155/2016/9614842 54. Hehn, T., & Manoli, Y. (2015). CMOS Circuits for Piezoelectric Energy Harvesters (Vol. 38). Dordrecht: Springer Netherlands. doi:10.1007/978-94-0179288-2 55. du Toit, N. E., Wardle, B. L., & Sang-Gook, K. (2005). Design Consıderations For Mems-Scale Piezoelectric Mechanical Vibration Energy Harvesters. Integrated Ferroelectrics, 71(1), 121–160. doi:10.1080/10584580590964574 56. du Toit, N. E. (2005). Modeling and Design of a MEMS Piezoelectric Vibration Energy Harvester. Design. 57. Erturk, A., & Inman, D. J. (2011). Piezoelectric Energy Harvesting. Chichester, UK: John Wiley & Sons, Ltd. doi:10.1002/9781119991151 58. Priya, S., & Inman, D. J. (2009). Energy Harvesting Technologies. (S. Priya & D. J. Inman, Eds.)Energy Harvesting Technologies. Boston, MA: Springer US. doi:10.1007/978-0-387-76464-1 59. Midé Technology Corporation. (2016). Volture Piezo Protection Advantage (PPA) Products Datasheet and User Manual. Medford. MA. Retrieved from https://cdn.shopify.com/s/files/1/0964/1872/files/ppa-piezo-productdatasheet.pdf?__hssc=190384372.7.1474317704847&__hstc=190384372.3e5080 d681d7c1f768ecaf6099d68c81.1461806530834.1471819316657.1474317704847 .30&__hsfp=2090415564&hsCtaTracking=6c786ff5-f2ff 60. Duman, F., Topsakal, F. Z., Pasahan, H., & Kucukcicibiyik, E. (2016). RTE 2000 81 Aracinin UIC-518 Standardina Gore Seyir Emniyeti Test Raporu. 61. Midé Technology Corporation. (2016). Vibration Energy Harvesting with Piezoelectrics. Midé Technology Corporation. Retrieved September 19, 2016, from http://www.mide.com/collections/vibration-energy-harvesting-withprotectedpiezos?__hssc=190384372.7.1474317704847&__hstc=190384372.3e5080d681d 7c1f768ecaf6099d68c81.1461806530834.1471819316657.1474317704847.30&_ _hsfp=2090415564&hsCtaTracking=1c27639e-3b9 62. Ludlow, C. (2016). Four Steps to Selecting a PiezoelectricEnergy Harvesting Device. Midé Technology Corporation. Retrieved September 19, 2016, from blog.mide.com/four-steps-to-selecting-a-piezoelectric-energy-harvesting-device 63. Hanly, S. (2016). Vibration Analysis: FFT, PSD, and Spectrogram Basics. Midé Technology Corporation. Retrieved September 19, 2016, from http://blog.mide.com/vibration-analysis-fft-psd-and-spectrogram 64. Hanly, S. (2016). Ever Wonder what the Resonance of a Bridge is? Midé Technology Corporation. Retrieved September 19, 2016, from http://blog.mide.com/university-drive-pedestrian-bridge-resonance 65. Hanly, S. (2016). MATLAB vs Python: Speed Test for Vibration Analysis. Midé Technology Corporation. Retrieved September 19, 2016, from http://blog.mide.com/matlab-vs-python-speed-for-vibration-analysis-freedownload 66. Scheidler, P. (2016). Frequency Leakage in Fourier Transforms. Midé Technology Corporation. Retrieved September 19, 2016, from http://blog.mide.com/fourier-transform-leakage 67. Hanly, S. (2016). Shock & Vibration Testing Overview. USA: Globalspec WEBINAR- Midé Technology Corporation. 68. Hanly, S. (2016). Shock & Vibration Overview. (S. Hanly, Ed.). Midé Technology Corporation. 69. Aravanis, T.-C. I., Sakellariou, J. S., & Fassois, S. D. (2016). Spectral analysis of railway vehicle vertical vibration under normal operating conditions, (August). doi:10.1080/23248378.2016.1221749 70. Leblanc, J., & Ave, B. (2015). Test Report Prepared for Mide, (978), 1–85. 71. Hanly, S. (2017). Top 8 Vibration Analysis. Midé Technology Corporation. 72. Midé Technology Corporation. (2017). SLAM STICK User Manual Vibration – Temperature – Pressure Data Logger. Midé Technology Corporation. 73. Midé Technology Corporation. (2017). Slam Stick Quick Start Guide. Midé Technology Corporation. doi:10.1160/TH10-05-0297 74. Court, J. (2017). Estimating Location with Pressure Data and Dynamic Time Warping. Midé Technology Corporation. 75. For, P., Field, M., Field, E., Field, M., Field, E., & By, P. (n.d.). NATIONAL TECHNICAL SYSTEMS (NTS) Customer: Mide Technology Corporation 82 Product: Slam Stick X, 032030. 76. Silva, C. W. De, & Group, F. (n.d.). Vibration and Shock Edited by. 77. Sinha, J. K. (n.d.). Vibration Analysis, Instruments, and Signal Processing. 78. Rao, S. S., & Hall, P. (n.d.). Mechanical Vibrations Fifth Edition. 79. Fundamentals of Mechanical Vibration 2nd by Kelly XXX.pdf. (n.d.). 80. Midé Technology Corporation. (2017). Piezo Product: PPA-9001 Clamp Kit. Midé Technology Corporation. Retrieved from http://www.mide.com/products/piezo-product-ppa-9001-clamp-kit 81. Hwang, S. J., Jung, H. J., Kim, J. H., Ahn, J. H., Song, D., Song, Y., … Sung, T. H. (2015). Designing and manufacturing a piezoelectric tile for harvesting energy from footsteps. Current Applied Physics, 15(6), 669–674. doi:http://dx.doi.org/10.1016/j.cap.2015.02.009 82. Caliò, R., Rongala, U., Camboni, D., Milazzo, M., Stefanini, C., de Petris, G., & Oddo, C. (2014). Piezoelectric Energy Harvesting Solutions. Sensors, 14(3), 4755–4790. doi:10.3390/s140304755 83. Erturk, A. (2009). Electromechanical Modeling of Piezoelectric Energy Harvesters. Virginia Polytechnic Institute and State University. 84. Hanly, S. (2017). Your Best 6 Options for Vibration Analysis Programming. Midé Technology Corporation. 85. Midé Technology Corporation. (2017). SLAM STICK Shock & Vibration Data Loggers. Midé Technology Corporation. 83 ÖZGEÇMİŞ PERSONAL INFORMATION Full Name Country Date of Birth Place of Birth Gender Marital status Citizenship : Nazenin Gure : Turkey : 20. 10. 1987 : Van (Turkey) : Female : Single : Republic of Turkey – T.C. CONTACT INFORMATION Address Phone Mobile Phone e-mail : Yali St. Cicekdagi Rd. No: 5/12 Maltepe-34844 Istanbul/TURKEY : +90 216 457 02 90 : +90 535 926 10 78 : [email protected] / [email protected] EDUCATION Primary Education High School– University Scholarship Work Experience B.Sc. Thesis M.Sc. Thesis Computer Skills : Private 75. Year Dokuz Eylul University Foundation (DEVAK), 2002 : Besiktas High School, 2005 : Marmara University, Faculty of Engineering Environmental Engineering Department (English Education), 2011 Mechanical Engineering Department (English Education), M.Sc. : Scientific and Technological Research Council (TUBITAK, 2210-C) (http://www.tubitak.gov.tr/sites/default/files/2210c_2014-2_listeler.pdf) : Part Time Student Assistant (Sep’11-Jun’14) Marmara University, Faculty of Engineering, Civil Engineering Dep. Advisor: Dr. Vail Karakale (Waiel Mowrtage) CEO (Sep 2015- Still) founder of ENHAS Research and Development Energy Systems Industry and Commerce Limited Company : Phosphorus Removal from Wastewater by Ultrasound Application : Integration of Mechanical Filters on Vibrational Energy Harvesters : VB, Fortran, MATLAB, Simscape, Ansys, Gambit, Fluent, Multisim, NI AWR Design Environment - Microwave Office, Labview, AutoCAD 84 PROJECT: Technological Entrepreneurship Industry Support (TGSD) by Republic of Turkey’s Ministry of Science, Industry, and Technology (MoSIT) for: “R&D of Novel Vibrational Energy Harvesters (VEHs) and Integration of Mechanical Filters on VEHs”, (Sep 2015 to Sep 2016). RESEARCHES: 1) N. GURE, A. KAR, E. TACGIN, A. SISMAN and N. M. TABATABAEI, Chapter A: “Hybrid Energy Harvesters” for the book on “Energy Harvesting and Energy Efficiency: Technology, Methods and Applications” (will be published by Springer in Feb 2017). 2) N. GURE, M. YILMAZ, “Alternative Solution via Car Window Filming Implementation to Combat Global Warming and Resulted Benefits around Geographic Europe and the European Union”, Special Issue on International Journal of Global Warming (IJGW), Inderscience (Accepted on 26 Aug 2015, will be published for Vol 10, No 1, 2 May 2016). (for abstract please see: http://www.inderscience.com/info/ingeneral/forthcoming.php?jcode=ijgw) 3) Bilge ALPASLAN KOCAMEMİ, Nazenin GURE, Feraye SARIALIOGLU, Cansu KUZEY, Ahmet Mete SAATCI, "Application of Low Intensity Ultrasound to Enhance Biological Phosphorous Removal", International Conference on Civil and Environmental Engineering (ICOCEE, Cappadocia2015), Cappadocia, TURKEY, 20-23 May, 2015. (http://www.icocee.org) 4) Nazenin Gure, Mustafa Yilmaz, “Alternative Way to Reduce Vehicle Emissions in Summer with the help of Car Window Filming and Car Window Filming's Economic Benefits over WA, NY, NC, U.S.A. and Istanbul, TURKEY”, Journal of Energy and Power Sources, Vol. 2, No. 1, Jan. 2015. (http://www.ethanpublishing.com/uploadfile/2015/0202/20150202033958640.pdf) CONFERENCE PRESENTATIONS (Presenter*): Poster : Nazenin Gure* and Mustafa Yilmaz, “Reducing Vehicle Emissions via Car Window Filming & Its Economic Benefits over WA, NY, NC, U.S.A. and Istanbul, TR”, The 12th Annual CMAS Conference, Chapel Hill, NC, USA, October 28-30, 2013. (https://www.cmascenter.org/conference/2013/abstracts/gure_alternative_w ay_2013.pdf) Oral : N. GURE*, M. YILMAZ, “Car Window Filming Effect: Reduced Fuel Consumption, Reduced Vehicle Emissions & Economic Contribution around Europe and EU in summer”, 13th International Conference on Clean Energy (ICCE 2014), Istanbul, TURKEY, June 8-12, 2014. 85 APPENDIX 3A: CHAPTER 3: Preliminary Evaluation of Train Vibration Data and the Selected VEHs ANALYSIS OF TRAIN VIBRATION SIGNAL DATA AND DETERMINATION OF THE TUNING FREQUENCIES [63–68, 70–75, 84, 85] Train vibration data is measured and recorded at the third train bogie with the rate of 300 samples per second. Regarding Nyquist theorem, since known aliasing frequency is 100 Hz, oversampling frequency is selected as three-times of the aliasing frequency. As a result, during acceleration measurements are filtered with a low pass filter having the cut-off frequency of 100 Hz. The analyzed vibration data is selected among other 16 different data sets for having the greatest excitation amplitude. Imported actual acceleration data is first filtered with 8th order Butterworth high-pass filter and then integrated to evaluate the velocity data. For velocity data to be integrated, it is again filtered with 8th order Butterworth high-pass filter and then integrated to evaluate the displacement data. The high pass cut-off frequency of 4.5Hz is selected for applicable tuning range of piezo-beams, and more accurate and clear filtered output. Next, FFT and PSD [69] of the calculated displacement and acceleration data are estimated and plotted as in related following figures. The complete input vibration analysis is composed of three different investigation of train acceleration belonging different train locations among 16 measurements. Train vibration data as analyzed in 3A.1. is selected for the determination of the tuning frequencies, which is decided upon dominant frequencies. The reason behind this decision is that the suspected wrong measurements of the initially analyzed train vibration signals. In chronological order those are: (1) Lateral acceleration data at the middle train body (Section 3A.4), (2) Vertical train vibration at the third train bogie (Section 3A.3), (3) the selected data for 3DOF VEH Vertical acceleration data at the middle train body (Section 3A.2) and lastly (4) the selected data for 2DOF VEH with the same location with (1) -lateral acceleration data at the middle train body, but measured at different test run (Section 3A.1). This is the reason why the resulted 86 examinations of (1) and (4) are close. One difference of (3) is that the original sampling frequency of 300 Hz is doubled up to 600 Hz by interpolation and then analyzed as in Section 3A.2. The reason is explained in Chapter 3, under Section 3.2. Evaluation of Train Vibration Acceleration Data, in detail. As seen in Sections 3A.2 and 3, the vertical vibration amplitudes are lower than lateral ones, as expected from train movement. Also, as seen in Figures 3A.2.10, 12, 15 and 17, there exists three dominant frequencies at around 5 Hz, 20 Hz and 80 Hz. This three input power peaks can be harvested by 3 DOF VEH. Whereas, for the lateral vibration input, there are two dominant frequencies -one for acceleration at 88 Hz and the other for displacement at 6 Hz- exist and this input can be harvested by 2 DOF VEH. In regard to this perspective, input train acceleration vibration signals are analyzed to find the most suitable dominant frequencies for vibration energy harvesting as follows: 3A.1. For 2 DOF VEH, Selected Train Vibration Signal Characteristics and Analysis: Lateral acceleration data at the middle train body FFTs of acceleration and displacement signals ……………………………..…92 PSDs of acceleration FFT and displacement FFTs: ........................................................ 97 Welch Power Spectral Estimator for Acceleration and displacement signals: ............. 102 RMS Values and Time-varying Plots of raw Acceleration signal, filtered Acceleration and Displacement: ................................................................................... 105 % publishformat_Train_Vibr_FFT_PSD_Welch_RMS.m MATLAB File clear all; close all; clc; filename = 'Tez_train_data.xlsx'; data =xlsread(filename); acc_signal =xlsread(filename,'B:B'); time =xlsread(filename,'A:A'); This program high pass (4.5 Hz) and low-pass (100 Hz) filters and integrates acceleration signal to displacement signal. Plots time varying signals, FFT, Welch PSD and RMS of Acceleration and Displacement signals. dt=1/300; % sampling rate 300 sample per second L=length(acc_signal); % number of data RTEALB3, bogie 3 vertical Fs=1/(time(2)-time(1)); %300Hz, Sampling frequency 87 %Designing Butterworth High-pass and Low-pass filters: [b,a]=butter(8,0.03,'high'); % designing butterworth highpass filter: high-pass cutoff frequency 0.03 * 150 = 4.5 hz [bb,aa]=butter(8,0.6667);% low-pass cutoff frequency 100/150 * 150 = 4.5 hz % acceleration signal highpass filtered at 4.5Hz: accf=filter(b,a,acc_signal); % acc signal low pass filtered at 100Hz: accfl=filter(bb,aa,acc_signal); % acc signal low&high pass filtered between: 4.5 to 100Hz: acc_hp_lp_filtered=filter(bb,aa,accf); %(4.5Hz-100Hz). %Integrating acceleration signal to gain displacement: veli=1000*cumsum(acc_hp_lp_filtered)*dt; % filtered acceleration integrated & conv to mm velf=filter(b,a,veli); %integrated velocity highpass filtered disp=cumsum(velf)*dt; % filtered velocity integrated %Plotting Time varying- Acceleration, Velocity and Displacement Signals: %Time varying- Acceleration in m/s^2 and 'g': figure(1) plot(time, accfl,'m')%magenta color title('Filtered Acceleration Signal','fontsize',11,'Fontname','Timesnewroman'); xlabel('Time (Sec)','fontsize',11,'Fontname','Timesnewroman'); ylabel('Acceleration (m/s^2)','fontsize',11,'Fontname','Timesnewroman'); set(gca,'fontsize',11,'Fontname','Timesnewroman'); grid on fprintf(' \n Figure 1. Time-varying Filtered Input Acceleration Signal(m/s^2) \n ') 88 Figure 3A.1.1. Time-varying Filtered Input Acceleration Signal (m/s2). figure(2) plot(time, accfl/(9.81),'m')%where g=9.81 m/s^2, magenta color title('Filtered Acceleration Signal','fontsize',11,'Fontname','Timesnewroman'); xlabel('Time (Sec)','fontsize',11,'Fontname','Timesnewroman'); ylabel('Acceleration (g)','fontsize',11,'Fontname','Timesnewroman'); set(gca,'fontsize',11,'Fontname','Timesnewroman'); grid on fprintf(' \n Figure 2. Time-varying Filtered Input Acceleration Signal(g) \n ') 89 Figure 3A.1.2. Time-varying Filtered Input Acceleration Signal (g). %Time varying- Velocity figure(3) plot(time, velf,'g')%green color title('Filtered Velocity','fontsize',11,'Fontname','Timesnewroman'); xlabel('Time (sec)','fontsize',11,'Fontname','Timesnewroman'); ylabel('Velocity (mm/sec)','fontsize',11,'Fontname','Timesnewroman'); set(gca,'fontsize',11,'Fontname','Timesnewroman'); grid on fprintf(' \n Figure 3. Time-varying Filtered Input Velocity(mm/sec) \n ') 90 Figure 3A.1.3. Time-varying Filtered Input Velocity (mm/sec). %Time varying- Displacement figure(4) plot(time, disp)%original blue color title('Displacement Signal','fontsize',11,'Fontname','Timesnewroman'); xlabel('Time (Sec)','fontsize',11,'Fontname','Timesnewroman'); ylabel('Displacement (mm)','fontsize',11,'Fontname','Timesnewroman'); set(gca,'fontsize',11,'Fontname','Timesnewroman'); grid on fprintf(' \n Figure 4. Time-varying Input Displacement (mm) \n ') 91 Figure 3A.1.4. Time-varying Input Displacement (mm). FFTs of acceleration and displacement signals FFT of Low-pass filtered Acceleration signal with f_cutoff= 100Hz. NFFT = 2^nextpow2(L); % Next power of 2 from length of y: it rounds up to the number such that will be as a power of 2 accflfft = fft(accfl,NFFT)/L; % FFT of Low-pass filtered Acceleration signal f = Fs/2*linspace(0,1,NFFT/2); % FFT freq axis, sampling frequency is Fs=300 %fft of Acceleration plot figure(5) plot(f,2*abs(accflfft(1:NFFT/2)),'m')%magenta color. For Single-Sided fft: 2*(abs value). xlim([0 100]) title_head = sprintf('Single-Sided Low-pass Filtered \n Acceleration Amplitude Spectrum'); title(title_head,'fontsize',11,'Fontname','Timesnewroman'); xlabel('Frequency (Hz)','fontsize',11,'Fontname','Timesnewroman'); ylabel('|Y(f)|, (m/s^2)','fontsize',11,'Fontname','Timesnewroman'); grid on fprintf(' \n Figure 5. Single-Sided low-pass Filtered Acceleration Amplitude Spectrum (m/s^2, Hz)\n ') 92 Figure 3A.1.5. Single-Sided low-pass Filtered Acceleration Amplitude Spectrum (m/s2, Hz). %the same fft of acc plot in 'dB' figure(6) plot(f,20*log10(2*abs(accflfft(1:NFFT/2))),'r')%red color, dB=20log10(A1/A2) & dB=10log10(Power1/Power2) xlim([0 100]) title_head = sprintf('Single-Sided Low-pass Filtered \n Acceleration Amplitude Spectrum'); title(title_head,'fontsize',11,'Fontname','Timesnewroman'); xlabel('Frequency (Hz)','fontsize',11,'Fontname','Timesnewroman'); ylabel('20*log10(|Y(f)|), (dB)','fontsize',11,'Fontname','Timesnewroman');% dB=20log10(A1/A2) & dB=10log10(Power1/Power2) set(gca,'fontsize',11,'Fontname','Timesnewroman'); grid on fprintf(' \n Figure 6. Single-Sided low-pass Filtered Acceleration Amplitude Spectrum (dB, Hz) \n ') 93 Figure 3A.1.6. Single-Sided low-pass Filtered Acceleration Amplitude Spectrum (dB, Hz) FFT of Position signal: dispfft = fft(disp,NFFT)/L; % in (mm)s. FFT of displacement signal %fft of position plot: figure(7) plot(f,2*abs(dispfft(1:NFFT/2)))%For Single-Sided fft: 2*(abs value) xlim([0 100]) title_head = sprintf('Single-Sided \n Displacement Amplitude Spectrum'); title(title_head,'fontsize',11,'Fontname','Timesnewroman'); xlabel('Frequency (Hz)','fontsize',11,'Fontname','Timesnewroman'); ylabel('|U(f)|, (mm)','fontsize',11,'Fontname','Timesnewroman'); set(gca,'fontsize',11,'Fontname','Timesnewroman'); grid on fprintf(' \n Figure 7. Single-Sided Displacement Amplitude Spectrum (mm, Hz) \n ') 94 Figure 3A.1.7. Single-Sided Displacement Amplitude Spectrum (mm, Hz). %ZOOMED fft of position plot: figure(8) plot(f,2*abs(dispfft(1:NFFT/2)))%For Single-Sided fft: 2*(abs value) axis([4 8 0 0.035]) title_head = sprintf('ZOOMED Single-Sided \n Displacement Amplitude Spectrum'); title(title_head,'fontsize',11,'Fontname','Timesnewroman'); xlabel('ZOOMED Frequency (Hz)','fontsize',11,'Fontname','Timesnewroman'); ylabel('|U(f)|, (mm)','fontsize',11,'Fontname','Timesnewroman'); set(gca,'fontsize',11,'Fontname','Timesnewroman'); grid on fprintf(' \n Figure 8. Zoomed Single-Sided Displacement Amplitude Spectrum (mm, Hz) \n ') 95 Figure 3A.1.8. Zoomed Single-Sided Displacement Amplitude Spectrum (mm, Hz). %the same FFT of position plot in 'dB' figure(9) plot(f,20*log10(2*abs(dispfft(1:NFFT/2))), 'c')%cyan color, dB=20log10(A1/A2) & dB=10log10(Power1/Power2) xlim([0 100]) title('Single-Sided Displacement Amplitude Spectrum','fontsize',11,'Fontname','Timesnewroman') xlabel('Frequency (Hz)','fontsize',11,'Fontname','Timesnewroman'); ylabel('20*log10(|U(f)|), (dB)','fontsize',11,'Fontname','Timesnewroman');% dB=20log10(A1/A2) & dB=10log10(Power1/Power2), set(gca,'fontsize',11,'Fontname','Timesnewroman'); grid on fprintf(' \n Figure 9. Single-Sided Displacement Amplitude Spectrum (dB, Hz) \n ') 96 Figure 3A.1.9. Single-Sided Displacement Amplitude Spectrum (dB, Hz). PSDs of acceleration FFT and displacement FFTs: PSD of the acceleration FFT: xdft_acc = fft(accfl); %FFT of the acc is selected for PSD evaluation xdft_acc = xdft_acc(1:(L+1)/2);%where L=length(accfl) not length(acc_signal) not length(accfft) psdx_acc = (1/(Fs*L)) * abs(xdft_acc).^2; %Power is directly proportional with the square of the absolute value of the position amplitude psdx_acc(2:end-1) = 2*psdx_acc(2:end-1); %dB/Hz freq_acc = 0:Fs/L:Fs/2; % PSD of FFT of Acceleration Plot figure(10) plot(freq_acc,psdx_acc,'m')%magenta color xlim([0 100]) grid on title_head = sprintf('Periodogram Using FFT of the \n Low-pass Filtered Acceleration Signal'); title(title_head,'fontsize',11,'Fontname','Timesnewroman'); xlabel('Frequency (Hz)','fontsize',11,'Fontname','Timesnewroman'); ylabel('Power/Frequency ( ((m/s^2)^2)/Hz )','fontsize',11,'Fontname','Timesnewroman'); set(gca,'fontsize',11,'Fontname','Timesnewroman'); 97 fprintf(' \n Figure 10. Periodogram Using FFT of the low-pass filtered Acceleration Signal (((m/s^2)^2)/Hz, Hz) \n ') Figure 3A.1.10. Periodogram Using FFT of the low-pass filtered Acceleration Signal (((m/s2)2)/Hz, Hz). %the same PSD of Acceleration Plot in 'dB' figure(11) plot(freq_acc,10*log10(psdx_acc),'r')%magenta color, dB=20log10(A1/A2) & dB=10log10(Power1/Power2) %xlim([0 100]) grid on title_head = sprintf('PSD Using FFT of the \n Low-pass Filtered Acceleration Signal'); title(title_head,'fontsize',11,'Fontname','Timesnewroman'); xlabel('Frequency (Hz)','fontsize',11,'Fontname','Timesnewroman'); ylabel('Power/Frequency (dB/Hz)','fontsize',11,'Fontname','Timesnewroman'); set(gca,'fontsize',11,'Fontname','Timesnewroman'); fprintf(' \n Figure 11. PSD Using FFT of the low-pass filtered Acceleration Signal (dB/Hz, Hz) \n ') 98 Figure 3A.1.11. PSD Using FFT of the low-pass filtered Acceleration Signal (dB/Hz, Hz). PSD of position signal calculation for position: xdft_disp = fft(disp/1000); %in meters , FFT of the acc is selected for PSD evaluation xdft_disp = xdft_disp(1:(L+1)/2);%where L=length(accfl) not length(acc_signal) not length(accfft) psdx_disp = (1/(Fs*L)) * abs(xdft_disp).^2; %Power is directly proportional with the square of the absolute value of the position amplitude psdx_disp(2:end-1) = 2*psdx_disp(2:end-1); %dB/Hz freq_disp = 0:Fs/L:Fs/2; % PSD of FFT of the position signal Plot figure(12) plot(freq_disp,(psdx_disp))%original blue color, dB=20log10(A1/A2) & dB=10log10(Power1/Power2) xlim([0 100]) ylim([0 1.21e-7]) title('Periodogram Using FFT of the Displacement','fontsize',11,'Fontname','Timesnewroman'); xlabel('Frequency (Hz)','fontsize',11,'Fontname','Timesnewroman'); ylabel('Power/Frequency ((m^2)/Hz)','fontsize',11,'Fontname','Timesnewroman'); set(gca,'fontsize',11,'Fontname','Timesnewroman'); grid on fprintf(' \n Figure 12. Periodogram Using FFT of Displacement ((m^2)/Hz, Hz) \n ') 99 Figure 3A.1.12. Periodogram Using FFT of Displacement ((m2)/Hz, Hz). ZOOMED PSD of FFT of the position signal Plot figure(13) plot(freq_disp,psdx_disp)%original blue color, dB=20log10(A1/A2) & dB=10log10(Power1/Power2) axis([4 8 0 1.21e-7]) title_head = sprintf('ZOOMED Periodogram Using \n FFT of the Displacement'); title(title_head,'fontsize',11,'Fontname','Timesnewroman'); xlabel('ZOOMED Frequency (Hz)','fontsize',11,'Fontname','Timesnewroman'); ylabel('Power/Frequency ((m^2)/Hz)','fontsize',11,'Fontname','Timesnewroman'); set(gca,'fontsize',11,'Fontname','Timesnewroman'); grid on fprintf(' \n Figure 13. Zoomed Periodogram Using FFT of Displacement ((m^2)/Hz, Hz) \n ') 100 Figure 3A.1.13. Zoomed Periodogram Using FFT of Displacement ((m2)/Hz, Hz). %the same PSD of Acceleration Plot in 'dB' figure(14) plot(freq_disp,10*log10(psdx_disp),'c')%magenta color %xlim([0 100]) title('Periodogram Using FFT of the Displacement','fontsize',11,'Fontname','Timesnewroman') xlabel('Frequency (Hz)','fontsize',11,'Fontname','Timesnewroman'); ylabel('Power/Frequency (dB/Hz)','fontsize',11,'Fontname','Timesnewroman'); set(gca,'fontsize',11,'Fontname','Timesnewroman'); grid on fprintf(' \n Figure 14. PSD Using FFT of Displacement (dB/Hz, Hz) \n ') 101 Figure 3A.1.14. PSD Using FFT of Displacement (dB/Hz, Hz). Welch Power Spectral Estimator for Acceleration and displacement signals: Create a Welch spectral estimator for acc and disp. h_welch= spectrum.welch; % Welch PSD of Raw and Filtered Acceleration: %One-sided PSD of welch of the raw Acceleration signal: Hpsd_raw_acc=psd(h_welch,acc_signal,'Fs',Fs); %One-sided PSD of welch of the low-pass filtered Acceleration signal: Hpsd_filtered_acc=psd(h_welch,accflfft,'Fs',Fs); % PSD of Welch spectral of the raw Acceleration Signal Plot: figure (15) plot(Hpsd_raw_acc) xlabel('Frequency, (Hz)','fontsize',11,'Fontname','Timesnewroman'); yaxis_head = sprintf('Raw Acceleration Signal \n Welch Power/Freq, (dB/Hz)'); ylabel(yaxis_head,'fontsize',11,'Fontname','Timesnewroman'); title('Welch PSD of Raw Train Acceleration Signal','fontsize',11,'Fontname','Timesnewroman'); set(gca,'fontsize',11,'Fontname','Timesnewroman'); grid on; fprintf(' \n Figure 15. Welch Power Spectrum of Raw Train Acceleration Signal(dB/Hz, Hz) \n ') 102 Figure 3A.1.15. Welch Power Spectrum of Raw Train Acceleration Signal (dB/Hz, Hz). PSD of Welch spectral of low-pass filtered Acceleration Signal Plot: figure (16) plot(Hpsd_filtered_acc) xlim([0 150]) xlabel('Frequency, (Hz)','fontsize',11,'Fontname','Timesnewroman'); yaxis_head = sprintf('Train Low-pass Filtered Acc Signal \n Welch Power/Frequency, (dB/Hz)'); ylabel(yaxis_head,'fontsize',11,'Fontname','Timesnewroman'); title('Welch PSD of Train Low-pass Filtered Acc Signal','fontsize',11,'Fontname','Timesnewroman'); set(gca,'fontsize',11,'Fontname','Timesnewroman'); grid on; fprintf(' \n Figure 16. Welch Power Spectrum of Train Filtered Acceleration Signal (dB/Hz, Hz) \n ') 103 Figure 3A.1.16. Welch Power Spectrum of Train Filtered Acceleration Signal (dB/Hz, Hz). Welch PSD of Displacement: 104 %One-sided PSD of welch of Displacement: Hpsd_disp=psd(h_welch,disp,'Fs',Fs);% Calculate and plot the one-sided PSD. % PSD of Welch spectral of Displacement Plot: figure (17) plot(Hpsd_disp) xlabel('Frequency, (Hz)','fontsize',11,'Fontname','Timesnewroman'); yaxis_head = sprintf('Train Displacement Welch Power/Frequency, \n (dB/Hz)'); ylabel(yaxis_head,'fontsize',11,'Fontname','Timesnewroman'); title('One-sided Welch PSD of Train Displacement','fontsize',11,'Fontname','Timesnewroman'); set(gca,'fontsize',11,'Fontname','Timesnewroman'); grid on; fprintf(' \n Figure 17. Welch Power Spectrum of Train Displacement(dB/Hz, Hz) \n ') Figure 3A.1.17. Welch Power Spectrum of Train Displacement (dB/Hz, Hz). RMS Values and Time-varying Plots of raw Acceleration signal, filtered Acceleration and Disp: %RMS Values of raw Acceleration signal, filtered Acceleration and Disp: AccRMSvalue=rms(acc_signal); fprintf(' \n RMS value of Train Raw Acceleration (m/s^2)= %.3f \n ',AccRMSvalue) fprintf(' \n RMS value of Train Raw Acceleration (dB)= %.3f \n ',20*log10(AccRMSvalue)) 105 FilteredAccRMSvalue=rms(accfl); fprintf(' \n RMS value of Train Low Pass Filtered Acceleration (m/s^2)= %.3f \n ',FilteredAccRMSvalue) fprintf(' \n RMS value of Train Low Pass Filtered Acceleration (dB)= %.3f \n ',20*log10(FilteredAccRMSvalue)) DispRMSvalue=rms(disp); fprintf(' \n RMS value of Train Displacement (m/s^2)= %.3f \n ',DispRMSvalue) fprintf(' \n RMS value of Train Displacement (dB)= %.3f \n ',20*log10(DispRMSvalue)) %RMS Plot of raw & filtered Acceleration signal REF: http://bit.ly/2bnXIW1 or http://blog.Midé.com/matlab-vs-python-speed-for-vibration-analysis-free-download N = length(acc_signal); ww = floor(Fs); %width of the window for computing RMS steps = floor(N/ww); %Number of steps for RMS t_RMS = zeros(steps,1); %Create time array for RMS time values raw_acc_RMS = zeros(steps,1); %Create raw acc array for RMS values filtered_acc_RMS=zeros(steps,1); %Create filtered acc array for RMS values for i=1:steps range = ((i-1)*ww+1):(i*ww); t_RMS(i) = mean(time(range)); raw_acc_RMS(i) = sqrt(mean(acc_signal(range).^2)); filtered_acc_RMS(i) = sqrt(mean(accfl(range).^2)); end %RMS Plot of raw Acceleration signal: figure(18) plot(t_RMS,raw_acc_RMS,'m') xlabel('Time (s)','fontsize',11,'Fontname','Timesnewroman'); ylabel('RMS raw Acceleration Signal (m/s^2)','fontsize',11,'Fontname','Timesnewroman'); title_head = sprintf('Moving-RMS of the \n Train Raw Acceleration Signal w/ RMS= %.2f',AccRMSvalue); title(title_head,'fontsize',11,'Fontname','Timesnewroman'); set(gca,'fontsize',11,'Fontname','Timesnewroman'); grid on; fprintf(' \n Figure 18. Moving-RMS of the Train Raw Acceleration Signal (m/s^2, Hz) \n ') RMS value of Train Raw Acceleration (m/s^2)= 12.735 RMS value of Train Raw Acceleration (dB)= 22.100 RMS value of Train Low-pass Filtered Acceleration (m/s^2)= 11.243 RMS value of Train Low-pass Filtered Acceleration (dB)= 21.018 RMS value of Train Displacement (m/s^2)= 0.252 RMS value of Train Displacement (dB)= -11.965 106 Figure 3A.1.18. Moving-RMS of the Train Raw Acceleration Signal (m/s^2, Hz). %RMS Plot of filtered Acceleration signal: figure(19) plot(t_RMS,filtered_acc_RMS,'r') xlabel('Time (s)','fontsize',11,'Fontname','Timesnewroman'); ylabel('RMS Low-pass Filtered Acceleration (m/s^2)','fontsize',11,'Fontname','Timesnewroman'); title_head = sprintf('Moving-RMS of the \n Low-pass Filtered Train Acceleration Signal \n w/ RMS= %.2f',FilteredAccRMSvalue); title(title_head,'fontsize',11,'Fontname','Timesnewroman'); set(gca,'fontsize',11,'Fontname','Timesnewroman'); grid on; fprintf(' \n Figure 19. Moving-RMS of the Filtered Train Acceleration Signal (m/s^2, Hz) \n ') 107 Figure 3A.1.19. Moving-RMS of the Filtered Train Acceleration Signal (m/s2, Hz). %RMS Plot of Position signal disp_RMS = zeros(steps,1); %Create array for RMS values regarding time for i=1:steps range = ((i-1)*ww+1):(i*ww); t_RMS(i) = mean(time(range)); disp_RMS(i) = sqrt(mean(disp(range).^2)); end figure(20) plot(t_RMS,disp_RMS) xlabel('Time (s)','fontsize',11,'Fontname','Timesnewroman'); ylabel('RMS Displacement (mm)','fontsize',11,'Fontname','Timesnewroman'); title_head = sprintf('Moving-RMS of the Train Displacement \n w/ RMS= %.2f',DispRMSvalue); title(title_head,'fontsize',11,'Fontname','Timesnewroman'); set(gca,'fontsize',11,'Fontname','Timesnewroman'); grid on; fprintf(' \n Figure 20. Moving-RMS of Train Input Displacement (m/s^2, Hz) \n ') 108 Figure 3A.1.20. Moving-RMS of Train Input Displacement (m/s2, Hz). 3A.2. For 3 DOF VEH, Selected Train Vibration Signal Characteristics and Analysis Vertical acceleration data at the middle train body (Interpolated, fs=600Hz) FFTs of acceleration and displacement signals ..................................................... 114 FFT of Position signal:........................................................................................... 116 ZOOMED FFT of position plot: ............................................................................ 117 PSDs of acceleration FFT and displacement FFTs: ............................................... 119 PSD of position signal calculation for position: .................................................... 121 ZOOMED PSD of FFT of the position signal Plot ............................................... 122 Welch Power Spectral Estimator for Acceleration and displacement signals: ...... 124 PSD of Welch spectral of Low Pass filtered Acceleration Signal Plot: ................ 125 Welch PSD of Displacement : ............................................................................... 126 RMS Values and Time-varying Plots of raw Acc signal, filtered Acc and Disp: .............................................................................................................. 127 109 % Train_Vibr_FFT_PSD_Welch_RMS.m MATLAB File clear all; close all; clc; filename = 'New_Train_Acc_Data_cut_raw_60sec600Hz.xlsx'; data =xlsread(filename); acc_signal =xlsread(filename,'B:B'); time =xlsread(filename,'A:A'); This program high pass (4,5 Hz) and low pass (100 Hz) filters and integrates acceleration signal to displacement signal. Plots time varying signals, FFT, Welch PSD and RMS of Acceleration and Displacement signals. dt=1/600; % sampling rate 600 sample per second L=length(acc_signal); % number of data RTEALB3, bogie 3 vertical Fs=1/(time(2)-time(1)); %600Hz, Sampling frequency %Designing Butterworth High-pass and Low Pass filters: [b,a]=butter(8,0.015,'high'); % designing butterworth highpass filter: high-pass cutoff frequency 0.015 * 300 = 4.5 hz [bb,aa]=butter(8,0.33334);% Low Pass cutoff frequency 100/150 * 300 = 4.5 hz % acceleration signal highpass filtered at 4.5Hz: accf=filter(b,a,acc_signal); % acc signal Low pass filtered at 100Hz: accfl=filter(bb,aa,acc_signal); % acc signal Low&high pass filtered between: 4.5 to 100Hz: acc_hp_lp_filtered=filter(bb,aa,accf); %(4.5Hz-100Hz). %Integrating acceleration signal to gain displacement: veli=1000*cumsum(acc_hp_lp_filtered)*dt; % filtered acceleration integrated & conv to mm velf=filter(b,a,veli); %integrated velocity highpass filtered disp=cumsum(velf)*dt; % filtered velocity integrated %Plotting Time varying- Acceleration, Velocity and Displacement Signals: %Time varying- Acceleration in m/s^2 and 'g': figure(1) plot(time, accfl,'m')%magenta color title('Filtered Acceleration Signal','fontsize',11,'Fontname','Timesnewroman'); xlabel('Time (Sec)','fontsize',11,'Fontname','Timesnewroman'); ylabel('Acceleration (m/s^2)','fontsize',11,'Fontname','Timesnewroman'); set(gca,'fontsize',11,'Fontname','Timesnewroman'); grid on fprintf(' \n Figure 3A.2.1. Time-varying Filtered Input Acceleration Signal(m/s^2) \n ') 110 Figure 3A.2.1. Time-varying Filtered Input Acceleration Signal(m/s^2). figure(2) plot(time, accfl/(9.81),'m')%where g=9.81 m/s^2, magenta color title('Filtered Acceleration Signal','fontsize',11,'Fontname','Timesnewroman'); xlabel('Time (Sec)','fontsize',11,'Fontname','Timesnewroman'); ylabel('Acceleration (g)','fontsize',11,'Fontname','Timesnewroman'); set(gca,'fontsize',11,'Fontname','Timesnewroman'); grid on fprintf(' \n Figure 3A.2.2. Time-varying Filtered Input Acceleration Signal(g) \n ') 111 Figure 3A.2.2. Time-varying Filtered Input Acceleration Signal (g). %Time varying- Velocity figure(3) plot(time, velf,'g')%green color title('Filtered Velocity','fontsize',11,'Fontname','Timesnewroman'); xlabel('Time (sec)','fontsize',11,'Fontname','Timesnewroman'); ylabel('Velocity (mm/sec)','fontsize',11,'Fontname','Timesnewroman'); set(gca,'fontsize',11,'Fontname','Timesnewroman'); grid on fprintf(' \n Figure 3A.2.3. Time-varying Filtered Input Velocity(mm/sec) \n ') 112 Figure 3A.2.3. Time-varying Filtered Input Velocity(mm/sec). %Time varying- Displacement figure(4) plot(time, disp)%original blue color title('Displacement Signal','fontsize',11,'Fontname','Timesnewroman'); xlabel('Time (Sec)','fontsize',11,'Fontname','Timesnewroman'); ylabel('Displacement (mm)','fontsize',11,'Fontname','Timesnewroman'); set(gca,'fontsize',11,'Fontname','Timesnewroman'); grid on fprintf(' \n Figure 3A.2.4. Time-varying Input Displacement (mm) \n ') 113 Figure 3A.2.4. Time-varying Input Displacement (mm). FFTs of acceleration and displacement signals FFT of Low Pass filtered Acceleration signal with f_cutoff= 100Hz. NFFT = 2^nextpow2(L); % Next power of 2 from length of y: it rounds up to the number such that will be as a power of 2 accflfft = fft(accfl,NFFT)/L; % FFT of Low Pass filtered Acceleration signal f = Fs/2*linspace(0,1,NFFT/2); % FFT freq axis, sampling frequency is Fs=600 %fft of Acceleration plot figure(5) plot(f,2*abs(accflfft(1:NFFT/2)),'m')%magenta color. For Single-Sided fft: 2*(abs value). xlim([3 100]) title_head = sprintf('Single-Sided Low Pass Filtered \n Acceleration Amplitude Spectrum'); title(title_head,'fontsize',11,'Fontname','Timesnewroman'); xlabel('Frequency (Hz)','fontsize',11,'Fontname','Timesnewroman'); ylabel('|Y(f)|, (m/s^2)','fontsize',11,'Fontname','Timesnewroman'); grid on fprintf(' \n Figure 3A.2.5. Single-Sided Low Pass Filtered Acceleration Amplitude Spectrum (m/s^2, Hz)\n ') 114 Figure 3A.2.5. Single-Sided Low Pass Filtered Acceleration Amplitude Spectrum (m/s^2, Hz). %the same fft of acc plot in 'dB' figure(6) plot(f,20*log10(2*abs(accflfft(1:NFFT/2))),'r')%red color, dB=20log10(A1/A2) & dB=10log10(Power1/Power2) xlim([4 100]) title_head = sprintf('Single-Sided Low Pass Filtered \n Acceleration Amplitude Spectrum'); title(title_head,'fontsize',11,'Fontname','Timesnewroman'); xlabel('Frequency (Hz)','fontsize',11,'Fontname','Timesnewroman'); ylabel('20*log10(|Y(f)|), (dB)','fontsize',11,'Fontname','Timesnewroman');% dB=20log10(A1/A2) & dB=10log10(Power1/Power2) set(gca,'fontsize',11,'Fontname','Timesnewroman'); grid on fprintf(' \n Figure 3A.2.6. Single-Sided Low Pass Filtered Acceleration Amplitude Spectrum (dB, Hz) \n ') 115 Figure 3A.2.6. Single-Sided Low Pass Filtered Acceleration Amplitude Spectrum (dB, Hz). FFT of Position signal: dispfft = fft(disp,NFFT)/L; % in (mm)s. FFT of displacement signal %fft of position plot: figure(7) plot(f,2*abs(dispfft(1:NFFT/2)))%For Single-Sided fft: 2*(abs value) xlim([0 100]) title_head = sprintf('Single-Sided \n Displacement Amplitude Spectrum'); title(title_head,'fontsize',11,'Fontname','Timesnewroman'); xlabel('Frequency (Hz)','fontsize',11,'Fontname','Timesnewroman'); ylabel('|U(f)|, (mm)','fontsize',11,'Fontname','Timesnewroman'); set(gca,'fontsize',11,'Fontname','Timesnewroman'); grid on fprintf(' \n Figure 3A.2.7. Single-Sided Displacement Amplitude Spectrum (mm, Hz) \n ') 116 Figure 3A.2.7. Single-Sided Displacement Amplitude Spectrum (mm, Hz). ZOOMED FFT of position plot: % figure(8) plot(f,2*abs(dispfft(1:NFFT/2)))%For Single-Sided fft: 2*(abs value) axis([3.5 25 0 0.02]) title_head = sprintf('ZOOMED Single-Sided \n Displacement Amplitude Spectrum'); title(title_head,'fontsize',11,'Fontname','Timesnewroman'); xlabel('ZOOMED Frequency (Hz)','fontsize',11,'Fontname','Timesnewroman'); ylabel('|U(f)|, (mm)','fontsize',11,'Fontname','Timesnewroman'); set(gca,'fontsize',11,'Fontname','Timesnewroman'); grid on fprintf(' \n Figure 3A.2.8. Zoomed Single-Sided Displacement Amplitude Spectrum (mm, Hz) \n ') 117 Figure 3A.2.8. Zoomed Single-Sided Displacement Amplitude Spectrum (mm, Hz) %the same fft of position plot in 'dB' figure(9) plot(f,20*log10(2*abs(dispfft(1:NFFT/2))), 'c')%cyan color, dB=20log10(A1/A2) & dB=10log10(Power1/Power2) xlim([3 100]) title('Single-Sided Displacement Amplitude Spectrum','fontsize',11,'Fontname','Timesnewroman') xlabel('Frequency (Hz)','fontsize',11,'Fontname','Timesnewroman'); ylabel('20*log10(|U(f)|), (dB)','fontsize',11,'Fontname','Timesnewroman');% dB=20log10(A1/A2) & dB=10log10(Power1/Power2), set(gca,'fontsize',11,'Fontname','Timesnewroman'); grid on fprintf(' \n Figure 3A.2.9. Single-Sided Displacement Amplitude Spectrum (dB, Hz) \n ') 118 Figure 3A.2.9. Single-Sided Displacement Amplitude Spectrum (dB, Hz). PSDs of acceleration FFT and displacement FFTs: PSD of the acceleration FFT: xdft_acc = fft(accfl); %FFT of the acc is selected for PSD evaluation xdft_acc = xdft_acc(1:(L+1)/2);%where L=length(accfl) not length(acc_signal) not length(accfft) psdx_acc = (1/(Fs*L)) * abs(xdft_acc).^2; %Power is directly proportional with the square of the absolute value of the position amplitude psdx_acc(2:end-1) = 2*psdx_acc(2:end-1); %dB/Hz freq_acc = 0:Fs/L:Fs/2; % PSD of FFT of Acceleration Plot figure(10) plot(freq_acc,psdx_acc,'m')%magenta color xlim([3 100]) grid on title_head = sprintf('Periodogram Using FFT of the \n Low Pass Filtered Acceleration Signal'); title(title_head,'fontsize',11,'Fontname','Timesnewroman'); xlabel('Frequency (Hz)','fontsize',11,'Fontname','Timesnewroman'); ylabel('Power/Frequency ( ((m/s^2)^2)/Hz )','fontsize',11,'Fontname','Timesnewroman'); set(gca,'fontsize',11,'Fontname','Timesnewroman'); 119 fprintf(' \n Figure 3A.2.10. Periodogram Using FFT of the Low Pass filtered Acceleration Signal (((m/s^2)^2)/Hz, Hz) \n ') Figure 3A.2.10. Periodogram Using FFT of the Low Pass filtered Acceleration Signal (((m/s^2)^2)/Hz, Hz). %the same PSD of Acceleration Plot in 'dB' figure(11) plot(freq_acc,10*log10(psdx_acc),'r')%magenta color, dB=20log10(A1/A2) & dB=10log10(Power1/Power2) xlim([3 100]) grid on title_head = sprintf('PSD Using FFT of the \n Low Pass Filtered Acceleration Signal'); title(title_head,'fontsize',11,'Fontname','Timesnewroman'); xlabel('Frequency (Hz)','fontsize',11,'Fontname','Timesnewroman'); ylabel('Power/Frequency (dB/Hz)','fontsize',11,'Fontname','Timesnewroman'); set(gca,'fontsize',11,'Fontname','Timesnewroman'); fprintf(' \n Figure 3A.2.11. PSD Using FFT of the Low Pass filtered Acceleration Signal (dB/Hz, Hz) \n ') 120 Figure 3A.2.11. PSD Using FFT of the Low Pass filtered Acceleration Signal (dB/Hz, Hz). PSD of position signal calculation for position: xdft_disp = fft(disp/1000); %in meters , FFT of the acc is selected for PSD evaluation xdft_disp = xdft_disp(1:(L+1)/2);%where L=length(accfl) not length(acc_signal) not length(accfft) psdx_disp = (1/(Fs*L)) * abs(xdft_disp).^2; %Power is directly proportional with the square of the absolute value of the position amplitude psdx_disp(2:end-1) = 2*psdx_disp(2:end-1); %dB/Hz freq_disp = 0:Fs/L:Fs/2; % PSD of FFT of the position signal Plot figure(12) plot(freq_disp,(psdx_disp))%original blue color, dB=20log10(A1/A2) & dB=10log10(Power1/Power2) xlim([3 100]) %ylim([0 1.21e-7]) title('Periodogram Using FFT of the Displacement','fontsize',11,'Fontname','Timesnewroman'); xlabel('Frequency (Hz)','fontsize',11,'Fontname','Timesnewroman'); ylabel('Power/Frequency ((m^2)/Hz)','fontsize',11,'Fontname','Timesnewroman'); set(gca,'fontsize',11,'Fontname','Timesnewroman'); grid on 121 fprintf(' \n Figure 3A.2.12. Periodogram Using FFT of Displacement ((m^2)/Hz, Hz) \n ') Figure 3A.2.12. Periodogram Using FFT of Displacement ((m^2)/Hz, Hz). ZOOMED PSD of FFT of the position signal Plot figure(13) plot(freq_disp,psdx_disp)%original blue color, dB=20log10(A1/A2) & dB=10log10(Power1/Power2) axis([4 25 0 0.2e-6]) title_head = sprintf('ZOOMED Periodogram Using \n FFT of the Displacement'); title(title_head,'fontsize',11,'Fontname','Timesnewroman'); xlabel('ZOOMED Frequency (Hz)','fontsize',11,'Fontname','Timesnewroman'); ylabel('Power/Frequency ((m^2)/Hz)','fontsize',11,'Fontname','Timesnewroman'); set(gca,'fontsize',11,'Fontname','Timesnewroman'); grid on fprintf(' \n Figure 3A.2.13. Zoomed Periodogram Using FFT of Displacement ((m^2)/Hz, Hz) \n ') 122 Figure 3A.2.13. Zoomed Periodogram Using FFT of Displacement ((m^2)/Hz, Hz). %the same PSD of Acceleration Plot in 'dB' figure(14) plot(freq_disp,10*log10(psdx_disp),'c')%magenta color xlim([3 100]) title('Periodogram Using FFT of the Displacement','fontsize',11,'Fontname','Timesnewroman') xlabel('Frequency (Hz)','fontsize',11,'Fontname','Timesnewroman'); ylabel('Power/Frequency (dB/Hz)','fontsize',11,'Fontname','Timesnewroman'); set(gca,'fontsize',11,'Fontname','Timesnewroman'); grid on fprintf(' \n Figure 3A.2.14. PSD Using FFT of Displacement (dB/Hz, Hz) \n ') 123 Figure 3A.2.14. PSD Using FFT of Displacement (dB/Hz, Hz). Welch Power Spectral Estimator for Acceleration and displacement signals: Create a Welch spectral estimator for acc and disp. h_welch= spectrum.welch; % Welch PSD of Raw and Filtered Acceleration: %One-sided PSD of welch of the raw Acceleration signal: Hpsd_raw_acc=psd(h_welch,acc_signal,'Fs',Fs); %One-sided PSD of welch of the Low Pass filtered Acceleration signal: Hpsd_filtered_acc=psd(h_welch,accflfft,'Fs',Fs); % PSD of Welch spectral of the raw Acceleration Signal Plot: figure (15) plot(Hpsd_raw_acc) xlim([3 100]) xlabel('Frequency, (Hz)','fontsize',11,'Fontname','Timesnewroman'); yaxis_head = sprintf('Raw Acceleration Signal \n Welch Power/Freq, (dB/Hz)'); ylabel(yaxis_head,'fontsize',11,'Fontname','Timesnewroman'); title('Welch PSD of Raw Train Acceleration Signal','fontsize',11,'Fontname','Timesnewroman'); set(gca,'fontsize',11,'Fontname','Timesnewroman'); grid on; fprintf(' \n Figure 3A.2.15. Welch Power Spectrum of Raw Train Acceleration Signal(dB/Hz, Hz) \n ') 124 Figure 3A.2.15. Welch Power Spectrum of Raw Train Acceleration Signal(dB/Hz, Hz). PSD of Welch spectral of Low Pass filtered Acceleration Signal Plot: figure (16) plot(Hpsd_filtered_acc) xlim([3 100]) xlabel('Frequency, (Hz)','fontsize',11,'Fontname','Timesnewroman'); yaxis_head = sprintf('Train Low Pass Filtered Acc Signal \n Welch Power/Frequency, (dB/Hz)'); ylabel(yaxis_head,'fontsize',11,'Fontname','Timesnewroman'); title('Welch PSD of Train Low Pass Filtered Acc Signal','fontsize',11,'Fontname','Timesnewroman'); set(gca,'fontsize',11,'Fontname','Timesnewroman'); grid on; fprintf(' \n Figure 3A.2.16. Welch Power Spectrum of Train Filtered Acc Signal (dB/Hz, Hz) \n ') 125 Figure 3A.2.16. Welch Power Spectrum of Train Filtered Acc Signal (dB/Hz, Hz). Welch PSD of Displacement : %One-sided PSD of welch of Displacement: Hpsd_disp=psd(h_welch,disp,'Fs',Fs);% Calculate and plot the one-sided PSD. % PSD of Welch spectral of Displacement Plot: figure (17) plot(Hpsd_disp) xlim([3 100]) xlabel('Frequency, (Hz)','fontsize',11,'Fontname','Timesnewroman'); yaxis_head = sprintf('Train Displacement Welch Power/Frequency, \n (dB/Hz)'); ylabel(yaxis_head,'fontsize',11,'Fontname','Timesnewroman'); title('One-sided Welch PSD of Train Displacement','fontsize',11,'Fontname','Timesnewroman'); set(gca,'fontsize',11,'Fontname','Timesnewroman'); grid on; fprintf(' \n Figure 3A.2.17. Welch Power Spectrum of Train Displacement(dB/Hz, Hz) \n ') 126 Figure 3A.2.17. Welch Power Spectrum of Train Displacement (dB/Hz, Hz). RMS Values and Time-varying Plots of raw Acc signal, filtered Acc and Disp: %RMS Values of raw Acc signal, filtered Acc and Disp: AccRMSvalue=rms(acc_signal); fprintf(' \n RMS value of Train Raw Acceleration (m/s^2)= %.3f \n ',AccRMSvalue) fprintf(' \n RMS value of Train Raw Acceleration (dB)= %.3f \n ',20*log10(AccRMSvalue)) FilteredAccRMSvalue=rms(accfl); fprintf(' \n RMS value of Train Low Pass Filtered Acceleration (m/s^2)= %.3f \n ',FilteredAccRMSvalue) fprintf(' \n RMS value of Train Low Pass Filtered Acceleration (dB)= %.3f \n ',20*log10(FilteredAccRMSvalue)) DispRMSvalue=rms(disp); fprintf(' \n RMS value of Train Displacement (m/s^2)= %.3f \n ',DispRMSvalue) fprintf(' \n RMS value of Train Displacement (dB)= %.3f \n ',20*log10(DispRMSvalue)) %RMS Plot of raw & filtered Acceleration signal REF: http://bit.ly/2bnXIW1 or http://blog.Midé.com/matlab-vs-python-speed-for-vibration-analysis-free-download N = length(acc_signal); ww = floor(Fs); %width of the window for computing RMS steps = floor(N/ww); %Number of steps for RMS t_RMS = zeros(steps,1); %Create time array for RMS time values raw_acc_RMS = zeros(steps,1); %Create raw acc array for RMS values 127 filtered_acc_RMS=zeros(steps,1); %Create filtered acc array for RMS values for i=1:steps range = ((i-1)*ww+1):(i*ww); t_RMS(i) = mean(time(range)); raw_acc_RMS(i) = sqrt(mean(acc_signal(range).^2)); filtered_acc_RMS(i) = sqrt(mean(accfl(range).^2)); end %RMS Plot of raw Acceleration signal: figure(18) plot(t_RMS,raw_acc_RMS,'m') xlabel('Time (s)','fontsize',11,'Fontname','Timesnewroman'); ylabel('RMS raw Acceleration Signal (m/s^2)','fontsize',11,'Fontname','Timesnewroman'); title_head = sprintf('Moving-RMS of the \n Train Raw Acceleration Signal w/ RMS= %.2f',AccRMSvalue); title(title_head,'fontsize',11,'Fontname','Timesnewroman'); set(gca,'fontsize',11,'Fontname','Timesnewroman'); grid on; fprintf(' \n Figure 3A.2.18. Moving-RMS of the Train Raw Acceleration Signal (m/s^2, Hz) \n ') RMS value of Train Raw Acceleration (m/s^2)= 1.891 RMS value of Train Raw Acceleration (dB)= 5.532 RMS value of Train Low Pass Filtered Acceleration (m/s^2)= 1.518 RMS value of Train Low Pass Filtered Acceleration (dB)= 3.628 RMS value of Train Displacement (m/s^2)= 0.108 RMS value of Train Displacement (dB)= -19.350 128 Figure 3A.2.18. Moving-RMS of the Train Raw Acceleration Signal (m/s^2, Hz) %RMS Plot of filtered Acceleration signal: figure(19) plot(t_RMS,filtered_acc_RMS,'r') xlabel('Time (s)','fontsize',11,'Fontname','Timesnewroman'); ylabel('RMS Low Pass Filtered Acceleration (m/s^2)','fontsize',11,'Fontname','Timesnewroman'); title_head = sprintf('Moving-RMS of the \n Low Pass Filtered Train Acceleration Signal \n w/ RMS= %.2f',FilteredAccRMSvalue); title(title_head,'fontsize',11,'Fontname','Timesnewroman'); set(gca,'fontsize',11,'Fontname','Timesnewroman'); grid on; fprintf(' \n Figure 3A.2.19. Moving-RMS of the Filtered Train Acceleration Signal (m/s^2, Hz) \n ') 129 Figure 3A.2.19. Moving-RMS of the Filtered Train Acceleration Signal (m/s^2, Hz) %RMS Plot of Position signal disp_RMS = zeros(steps,1); %Create array for RMS values regarding time for i=1:steps range = ((i-1)*ww+1):(i*ww); t_RMS(i) = mean(time(range)); disp_RMS(i) = sqrt(mean(disp(range).^2)); end figure(20) plot(t_RMS,disp_RMS) xlabel('Time (s)','fontsize',11,'Fontname','Timesnewroman'); ylabel('RMS Displacement (mm)','fontsize',11,'Fontname','Timesnewroman'); title_head = sprintf('Moving-RMS of the Train Displacement \n w/ RMS= %.2f',DispRMSvalue); title(title_head,'fontsize',11,'Fontname','Timesnewroman'); set(gca,'fontsize',11,'Fontname','Timesnewroman'); grid on; fprintf(' \n Figure 3A.2.20. Moving-RMS of Train Input Displacement (m/s^2, Hz) \n ') 130 Figure 3A.2.20. Moving-RMS of Train Input Displacement (m/s^2, Hz) 131 3A.3. Previously Investigated Train Vibration Signal Characteristics and Analysis Vertical train vibration at the third train bogie clc% acctodisp.m MATLAB File % This program high pass filteres and integrates acceleration signal to displacement signal and plots time varying signals, FFT and PSD of displacement signal dt=1/300; % sampling rate 300 sample per second L=262144; % number of data RTEALB3, bogie 3 vertical Fs=300; % Sampling frequency [b,a]=butter(8,0.03,'high'); % designing butterworth high pass filter %order 8th order % high-pass cutoff frequency 0.03 * 150 = 4.5 hz accf=filter(b,a,RTEALB3); % acceleration signal samples are high pass filtered, (m/s^2) veli=1000*cumsum(accf)*dt; % acceleration integrated & converted to mm (i for integrated), (mm/s) velf=filter(b,a,veli); %integrated velocity high pass filtered (f for filtered), (mm/s) disp=cumsum(velf)*dt; % filtered velocity integrated, (mm) figure(1) plot(Time, accf, 'm') title('Filtered Acceleration Signal','fontsize',12) xlabel('Time (Sec)','fontsize',12) ylabel('Acceleration (m/s^2)','fontsize',12) grid on figure(2) plot(Time, accf/(9.81) , 'm') %where g=9.81 m/s^2 title('Filtered Acceleration Signal','fontsize',12) xlabel('Time (Sec)','fontsize',12) ylabel('Acceleration (g)','fontsize',12) grid on figure(3) plot(Time, velf, 'g') title('Filtered Velocity','fontsize',12) xlabel('Time (sec)','fontsize',12) ylabel('Velocity (mm/sec)','fontsize',12) grid on figure(4) plot(Time, disp) title('Displacement Signal','fontsize',12) xlabel('Time (Sec)','fontsize',12) ylabel('Displacement (mm)','fontsize',12) grid on 132 %% FFTs of acceleration and displacement signals NFFT = 2^nextpow2(L); f = Fs/2*linspace(0,1,NFFT/2); % FFT freq axis, sampling frequency is Fs=300 % FFT of acceleration signal % Next power of 2 from length of y accfft = fft(accf,NFFT)/L; % FFT of displacement signal figure(5) plot(f,2*abs(accfft(1:NFFT/2)), 'm') xlim([0 100]) title('Single-Sided Acceleration Amplitude Spectrum','fontsize',12) xlabel('Frequency (Hz)','fontsize',12) ylabel('|Y(f)|','fontsize',12) grid on %the same fft of acc plot in 'dB' figure(6) plot(f,20*log10(2*abs(accfft(1:NFFT/2))) , 'r') xlim([0 100]) title('Single-Sided Acceleration Amplitude Spectrum','fontsize',12) xlabel('Frequency (Hz)','fontsize',12) ylabel('dB, 20*log10(|Y(f)|)','fontsize',12)% dB=20log10(A1/A2) & dB=10log10(Power1/Power2) grid on % FFT of position signal dispfft = fft(disp,NFFT)/L; % FFT of displacement signal figure(7) plot(f,2*abs(dispfft(1:NFFT/2))) xlim([0 100]) title('Single-Sided Displacement Amplitude Spectrum','fontsize',12) xlabel('Frequency (Hz)','fontsize',12) ylabel('|U(f)|','fontsize',12) grid on %the same fft of position plot in 'dB' figure(8) plot(f,20*log10(2*abs(dispfft(1:NFFT/2))), 'c') xlim([0 100]) title('Single-Sided Displacement Amplitude Spectrum','fontsize',12) xlabel('Frequency (Hz)','fontsize',12) ylabel('dB, 20*log10(|U(f)|)','fontsize',12)% dB=20log10(A1/A2) & dB=10log10(Power1/Power2) grid on %% PSDs of acceleration and displacement signals % PSD of acceleration signal % calculation for acceleration: xdft_acc = accfft; %FFT of the acc is selected for PSD evaluation xdft_acc = xdft_acc(1:L/2+1); psdx_acc = (1/(Fs*L)) * abs(xdft_acc).^2; %Power is directly proportional with the square of the absolute value of the position amplitude 133 psdx_acc(2:end-1) = 2*psdx_acc(2:end-1); %dB freq_acc = 0:Fs/length(accfft):Fs/2; % PSD of acceleration signal Plot figure(9) plot(freq_acc,(psdx_acc) , 'r') xlim([0 100]) grid on title('Periodogram Using FFT of the Filtered Acceleration','fontsize',12)xlabel('Frequency (Hz)','fontsize',12) ylabel('Power (dB)','fontsize',12) %ylabel('Power/Frequency (dB)/Hz','fontsize',12) set(gca,'fontsize',12) % PSD of position signal % calculation for position: xdft_disp = dispfft; %FFT of the position is selected for PSD evaluation xdft_disp = xdft_disp(1:L/2+1); psdx_disp = (1/(Fs*L)) * abs(xdft_disp).^2; %Power is directly proportional with the square of the absolute value of the position amplitude psdx_disp(2:end-1) = 2*psdx_disp(2:end-1); %dB freq_disp = 0:Fs/length(dispfft):Fs/2; % PSD of position signal Plot figure(10) plot(freq_disp,(psdx_disp)) xlim([0 100]) grid on title('Periodogram Using FFT of the Integrated Displacement','fontsize',12)xlabel('Frequency (Hz)','fontsize',12) ylabel('Power (dB)','fontsize',12) %ylabel('Power/Frequency (dB)/Hz','fontsize',12) set(gca,'fontsize',12) 134 Figure 3A.3.1. High pass filtered actual acceleration data in the units of m/s2 (top) and the gravity, g (bottom). 135 Figure 3A.3.2. High pass filtered velocity (m/s) that is gained after integrating filtered acceleration. Figure 3A.3.3. Displacement (mm) that is gained after integrating filtered velocity. 136 Figure 3A.3.4. Single-sided acceleration amplitude spectrum of the vertical acceleration at the third train bogie, in the units of m/s2 (top) and dB (bottom). 137 Figure 3A.3.5. Single-sided displacement amplitude spectrum of the integrated vertical displacement at the third train bogie, in the units of mm (top) and dB (bottom). The bottom plots in Figure 3A.3.4 and 3A.3.5, the change in the amplitude is the main point rather than the acceleration/displacement amplitudes in dB due to very low magnitudes of amplitudes as a result of FFT. Energy drops as the frequency increases are clearly seen in bottom plots. 138 Figure 3A.3.6. Periodogram of filtered vertical acceleration amplitude (dB) spectrum at the third train bogie, in the range of 0-100 Hz (top) and zoomed around 10-100 Hz (bottom). 139 Figure 3A.3.7. Periodogram integrated vertical displacement amplitude (dB) spectrum at the third train bogie, in the range of 0-100 Hz (top) and zoomed around 0-10 Hz (bottom). For better comparison of the frequencies having the greatest power, in addition to 140 power spectrum densities in Figure 3A.3.6 and 7, zoomed FFT plots of the acceleration and displacement (Figure 3A.3.4 and 3A.3.5) are shown in Figure 3A.3.8 and 9. Figure 8 and 9 also indicate that the frequency distribution exactly matches in both units. Figure 3A.3.8. Single-sided acceleration amplitude spectrum zoomed for the range of 5-100 Hz, in the units of m/s2 (top) and dB (bottom). 141 Figure 3A.3.9. Single-sided displacement amplitude spectrum zoomed for the range of 4-10 Hz, in the units of mm (top) and dB (bottom). Zoomed figures (Figure 3A.3.6-10) indicate that the frequencies having greatest 142 power matches exactly the same with FFT and PSD. In Table 3A.3.1, the frequencies among with the related powers (dB) are listed for input acceleration and displacement in order to select tuning frequencies. Since FFT and PSD gives the same frequencies, one sample set of each is listed for input excitation and displacement. Table 3A.3.1. The frequencies own displacement spectrum. Source: FFT of Acceleration Frequencies Magnitude (Hz) (dB) f1 17.21 -31.16 f2 19.22 -30.21 f3 39.64 -33.25 f4 62.31 -30.3 f5 78.92 -33.44 f6 99.48 -32.58 the greatest power in input excitation and PSD of Displacement Frequencies Magnitude (Hz) (mm,10^-3) 5.137 2.74 5.517 2.879 5.723 2.793 7.104 2.27 7.671 2.15 8.448 2.026 3A.4. Previously Investigated Train Vibration Signal Characteristics and Analysis Lateral acceleration data at the middle train body In similar perspective with sections 3A.1 and 2., FFT and PSD are performed on the lateral acceleration data at the middle train body as seen in Figure 3A.3.10 and 11. In these figures, frequency axis is set to half of the sampling frequency of 150 Hz; however, maximum sampling frequency of 100 Hz is set as a limit. Figure 3A.3.10. FFT of train lateral acceleration data. 143 Figure 3A.3.11. Zoomed FFT of train lateral acceleration data. Frequencies having peak magnitudes can be selected. 144 Figure 3A.3.12. Power spectrum density of train lateral acceleration data. The most frequent frequencies own the greatest power. Figure 13. Zoomed PSD of train lateral acceleration data. Dominant frequencies having peak powers can be selected. 145 According to Figure 3A.3.11, the frequencies having highest energy to be harvested are close to 83.5 Hz, 76 Hz, 73 Hz, 90.2 Hz, 78.5 Hz, 92 Hz, 75 Hz and 79.5 Hz. Although, these frequencies possess the greatest energies, it is important to keep in mind that the half-power bandwidth of the harvesters and the first mode frequency results the resonance thus, greatest power. Consequently, it would be wise to pick the frequencies in ascending order and eliminating the close values of frequencies. In order to most efficiently harvest energy from train lateral movement, the VEH mode frequencies can be 83.5 Hz, 90.2 Hz and 92 Hz. On the other hand, the ascending order of the following frequencies can also be considered: 76 Hz, 78.5 Hz (only for very narrow bandwidth), 83.5 Hz, 90.2 Hz and 92 Hz. Table 3A.3.2. The frequencies own the greatest power in input excitation spectrum. Source: f1 f2 f3 f4 FFT of Acceleration Frequencies (Hz) Magnitude (m/s2) 76 0.55 83.5 0.6 90.2 0.49 92 0.48 146 Appendix 3B Datasheet Details of the Evaluated Midé Volture Piezoelectric VEHs Detail information about the energy harvesters used in the thesis is given below[59, 61, 62, 70–75, 84, 85]. Figure 4A.1. Layers of Midé Volture Piezo Protection Advantage VEHs. Figure 4A.2. Midé Volture PPA models. 147 Figure 4A.3. Selected Midé Volture Models: PPA 1011 and 2011. Figure 4A.4. PPA 1011 and 2011 piezoelectricity and VEH parameters used in calculations. 148 PPA-1001 VEH 149 PPA-1021 Energy Harvester 150 PPA-1011 Energy Harvester 151 152 PPA-2011 Energy Harvester 153 154 APPENDIX 4 MATLAB Programs for Evaluation of Mathematical Model of Energy Harvesters with Various Natural Frequencies PPA-2011, Clamped at 0 and Input Vibration is at 60 Hz and 0.25g % called Func60HzPPA2011Compare.m % MIDÉ PPA-2011 selected energy harvester function dy = Func60HzPPA2011Compare(t,y,accel) % ENTER input: mu1=1.25 ; Amp=0.25*9.81; f=24;%Hz Req=24e3;%ohm mtip=25.3e-3;%kg mb=4e-3;% kg beam mass,no tip mass added weıght m_eq=0.607e-3;%kg Qt=15.1; Qm=30; %1 zeta=1/(2*Qm); %2 %zeta=1/(2*Qt); %3 %zeta=1/(2*Qt)+1/(2*Qm); meff=mtip+m_eq; % kg wn=f*2*pi; % rad/sec, nat freq 60 Hz %if Amp<=2.5 %~=0.25*9.81 %zeta=1/(2*Qt); % else %zeta=1/(2*Qt)+1/(2*Qm);%Q AT CLAMP0=17.6 % end Cp=190.0E-9; d31= -320E-12; c11e= 63e+9; bp= 20.8e-3; hp= 0.18e-3; 155 Lp= 40.0e-3; Ap=bp*hp; Gamma=d31*c11e*Ap/Lp; dy(1) = y(2); dy(2) = -mu1*accel-2*zeta*wn*y(2)-wn^2*y(1)-Gamma/meff*y(3); dy(3)= -1/(Req*Cp)*y(3)+(Gamma/Cp)*y(2); dy=dy'; Called Function for PPA-2011, Clamped at 0 and Input Vibration is at 60 Hz and 0.25g % solving equations using Runga Kutta 4-5 % called PPA2011_Clamp0_freq60Hz_FuncEqSim.m % calls Func60HzPPA2011Compare.m % MIDÉ PPA-2011 % ENTER input: Amp=0.25*9.81; f=24;%Hz Req=24e3;%ohm mtip=25.3e-3;%kg fprintf('\n \n'); m_eq=0.607e-3;%kg meff=mtip+m_eq; % kg y0=[0 0 0]; %initial conditions in column form timespan=[0:1/600:0.95]'; % time axis, interpolated time dt=1/600 sec, row format u=Amp*sin(f*2*pi*timespan); % acceleration input Velocity= Amp/(2*pi*f)*cos(2*pi*f*timespan); % base velocity nTime = length(timespan); y_list = zeros(nTime, length(y0)); y_list(1, :) = y0; for iTime = 2:nTime [T,Y]=ode45(@Func60HzPPA2011Compare,timespan(iTime-1:iTime),y0,[],u(iTime)); y0 = Y(end, :); y_list(iTime, :) = y0; 156 end % at the end of solution, solved variables are in % time is in timespan ........ y_list, v= y_list(:,3); % Voltage generated (volt) power_out = v.^2/Req; % Power at every instant (watt) speed=y_list(:,2) + Velocity; % absolute velocity of mass power_in=(mb+mtip)*u.*speed; dispm=y_list(:,1); RMS_disp=rms(dispm); fprintf('%s %2.3f %s \n',num2str('RMS_disp = '),RMS_disp*1000, ' mm'); fprintf('%s %2.3f %s \n',num2str('Peak to Peak_disp = '),RMS_disp*1000*2, ' mm'); fprintf('%s %2.0f %s \n',num2str('Experimental Value = 4.8 mm and % Err='),abs(4.8RMS_disp*1000*2)/4.8*100,'%'); fprintf('%s %2.0f %s \n',num2str('Ropt Code Output = 1.194 mm and % Difference='),abs( 1.194 -RMS_disp*1000)/ 1.194 *100,'%'); RMS_Pout=rms(power_out)*1000; fprintf('%s %6.9f %s \n',num2str('RMS_Pout = '),RMS_Pout, ' mW'); fprintf('%s %2.0f %s \n',num2str('Experimental Value = 4.1 mW and % Err='),abs(4.1RMS_Pout)/4.1*100,'%'); fprintf('%s %2.0f %s \n',num2str('Ropt Code Output = 2.602777670 mW and % Difference='),abs( 2.602777670 -RMS_Pout)/ 2.602777670 *100,'%'); RMS_volt=rms(v); fprintf('%s %2.3f %s \n',num2str('RMS_volt = '),RMS_volt,' Volt'); fprintf('%s %2.0f %s \n',num2str('Experimental Value = 9.9 V and % Err='),abs(9.9RMS_volt)/9.9*100,'%'); fprintf('%s %2.0f %s \n',num2str('Ropt Code Output = 6.730 V and % Difference='),abs(6.730 -RMS_volt)/6.730 *100,'%'); RMS_Pin=rms(power_in)*1000; fprintf('%s %2.3f %s \n',num2str('RMS_Pin = '),RMS_Pin, ' mW'); Power_Ratio= RMS_Pout/RMS_Pin*100; fprintf('%s %2.3f \n',num2str('Power_Ratio1 (%) = '),Power_Ratio); 157 APPENDIX 5A Chapter 5: Results and Discussion MODAL ANALYSIS of PPA 1001 & 1021 in SAP2000 and Ansys, and VIBRATION BEHAVIOUR of PPA 1021 5A.1. ANSYS SIMULATIONS 5A.1.1. RESPONSE TO THE HARMONIC BASE INPUT First of all, PPA-1021 is modelled regarding all structural layers in realistic dimensions but the thickness of the glue in between layers. Then, PPA-1021’s modal analysis for the selected tuning frequency of 8.5 Hz (tip mass of 88.6 g) is completed and at the respective mode frequencies of 8.5, 109.62, 541.4, 1040.5, 1108.6 and 1731.5 Hz, 1mm amplitude of harmonic base excitation is applied. Mode response for the first six modes among infinitely many are as follows: The first mode: Regarding the dominant frequency of displacement, PPA-1021 is tuned to 8.5 Hz and the harmonic vibration input is given at the base of the beam with the displacement amplitude of 1mm. Fist mode response is given in Figure 5A.1. Figure 5A.1. Mode response of PPA 1021 at 8.5 Hz under 1mm of harmonic base excitation, longitudinal view is at the bottom and the units are in mm. 158 The 2nd mode: The same harvester structure is subjected to the second mode frequency of 109.62 Hz at the same displacement amplitude of 1mm. Second mode response is given in Figure 5A.2-4. Figure 5A.2. Mode response of PPA 1021-tuned to 8.5 Hz- at 109.62 Hz under 1mm of harmonic base excitation, the units are in mm. Figure 5A.3. Longitudinal view of the mode response of PPA 1021-tuned to 8.5 Hz- at 109.62 Hz under 1mm of harmonic base excitation, the units are in mm. Figure 5A.4. Cross-sectional view of the mode response of PPA 1021-tuned to 8.5 Hz- at 109.62 Hz under 1mm of harmonic base excitation, the units are in mm. 159 The 3rd mode: The same harvester structure is subjected to the third mode frequency of 541.4 Hz at the same displacement amplitude of 1mm. Third mode response is given in Figure 5A.5-7. Figure 5A.5. Mode response of PPA 1021-tuned to 8.5 Hz- at 541.4 Hz under 1mm of harmonic base excitation, the units are in mm. Figure 5A.6. Longitudinal view of the mode response of PPA 1021-tuned to 8.5 Hz- at 541.4 Hz under 1mm of harmonic base excitation, the units are in mm. 160 Figure 5A.7. Cross-sectional view of the mode response of PPA 1021-tuned to 8.5 Hz- at 541.4 Hz under 1mm of harmonic base excitation, the units are in mm. The 4th mode: The same harvester structure is subjected to the fourth mode frequency of 1040.5 Hz at the same displacement amplitude of 1mm. Fourth mode response is given in Figure 5A.8-10. Figure 5A.5. Mode response of PPA 1021-tuned to 8.5 Hz- at 1040.5 Hz under 1mm of harmonic base excitation, the units are in mm. 161 Figure 5A.9. Longitudinal view of the mode response of PPA 1021-tuned to 8.5 Hz- at 1040.5 Hz under 1mm of harmonic base excitation, the units are in mm. Figure 5A.10. Cross-sectional view of the mode response of PPA 1021-tuned to 8.5 Hz- at 1040.5 Hz under 1mm of harmonic base excitation, the units are in mm. 162 The 5th mode: The same harvester structure is subjected to the fifth mode frequency of 1108.6 Hz at the same displacement amplitude of 1mm. Fifth mode response is given in Figure 5A.11-13. Please see the explanation in section named Note on 5th and 6th modes. Figure 5A.11. Mode response of PPA 1021-tuned to 8.5 Hz- at 1108.6 Hz under 1mm of harmonic base excitation, the units are in mm. Figure 5A.12. Longitudinal view of the mode response of PPA 1021-tuned to 8.5 Hz- at 1108.6 Hz under 1mm of harmonic base excitation, the units are in mm. 163 Figure 5A.13. Cross-sectional view of the mode response of PPA 1021-tuned to 8.5 Hz- at 1108.6 Hz under 1mm of harmonic base excitation, the units are in mm. The 6th mode: The same harvester structure is subjected to the sixth mode frequency of 1731.5 Hz at the same displacement amplitude of 1mm. Sixth mode response is given in Figure 5A.14-16. Please see the explanation in section named Note on 5th and 6th modes. Figure 5A.14-a. Mode response of PPA 1021-tuned to 8.5 Hz- at 1731.5 Hz under 1mm of harmonic base excitation, the units are in mm. 164 Figure 5A.14-b. Mode response of PPA 1021-tuned to 8.5 Hz- at 1731.5 Hz under 1mm of harmonic base excitation, the units are in mm. Figure 5A.15. Longitudinal view of the mode response of PPA 1021-tuned to 8.5 Hz- at 1731.5 Hz under 1mm of harmonic base excitation, the units are in mm. 165 Figure 5A.16. Cross-sectional view of the mode response of PPA 1021-tuned to 8.5 Hz- at 1731.5 Hz under 1mm of harmonic base excitation, the units are in mm. Note on 5th and 6th modes: After the fifth mode, modal analysis results turn out to be unrealistic as seen from the maximum deflection value to have the tendency to reach 2 m in fifth mode and 1.2 m in the sixth mode. Thus, they are investigated to see the mode shapes but not for the mode deflection values. 5A.1.2. RESPONSE TO THE RANDOM VIBRATION INPUT In harmonic vibration input, the input frequency is predefined at the exact mode frequencies. To see harvester behavior other than modal response, random vibration input at the maximum amplitude of 1mm is given. Static deflection under the tuning mass of 88.7g is in Figure 5A.17. Frequency response plot of PPA 1021-tuned to 8.5 Hz- is in Figure 5A.18 to indicate the expected output in wide frequency range. The vibration response analysis for 1mm of maximum input displacement amplitude is shown in Figures 5A.19-21. Since random vibration input is given, harvester response when the harvester is tuned to other dominant frequency of train vibration is also investigated. Therefore, the same harvester is tuned to 22 Hz with 12.7g of tip mass and its vibration response to random vibration input with a maximum input acceleration amplitude of 1g is shown in Figures 5A.22-24. Figure 5A.17. Static deflection of PPA 1021 when tuned to 8.5 Hz with a tip mass of 88.7 g. 166 Figure 5A.18. Frequency response of PPA 1021-tuned to 8.5 Hz. Figure 5A.19. Vibration response of PPA 1021-tuned to 8.5 Hz- under random vibration with maximum input displacement amplitude of 1mm, units are in mm. 167 Figure 5A.20. Longitudinal view of PPA 1021-tuned to 8.5 Hz- vibration response under random vibration with maximum input displacement amplitude of 1mm, units are in mm. Figure 5A.21. Cross-sectional view of PPA 1021-tuned to 8.5 Hz- vibration response under random vibration with maximum input displacement amplitude of 1mm, units are in mm. Figure 5A.22. Vibration response of PPA 1021-tuned to 22 Hz- under random vibration with maximum input acceleration amplitude of 1g, units are in mm. 168 Figure 5A.23. Longitudinal view of PPA 1021-tuned to 22 Hz- vibration response under random vibration with maximum input acceleration amplitude of 1g, units are in mm. Figure 5A.24. Cross-sectional view of PPA 1021-tuned to 22 Hz- vibration response under random vibration with maximum input acceleration amplitude of 1g, units are in mm. 5A.2. SAP2000 SIMULATIONS Modal analysis is initially studied in SAP2000 for PPA-1001. However, it is seen that rough one-layer modelling with estimated composite volumetric elasticity modulus resulted errors especially in higher modes. At first, harvester was modelled as a whole but it is seen that the natural frequency was much higher than the 98.9 Hz for 6.0 clamp position of PPA-1011 (without any tip mass). Therefore, it is attempted to use the lumped mass model. In this case the total mass of the harvester is divided in to the small masses to fill the nodes when the harvester is divided into 20x20 planar grids. This model is given in Figure 5A.25. In this modelling case, the natural frequency converged and simulation resulted as 71.3 Hz. Other mode frequencies are listed in Table 5A.1. The first mode is as expected and other mode shapes are given in the following figures (Figure 5A.26-42). 169 Figure 5A.25. Lumped mass model for 441 nodes (20x20 grids) of PPA-1001 at Clamp 6 with no tip mass. Table 5A.1. Modal analysis in SAP2000 and resulted mode frequencies. Frequency Frequency Mode No Mode No Mode No Frequency Hz Hz Hz 1 71.3 11 2464.1 21 5184.5 2 213.6 12 2662.3 22 5768.7 3 440.5 13 2819.5 23 5989.2 4 724.4 14 3578.7 24 6042.1 5 874.0 15 3755.0 25 6599.3 6 1245.5 16 3991.2 26 6881.6 7 1472.2 17 4188.9 27 7066.3 8 1506.8 18 4273.5 28 7153.1 9 2218.7 19 4865.2 29 7704.6 10 2349.8 20 5039.0 30 8221.7 Figure 5A.26. First mode shape of PPA-1001 at Clamp 6 with no tip mass. 170 Figure 5A.27. Second mode of PPA-1001 at Clamp 6 with no tip mass. Figure 5A.28. 3rd mode of PPA-1001 at Clamp 6 with no tip mass. Figure 5A.29. 4th mode of PPA-1001 at Clamp 6 with no tip mass. 171 Figure 5A.30. 5th mode of PPA-1001 at Clamp 6 with no tip mass. Figure 5A.31. 6th mode of PPA-1001 at Clamp 6 with no tip mass. Figure 5A.32. 7th mode of PPA-1001 at Clamp 6 with no tip mass. 172 Figure 5A.33. 8th mode of PPA-1001 at Clamp 6 with no tip mass. Figure 5A.34. 9th mode of PPA-1001 at Clamp 6 with no tip mass. Figure 5A.35. 10th mode of PPA-1001 at Clamp 6 with no tip mass. 173 Figure 5A.36. 11th mode of PPA-1001 at Clamp 6 with no tip mass. Figure 5A.37. 12th mode of PPA-1001 at Clamp 6 with no tip mass. Figure 5A.38. 13th mode of PPA-1001 at Clamp 6 with no tip mass. 174 Figure 5A.39. 15th mode of PPA-1001 at Clamp 6 with no tip mass. Figure 5A.40. 21th mode of PPA-1001 at Clamp 6 with no tip mass. Figure 5A.41. 29th mode of PPA-1001 at Clamp 6 with no tip mass. 175 Figure 5A.42. 30th mode of PPA-1001 at Clamp 6 with no tip mass. It is seen that all the simulated mode shapes of the tip massless cases in Sap2000 differs from the tip mass added results of other harvester model. Although the added tip mass in Ansys differs than the analytically calculated tip mass value or the related natural frequency, the difference is always less than 5% and far better than SAP2000 having the error of approximately 20%. Therefore, future studies will be carried out in Ansys. 176 APPENDIX 5B Chapter 5: Results and Discussion VALIDATION & SENSITIVITY ANALYSIS 5B.1. VALIDATING MATLAB CODE OUTPUT WITH EXPERIMENTS To begin with, one major check is conducted over the optimum load. If our approach is correct, then the experımentally found optimum load should give peak on the output power. It is also seen that the optimum load formulas presented in literature [54–58, 82, 83] does neither hold the values of the experiments nor gives reasonable output powers as a result of the evaluation of our proposed approach. As seen in Table 1, Matlab function output is listed and compared with the Midé’s experiment results. For the complete comparision with Midé’s tests for the middle clamp location, constitutive set of differential equations are solved for the 147 Hz, 60 Hz at 0.25g, 0.5g, 1g and 2g, and for 21 Hz at 0.25g, 0.5g and 1g. Relevant optimum load value that is used in experiments are selected for the solutions in MATLAB. At this point, calculation results are highly dependent on the damping coeficient. Additionally, for the correct estimation, output power trend is expected to have its maximum value at optimuml oad used in experiments. Therefore, sensitivity analysis is also run for different R loads at constant two seperate damping coefficients of transducerdamping alone and sum oftransducer and mechanical damping. As expected, evaluations for both damping coefficient have given the expected result and trends. The sensitivity analysis is given in Tables 2-6 and Figures 1-6 for the optimum load resistance. In Table 7-9, overall sensitivity analysis of load and damping coefficient on output power is listed. It is seen that except 0.25g input case, transducer damping coefficient alone gives very high output powers. On the other hand, total damping coefficient as the sum of mechanical and transducer damping gives more accurate results. Thus, the function is updated with the if block. In comparision with the other proposed analitical formulas in literature [54–58, 82, 83], the our approach on solving set of differential constitutive equations results vastly better in terms of the minimum error of the previously explained comparision with experiment data. However, even in our proposed approach, there exists only very few cases come precisely close with the experimental results. Thus, the appraoch needs to be corrected due to the nonlinearities in application. The same issue is also covered by Erturk, and he stated the input amplitude is one of the major factor and derives amplitude correction factor if there is no tip mass and for added tip mass, he suggests mass correction factor. As mentioned in Section 2.3, mass term in the constitutive set of equations are multiplied with the correction factor and for no added tip mass, amplitude correction 177 factor is multiplied with base amplitude value [83]. In constitutive equations, the only and obvious one place is shown as in Eqs (2.3.6, 2.4.1-3) and substituted as µ. Moreover, even when all these corrections are made, amplitude and tip mass are not the only factors affecting the change in the output power of the plate model piezoelectric energy harvesters. In this study, sensitivity analysis indicates that in addition to input amplitude and tip mass, damping coefficient and correction factor are also the terms needs to be arranged regarding these two corrected major components. The common sense in the correction depends basically on the importance of tip mass and the input amplitude for directly affecting the output voltage and power. When the input mass does not exist, total damping or transducer damping alone is too much and effective mass is already very low due to the nature of the harvester structure, which leads very high mass correction factor to be effective. On the other hand, when the tip mass is high, mechanical damping alone is very low and as the amplitude increases, the total damping and low mass correction factor gives better results. For having many terms affecting the outputs, instead of the derivation of correction factor formula including tip mass, acceleration amplitude and damping coefficient, via trial and error, the optimum values of all these mentioned parameters are tried to be found for the fixed tuned frequencies, namely tip mass values and acceleration amplitudes. Overall sensitivity analysis results for the input accelerations of 0.25g, 0.5g, 1g, and 2g for the tuning masses of 25.3g, 2.7g and no tip mass for the relative tuning frequencies of 21 Hz, 60 Hz and 145 Hz, for the mechanical damping alone, transducer damping alone and the total damping, over the varying coefficient factors are listed in Table 10 and summarized graphs are in Figures 7-9. 178 Table 5B.1. Comparision of resulted outputs in MATLAB for the transducer alone damping case (Q=17.6) and Midé’s experiment results from Volture PPA manual. INPUT USED IN CODES AND TESTS MIDÉ SPECSHEET Test Q=17.6,f=147,Amp= 0.25g,mtip=0, Result meff=0.614/1000 kg and Req=12.1e3ohm PPA1011_Clamp0_freq147Hz_FuncEqSim RMS_disp = 0.021 mm Peak to Peak_disp = 0.043 mm RMS_Pout = 0.006474 mW RMS_volt = 0.251 Volt RMS_Pin = 0.017 mW Power_Ratio1 (%) = 38.037 Amp=0.25*9.81; Req=25e3;%ohm f=60;%Hz ,mtip=2.7e-3;%kg m_eq=0.614e-3;%kg PPA1011_Clamp0_freq60Hz_FuncEqSim RMS_disp = 0.132 mm Peak to Peak_disp = 0.265 mm RMS_Pout = 0.102379 mW RMS_volt = 1.424 Volt RMS_Pin = 0.306 mW Power_Ratio1 (%) = 33.408 Amp=0.25*9.81; Req=76.6e3;%ohm f=20.8;%Hz , mtip=25.3e-3;%kg m_eq=0.614e-3;%kg PPA1011_Clamp0_freq21Hz_FuncEqSim RMS_disp = 0.981 mm 0.9 0.1 1.1 Test Result 2.7 0.4 3.3 Test Result MIDÉ SPECSHEET Err Q=17.6,f=146,Amp= 0.5g,mtip=0, meff=0.614/1000 kg and Req=12.6e3ohm 95% 94% 77% RMS_disp = 0.043 mm Peak to Peak_disp = 0.086 mm RMS_Pout = 0.026014 mW RMS_volt = 0.514 Volt RMS_Pin = 0.068 mW Power_Ratio1 (%) = 38.529 1.4 0.3 1.8 Err Req=15.8e3;%ohm, Amp=0.5*9.81; f=60;%Hz, mtip=2.6e-3;%kg m_eq=0.614e-3;%kg Test Result 90% 74% 57% RMS_disp = 0.279 mm Peak to Peak_disp = 0.557 mm RMS_Pout = 0.400569 mW RMS_volt = 2.237 Volt RMS_Pin = 1.308 mW Power_Ratio1 (%) = 30.622 3.5 1.1 4.2 Err Amp=0.5*9.81; Req=41.2e3;%ohm f=21;%Hz , mtip=25.3e-3;%kg m_eq=0.614e-3;%kg Test Result Peak to Peak_disp = 1.962 mm 9.7 80% RMS_Pout = 2.063788 mW RMS_volt = 10.909 Volt RMS_Pin = 6.702 mW Power_Ratio1 (%) = 30.793 2.4 13.8 14% 21% RMS_disp = 2.077 mm Peak to Peak_disp = 4.154 mm RMS_Pout = 7.886170 mW RMS_volt = 15.597 Volt RMS_Pin = 29.454 mW Power_Ratio1 (%) = 26.774 Test Result 12.9 5.4 14.9 MIDÉ SPECSHEET Err Q=17.6,f=146,Amp= 1g,mtip=0, meff=0.614/1000 kg and Req=10.2e3ohm 94% 91% 71% RMS_disp = 0.087 mm Peak to Peak_disp = 0.175 mm RMS_Pout = 0.108060 mW RMS_volt = 0.943 Volt RMS_Pin = 0.285 mW Power_Ratio1 (%) = 37.863 Err Req=19.5e3;%ohm, Amp=1*9.81; f=60;%Hz, mtip=2.6e-3;%kg m_eq=0.614e-3;%kg 84% 64% 47% RMS_disp = 0.539 mm Peak to Peak_disp = 1.078 mm RMS_Pout = 1.627338 mW RMS_volt = 5.013 Volt RMS_Pin = 4.969 mW Power_Ratio1 (%) = 32.748 Err Amp=1*9.81; Req=33.7e3;%ohm Test f=21;%Hz , mtip=25.3e-3;%kg Result m_eq=0.614e-3;%kg RMS_disp = 4.279 mm Peak to Peak_disp = 8.558 100% mm RMS_Pout = 29.710363 46% mW 5% RMS_volt = 27.338 Volt RMS_Pin = 122.084 mW Power_Ratio1 (%) = 24.336 179 MIDÉ SPECSHEET Err Q=17.6,f=145,Amp= 2g,mtip=0, meff=0.614/1000 kg and Req=10.1e3ohm 2.2 0.7 2.7 92% 85% 65% RMS_disp = 0.177 mm Peak to Peak_disp = 0.354 mm RMS_Pout = 0.438327 mW RMS_volt = 1.891 Volt RMS_Pin = 1.152 mW Power_Ratio1 (%) = 38.053 Test Result Err Test Result 7 3.2 2.8 85% 49% 79% Err 34.1 100% 16 23.2 86% 18% Req=17.3e3; Amp=2*9.81; f=60;%Hz , Q=17.6; mtip=2.6e3;%kg, m_eq=0.614e-3;%kg RMS_disp = 1.098 mm Peak to Peak_disp = 2.195 mm RMS_Pout = 6.472823 mW RMS_volt = 9.414 Volt RMS_Pin = 20.467 mW Power_Ratio1 (%) = 31.626 Test Result Err 3.7 2.1 4.6 90% 79% 59% Test Result Err 10.1 9.6 12.8 78% 33% 26% Table 5B.2. The damping coefficient effect on output power is analyzed for 1g input acceleration amplitude when PPA1011 is tuned to input frequency of 21 Hz . SENSITIVITY ANALYSIS: 1a) Adding Mechanical Damping: Zeta = Zeta_mech + Zeta_transducer Amp=0.25*9.81; Req=76.6e3;%ohm f=20.8;%Hz , mtip=25.3e-3;%kg, m_eq=0.614e-3;%kg RMS_disp = 0.750 mm Peak to Peak_disp = 1.499 mm Experimental Value = 12.9 mm and % Err= 88 % Ropt Code Output = 4.154 mm and % Err= 64 % RMS_Pout = 1.179160 mW Experimental Value = 5.4 mW and % Err= 78 % Amp=0.5*9.81; Req=41.2e3;%ohm f=21;%Hz , mtip=25.3e-3;%kg, m_eq=0.614e-3;%kg RMS_disp = 1.555 mm Peak to Peak_disp = 3.110 mm Experimental Value = 12.9 mm and % Err= 76 % Ropt Code Output = 4.154 mm and % Err= 25 % RMS_Pout = 4.318830 mW Experimental Value = 5.4 mW and % Err= 20 % Amp=1*9.81; Req=33.7e3;%ohm f=21;%Hz , mtip=25.3e-3;%kg, m_eq=0.614e-3;%kg RMS_disp = 3.181 mm Peak to Peak_disp = 6.361 mm Experimental Value = 34.1 mm and % Err= 81 % Ropt Code Output = 8.6 mm and % Err= 26 % RMS_Pout = 16.011037 mW Experimental Value = 16.0 mW and % Err= 0 % Ropt Code Output = 7.886170 mW and % Err= 85 % Ropt Code Output = 7.886170 mW and % Err= 45 % Ropt Code Output = 29.7 mW and % Err= 46 % RMS_volt = 8.327 Volt Experimental Value = 14.9 V and % Err= 44 % Ropt Code Output = 15.597 V and % Err= 47 % RMS_Pin = 5.183 mW Power_Ratio1 (%) = 22.752 RMS_volt = 11.678 Volt Experimental Value = 14.9 V and % Err= 22 % Ropt Code Output = 15.597 V and % Err= 25 % RMS_Pin = 22.147 mW Power_Ratio1 (%) = 19.501 RMS_volt = 20.320 Volt Experimental Value = 23.2 V and % Err= 12 % Ropt Code Output = 27.338 V and % Err= 26 % RMS_Pin = 90.989 mW Power_Ratio1 (%) = 17.597 SENSITIVITY ANALYSIS: 1b) ONLY Mechanical Damping: Zeta = Zeta_mech Amp=0.25*9.81; Req=76.6e3;%ohm f=20.8;%Hz , mtip=25.3e-3;%kg m_eq=0.614e-3;%kg RMS_disp = 1.246 mm Peak to Peak_disp = 2.492 mm Experimental Value = 9.7 mm and % Err= 74 % RMS_Pout = 3.410713 mW Experimental Value = 2.4 mW and % Err= 42 % RMS_volt = 13.867 Volt Experimental Value = 13.8 V and % Err= 0 % RMS_Pin = 8.384 mW Power_Ratio1 (%) = 40.681 Amp=0.5*9.81; Req=41.2e3;%ohm f=21;%Hz , mtip=25.3e-3;%kg m_eq=0.614e-3;%kg RMS_disp = 2.696 mm Peak to Peak_disp = 5.391 mm Experimental Value = 12.9 mm and % Err= 58 % RMS_Pout = 13.656641 mW Experimental Value = 5.4 mW and % Err= 153 % RMS_volt = 20.236 Volt Experimental Value = 14.9 V and % Err= 36 % RMS_Pin = 38.033 mW Power_Ratio1 (%) = 35.908 Amp=1*9.81; Req=33.7e3;%ohm f=21;%Hz , mtip=25.3e-3;%kg m_eq=0.614e-3;%kg RMS_disp = 5.596 mm Peak to Peak_disp = 11.192 mm Experimental Value = 12.9 mm and % Err= 13 % RMS_Pout = 52.369289 mW Experimental Value = 5.4 mW and % Err= 870 % RMS_volt = 35.748 Volt Experimental Value = 14.9 V and % Err= 140 % RMS_Pin = 159.169 mW Power_Ratio1 (%) = 32.902 SENSITIVITY ANALYSIS: 1c) ONLY Transducer Damping: Zeta = Zeta_transducer Amp=0.25*9.81; Req=76.6e3;%ohm f=20.8;%Hz , mtip=25.3e-3;%kg m_eq=0.614e-3;%kg RMS_disp = 0.981 mm Peak to Peak_disp = 1.962 mm Experimental Value = 9.7 mm and % Err= 80 % RMS_Pout = 2.063788 mW Experimental Value = 2.4 mW and % Err= 14 % RMS_volt = 10.909 Volt Experimental Value = 13.8 V and % Err= 21 % RMS_Pin = 6.702 mW Power_Ratio1 (%) = 30.793 Amp=0.5*9.81; Req=41.2e3;%ohm f=21;%Hz , mtip=25.3e-3;%kg m_eq=0.614e-3;%kg RMS_disp = 2.077 mm Peak to Peak_disp = 4.154 mm Experimental Value = 12.9 mm and % Err= 68 % RMS_Pout = 7.886170 mW Experimental Value = 5.4 mW and % Err= 46 % RMS_volt = 15.597 Volt Experimental Value = 14.9 V and % Err= 5 % RMS_Pin = 29.454 mW Power_Ratio1 (%) = 26.774 Amp=1*9.81; Req=33.7e3;%ohm f=21;%Hz , mtip=25.3e-3;%kg m_eq=0.614e-3;%kg RMS_disp = 4.279 mm Peak to Peak_disp = 8.558 mm Experimental Value = 34.1 mm and % Err= 75 % RMS_Pout = 29.710363 mW Experimental Value = 16.0 mW and % Err= 86 % RMS_volt = 27.338 Volt Experimental Value = 23.2 V and % Err= 18 % RMS_Pin = 122.084 mW Power_Ratio1 (%) = 24.336 180 Table 5B.3. The load effect on output power is analyzed for 1g input acceleration amplitude when PPA1011 is tuned to input frequency of 21 Hz (damping coefficient is taken as transducer damping alone). SENSITIVITY ANALYSIS: 2a) Changing Req,for Ropt=33.7 kOhm case For Zeta= Zeta_transducer Alone Req = 30.0 kOhm Experimental Ropt=33.7 kOhm and % Difference= 11 % RMS_disp = 4.354 mm Peak to Peak_disp = 8.708 mm Experimental Value = 34.1 mm and % Err= 74 % Ropt Code Output = 8.6 mm and % Err= 2 % RMS_Pout = 28.413387 mW Experimental Value = 16.0 mW and % Err= 78 % Req = 20.0 kOhm Experimental Ropt=33.7 kOhm and % Difference= 41 % RMS_disp = 4.611 mm Peak to Peak_disp = 9.221 mm Experimental Value = 34.1 mm and % Err= 73 % Ropt Code Output = 8.6 mm and % Err= 8 % RMS_Pout = 23.120622 mW Experimental Value = 16.0 mW and % Err= 45 % Req = 10.0 kOhm Experimental Ropt=33.7 kOhm and % Difference= 70 % RMS_disp = 4.957 mm Peak to Peak_disp = 9.914 mm Experimental Value = 34.1 mm and % Err= 71 % Ropt Code Output = 8.6 mm and % Err= 16 % RMS_Pout = 14.168769 mW Experimental Value = 16.0 mW and % Err= 11 % Req = 0.1 kOhm Experimental Ropt=33.7 kOhm and % Difference= 100 % RMS_disp = 5.393 mm Peak to Peak_disp = 10.786 mm Experimental Value = 34.1 mm and % Err= 68 % Ropt Code Output = 8.6 mm and % Err= 26 % RMS_Pout = 0.172482 mW Experimental Value = 16.0 mW and % Err= 99 % Ropt Code Output = 29.7 mW and % Err= 4 % Ropt Code Output = 29.7 mW and % Err= 22 % Ropt Code Output = 29.7 mW and % Err= 52 % Ropt Code Output = 29.7 mW and % Err= 99 % RMS_volt = 25.202 Volt Experimental Value = 23.2 V and % Err= 9 % Ropt Code Output = 27.338 V and % Err= 8 % RMS_Pin = 124.554 mW Power_Ratio1 (%) = 22.812 Req = 30.0 kOhm Experimental Ropt=33.7 kOhm and % Difference= 11 % RMS_disp = 4.354 mm Peak to Peak_disp = 8.708 mm Experimental Value = 34.1 mm and % Err= 74 % Ropt Code Output = 8.6 mm and % Err= 2 % RMS_Pout = 28.413387 mW Experimental Value = 16.0 mW and % Err= 78 % RMS_volt = 18.506 Volt Experimental Value = 23.2 V and % Err= 20 % Ropt Code Output = 27.338 V and % Err= 32 % RMS_Pin = 132.652 mW Power_Ratio1 (%) = 17.430 Req = 40.0 kOhm Experimental Ropt=33.7 kOhm and % Difference= 19 % RMS_disp = 4.172 mm Peak to Peak_disp = 8.344 mm Experimental Value = 34.1 mm and % Err= 76 % Ropt Code Output = 8.6 mm and % Err= 2 % RMS_Pout = 31.312319 mW Experimental Value = 16.0 mW and % Err= 96 % RMS_volt = 10.203 Volt Experimental Value = 23.2 V and % Err= 56 % Ropt Code Output = 27.338 V and % Err= 63 % RMS_Pin = 143.123 mW Power_Ratio1 (%) = 9.900 Req = 50.0 kOhm Experimental Ropt=33.7 kOhm and % Difference= 48 % RMS_disp = 4.047 mm Peak to Peak_disp = 8.095 mm Experimental Value = 34.1 mm and % Err= 76 % Ropt Code Output = 8.6 mm and % Err= 5 % RMS_Pout = 32.696183 mW Experimental Value = 16.0 mW and % Err= 104 % RMS_volt = 0.112 Volt Experimental Value = 23.2 V and % Err= 100 % Ropt Code Output = 27.338 V and % Err= 100 % RMS_Pin = 155.924 mW Power_Ratio1 (%) = 0.111 Req = 100.0 kOhm Experimental Ropt=33.7 kOhm and % Difference= 197 % RMS_disp = 3.864 mm Peak to Peak_disp = 7.728 mm Experimental Value = 34.1 mm and % Err= 77 % Ropt Code Output = 8.6 mm and % Err= 10 % RMS_Pout = 30.976989 mW Experimental Value = 16.0 mW and % Err= 94 % Ropt Code Output = 29.7 mW and % Err= 4 % Ropt Code Output = 29.7 mW and % Err= 5 % Ropt Code Output = 29.7 mW and % Err= 10 % Ropt Code Output = 29.7 mW and % Err= 4 % RMS_volt = 25.202 Volt Experimental Value = 23.2 V and % Err= 9 % Ropt Code Output = 27.338 V and % Err= 8 % RMS_Pin = 124.554 mW Power_Ratio1 (%) = 22.812 Req = 200.0 kOhm Experimental Ropt=33.7 kOhm and % Difference= 493 % RMS_disp = 3.993 mm Peak to Peak_disp = 7.985 mm Experimental Value = 34.1 mm and % Err= 77 % Ropt Code Output = 8.6 mm and % Err= 7 % RMS_Pout = 22.766415 mW Experimental Value = 16.0 mW and % Err= 42 % RMS_volt = 30.616 Volt Experimental Value = 23.2 V and % Err= 32 % Ropt Code Output = 27.338 V and % Err= 12 % RMS_Pin = 118.439 mW Power_Ratio1 (%) = 26.437 Req = 300.0 kOhm Experimental Ropt=33.7 kOhm and % Difference= 790 % RMS_disp = 4.124 mm Peak to Peak_disp = 8.247 mm Experimental Value = 34.1 mm and % Err= 76 % Ropt Code Output = 8.6 mm and % Err= 4 % RMS_Pout = 17.434261 mW Experimental Value = 16.0 mW and % Err= 9 % RMS_volt = 35.033 Volt Experimental Value = 23.2 V and % Err= 51 % Ropt Code Output = 27.338 V and % Err= 28 % RMS_Pin = 113.875 mW Power_Ratio1 (%) = 28.712 Req = 500.0 kOhm Experimental Ropt=33.7 kOhm and % Difference= 1384 % RMS_disp = 4.275 mm Peak to Peak_disp = 8.550 mm Experimental Value = 34.1 mm and % Err= 75 % Ropt Code Output = 8.6 mm and % Err= 0 % RMS_Pout = 11.721797 mW Experimental Value = 16.0 mW and % Err= 27 % RMS_volt = 48.377 Volt Experimental Value = 23.2 V and % Err= 109 % Ropt Code Output = 27.338 V and % Err= 77 % RMS_Pin = 103.697 mW Power_Ratio1 (%) = 29.873 Req = 1000.0 kOhm Experimental Ropt=33.7 kOhm and % Difference= 2867 % RMS_disp = 4.420 mm Peak to Peak_disp = 8.841 mm Experimental Value = 34.1 mm and % Err= 74 % Ropt Code Output = 8.6 mm and % Err= 3 % RMS_Pout = 6.390188 mW Experimental Value = 16.0 mW and % Err= 60 % Ropt Code Output = 29.7 mW and % Err= 23 % Ropt Code Output = 29.7 mW and % Err= 41 % Ropt Code Output = 29.7 mW and % Err= 61 % Ropt Code Output = 29.7 mW and % Err= 78 % RMS_volt = 58.656 Volt Experimental Value = 23.2 V and % Err= 153 % Ropt Code Output = 27.338 V and % Err= 115 % RMS_Pin = 101.370 mW Power_Ratio1 (%) = 22.459 RMS_volt = 62.821 Volt Experimental Value = 23.2 V and % Err= 171 % Ropt Code Output = 27.338 V and % Err= 130 % RMS_Pin = 102.240 mW Power_Ratio1 (%) = 17.052 RMS_volt = 66.432 Volt Experimental Value = 23.2 V and % Err= 186 % Ropt Code Output = 27.338 V and % Err= 143 % RMS_Pin = 103.965 mW Power_Ratio1 (%) = 11.275 RMS_volt = 69.292 Volt Experimental Value = 23.2 V and % Err= 199 % Ropt Code Output = 27.338 V and % Err= 153 % RMS_Pin = 106.018 mW Power_Ratio1 (%) = 6.027 181 Optimum Load Effect on Output Power 35 Power Output (mW) 30 25 20 15 10 5 0 0 100 200 300 400 500 Req (kOhm) 600 700 800 900 1000 Optimum Load Effect on Output Power 35 Power Output (mW) 30 25 20 15 10 y = -6E-13x6 + 9E-10x5 - 5E-07x4 + 0,0002x3 - 0,0229x2 + 1,4734x + 0,8189 R² = 0,9965 5 0 0 50 100 150 200 250 Req (kOhm) 300 350 400 450 500 Figure 5B.1. Analysis of optimum load on output power for 1g input acceleration amplitude when PPA1011 is tuned to input frequency of 21 Hz (damping coefficient is taken as transducer damping alone). . 182 Table 5B.4. The load effect on output power is analyzed for 1g input acceleration amplitude when PPA1011 is tuned to input frequency of 21 Hz (sum of mechanical and transducer damping coefficients are used). SENSITIVITY ANALYSIS: 2b) Changing Req,for Ropt=33.7 kOhm case For Zeta = Zeta_mech + Zeta_transducer Req = 30.0 kOhm Experimental Ropt=33.7 kOhm and % Difference= 11 % RMS_disp = 3.223 mm Peak to Peak_disp = 6.445 mm Experimental Value = 34.1 mm and % Err= 81 % Ropt Code Output = 6.361 mm and % Err= 1 % RMS_Pout = 15.168565 mW Experimental Value = 16.0 mW and % Err= 5 % Req = 20.0 kOhm Experimental Ropt=33.7 kOhm and % Difference= 41 % RMS_disp = 3.365 mm Peak to Peak_disp = 6.730 mm Experimental Value = 34.1 mm and % Err= 80 % Ropt Code Output = 6.361 mm and % Err= 6 % RMS_Pout = 11.964800 mW Experimental Value = 16.0 mW and % Err= 25 % Req = 10.0 kOhm Experimental Ropt=33.7 kOhm and % Difference= 70 % RMS_disp = 3.553 mm Peak to Peak_disp = 7.105 mm Experimental Value = 34.1 mm and % Err= 79 % Ropt Code Output = 6.361 mm and % Err= 12 % RMS_Pout = 7.042754 mW Experimental Value = 16.0 mW and % Err= 56 % Req = 0.1 kOhm Experimental Ropt=33.7 kOhm and % Difference= 100 % RMS_disp = 3.782 mm Peak to Peak_disp = 7.564 mm Experimental Value = 34.1 mm and % Err= 78 % Ropt Code Output = 6.361 mm and % Err= 19 % RMS_Pout = 0.081659 mW Experimental Value = 16.0 mW and % Err= 99 % Ropt Code Output =16.011037 mW and % Err= 5 % Ropt Code Output =16.011037 mW and % Err= 25 % Ropt Code Output =16.011037 mW and % Err= 56 % Ropt Code Output =16.011037 mW and % Err= 99 % RMS_volt = 18.652 Volt Experimental Value = 23.2 V and % Err= 20 % Ropt Code Output = 20.320 V and % Err= 8 % RMS_Pin = 92.375 mW Power_Ratio1 (%) = 16.421 Req = 30.0 kOhm Experimental Ropt=33.7 kOhm and % Difference= 11 % RMS_disp = 3.223 mm Peak to Peak_disp = 6.445 mm Experimental Value = 34.1 mm and % Err= 81 % Ropt Code Output = 6.361 mm and % Err= 1 % RMS_Pout = 15.168565 mW Experimental Value = 16.0 mW and % Err= 5 % RMS_volt = 13.505 Volt Experimental Value = 23.2 V and % Err= 42 % Ropt Code Output = 20.320 V and % Err= 34 % RMS_Pin = 96.876 mW Power_Ratio1 (%) = 12.351 Req = 40.0 kOhm Experimental Ropt=33.7 kOhm and % Difference= 19 % RMS_disp = 3.120 mm Peak to Peak_disp = 6.241 mm Experimental Value = 34.1 mm and % Err= 82 % Ropt Code Output = 6.361 mm and % Err= 2 % RMS_Pout = 17.108229 mW Experimental Value = 16.0 mW and % Err= 7 % RMS_volt = 0.079 Volt Experimental Value = 23.2 V and % Err= 100 % Ropt Code Output = 20.320 V and % Err= 100 % RMS_Pin = 109.286 mW Power_Ratio1 (%) = 0.075 Req = 100.0 kOhm Experimental Ropt=33.7 kOhm and % Difference= 197 % RMS_disp = 2.955 mm Peak to Peak_disp = 5.909 mm Experimental Value = 34.1 mm and % Err= 83 % Ropt Code Output = 6.361 mm and % Err= 7 % RMS_Pout = 17.791534 mW Experimental Value = 16.0 mW and % Err= 11 % Ropt Code Output =16.011037 mW and % Err= 5 % Ropt Code Output =16.011037 mW and % Err= 7 % Ropt Code Output =16.011037 mW and % Err= 13 % Ropt Code Output =16.011037 mW and % Err= 11 % RMS_volt = 18.652 Volt Experimental Value = 23.2 V and % Err= 20 % Ropt Code Output = 20.320 V and % Err= 8 % RMS_Pin = 92.375 mW Power_Ratio1 (%) = 16.421 Req = 200.0 kOhm Experimental Ropt=33.7 kOhm and % Difference= 493 % RMS_disp = 3.048 mm Peak to Peak_disp = 6.096 mm Experimental Value = 34.1 mm and % Err= 82 % Ropt Code Output = 6.361 mm and % Err= 4 % RMS_Pout = 13.044767 mW Experimental Value = 16.0 mW and % Err= 18 % RMS_volt = 22.899 Volt Experimental Value = 23.2 V and % Err= 1 % Ropt Code Output = 20.320 V and % Err= 13 % RMS_Pin = 88.937 mW Power_Ratio1 (%) = 19.236 Req = 300.0 kOhm Experimental Ropt=33.7 kOhm and % Difference= 790 % RMS_disp = 3.133 mm Peak to Peak_disp = 6.265 mm Experimental Value = 34.1 mm and % Err= 82 % Ropt Code Output = 6.361 mm and % Err= 2 % RMS_Pout = 9.888256 mW Experimental Value = 16.0 mW and % Err= 38 % RMS_volt = 37.005 Volt Experimental Value = 23.2 V and % Err= 60 % Ropt Code Output = 20.320 V and % Err= 82 % RMS_Pin = 80.929 mW Power_Ratio1 (%) = 21.984 Req = 1000.0 kOhm Experimental Ropt=33.7 kOhm and % Difference= 2867 % RMS_disp = 3.314 mm Peak to Peak_disp = 6.629 mm Experimental Value = 34.1 mm and % Err= 81 % Ropt Code Output = 6.361 mm and % Err= 4 % RMS_Pout = 3.524069 mW Experimental Value = 16.0 mW and % Err= 78 % Ropt Code Output =16.011037 mW and % Err= 19 % Ropt Code Output =16.011037 mW and % Err= 38 % Ropt Code Output =16.011037 mW and % Err= 59 % Ropt Code Output =16.011037 mW and % Err= 78 % RMS_volt = 44.795 Volt Experimental Value = 23.2 V and % Err= 93 % Ropt Code Output = 20.320 V and % Err= 120 % RMS_Pin = 80.453 mW Power_Ratio1 (%) = 16.214 RMS_volt = 47.739 Volt Experimental Value = 23.2 V and % Err= 106 % Ropt Code Output = 20.320 V and % Err= 135 % RMS_Pin = 81.450 mW Power_Ratio1 (%) = 12.140 RMS_volt = 51.960 Volt Experimental Value = 23.2 V and % Err= 124 % Ropt Code Output = 20.320 V and % Err= 156 % RMS_Pin = 84.476 mW Power_Ratio1 (%) = 4.172 RMS_volt = 7.311 Volt Experimental Value = 23.2 V and % Err= 68 % Ropt Code Output = 20.320 V and % Err= 64 % RMS_Pin = 102.562 mW Power_Ratio1 (%) = 6.867 Req = 50.0 kOhm Experimental Ropt=33.7 kOhm and % Difference= 48 % RMS_disp = 3.050 mm Peak to Peak_disp = 6.100 mm Experimental Value = 34.1 mm and % Err= 82 % Ropt Code Output = 6.361 mm and % Err= 4 % RMS_Pout = 18.168326 mW Experimental Value = 16.0 mW and % Err= 14 % RMS_volt = 26.405 Volt Experimental Value = 23.2 V and % Err= 14 % Ropt Code Output = 20.320 V and % Err= 30 % RMS_Pin = 86.370 mW Power_Ratio1 (%) = 21.035 Req = 500.0 kOhm Experimental Ropt=33.7 kOhm and % Difference= 1384 % RMS_disp = 3.227 mm Peak to Peak_disp = 6.454 mm Experimental Value = 34.1 mm and % Err= 81 % Ropt Code Output = 6.361 mm and % Err= 1 % RMS_Pout = 6.557519 mW Experimental Value = 16.0 mW and % Err= 59 % RMS_volt = 50.154 Volt Experimental Value = 23.2 V and % Err= 116 % Ropt Code Output = 20.320 V and % Err= 147 % RMS_Pin = 82.913 mW Power_Ratio1 (%) = 7.909 183 Optimum Load Effect on Output Power 20 18 Power Output (mW) 16 14 12 10 8 6 4 2 0 0 100 200 300 400 500 Req (kOhm) 600 700 800 900 1000 Optimum Load Effect on Output Power 20 18 Power Output (mW) 16 14 12 10 8 6 4 y = -5E-14x6 + 1E-10x5 - 1E-07x4 + 4E-05x3 - 0,0082x2 + 0,6764x + 0,6022 R² = 0,9884 2 0 0 50 100 150 200 250 Req (kOhm) 300 350 400 450 500 Figure 2. The analysis of load effect on output power for 1g input acceleration amplitude when PPA1011 is tuned to input frequency of 21 Hz (sum of mechanical and transducer damping coefficients are used). 184 Table 5B.5. The load effect on output power is analyzed for 0.25g input acceleration amplitude when PPA1011 is tuned to input frequency of 20.8 Hz (damping coefficient is taken as transducer damping alone). SENSITIVITY ANALYSIS: 2c) Changing Req,for Ropt=76.6 kOhm case For Zeta= Zeta_transducer Alone Req = 60.0 kOhm Experimental Ropt=76.6 kOhm and % Difference= 22 % RMS_disp = 1.001 mm Peak to Peak_disp = 2.002 mm Experimental Value = 9.7 mm and % Err= 79 % Ropt Code Output = 1.962 mm and % Err= 2 % RMS_Pout = 2.091400 mW Experimental Value = 2.4 mW and % Err= 13 % Req = 50.0 kOhm Experimental Ropt=76.6 kOhm and % Difference= 35 % RMS_disp = 1.023 mm Peak to Peak_disp = 2.046 mm Experimental Value = 9.7 mm and % Err= 79 % Ropt Code Output = 1.962 mm and % Err= 4 % RMS_Pout = 2.063925 mW Experimental Value = 2.4 mW and % Err= 14 % Req = 10.0 kOhm Experimental Ropt=76.6 kOhm and % Difference= 87 % RMS_disp = 1.257 mm Peak to Peak_disp = 2.515 mm Experimental Value = 9.7 mm and % Err= 74 % Ropt Code Output = 1.962 mm and % Err= 28 % RMS_Pout = 0.898158 mW Experimental Value = 2.4 mW and % Err= 63 % Req = 0.1 kOhm Experimental Ropt=76.6 kOhm and % Difference= 100 % RMS_disp = 1.369 mm Peak to Peak_disp = 2.738 mm Experimental Value = 9.7 mm and % Err= 72 % Ropt Code Output = 1.962 mm and % Err= 40 % RMS_Pout = 0.010966 mW Experimental Value = 2.4 mW and % Err= 100 % Ropt Code Output =2.063788 mW and % Err= 1 % Ropt Code Output =2.063788 mW and % Err= 0 % Ropt Code Output =2.063788 mW and % Err= 56 % Ropt Code Output =2.063788 mW and % Err= 99 % RMS_volt = 9.708 Volt Experimental Value = 13.8 V and % Err= 30 % Ropt Code Output = 10.909 V and % Err= 11 % RMS_Pin = 6.945 mW Power_Ratio1 (%) = 30.113 RMS_volt = 8.793 Volt Experimental Value = 13.8 V and % Err= 36 % Ropt Code Output = 10.909 V and % Err= 19 % RMS_Pin = 7.161 mW Power_Ratio1 (%) = 28.823 RMS_volt = 2.567 Volt Experimental Value = 13.8 V and % Err= 81 % Ropt Code Output = 10.909 V and % Err= 76 % RMS_Pin = 9.011 mW Power_Ratio1 (%) = 9.967 RMS_volt = 0.028 Volt Experimental Value = 13.8 V and % Err= 100 % Ropt Code Output = 10.909 V and % Err= 100 % RMS_Pin = 9.822 mW Power_Ratio1 (%) = 0.112 Req = 70.0 kOhm Experimental Ropt=76.6 kOhm and % Difference= 9 % RMS_disp = 0.987 mm Peak to Peak_disp = 1.974 mm Experimental Value = 9.7 mm and % Err= 80 % Ropt Code Output = 1.962 mm and % Err= 1 % RMS_Pout = 2.082451 mW Experimental Value = 2.4 mW and % Err= 13 % Req = 80.0 kOhm Experimental Ropt=76.6 kOhm and % Difference= 4 % RMS_disp = 0.979 mm Peak to Peak_disp = 1.958 mm Experimental Value = 9.7 mm and % Err= 80 % Ropt Code Output = 1.962 mm and % Err= 0 % RMS_Pout = 2.051392 mW Experimental Value = 2.4 mW and % Err= 15 % Req = 90.0 kOhm Experimental Ropt=76.6 kOhm and % Difference= 17 % RMS_disp = 0.975 mm Peak to Peak_disp = 1.949 mm Experimental Value = 9.7 mm and % Err= 80 % Ropt Code Output = 1.962 mm and % Err= 1 % RMS_Pout = 2.007255 mW Experimental Value = 2.4 mW and % Err= 16 % Req = 100.0 kOhm Experimental Ropt=76.6 kOhm and % Difference= 31 % RMS_disp = 0.973 mm Peak to Peak_disp = 1.946 mm Experimental Value = 9.7 mm and % Err= 80 % Ropt Code Output = 1.962 mm and % Err= 1 % RMS_Pout = 1.955767 mW Experimental Value = 2.4 mW and % Err= 19 % Ropt Code Output =2.063788 mW and % Err= 1 % Ropt Code Output =2.063788 mW and % Err= 1 % Ropt Code Output =2.063788 mW and % Err= 3 % Ropt Code Output =2.063788 mW and % Err= 5 % RMS_volt = 10.471 Volt Experimental Value = 13.8 V and % Err= 24 % Ropt Code Output = 10.909 V and % Err= 4 % RMS_Pin = 6.785 mW Power_Ratio1 (%) = 30.693 RMS_volt = 11.117 Volt Experimental Value = 13.8 V and % Err= 19 % Ropt Code Output = 10.909 V and % Err= 2 % RMS_Pin = 6.665 mW Power_Ratio1 (%) = 30.777 RMS_volt = 11.668 Volt Experimental Value = 13.8 V and % Err= 15 % Ropt Code Output = 10.909 V and % Err= 7 % RMS_Pin = 6.577 mW Power_Ratio1 (%) = 30.521 RMS_volt = 12.144 Volt Experimental Value = 13.8 V and % Err= 12 % Ropt Code Output = 10.909 V and % Err= 11 % RMS_Pin = 6.511 mW Power_Ratio1 (%) = 30.040 Req = 200.0 kOhm Experimental Ropt=76.6 kOhm and % Difference= 161 % RMS_disp = 1.004 mm Peak to Peak_disp = 2.008 mm Experimental Value = 9.7 mm and % Err= 79 % Ropt Code Output = 1.962 mm and % Err= 2 % RMS_Pout = 1.439929 mW Experimental Value = 2.4 mW and % Err= 40 % Req = 300.0 kOhm Experimental Ropt=76.6 kOhm and % Difference= 292 % RMS_disp = 1.037 mm Peak to Peak_disp = 2.074 mm Experimental Value = 9.7 mm and % Err= 79 % Ropt Code Output = 1.962 mm and % Err= 6 % RMS_Pout = 1.104031 mW Experimental Value = 2.4 mW and % Err= 54 % Req = 500.0 kOhm Experimental Ropt=76.6 kOhm and % Difference= 553 % RMS_disp = 1.075 mm Peak to Peak_disp = 2.151 mm Experimental Value = 9.7 mm and % Err= 78 % Ropt Code Output = 1.962 mm and % Err= 10 % RMS_Pout = 0.743270 mW Experimental Value = 2.4 mW and % Err= 69 % Req = 1000.0 kOhm Experimental Ropt=76.6 kOhm and % Difference= 1205 % RMS_disp = 1.113 mm Peak to Peak_disp = 2.225 mm Experimental Value = 9.7 mm and % Err= 77 % Ropt Code Output = 1.962 mm and % Err= 13 % RMS_Pout = 0.405694 mW Experimental Value = 2.4 mW and % Err= 83 % Ropt Code Output =2.063788 mW and % Err= 30 % Ropt Code Output =2.063788 mW and % Err= 47 % Ropt Code Output =2.063788 mW and % Err= 64 % Ropt Code Output =2.063788 mW and % Err= 80 % RMS_volt = 14.738 Volt Experimental Value = 13.8 V and % Err= 7 % Ropt Code Output = 10.909 V and % Err= 35 % RMS_Pin = 6.356 mW Power_Ratio1 (%) = 22.656 RMS_volt = 15.793 Volt Experimental Value = 13.8 V and % Err= 14 % Ropt Code Output = 10.909 V and % Err= 45 % RMS_Pin = 6.408 mW Power_Ratio1 (%) = 17.230 RMS_volt = 17.441 Volt Experimental Value = 13.8 V and % Err= 26 % Ropt Code Output = 10.909 V and % Err= 60 % RMS_Pin = 6.644 mW Power_Ratio1 (%) = 6.106 RMS_volt = 16.712 Volt Experimental Value = 13.8 V and % Err= 21 % Ropt Code Output = 10.909 V and % Err= 53 % RMS_Pin = 6.515 mW Power_Ratio1 (%) = 11.409 185 Optimum Load Effect on Output Power 2,5 Power Output (mW) 2 1,5 1 0,5 0 0 100 200 300 400 500 Req (kOhm) 600 700 800 900 1000 Optimum Load Effect on Output Power 2,5 Power Output (mW) 2 1,5 1 0,5 y = -3E-14x6 + 5E-11x5 - 3E-08x4 + 9E-06x3 - 0,0014x2 + 0,0889x + 0,0554 R² = 0,9975 0 0 50 100 150 200 250 Req (kOhm) 300 350 400 450 500 Figure 5B.3. The analysis of load on output power for 0.25g input acceleration amplitude when PPA1011 is tuned to input frequency of 20.8 Hz (damping coefficient is taken as transducer damping alone). 186 Table 5B.6. The load effect on output power is analyzed for 0.25g input acceleration amplitude when PPA1011 is tuned to input frequency of 20.8 Hz (sum of mechanical and transducer damping coefficients are used). SENSITIVITY ANALYSIS: 2d) Changing Req,for Ropt=76.6 kOhm case For Zeta = Zeta_mech + Zeta_transducer Req = 60.0 kOhm Experimental Ropt=76.6 kOhm and % Difference= 22 % RMS_disp = 0.760 mm Peak to Peak_disp = 1.521 mm Experimental Value = 9.7 mm and % Err= 84 % Ropt Code Output = 1.962 mm and % Err= 23 % RMS_Pout = 1.179644 mW Experimental Value = 2.4 mW and % Err= 51 % Req = 50.0 kOhm Experimental Ropt=76.6 kOhm and % Difference= 35 % RMS_disp = 0.772 mm Peak to Peak_disp = 1.545 mm Experimental Value = 9.7 mm and % Err= 84 % Ropt Code Output = 1.962 mm and % Err= 21 % RMS_Pout = 1.150017 mW Experimental Value = 2.4 mW and % Err= 52 % Req = 10.0 kOhm Experimental Ropt=76.6 kOhm and % Difference= 87 % RMS_disp = 0.903 mm Peak to Peak_disp = 1.805 mm Experimental Value = 9.7 mm and % Err= 81 % Ropt Code Output = 1.962 mm and % Err= 8 % RMS_Pout = 0.447346 mW Experimental Value = 2.4 mW and % Err= 81 % Req = 0.1 kOhm Experimental Ropt=76.6 kOhm and % Difference= 100 % RMS_disp = 0.961 mm Peak to Peak_disp = 1.922 mm Experimental Value = 9.7 mm and % Err= 80 % Ropt Code Output = 1.962 mm and % Err= 2 % RMS_Pout = 0.005200 mW Experimental Value = 2.4 mW and % Err= 100 % Ropt Code Output =2.063788 mW and % Err= 43 % Ropt Code Output =2.063788 mW and % Err= 44 % Ropt Code Output =2.063788 mW and % Err= 78 % Ropt Code Output =2.063788 mW and % Err= 100 % RMS_volt = 7.367 Volt Experimental Value = 13.8 V and % Err= 47 % Ropt Code Output = 10.909 V and % Err= 32 % RMS_Pin = 5.315 mW Power_Ratio1 (%) = 22.196 RMS_volt = 6.636 Volt Experimental Value = 13.8 V and % Err= 52 % Ropt Code Output = 10.909 V and % Err= 39 % RMS_Pin = 5.435 mW Power_Ratio1 (%) = 21.159 RMS_volt = 1.841 Volt Experimental Value = 13.8 V and % Err= 87 % Ropt Code Output = 10.909 V and % Err= 83 % RMS_Pin = 6.463 mW Power_Ratio1 (%) = 6.922 RMS_volt = 0.020 Volt Experimental Value = 13.8 V and % Err= 100 % Ropt Code Output = 10.909 V and % Err= 100 % RMS_Pin = 6.889 mW Req = 70.0 kOhm Experimental Ropt=76.6 kOhm and % Difference= 9 % RMS_disp = 0.753 mm Peak to Peak_disp = 1.505 mm Experimental Value = 9.7 mm and % Err= 84 % Ropt Code Output = 1.962 mm and % Err= 23 % RMS_Pout = 1.184804 mW Experimental Value = 2.4 mW and % Err= 51 % Req = 80.0 kOhm Experimental Ropt=76.6 kOhm and % Difference= 4 % RMS_disp = 0.748 mm Peak to Peak_disp = 1.497 mm Experimental Value = 9.7 mm and % Err= 85 % Ropt Code Output = 1.962 mm and % Err= 24 % RMS_Pout = 1.174194 mW Experimental Value = 2.4 mW and % Err= 51 % Req = 90.0 kOhm Experimental Ropt=76.6 kOhm and % Difference= 17 % RMS_disp = 0.746 mm Peak to Peak_disp = 1.493 mm Experimental Value = 9.7 mm and % Err= 85 % Ropt Code Output = 1.962 mm and % Err= 24 % RMS_Pout = 1.153662 mW Experimental Value = 2.4 mW and % Err= 52 % Req = 100.0 kOhm Experimental Ropt=76.6 kOhm and % Difference= 31 % RMS_disp = 0.746 mm Peak to Peak_disp = 1.492 mm Experimental Value = 9.7 mm and % Err= 85 % Ropt Code Output = 1.962 mm and % Err= 24 % RMS_Pout = 1.127110 mW Experimental Value = 2.4 mW and % Err= 53 % Ropt Code Output =2.063788 mW and % Err= 43 % Ropt Code Output =2.063788 mW and % Err= 43 % Ropt Code Output =2.063788 mW and % Err= 44 % Ropt Code Output =2.063788 mW and % Err= 45 % RMS_volt = 7.977 Volt Experimental Value = 13.8 V and % Err= 42 % Ropt Code Output = 10.909 V and % Err= 27 % RMS_Pin = 5.227 mW Power_Ratio1 (%) = 22.668 RMS_volt = 8.492 Volt Experimental Value = 13.8 V and % Err= 38 % Ropt Code Output = 10.909 V and % Err= 22 % RMS_Pin = 5.163 mW Power_Ratio1 (%) = 22.741 RMS_volt = 8.929 Volt Experimental Value = 13.8 V and % Err= 35 % Ropt Code Output = 10.909 V and % Err= 18 % RMS_Pin = 5.118 mW Power_Ratio1 (%) = 22.541 Req = 200.0 kOhm Experimental Ropt=76.6 kOhm and % Difference= 161 % RMS_disp = 0.769 mm Peak to Peak_disp = 1.538 mm Experimental Value = 9.7 mm and % Err= 84 % Ropt Code Output = 1.962 mm and % Err= 22 % RMS_Pout = 0.828504 mW Experimental Value = 2.4 mW and % Err= 65 % Req = 300.0 kOhm Experimental Ropt=76.6 kOhm and % Difference= 292 % RMS_disp = 0.790 mm Peak to Peak_disp = 1.580 mm Experimental Value = 9.7 mm and % Err= 84 % Ropt Code Output = 1.962 mm and % Err= 19 % RMS_Pout = 0.629002 mW Experimental Value = 2.4 mW and % Err= 74 % Req = 500.0 kOhm Experimental Ropt=76.6 kOhm and % Difference= 553 % RMS_disp = 0.814 mm Peak to Peak_disp = 1.628 mm Experimental Value = 9.7 mm and % Err= 83 % Ropt Code Output = 1.962 mm and % Err= 17 % RMS_Pout = 0.417792 mW Experimental Value = 2.4 mW and % Err= 83 % RMS_volt = 9.304 Volt Experimental Value = 13.8 V and % Err= 33 % Ropt Code Output = 10.909 V and % Err= 15 % RMS_Pin = 5.086 mW Power_Ratio1 (%) = 22.160 >> PPA1011_Clamp0_freq21Hz_FuncEqSim Req = 1000.0 kOhm Experimental Ropt=76.6 kOhm and % Difference= 1205 % RMS_disp = 0.836 mm Peak to Peak_disp = 1.673 mm Experimental Value = 9.7 mm and % Err= 83 % Ropt Code Output = 1.962 mm and % Err= 15 % RMS_Pout = 0.224841 mW Experimental Value = 2.4 mW and % Err= 91 % Ropt Code Output =2.063788 mW and % Err= 60 % Ropt Code Output =2.063788 mW and % Err= 70 % Ropt Code Output =2.063788 mW and % Err= 80 % Ropt Code Output =2.063788 mW and % Err= 89 % RMS_volt = 11.277 Volt Experimental Value = 13.8 V and % Err= 18 % Ropt Code Output = 10.909 V and % Err= 3 % RMS_Pin = 5.052 mW Power_Ratio1 (%) = 16.399 RMS_volt = 12.028 Volt Experimental Value = 13.8 V and % Err= 13 % Ropt Code Output = 10.909 V and % Err= 10 % RMS_Pin = 5.115 mW Power_Ratio1 (%) = 12.298 RMS_volt = 13.110 Volt Experimental Value = 13.8 V and % Err= 5 % Ropt Code Output = 10.909 V and % Err= 20 % RMS_Pin = 5.307 mW Power_Ratio1 (%) = 4.237 RMS_volt = 12.646 Volt Experimental Value = 13.8 V and % Err= 8 % Ropt Code Output = 10.909 V and % Err= 16 % RMS_Pin = 5.207 mW Power_Ratio1 (%) = 8.023 187 Optimum Load Effect on Output Power Power Output (mW) 2,5 2 1,5 1 0,5 0 0 200 400 600 Req (kOhm) 800 1000 Optimum Load Effect on Output Power Power Output (mW) 2,5 y = -5E-15x6 + 1E-11x5 - 1E-08x4 + 4E-06x3 - 0,0007x2 + 0,0536x + 0,0119 R² = 0,8199 2 1,5 1 0,5 0 0 100 200 300 Req (kOhm) 400 500 Figure 5B.4. The analysis of load on output power for 0.25g input acceleration amplitude when PPA1011 is tuned to input frequency of 20.8 Hz (sum of mechanical and transducer damping coefficients are used). 188 Optimum Load Effect on Output Power 9 Power Output (mW) 8 7 6 5 4 3 2 1 0 0 200 400 600 Req (kOhm) 800 1000 Power Output (mW) Optimum Load Effect on Output Power 10 9 8 7 6 5 4 3 2 1 0 y = -1E-13x6 + 2E-10x5 - 1E-07x4 + 4E-05x3 - 0,0057x2 + 0,3654x + 0,2365 R² = 0,9965 0 100 200 300 Req (kOhm) 400 500 Figure 5B.5. The analysis of load on output power for 0.5g input acceleration amplitude when PPA1011 is tuned to input frequency of 21 Hz (damping coefficient is taken as transducer damping alone). 189 Optimum Load Effect on Output Power Power Output (mW) 5 4 3 2 1 0 0 200 400 600 Req (kOhm) 800 1000 Optimum Load Effect on Output Power Power Output (mW) 5 4 3 2 1 y = -7E-14x6 + 1E-10x5 - 6E-08x4 + 2E-05x3 - 0,0029x2 + 0,1934x + 0,0586 R² = 0,9991 0 0 100 200 300 Req (kOhm) 400 500 Figure 5B.6. The analysis of load on output power for 0.5g input acceleration amplitude when PPA1011 is tuned to input frequency of 21 Hz (sum of mechanical and transducer damping coefficients are used). 190 Table 5B.7. Sensitivity analysis chart showing theeffects of of optimum load and damping coefficient on output power for 1g input acceleration amplitude when PPA1011 is tuned to input frequency of 21 Hz. 1g, 21 Hz, Req=33.6kΩ, Pout_exp=16.0 mW 33.6 kΩ Req (kΩ) 0.1 10 20 30 40 50 100 200 300 400 500 1000 transducer damping alone (2a) 29.710363 Pout (mW) 0.172482 14.168769 23.120622 28.413387 31.312319 32.696183 30.976989 22.766415 17.434261 14.036419 11.72179 6.390188 sum of mechanical and transducer damping (2b) 16.0 Pout (mW) 0.081659 7.042754 11.9648 15.16857 17.10823 18.16833 17.79153 13.04477 9.888256 7.898237 6.557519 3.524069 Table 5B.8. Sensitivity analysis chart showing theeffects of of optimum load and damping coefficient on output power for 0.25g input acceleration amplitude when PPA1011 is tuned to input frequency of 20.8 Hz. 76.6 kΩ Req (kΩ) 0.1 10 50 60 70 80 90 100 200 300 400 500 1000 0.25g, 20.8 Hz, Req=76.6 kΩ, Pout_exp=2.4 mW transducer damping alone sum of mechanical and transducer damping (2c) (2d) 2.06 1.18 Pout (mW) Pout (mW) 0.010966 0.0052 0.898158 0.447346 2.063925 1.150017 2.0914 2.063788 2.082451 1.184804 2.051392 1.174194 2.007255 1.153662 1.955767 1.12711 1.439929 0.828504 1.104031 0.629002 0.889561 0.502893 0.74327 0.417792 0.405694 0.224841 191 Table 5B.9. Sensitivity analysis chart showing theeffects of of optimum load and damping coefficient on output power for 0.25g input acceleration amplitude when PPA1011 is tuned to input frequency of 21 Hz. 0.5g, 21 Hz, Req= 41.2 kΩ, Pout_exp=5.4 mW 41.2 kΩ Req (kΩ) 0.1 10 20 30 40 50 60 70 80 90 100 200 300 400 500 1000 transducer damping alone 7.88617 Pout (mW) 0.04312 3.542192 5.780156 7.103347 7.82808 8.174046 8.284082 8.248758 8.125164 7.949364 7.744247 5.691604 4.358565 3.509105 2.930449 1.597547 sum of mechanical and transducer damping 4.31883 Pout (mW) 0.020416 1.760688 2.9912 3.792141 4.277057 4.542081 4.659162 4.678977 4.636138 4.553927 4.447884 3.261192 2.472064 1.974559 1.63938 0.881017 192 Table 5B.10. Overall sensitivity analysis results for the input accelerations of 0.25g, 0.5g, 1g, and 2g for the tuning masses of 25.3g, 2.7g and no tip mass for the relative tuning frequencies of 21 Hz, 60 Hz and 145 Hz Exp Midé rmsPout=2.5 mW Mass Corr Factor f=21 Hz, mtip=25.3e-3 kg, m_eq=0.614e -3kg mu 1 1.1 1.2 1.3 Confidence Interval: Amp=0.25g, Req=76.6e3ohm Exp Midé rms Pout at 0.25g Amp=0.25g, Amp=0.25g, Amp=0.25g, Zmech Ztransd Zm+Zt 3.4 2.1 1.2 4.1 2.5 1.4 4.9 3.0 1.7 5.8 3.5 2.0 2.5 2.5 2.5 2.5 mu= 1.1, Ztransducer Exp Midé rmsPout=0.4 mW Mass Corr Factor f=60 Hz, mtip=2.7e-3 kg, m_eq=0.614e -3kg mu 1 1.1 1.2 1.3 Confidence Interval: Mass Corr Factor Amp=0.25g, Req=25e3ohm Amp=0.25g, Amp=0.25g, Amp=0.25g, Zmech Ztransd Zm+Zt 0.18 0.10 0.05 0.21 0.12 0.07 0.25 0.14 0.08 0.30 0.17 0.09 0.40 0.40 0.40 0.40 mu= 1.5, Zmech mu f=147,146,145 Hz, mtip0 kg, m_eq=0.614e -3kg 1 1.1 1.2 1.3 1.5 1.6 1.8 2.4 3 3.2 3.8 Confidence Interval: Amp=0.25g, Req=12.1e3ohm Amp=0.25g, Amp=0.25g, Amp=0.25g, Zmech Ztransd Zm+Zt 0.011 0.006 0.003 0.014 0.008 0.004 0.017 0.009 0.005 0.019 0.011 0.006 0.026 0.014 0.011 0.029 0.037 0.021 0.011 0.066082 0.036782 0.020383 0.103495 0.057472 0.036237 mu= 3.0, Zmech Exp Midé rmsPout=-- mW Exp Midé Exp Midé rmsPout=16.0 mW Exp Midé Exp Midé rms Pout Amp=1.0g, Req=33.7e3 rms Pout at rms Pout Amp=0.5g, Req=41.2e3 ohm Amp=2.0g, Req=--e3 ohm at 0.5g 1g at 2g ohm Amp=0.5g, Amp=0.5g, Amp=0.5g, Amp=1g, Amp=1g, Amp=1g, Amp=2.0g, Amp=2.0g, Amp=2.0g, Zmech Ztransd Zm+Zt Zmech Ztransd Zm+Zt Zmech Ztransd Zm+Zt 13.7 7.9 4.3 5.4 52.4 29.7 16.0 16.0 16.5 9.5 5.2 5.4 63.4 35.9 19.4 16.0 19.7 11.4 6.2 5.4 75.4 42.8 23.1 16.0 23.1 13.3 7.3 5.4 88.5 50.2 27.1 16.0 mu=1.1, Ztotal Exp Midé rms Pout at 0.25g Exp Midé rmsPout=0.1 mW Exp Midé rmsPout=5.4 mW 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 - Exp Midé rmsPout=1.1 mW Exp Midé rmsPout=9.6 mW Exp Midé Exp Midé rmsPout=3.2 mW Exp Midé Exp Midé rms Pout Amp=1.0g, Req=19.5e3 rms Pout at rms Pout Amp=0.5g, Req=15.8e3 ohm Amp=2.0g, Req=17.3e3 ohm at 0.5g 1g at 2g ohm Amp=0.5g, Amp=0.5g, Amp=0.5g, Amp=1g, Amp=1g, Amp=1g, Amp=2.0g, Amp=2.0g, Amp=2.0g, Zmech Ztransd Zm+Zt Zmech Ztransd Zm+Zt Zmech Ztransd Zm+Zt 0.8 0.4 0.2 1.10 3.0 1.6 0.9 3.20 12.0 6.5 3.4 9.60 0.9 0.5 0.3 1.10 3.6 2.0 1.0 3.20 14.5 7.8 4.1 9.60 1.1 0.6 0.3 1.10 4.3 2.3 1.2 3.20 17.3 9.3 4.9 9.60 1.3 0.7 0.4 1.10 5.0 2.8 1.5 3.20 20.3 10.9 5.7 9.60 mu=1.2, Zmech Exp Midé rms Pout at 0.25g mu=1.0, Ztotal mu=1.04, Zmech mu=1.21, Ztransd Exp Midé rmsPout=0.3 mW Exp Midé rmsPout=2.1 mW Exp Midé Exp Midé rmsPout= 0.7 mW Exp Midé Exp Midé rms Pout Amp=1.0g, Req=10.2e3 rms Pout at rms Pout Amp=0.5g, Req=12.6e3 ohm Amp=2.0g, Req=10.1 ohm at 0.5g 1g at 2g ohm Amp=0.5g, Amp=0.5g, Amp=0.5g, Amp=1g, Amp=1g, Amp=1g, Amp=2.0g, Amp=2.0g, Amp=2.0g, Zmech Ztransd Zm+Zt Zmech Ztransd Zm+Zt Zmech Ztransd Zm+Zt 0.047 0.026 0.014 0.3 0.20 0.11 0.06 0.7 0.81 0.44 0.23 2.1 0.056 0.031 0.020 0.3 0.24 0.13 0.07 0.7 0.97 0.53 0.28 2.1 0.079 0.037 0.023 0.3 0.29 0.16 0.08 0.7 1.16 0.63 0.33 2.1 0.091 0.044 0.027 0.3 0.34 0.18 0.10 0.7 1.36 0.74 0.39 2.1 0.105 0.059 0.031 0.3 0.45 0.25 0.13 0.7 1.81 0.99 0.52 2.1 0.3 0.51 0.7 2.06 2.1 0.3 0.64 0.7 2.1 0.269 0.150 0.080 0.3 0.7 2.1 0.234 0.3 0.7 2.1 0.266 0.142 0.3 0.7 2.1 0.200 0.3 0.7 2.1 mu=2.6, Zmech 193 mu=1.8, Zmech mu=1.6, Zmech Figure 5B.7. Sensitivity analysis for output power at the input frequency of 21 Hz, tip mass of 25.3 g and the effective mass of 0.614 g. 194 Figure 5B.8. Sensitivity analysis for output power at the input frequency of 60 Hz, tip mass of 2.7 g and the effective mass of 0.614 g. 195 Figure 5B.9. Sensitivity analysis for output power at the input frequency of 146 Hz in average, no tip mass and the effective mass of 0.614 g. 196 Figure 7, 8 and 9 enable the determination of confidence intervals for the better estimation and those intervals are listed in Table 10. For the precision, the usage of curve fit formulas sometimes needed and confidence intervals are gathered. The evaluations from curve fits are listed in Table 11 and 12. Table 5B.11. Curve fit evaluation findings for the input frequency of 60 Hz, tip mass of 2.7 g and the effective mass of 0.614 g. Zmech, Zmech, Zmech, Ztransd, 0.25g 0.5g 1g 2g Correction Factor Power output matching with the experimental value 1.5 - 1.04 1.21 0.40 - 3.20 9.48 Table 5B.12. Curve fit evaluation findings for the input frequency of 146 Hz in average, no tip mass and the effective mass of 0.614 g. Zmech, 0.25g Correction Factor Power output matching with the experimental value Zmech, Zmech, 0.5g 1g Zmech, 2g 3 2.6 1.9 1.7 0.10 0.31 0.72 2.33 The same methodology on correction factor and the selection of the damping coefficient is also followed for the Midé’s counterpart piezoelectric energy harvester model PPA 2011. The resulted output and its precision are listed in Table 13. As highlighted with red, the output power of the final case (2g amplitude input at 23 Hz with 25.3g of tip mass) is overestimated by 11% for output power even when no correction is made. Since correction factor is not stated as lower than 1 by Erturk [83], it is left as its original estimated results. In Tables 5B.14 and 15, selected damping and correction factors of PPA-1011 and 2011 are listed as a summary. It is seen that since PPA-2011 is improved version of PPA1011, the output powers are greater. Thus, damping coefficients of PPA2011 and 1011 differs for the similar inputs and correction factors are greater for PPA 2011. For no tip mass case, amplitude correction factor is also considered along with the mass correction factor in the model of constitutive set of equations both takes place in the same and only place with 𝑚𝑤𝑏̈ term. Though, in this study the chosen method to find the correction factor is trial and error since the empirically driven formula does not hold. The correction factor of maximum 3 also gives clue that it is very close to the summation of the amplitude and mass correction factors. Thus, for improved derivations, this observation is expected to be used in future studies. 197 Table 5B.13. For PPA-2011 Midé energy harvester, output power corrections are made upon the correction factors and the relative damping coefficient as both stated for each case. At the same damping coefficient, the difference from the uncorrected estimation (correction factor of 1) is stated as well as the tip displacement, power and voltage output errors from the experimental data. Q=15.1,f=154,Amp= 0.25g,mtip=0, meff=0.607/1000 kg and Req=7e3ohm Corr Factor =3, Damping Coeff=Zmech RMS_disp = 0.093 mm Peak to Peak_disp = 0.185 mm Experimental Value = 0.6 mm and % Err= 69 % Ropt Code Output =0.044 mm and % Difference= 110 % RMS_Pout = 0.095208 mW Experimental Value = 0.1 mW and % Err= 5 % Ropt Code Output =0.021589 mW and % Difference= 341 % RMS_volt = 0.727 Volt Experimental Value = 0.9 V and % Err= 19 % Ropt Code Output = 0.346 V and % Difference= 110 % RMS_Pin = 0.586 mW Power_Ratio1 (%) = 16.244 Q=15.1,f=60,Amp= 0.25g,mtip=3.5, meff=0.607/1000 kg and Req=10.5e3ohm Corr Factor =1.54. Damping Coeff=Zmech RMS_disp = 0.338 mm Peak to Peak_disp = 0.677 mm Experimental Value = 1.6mm and % Err= 90 % Ropt Code Output = 0.220 mm and % Difference= 54 % RMS_Pout = 0.499095529 mW Experimental Value = 0.5 mW and % Err= 0 % Ropt Code Output =0.210446757 mW and % Difference= 137 % RMS_volt = 2.017 Volt Experimental Value = 2.3 V and % Err= 12 % Ropt Code Output = 1.309 V and % Difference= 54 % RMS_Pin = 1.839 mW Power_Ratio1 (%) = 27.142 Q=15.1,f=24,Amp= 0.25g,mtip=25.3, meff=0.607/1000 kg and Req=24e3ohm Corr Factor =1.25, Damping Coeff=Zmech RMS_disp = 1.493 mm Peak to Peak_disp = 2.985 mm Experimental Value = 4.8 mm and % Err= 38 % Ropt Code Output = 1.194 mm and % Difference= 25 % RMS_Pout = 4.066840109 mW Experimental Value = 4.1 mW and % Err= 1 % Ropt Code Output = 2.602777670 mW and % Difference= 56 % RMS_volt = 8.412 Volt Experimental Value = 9.9 V and % Err= 15 % Ropt Code Output = 6.730 V and % Difference= 25 % RMS_Pin = 13.699 mW Power_Ratio1 (%) = 29.688 Q=15.1,f=152,Amp= 0.5g,mtip=0, meff=0.607/1000 kg and Req=4e3ohm Corr Factor =2.95, Damping Coeff=Zmech RMS_disp = 0.192 mm Peak to Peak_disp = 0.384 mm Experimental Value = 1.0 mm and % Err= 62 % Ropt Code Output =0.065 mm and % Difference= 196 % RMS_Pout = 0.393048 mW Experimental Value = 0.4 mW and % Err= 2 % Ropt Code Output = 0.045165 mW and % Difference= 770 % RMS_volt = 1.123 Volt Experimental Value = 1.2 V and % Err= 6 % Ropt Code Output = 0.381 V and % Difference= 195 % RMS_Pin = 2.219 mW Power_Ratio1 (%) = 17.712 Q=15.1,f=60,Amp= 0.5g,mtip=3.4, meff=0.607/1000 kg and Req=9e3ohm Corr Factor =1.35, Damping Coeff=Zmech RMS_disp = 0.602 mm Peak to Peak_disp = 1.203 mm Experimental Value = 2.3 mm and % Err= 48 % Ropt Code Output = 0.446 mm and % Difference= 35 % RMS_Pout = 1.497366866 mW Experimental Value = 1.5 mW and % Err= 0 % Ropt Code Output =0.821600475 mW and % Difference= 82 % RMS_volt = 3.232 Volt Experimental Value = 3.7 V and % Err= 13 % Ropt Code Output = 2.394 V and % Difference= 35 % RMS_Pin = 6.523 mW Power_Ratio1 (%) = 22.955 Q=15.1,f=24,Amp= 0.5g,mtip=25.3, meff=0.607/1000 kg and Req=39.4e3ohm Corr Factor =1.57, Damping Coeff=Ztransducer RMS_disp = 2.477 mm Peak to Peak_disp = 4.954 mm Experimental Value = 7.9 mm and % Err= 37 % Ropt Code Output = 1.578 mm and % Difference= 57 % RMS_Pout = 11.451338641 mW Experimental Value = 11.5 mW and % Err= 0 % Ropt Code Output =4.645761954 mW and % Difference= 146 % RMS_volt = 17.638 Volt Experimental Value = 21.3 V and % Err= 17 % Ropt Code Output = 11.235 V and % Difference= 57 % RMS_Pin = 45.202 mW Power_Ratio1 (%) = 25.334 Q=15.1,f=149,Amp= 1g,mtip=0, meff=0.607/1000 kg and Req=3.3e3ohm Corr Factor =2.55, Damping Coeff=Zmech RMS_disp = 0.355 mm Peak to Peak_disp = 0.710 mm Experimental Value = 1.6 mm and % Err= 56 % Ropt Code Output =0.139 mm and % Difference= 155 % RMS_Pout = 1.209620 mW Experimental Value = 1.2 mW and % Err= 1 % Ropt Code Output =0.186024 mW and % Difference= 550 % RMS_volt = 1.790 Volt Experimental Value = 2 V and % Err= 11 % Q=15.1,f=147,Amp= 2g,mtip=0, meff=0.607/1000 kg and Req=5.1e3ohm Corr Factor =2.28, Damping Coeff=Zmech RMS_disp = 0.605 mm Peak to Peak_disp = 1.210 mm Experimental Value = 2.6 mm and % Err= 53 % Ropt Code Output =0.265 mm and % Difference= 128 % RMS_Pout = 3.956819 mW Experimental Value = 4 mW and % Err= 1 % Ropt Code Output =0.761161 mW and % Difference= 420 % RMS_volt = 4.017 Volt Experimental Value = 4.5 V and % Err= 11 % Ropt Code Output = 0.702 V and % Difference= 155 % RMS_Pin = 8.393 mW Power_Ratio1 (%) = 14.412 Q=15.1,f=60,Amp= 1g,mtip=3.3, meff=0.607/1000 kg and Req=14.7e3ohm Corr Factor =1.16, Damping Coeff=Zmech RMS_disp = 0.970 mm Peak to Peak_disp = 1.939 mm Experimental Value = 4.3 mm and % Err= 55 % Ropt Code Output = 0.836 mm and % Difference= 16 % RMS_Pout = 4.244322670 mW Experimental Value = 4.3 mW and % Err= 1 % Ropt Code Output =3.154223149 mW and % Difference= 35 % RMS_volt = 6.971 Volt Experimental Value = 7.9 V and % Err= 12 % Ropt Code Output = 6.009 V and % Difference= 16 % RMS_Pin = 19.663 mW Power_Ratio1 (%) = 21.586 Q=15.1,f=23.8,Amp= 1g,mtip=25.3, meff=0.607/1000 kg and Req=30.9e3ohm Corr Factor =1.28, Damping Coeff=Ztransducer RMS_disp = 4.126 mm Peak to Peak_disp = 8.253 mm Experimental Value = 12 mm and % Err= 31 % Ropt Code Output = 3.224 mm and % Difference= 28 % RMS_Pout = 30.972793165 mW Experimental Value = 31 mW and % Err= 0 % Ropt Code Output =18.904292703 mW and % Difference= 64 % RMS_volt = 26.884 Volt Experimental Value = 31 V and % Err= 13 % Ropt Code Output = 1.762 V and % Difference= 128 % RMS_Pin = 26.346 mW Power_Ratio1 (%) = 15.019 Q=15.1,f=60,Amp= 2g,mtip=3.4, meff=0.607/1000 kg and Req=18.2e3ohm Corr Factor =1.44, Damping Coeff=Ztransducer RMS_disp = 1.555 mm Peak to Peak_disp = 3.110 mm Experimental Value = 6.9 mm and % Err= 55 % Ropt Code Output = 1.080 mm and % Difference= 44 % RMS_Pout = 10.385457130 mW Experimental Value = 10.4 mW and % Err= 0 % Ropt Code Output =5.008418755 mW and % Difference= 107 % RMS_volt = 12.236 Volt Experimental Value = 13.7 V and % Err= 11 % Ropt Code Output = 8.497 V and % Difference= 44 % RMS_Pin = 65.798 mW Power_Ratio1 (%) = 15.784 Q=15.1,f=23,Amp= 2g,mtip=25.3, meff=0.607/1000 kg and Req=17.2e3ohm Corr Factor =1, Damping Coeff=Ztotal RMS_disp = 5.256 mm Peak to Peak_disp = 10.512 mm Experimental Value = 18.5 mm and % Err= 43 % Ropt Code Output = 21.003 V and % Difference= 28 % RMS_Pin = 148.381 mW Power_Ratio1 (%) = 20.874 198 RMS_Pout = 37.699007283 mW Experimental Value = 34 mW and % Err= 11 % RMS_volt = 22.338 Volt Experimental Value = 34 V and % Err= 34 % RMS_Pin = 374.640 mW Power_Ratio1 (%) = 10.063 Table 5B.14. Confidence interval table of all investigated cases for PPA-1011 power output estimation. Selected damping and correction factors are listed. 146 Hz, no tip mass 60 Hz, 2.7 g tip mass 21 Hz, 25.3 g tip mass Amp 0.25g 0.5g 1g 2g 0.25g 0.5g 1g 2g 0.25g 0.5g 1g 2g Damping Zmech Zmech Zmech Zmech Zmech Zmech Zmech Ztransducer Ztransducer Ztotal Ztotal Ztotal* coefficient 0.0167 0.0167 0.0167 0.0167 0.0167 0.0167 0.0167 0.0284 0.0284 0.0451 0.0451 0.0451* Correction 3.0 2.6 1.8 1.6 1.5 1.2 1.04 1.2 1.01 1.01 1.0 1.0* factor (*) After 1g amplitude input value, it is assumed no correction regarding as its previous behavior. Table 5B.15. Confidence interval table of all investigated cases for PPA-1011 power output estimation. Selected damping and correction factors are listed. 154-147 Hz, no tip mass 60 Hz, 3.5 g tip mass 24 Hz, 25.3 g tip mass Amp 0.25g 0.5g 1g 2g 0.25g 0.5g 1g 2g 0.25g 0.5g 1g 2g Zmech Zmech Zmech Zmech Ztransd. Zmech Ztransd. Ztransd. Ztotal* Damping Zmech Zmech Zmech coefficient 0.0167 0.0167 0.0167 0.0167 0.0167 0.0167 0.0167 0.0284 0.0167 0.0451 0.0451 0.0451* Correction 3.0 2.95 2.55 2.28 1.54 1.35 1.16 1.44 1.25 1.57 1.28 1.00* factor (*) At 2g amplitude input, no correction is made and the results are overestimated by 11% for output power. 199 5B.1.1. Determination of the Optimum Load As briefly covered optimum load is also investigated over the proposed formulations by Calio et al, du Toit and Erturk as in Eqs (1.1.1-4) determination. It is seen that Calio et al.’s proposed optimum load formula differs far from the equivalent loads used in experiments but the basis of the load estimation since duToit and Erturk use this basic 1 term with correction. It is also observed that this term is missing in du Toit’s 𝜔 𝐶 𝑛 𝑝 equation thus results in very low values in between 0.5-2. Du Toit’s formula is corrected by this term in Eq (1.1.4) and re-calculated. Evaluations along with the optimum load result in our approach are listed in Table 5B.16 and 17. 1  Abeam 2 33 RL   A  R   1  A. 33 L   A 33 Ropt  hp   h h piezo  p    hp 1 R opt .Calio = = A ii C p Ropt ,duToit  Ropt  (1.1.1)  4   4 m 2  2   2 1   2 6 2    4 m  2 1   k C  eff p   1 n C P  4  2     1     keff C p (1.1.2) 2  2    2 1   m 2     2  m  1     m 2 1    2 m 2  (1.1.3) where:  2 C p n 2 Ropt ,duToit ,corrected 1  n C p  4   4 m 2  2   2 1   2  6   4 m 2  2 1   keff C p     4  2    1       keff C p 2  2    (1.1.4) Confidence intervals are clarified in Table 5B.14 and 15. Referring Table 14 and 15, it is seen that for no tip mass and low tip mass, mechanical coupling factor is dominant with a small transducer damping factor transition up until the 25.3 g of tip mass at 0.25g for PPA-2011 and at 0.5g acceleration amplitude for PPA-1011. After this amplitude range, damping coefficient is stabilized for the sum of mechanical and transducer damping. In Table 5B.14 and 15, it is also seen that not only as the amplitude increases, 200 but also as tip mass increases, correction factor decreases to its ineffective value of 1. In other words, for PPA-1011, after around 0.5g amplitude at 25.3 g of tip mass, the model does not need any correction to gain the accurate power output. On the other hand, for PPA-2011, this is not the case, correction factor is always needed except the case with a tip mass of 25.3g at 2g. 201 Table 5B.16. Confidence interval table of all investigated cases for PPA-1011 power output estimation. Selected damping and correction factors are listed along with the optimum load investigation in comparision with the Midé’s experimental loads. 146 Hz, no tip mass 60 Hz, 2.7 g tip mass 21 Hz, 25.3 g tip mass Amp 0.25g 0.5g 1g 2g 0.25g 0.5g 1g 2g Damping Zmech Zmech Zmech Zmech Zmech Zmech Zmech coefficient 0.0167 0.0167 0.0167 0.0167 0.0167 0.0167 0.0167 0.0284 0.25g 0.5g 1g 2g Ztotal Ztotal Ztotal* 0.0167 0.0451 0.0451 0.0451* Ztransducer Ztransducer Correction 3.0 factor Req, exp -Midé 12,100 2.95 2.55 2.28 1.54 1.35 1.16 1.44 1.25 1.57 1.28 12,600 10,200 10,100 25,000 15,800 19,500 17,300 76,600 41,200 33,700 Ropt-code sweep 4,000 5,000 5,000 10,100 16,000 15,000 15,000 20,000 60,000 60,000 60,000 Ropt Calio 11,904 11,904 11,904 11,904 27,250 26,836 26,836 26,836 76,201 76,201 76,201 Ropt Calio,w 10,827 10,901 10,901 10,976 26,526 26,526 26,526 26,526 76,517 75,788 75,788 Ropt duToit Ropt duToit 0.867 0.859 1.097 1.124 1.552 1.159 1.159 1.053 0.819 0.973 0.973 10,322 10,220 13,064 13,378 42,288 31,111 31,111 28,271 62,431 74,150 74,150 9,388 9,359 11,963 12,336 41,163 30,751 30,751 27,945 62,689 73,748 73,748 Ropt Erturk 6,264 6,264 6,264 6,264 14,339 14,121 14,121 14,125 40,109 40,156 40,156 Ropt Erturk,w 5,697 5,736 5,736 5,776 13,958 13,958 13,958 13,962 40,275 39,938 39,938 corrected Ropt duToit corrected,w (*) After 1g value, it is assumed no correction regarding as its previous behavior. 202 1.00* 𝟏 𝝎𝒏 ⋅ 𝑪𝑷 𝟏 𝝎 ⋅ 𝑪𝑷 - 𝟏 𝝎𝒏 ⋅ 𝑪𝑷 𝟏 𝝎 ⋅ 𝑪𝑷 𝟏 𝝎𝒏 ⋅ 𝑪𝑷 𝟏 𝝎 ⋅ 𝑪𝑷 Table 5B.17. Confidence interval table of all investigated cases for PPA-2011 power output estimation. Selected damping and correction factors are listed along with the optimum load investigation in comparision with the Midé’s experimental loads. 154-147 Hz, no tip mass Amp 0.25g 0.5g 1g 2g 60 Hz, 3.5 g tip mass 0.25g 0.5g 1g 24 Hz, 25.3 g tip mass 2g 0.25g 0.5g 1g 2g Damping Zmech Zmech Zmech Zmech Zmech Zmech Zmech Ztransd. Zmech Ztransd. Ztransd. Ztotal* coefficient 0.0167 0.0167 0.0167 0.0167 0.0167 0.0167 0.0167 0.0284 0.0167 0.0284 0.0284 0.0451* 1.44 1.25 1.57 1.28 1.00* Correction 3.0 factor Req, exp -Midé 7,000 Ropt-code sweep 4,000 2.95 2.55 2.28 4,000 3,300 5,100 10,500 9,000 14,700 18,200 24,000 39,400 30,900 17,200 4,000 4,000 5,100 10,500 11,000 11,000 12,500 32,000 32,000 32,000 35,000 Ropt Calio 5,558 5,558 5,558 5,558 14,082 14,082 14,082 14,082 35,369 35,369 35,369 35,369 Ropt Calio,w 5,439 5,511 5,622 5,698 13,961 13,961 13,961 13,961 34,902 34,902 35,196 36,420 Ropt duToit Ropt duToit 1.525 1.010 0.624 0.681 1.013 1.000 1.241 1.083 0.935 0.856 8,477 5,611 3,470 3,787 14,266 14,266 14,266 14,081 43,898 38,308 33,083 30,284 8,296 5,563 3,509 3,882 14,143 14,143 14,143 13,959 43,319 37,802 32,920 31,184 Ropt Erturk 5,557 5,557 5,557 5,557 14,079 14,079 14,079 14,084 35,360 35,372 35,372 35,413 Ropt Erturk,w 5,438 5,510 5,620 5,697 13,958 13,958 13,958 13,962 34,894 34,905 35,198 36,465 1.54 1.013 1.35 1.013 1.16 corrected Ropt duToit corrected,w (*) At 2g amplitude input, no correction is made and the results are overestimated by 11% for output power. 203 Multiplied term 𝟏 𝝎𝒏 ⋅ 𝑪𝑷 𝟏 𝝎 ⋅ 𝑪𝑷 - 𝟏 𝝎𝒏 ⋅ 𝑪𝑷 𝟏 𝝎 ⋅ 𝑪𝑷 𝟏 𝝎𝒏 ⋅ 𝑪𝑷 𝟏 𝝎 ⋅ 𝑪𝑷