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عدد المساهمات : 18928 التقييم : 35294 تاريخ التسجيل : 01/07/2009 الدولة : مصر العمل : مدير منتدى هندسة الإنتاج والتصميم الميكانيكى
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أخواني في الله أحضرت لكم كتاب Frequency Analysis of Vibration Energy Harvesting Systems Xu Wang School of Engineering College of Science Engineering and Health RMIT University
![كتاب Frequency Analysis of Vibration Energy Harvesting Systems F_a_o_11](https://i.servimg.com/u/f61/15/36/74/14/f_a_o_11.jpg) و المحتوى كما يلي :
List of Figures Figure 1.1 A single degree of freedom spring-mass-dashpot system driven by (A) direct force and (B) inertial force. 3 Figure 1.2 Prediction of the mass relative displacement response from the base displacement excitation using Matlab Simulink transfer function method. 5 Figure 1.3 Prediction of the relative displacement response from the base acceleration excitation input using the Matlab Simulink integration method: (A) Integration schematic of Eq. (1.12); (B) Matlab Simulink schematic following Eq. (1.12). 6 Figure 1.4 Schematic of the Matlab Simulink state space method for prediction of the relative displacement response from the base displacement excitation. 8 Figure 1.5 A sine wave excitation signal input with a frequency of 275 Hz and RMS acceleration of 9.8 m/s2. 9 Figure 1.6 Parameter inputs of sine wave excitation acceleration for the source block of the Matlab Simulink. 10 Figure 1.7 The time trace of the relative displacement response. 11 Figure 1.8 A Matlab code for calculation of the relative displacement frequency response function amplitude versus frequency curve. 11 Figure 1.9 Relative displacement frequency response function (the relative displacement amplitude over the excitation acceleration amplitude). 12 Figure 2.1 Schematic of single degree of freedom piezoelectric vibration energy harvester system connected to a single electric load resistor (A) the harvester oscillator electromechanical system (B) energy harvesting circuit. 16 Figure 2.2 Simulation schematic for Eq. (2.10) in Matlab Simulink with a sine wave base excitation input and a sinusoidal voltage output at a frequency using the transfer function method. 20 Figure 2.3 (A) Relative acceleration integrated to the relative displacement; (B) Derivative of the output voltage integrated to the output voltage; (C) Simulation schematic for Eq. (2.11) with a sine wave base excitation input and a sinusoidal voltage output at a frequency using the integration method. 22 Figure 2.4 The Matlab Simulink state space method schematic for the prediction of the response relative displacement and output voltage from the base excitation acceleration. 24 Figure 2.5 Output sinusoidal voltage signal (volt) from an input excitation acceleration signal with a root mean square acceleration value of 1 g (9.8 m/s2) at an excitation frequency of 275 Hz. 25 Figure 2.6 The output voltage amplitude versus the base excitation acceleration amplitude at an excitation frequency of 275 Hz. 26 xiFigure 2.7 Harvested resonant power versus the base excitation acceleration amplitude at an excitation frequency of 275 Hz. 27 Figure 2.8 The output voltage amplitude versus the mechanical damping under an input excitation acceleration signal of an RMS value of 1 g (9.8 m/s2) at an excitation frequency of 275 Hz. 27 Figure 2.9 Harvested resonant power versus the mechanical damping under an input excitation acceleration signal of an RMS value of 1 g (9.8 m/s2) at an excitation frequency of 275 Hz. 28 Figure 2.10 The output voltage amplitude versus the electrical load resistance under an input excitation acceleration signal of an RMS value of 1 g (9.8 m/s2) at an excitation frequency of 275 Hz. 28 Figure 2.11 Harvested resonant power versus the electrical load resistance under an input excitation acceleration signal of an RMS value of 1 g (9.8 m/s2) at an excitation frequency of 275 Hz. 29 Figure 2.12 Harvested resonant power versus the force factor under an input excitation acceleration signal of an RMS value of 1 g (9.8 m/s2) at an excitation frequency of 275 Hz. 29 Figure 2.13 Matlab codes for simulation of the frequency response functions of the output relative displacement and voltage over the input excitation acceleration. 30 Figure 2.14 Amplitude of the relative displacement frequency response functiondthe output relative displacement divided by the input excitation acceleration. 31 Figure 2.15 Amplitude of the voltage frequency response functiondthe output voltage divided by the input excitation acceleration. 32 Figure 2.16 Harvested resonant power and output voltage versus frequency. 34 Figure 2.17 Dimensionless harvested resonant power versus dimensionless resistance and force factors for the single degree of freedom system connected to a load resistor. 36 Figure 2.18 Resonant energy harvesting efficiency versus dimensionless resistance and force factors for the single degree of freedom system connected to a load resistor. 38 Figure 3.1 Extraction and storage interface circuits for vibration energy harvesters, (A) standard; (B) synchronous electric charge extraction (SECE); (C) parallel synchronous switch harvesting on inductor (SSHI); (D) series SSHI Circuit. 44 Figure 3.2 Working principle of a full cycle of bridge rectification. (A) Positive half-cycle; (B) negative half-cycle; (C) positive output waveform. 45 Figure 3.3 The resonant energy harvesting efficiency versus dimensionless resistance and force factors for the single degree of freedom piezoelectric harvester connected to the four types of xii List of Figuresinterface circuits. (A) Standard; (B) synchronous electric charge extraction; (C) parallel synchronous switch harvesting on inductor (SSHI); (D) series SSHI. 56 Figure 3.4 The dimensionless harvested resonant power versus dimensionless resistance and force factors for the single degree of freedom piezoelectric harvester connected to the four types of interface circuits. (A) Standard; (B) synchronous electric charge extraction; (C) parallel synchronous switch harvesting on inductor (SSHI); (D) series SSHI. 57 Figure 3.5 Output voltage and harvested resonant power versus the input excitation acceleration. (A) Harvested resonant power versus the excitation acceleration; (B) output voltage versus the excitation acceleration. 61 Figure 3.6 (A) Output voltage and (B) harvested resonant power versus mechanical damping under the base acceleration of 9.8 m/s2. 62 Figure 3.7 Output voltage (A) and harvested resonant power (B) versus electric resistance under the base acceleration of 9.8 m/s2. 63 Figure 3.8 Harvested resonant power versus force factor under the base acceleration of 9.8 m/s2. 63 Figure 4.1 Schematic of a single degree of freedom electromagnetic vibration energy harvester connected to a single load resistor. 71 Figure 4.2 Simulation schematic for Eq. (4.9) in Matlab Simulink with a sine wave base excitation input and a sinusoidal voltage output at a frequency using the transfer function method. 72 Figure 4.3 (A) Relative acceleration integrated to the relative displacement. (B) Derivative of the output voltage integrated to the output voltage. (C) Simulation schematic for Eq. (4.10) with a sine wave base excitation input and a sinusoidal voltage output at a frequency using the time domain integration method. 73 Figure 4.4 Matlab codes for simulation of the frequency response functions of the output relative displacement and voltage over the base input excitation acceleration. 77 Figure 4.5 Resonant energy harvesting efficiency versus dimensionless resistance and equivalent force factors for the single degree of freedom electromagnetic harvester connected to a load resistor. 81 Figure 4.6 Dimensionless harvested resonant power versus dimensionless resistance and equivalent force factors for the single degree of freedom electromagnetic harvester connected to a load resistor. 82 Figure 4.7 The resonant energy harvesting efficiency versus dimensionless resistance and force factors for the single degree of freedom piezoelectric harvester connected to the four types of interface circuits. (A) Standard interface; (B) synchronous electric charge extraction; (C) parallel synchronous switch harvesting on inductor (SSHI); (D) series SSHI. 96 List of Figures xiiiFigure 4.8 The dimensionless harvested resonant power versus dimensionless resistance and force factors for the SDOF electromagnetic harvester connected to the four types of interface circuits. (A) Standard interface; (B) synchronous electric charge extraction; (C) parallel synchronous switch harvesting on inductor (SSHI); (D) series SSHI. 101 Figure 5.1 Dimensionless resonant energy harvesting efficiency of piezoelectric and electromagnetic vibration energy harvesters versus dimensionless resistance and force factors for the single degree of freedom harvesters connected to a load resistance; (A) piezoelectric; (B) electromagnetic. 109 Figure 5.2 Dimensionless harvested resonant power versus dimensionless resistance and force factors for the single degree of freedom piezoelectric and electromagnetic harvesters connected to a load resistor; (A) piezoelectric; (B) electromagnetic. 111 Figure 5.3 Dimensionless harvested resonant power versus the dimensionless force factor by fixing the dimensionless resistance of 0.5. 113 Figure 5.4 Dimensionless harvested resonant power versus the dimensionless force factor by fixing the dimensionless resistance of 1.5. 113 Figure 5.5 Dimensionless harvested resonant power versus the dimensionless resistance by fixing the dimensionless force factor of 0.5. 114 Figure 5.6 Dimensionless harvested resonant power versus the dimensionless resistance by fixing the dimensionless force factor of 1.5. 114 Figure 6.1 Schematic of a two degree of freedom piezoelectric vibration energy harvesting system model. 125 Figure 6.2 The dimensionless harvested resonant power and energy harvesting efficiency versus various mass ratio (MR ¼ m2/m1). 132 Figure 6.3 The dimensionless harvested resonant power and energy harvesting efficiency versus various stiffness ratio (KR ¼ k2/k1). 133 Figure 6.4 Case study of a quarter vehicle suspension model with piezoelectric element insert. 133 Figure 6.5 Simulation schematic for Eqs. (6.1) and (6.2) with a sine wave base excitation input and a sinusoidal voltage output at a frequency using the integration method. 135 Figure 6.6 Output voltage for the input excitation acceleration amplitude of 1 g (9.80 m/s2) and excitation frequency of 1.45 Hz. 135 Figure 6.7 Output power for the input excitation acceleration amplitude of 1 g (9.80 m/s2) and excitation frequency of 1.45 Hz. 136 Figure 6.8 Displacement amplitude ratios of mass 1 and mass 2 with respect to the input displacement amplitude versus frequency. 137 Figure 6.9 Output voltage and harvested resonant power versus the input excitation acceleration amplitude. 137 Figure 6.10 The output voltage and harvested resonant power versus frequency. 138 Figure 6.11 Output voltage and harvested resonant power versus electric load resistance. 139 xiv List of FiguresFigure 6.12 Output voltage and harvested resonant power versus the wheeletire damping. 140 Figure 6.13 Output voltage and harvested resonant power versus the suspension damping. 141 Figure 6.14 Output voltage and harvested resonant power versus the force factor. 142 Figure 6.15 Output voltage versus frequency for various wheeletire mass. 143 Figure 6.16 Output voltage versus frequency for various quarter vehicle masses. 144 Figure 6.17 Output voltage versus frequency for various wheeletire stiffness. 145 Figure 6.18 Output voltage versus frequency for various suspension stiffness. 145 Figure 6.19 The dimensionless harvested resonant power versus the stiffness ratio (k2/k1). 147 Figure 6.20 Output voltage versus frequency for various wheeletire damping coefficients. 148 Figure 6.21 Output voltage versus frequency for various suspension damping coefficients. 148 Figure 6.22 Dimensionless harvested resonant power versus the damping ratio (c1/c2). 149 Figure 7.1 Schematic of a two degree of freedom piezoelectric vibration energy harvester inserted with two piezoelectric patch elements. 157 Figure 7.2 The difference of the two dimensionless resonant frequencies versus the mass ratio M1 and dimensionless frequency ratio U1 under the synchronous changes of the coupling strength of the piezoelectric patches. (A) Lz12 1 ¼ Lz22 2 ¼ 0:02; (B) Lz12 1 ¼ Lz22 2 ¼ 5; (C) Lz12 1 ¼ Lz22 2 ¼ 10; and (D) L2 1 z1 ¼ L2 2 z2 ¼ 40. 162 Figure 7.3 The difference of the two dimensionless resonant frequencies versus the mass ratio M1 and the U1 under the coupling strength changes of the primary and auxiliary oscillator systems. (A) L2 1 z1 ¼ 0:02 Lz22 2 ¼ 5; (B) Lz12 1 ¼ 0:02 Lz22 2 ¼ 10; (C) Lz12 1 ¼ 0:02 Lz22 2 ¼ 40; and (D) L2 1 z1 ¼ 40 Lz22 2 ¼ 0:02. 163 Figure 7.4 The dimensionless harvested power of the two degree of freedom piezoelectric vibration energy harvester versus the dimensionless frequency for different dimensionless mass ratio (M1). (A) The dimensionless harvested power of the first piezo patch element; (B) the dimensionless harvested power of the second piezo patch element; and (C) the total dimensionless harvested power of the first and second piezo patch elements. DOF, degree of freedom. 164 Figure 7.5 The dimensionless harvested power of the two degree of freedom piezoelectric vibration energy harvester versus the dimensionless frequency for different U1. (A) The dimensionless harvested power of the first piezo patch element; (B) the dimensionless harvested power of the second piezo patch element; and (C) the total dimensionless harvested power. DOF, degree of freedom. 165 List of Figures xvFigure 7.6 Dimensionless harvested power of the two degree of freedom piezoelectric vibration energy harvester versus F and z1. (A) Dimensionless harvested power of the first piezo patch element; (B) dimensionless harvested power of the second piezo patch element; and (C) total dimensionless harvested power. 166 Figure 7.7 Dimensionless harvested power of the two degree of freedom piezoelectric vibration energy harvester versus F and z2. (A) dimensionless harvested power of the first piezo patch element; (B) dimensionless harvested power of the second piezo patch element; and (C) total dimensionless harvested power. 167 Figure 7.8 The energy harvesting efficiency of the first piezoelectric patch element versus F and M1 for different coupling strengths. (A) L2 1 z1 ¼ L2 2 z2 ¼ 0:02; (B) Lz12 1 ¼ Lz22 2 ¼ 5; (C) Lz12 1 ¼ Lz22 2 ¼ 10; and (D) L2 1 z1 ¼ L2 2 z2 ¼ 40. 168 Figure 7.9 The energy harvesting efficiency of the second piezoelectric patch element versus F and M1 for different coupling strengths. (A) L2 1 z1 ¼ L2 2 z2 ¼ 0:02; (B) Lz12 1 ¼ Lz22 2 ¼ 5; (C) Lz12 1 ¼ Lz22 2 ¼ 10; and (D) L2 1 z1 ¼ L2 2 z2 ¼ 40. 169 Figure 7.10 Schematic of a three degree of freedom piezoelectric vibration energy harvester inserted with three piezoelectric patch elements. 170 Figure 7.11 The dimensionless harvested power of the 3DOF piezoelectric vibration energy harvester versus the dimensionless frequency for different mass ratio M1. (A) Dimensionless harvested power of the first piezo patch element; (B) dimensionless harvested power of the second piezo patch element; (C) dimensionless harvested power of the third piezo patch element; and (D) total dimensionless harvested power. DOF, degree of freedom. 172 Figure 7.12 The dimensionless harvested power of the 3DOF system versus the dimensionless frequency for different U1. (A) dimensionless harvested power of the first piezo patch element; (B) dimensionless harvested power of the second piezo patch element; (C) dimensionless harvested power of the third piezo patch element; and (D) total dimensionless harvested power. DOF, degree of freedom. 173 Figure 7.13 Dimensionless harvested power of the 3DOF piezoelectric vibration energy harvester versus F and z1. (A) dimensionless harvested power of the first piezo patch element. (B) dimensionless harvested power of the second piezo patch element. (C) dimensionless harvested power of the third piezo patch element. (D) total dimensionless harvested power. DOF, degree of freedom. 174 Figure 7.14 Dimensionless harvested power of the 3DOF piezoelectric vibration energy harvester versus F and z2. (A) dimensionless harvested power of the first piezo patch element. (B) dimensionless harvested power of the second piezo patch element. xvi List of Figures(C) dimensionless harvested power of the third piezo patch element. (D) total dimensionless harvested power. DOF, degree of freedom. 175 Figure 7.15 The harvested efficiency of the 3DOF piezoelectric vibration energy harvester versus M1 and F. (A) the efficiency of the first piezo patch element. (B) the efficiency of the second piezo patch element. (C) The efficiency of the third piezo patch element. (D) Total efficiency. DOF, degree of freedom. 175 Figure 7.16 Schematic of a generalized multiple degree of freedom piezoelectric vibration energy harvester with piezoelectric elements between all two nearby oscillators. 177 Figure 7.17 The dimensionless harvested power and the harvested power density versus the number of degrees of freedom of piezoelectric vibration energy harvester (PVEH). 179 Figure 8.1 A cantilevered bimorph beam clamped by washers with a nut mass glued at the free end. 188 Figure 8.2 The bimorph cantilevered beam set up on the shaker for lab testing. 189 Figure 8.3 Polytec Laser Doppler Vibrometer system display. 190 Figure 8.4 The measured vibration spectrum and first natural frequency of 24.375 Hz for the cantilevered beam under a white noise random force excitation. 191 Figure 8.5 The predicted and measured voltage output versus excitation frequency for the PZT-5H cantilevered beam. 192 Figure 8.6 The predicted and measured harvested resonant power versus the excitation frequency for the PZT-5H cantilevered beam. 192 Figure 8.7 The predicted and measured resonant output voltage versus the external electric load resistance for the PZT-5H cantilevered beam. 193 Figure 8.8 The predicted and measured resonant output voltages versus the excitation acceleration amplitude for the PZT-5H cantilevered beam. 194 Figure 8.9 A two degree of freedom piezoelectric vibration energy harvester with one piezoelectric element was mounted on the shaker. 195 Figure 8.10 The predicted and experimentally measured voltage output values versus the excitation frequency for a two degree of freedom piezoelectric vibration energy harvester inserted with one piezoelectric element. 196 Figure 8.11 The predicted and experimentally measured voltage output values versus the external electric load resistance for a two degree of freedom piezoelectric vibration energy harvester inserted with one piezoelectric element. 197 Figure 8.12 The experimental setup of the two degree of freedom (2DOF) piezoelectric vibration energy harvester (PVEH) built with two piezoelectric elements. 198 Figure 8.13 The isolated tests for the primary and auxiliary oscillators of the two degree of freedom (2DOF) piezoelectric vibration energy harvester (PVEH) built with two piezoelectric elements. 199 List of Figures xviiFigure 8.14 The analytically predicted and experimentally measured voltage outputs versus the excitation frequency for a two degree of freedom piezoelectric vibration energy harvester inserted with two piezoelectric elements. (A) The analytically predicted and experimentally measured voltage output of the first piezo patch element; (B) The analytically predicted and experimentally measured voltage output of the second piezo patch element. 201 Figure 9.1 Statistical energy analysis model of a linear vibration energy harvesting system. 218 Figure 9.2 Dimensionless mean harvested resonant power of linear single degree of freedom piezoelectric and electromagnetic vibration energy harvesters for the cases of the weak and non-weak coupling (hM ¼ hE). 223 Figure 9.3 Resonant energy harvesting efficiency of linear single degree of freedom piezoelectric and electromagnetic vibration energy harvesters for the cases of weak and non-weak coupling (hM ¼ hE). 224 Figure 10.1 Time record, autocorrelation, and autospectrum functions of broadband random or white noise excitation input of the inertia force. 236 Figure 10.2 Time record, autocorrelation, and autospectrum functions of finite narrow bandwidth random excitation input of the inertia force. 238 Figure 10.3 Time record, autocorrelation, and autospectrum functions of harmonic excitation input of the inertia force. 240 Figure 10.4 Schematic for the direct methods for measurement of the mean input and harvested power and energy harvesting efficiency of a vibration energy harvester. DC, direct current; FFT, fast Fourier transform; FRFP, frequency response function of power fluctuation. 242 Figure 11.1 Hydraulic actuation type of Pelamis. 250 Figure 11.2 Wave energy converter types: (A) the PS Frog point absorber; (B) heaving buoy point absorber; (C) archimedes wave swing. 251 Figure 11.3 Rotational turbine type. 251 Figure 11.4 A single degree of freedom nonlinear oscillator in a cylindrical tube generator. 256 Figure 11.5 Simulation schematic for Eq. (11.7) for prediction of the oscillator relative displacement response (x-y) and output voltage v from a sine wave base excitation acceleration input y€ at a frequency using Matlab Simulink time domain integration method. 257 Figure 12.1 Schematic of a two degree of freedom electromagnetic vibration energy harvester oscillator system. 272 Figure 12.2 Simulation schematic for Eq. (12.3) for prediction of the oscillator relative displacement responses and output voltages from the base excitation acceleration using Matlab Simulink time domain integration method. 274 Figure 12.3 Schematic of a four degree of freedom electromagnetic vibration energy harvester oscillator system. 275 xviii List of FiguresFigure 12.4 Schematic of an N degree of freedom electromagnetic vibration energy harvester oscillator system. 280 Figure 12.5 Regenerative shock absorber built with a linear generator with longitudinal or transverse magnets and highly magnetic conductive casing. 282 Figure 12.6 Regenerative shock absorber built with the rack and pinion mechanism. 282 Figure 12.7 Regenerative shock absorber built with the ballescrew mechanism. 283 Figure 12.8 Structure frame built with the multiple degrees of freedom electromagnetic vibration energy harvester cells. 284 List of Figures xixList of Tables Table 1.1 The Parameters of the Single Degree of Freedom Spring-Mass-Dashpot System 9 Table 2.1 The Identified Single Degree of Freedom Piezoelectric System Parameters 20 Table 3.1 Formulas for Dimensionless Harvested Resonant Power and the Energy Harvesting Efficiency of a Piezoelectric Harvester for Four Different Interface Circuits 55 Table 3.2 Formulas for the Peak Dimensionless Harvested Resonant Power and Resonant Energy Harvesting Efficiency of a Piezoelectric Harvester With Four Different Interface Circuits With the Resistance Variation 58 Table 3.3 Formulas for the Peak Dimensionless Harvested Resonant Power and Resonant Energy Harvesting Efficiency of a Piezoelectric Harvester Connected With Four Different Interface Circuits With the Force Factor Variation 59 Table 3.4 Parameters of the Single Degree of Freedom Piezoelectric Harvester for the Case Study 60 Table 3.5 Harvested Resonant Power and Energy Harvesting Efficiency for a Single Degree of freedom Piezoelectric Harvester Connected With Different Interface Circuits Under a Base Acceleration of 9.8 m/s2 64 Table 4.1 Comparison of Dimensionless Harvested Power and Efficiency for Electromagnetic Vibration Energy Harvesters Connected With Four Different Interface Circuits 100 Table 5.1 Comparison of Electromagnetic and Piezoelectric Vibration Energy Harvesters Connected to a Single Load Resistor 112 Table 5.2 Comparison of Electromagnetic and Piezoelectric Vibration Energy Harvesters Connected to the Standard and SECE Interface Circuits 116 Table 5.3 Comparison of Electromagnetic and Piezoelectric Vibration Energy Harvesters Connected to Series and Parallel SSHI Interface Circuits 117 Table 6.1 Parameters of a Quarter Vehicle Suspension Model With Piezoelectric Insert 134 Table 7.1 System Parameters of a 2DOF Piezoelectric Vibration Energy Harvester 161 Table 7.2 Comparison of Harvesting Performance From 1DOF to 5DOF Piezoelectric Vibration Energy Harvester 178 Table 8.1 Piezoelectric Vibration Energy Harvester Parameters 189 Table 8.2 The Parameters of a Two Degree of Freedom Piezoelectric Vibration Energy Harvester With One Piezoelectric Element 196 Table 8.3 The Experimentally Identified Parameters of the 2DOF Piezoelectric Vibration Energy Harvester Built With Two Piezoelectric Elements 200 Table 9.1 Dimensionless Mean Harvested Resonant Power of Piezoelectric and Electromagnetic Vibration Energy Harvesters at the Circuit Oscillation Resonance 210 xxiTable 9.2 Dimensionless Mean Harvested Resonant Power of Piezoelectric and Electromagnetic Vibration Energy Harvesters Without the Circuit Oscillation Resonance 211 Table 9.3 Energy Harvesting Efficiency of Piezoelectric and Electromagnetic Vibration Energy Harvesters at the Circuit Oscillation Resonance 211 Table 9.4 Energy Harvesting Efficiency of Piezoelectric and Electromagnetic Vibration Energy Harvesters Without the Circuit Oscillation Resonance 21 Analysis of a Single Degree of Freedom Spring-MassDashpot System Using Transfer Function, Integration, State Space, and Frequency Response Methods 1 CHAPTER OUTLINE 1.1 Introduction . 1 1.2 Laplace Transform and Transfer Function Analysis Method 2 1.3 Time Domain Integration Method . 5 1.4 State Space Method . 6 1.5 Frequency Response Method . 8 1.6 An Example of the Time Domain Integration Simulation and Frequency Response Analysis Methods Using Matlab Simulink Program Codes . 8 1.6.1 Time Domain Integration Simulation Method Using the Matlab Simulink Program codes . 9 1.6.2 The Frequency Response Analysis Method Using a Matlab Code . 10 Nomenclature . 12 References . 13 Analysis of a Single Degree of Freedom Piezoelectric Vibration Energy Harvester System Using the Transfer Function, Integration, State Space, and Frequency Response Methods 2 CHAPTER OUTLINE 2.1 Introduction . 15 2.2 Analysis and Simulation of an Single Degree of Freedom Piezoelectric Vibration Energy Harvester Connected to a Single Load Resistance . 18 2.3 Laplace Transform and Transfer Function Analysis Method 19 2.4 Time Domain Integration Method . 21 2.5 State Space Method . 23 2.6 Frequency Response Analysis Method . 26 2.7 Dimensionless Frequency Response Analysis and Harvesting Performance Optimization . 35 Nomenclature . 40 References . 42 Analysis of Piezoelectric Vibration Energy Harvester System With Different Interface Circuits 3 CHAPTER OUTLINE 3.1 Introduction . 43 3.2 Standard Interface Circuit . 44 3.3 Synchronous Electric Charge Extraction Circuit 47 3.4 Parallel Synchronous Switch Harvesting on Inductor Circuit . 49 3.5 Series Synchronous Switch Harvesting on Inductor Circuit . 52 3.6 Analysis and Comparison 54 3.7 Case Study 60 3.8 Summary . 64 Nomenclature . 65 References . Analysis of Electromagnetic Vibration Energy Harvesters With Different Interface Circuits 4 CHAPTER OUTLINE 4.1 Introduction . 69 4.2 Dimensionless Analysis of Single Degree of Freedom Electromagnetic Vibration Energy Harvester Connected With a Single Load Resistance . 70 4.3 Laplace Transform and Transfer Function Method . 71 4.4 Time Domain Integration Method . 72 4.5 State Space Method . 74 4.6 Frequency Response Method . 75 4.7 Dimensionless Frequency Response Analysis and Harvesting Performance Optimization . 78 4.8 Dimensionless Analysis of Electromagnetic Vibration Energy Harvesters Connected With Energy Extraction and Storage Circuits 84 4.8.1 Standard Interface Circuit . 85 4.8.2 Synchronous Electric Charge Extraction Circuit . 88 4.8.3 Parallel Synchronous Switch Harvesting on Inductor Circuit . 91 4.8.4 Series Synchronous Switch Harvesting on Inductor Circuit . 95 4.9 Analysis and Comparison 99 4.10 Summary . 102 Nomenclature . 103 References . Similarity and Duality of Electromagnetic and Piezoelectric Vibration Energy Harvesters 5 CHAPTER OUTLINE 5.1 Introduction . 107 5.2 Dimensionless Comparison of SDOF Piezoelectric and Electromagnetic Vibration Energy Harvesters Connected With a Single Load Resistance 108 5.3 Dimensionless Comparison of SDOF Piezoelectric and Electromagnetic Vibration Energy Harvesters Connected With the Four Types of Interface Circuits . 115 5.4 Summary . 118 Nomenclature . 119 References A Study of a 2DOF Piezoelectric Vibration Energy Harvester and Its Application 6 H. Xiao, X. Wang RMIT University, Melbourne, VIC, Australia CHAPTER OUTLINE 6.1 Introduction . 123 6.2 Analysis and Simulation of a Two Degree of Freedom Piezoelectric Vibration Energy Harvester . 125 6.3 Dimensionless Analysis of a Weakly Coupled 2DOF Piezoelectric Vibration Energy Harvester Model 128 6.4 Case Study of a Quarter Vehicle Suspension Model and Simulation . 132 6.5 Summary . 148 Nomenclature . 150 References . A Study of Multiple Degree of Freedom Piezoelectric Vibration Energy Harvester 7 H. Xiao, X. Wang RMIT University, Melbourne, VIC, Australia CHAPTER OUTLINE 7.1 Introduction . 155 7.2 A Two Degree of Freedom Piezoelectric Vibration Energy Harvester Inserted With Two Piezoelectric Patch Elements . 156 7.3 A Three Degree of Freedom Piezoelectric Vibration Energy Harvester Inserted With Three Piezoelectric Patch Elements 167 7.4 A Generalized Multiple Degree of Freedom Piezoelectric Vibration Harvester . 173 7.5 Modal Analysis and Simulation of Multiple Degree of Freedom Piezoelectric Vibration Energy Harvester . 178 7.6 Summary . 180 Nomenclature . 181 References . Experimental Validation of Analytical Methods 8 H. Xiao, X. Wang RMIT University, Melbourne, VIC, Australia CHAPTER OUTLINE 8.1 Introduction . 187 8.2 Experimental Results of a Single Degree of Freedom Vibration Energy Harvester 188 8.2.1 Frequency Response Function . 190 8.3 Experimental Results of a Two Degree of Freedom Vibration Energy Harvesters With One and Two Piezoelectric Elements . 194 Nomenclature . Coupling Analysis of Linear Vibration Energy Harvesting Systems 9 CHAPTER OUTLINE 9.1 Introduction . 203 9.2 Coupling Analysis of a Linear Single Degree of Freedom Piezoelectric Vibration Energy Harvesting System Under a Harmonic Excitation . 205 9.3 Coupling Analysis of a Linear Single Degree of Freedom Electromagnetic Vibration Energy Harvesting System Under a Harmonic Excitation . 212 9.4 Coupling Analyses of Linear Piezoelectric and Electromagnetic Vibration Energy Harvesters Under Random Excitations . 217 9.5 Relationship Between the Vibration Energy Harvesting Performance and Critical Coupling Strength . 221 Nomenclature . 225 References Correlation and Frequency Response Analyses of Input and Harvested Power Under White Noise, Finite Bandwidth Random and Harmonic Excitations 10 CHAPTER OUTLINE 10.1 Introduction . 231 10.2 Correlation and Frequency Response Analysis of Power Variables . 233 10.3 Harvested Resonant Power and Energy Harvesting Efficiency Under White Noise Random Excitation 235 10.4 Harvested Resonant Power and Energy Harvesting Efficiency Under Finite Bandwidth Random Excitation . 237 10.5 Harvested Resonant Power and Energy Harvesting Efficiency Under a Harmonic Excitation 239 Nomenclature . 243 References . Ocean Wave Energy Conversion Analysis 11 CHAPTER OUTLINE 11.1 Introduction . 249 11.2 Analysis of a Single Degree of Freedom Nonlinear Oscillator in a Cylindrical Tube Generator Using the Time Domain Integration Method . 254 11.3 Analysis of a Single Degree of Freedom Nonlinear Oscillator in a Cylindrical Tube Generator Using the Harmonic Balance Method 258 11.4 Analysis of a Single Degree of Freedom Nonlinear Oscillator in a Cylindrical Tube Generator Using the Perturbation Method . 262 Nomenclature . 265 References Analysis of Multiple Degrees of Freedom Electromagnetic Vibration Energy Harvesters and Their Applications 12 CHAPTER OUTLINE 12.1 Analysis of a Two Degrees of Freedom Electromagnetic Vibration Energy Harvester Oscillator System . 271 12.2 Analysis of a Four Degree of Freedom Electromagnetic Vibration Energy Harvester Oscillator System . 276 12.3 Analysis of an N Degree of Freedom Electromagnetic Vibration Energy Harvester Oscillator System . 279 12.4 Fields of Application . 281 Nomenclature . 285 References . Index ‘Note: Page numbers followed by “f ” indicate figures and “t” indicate tables.’ A Analytical methods ANSYS modal analysis module, 188e190 bimorph cantilevered beam set up, 188, 189f single degree of freedom piezoelectric vibration energy harvester (SDOF PVEH), 194e195, 195f excitation amplitudes, 187 experimental results, 200 frequency response function alternative current (AC) voltage, 190e191 nonresonant frequencies, 191e193 predicted and measured harvested resonant power vs. excitation frequency, 191, 192f predicted and measured resonant output voltages vs. excitation acceleration amplitude, 193e194, 194f predicted and measured resonant output voltage vs. external electric load resistance, 193, 193f predicted and measured voltage output vs. excitation frequency, 191, 192f SDOF PVEH, 194 shaker amplifier gain, 190e191 harvested resonant voltage and power, 190 Laser Vibrometer system, 195 parameters, 188, 189t, 195, 196t, 198e199, 200t Polytec Laser Doppler Vibrometer System display, 190, 190f primary and auxiliary oscillators, isolated tests, 197e198, 199f PZT-5H piezoelectric material, 187 short circuit stiffness, 188e190 two degree of freedom piezoelectric vibration energy harvester (2DOF PVEH), 197e198, 198f voltage outputs vs. excitation frequency, 199e200, 201f voltage output values vs. excitation frequency, 195e197, 196f vs. external electric load resistance, 195e197, 197f washers, 188, 188f white noise random signal, 190, 191f B Buoy-type wave energy harvester, 252e254 C Correlation and frequency response analyses dynamic differential equations, 231e232 Fourier and Laplace transforms, 231e232 harvested resonant power and energy harvesting efficiency finite bandwidth random excitation, 237e239, 238f harmonic excitation, 239e243, 240f, 242f white noise random excitation, 235e237, 236f linear vibration energy harvesting systems, 231e232 power fluctuation frequency, 232e233 power variables autospectral densities, 233e235 cross-spectral density function, 233e235 Gaussian random variables, 233 output electric power, 233 spring-mass-dashpot system, 232e233 Coupling loss factor, 222 D Damping loss factor, 204e205 Dimensionless harvested resonant power, 54e56, 55t vs. dimensionless force factor, 112, 113f vs. dimensionless resistance, 112, 114f vs. dimensionless resistance and dimensionless force factor, 56e57, 57f Dimensionless mean harvested resonant power at circuit oscillation resonance, 210e212, 210t without the circuit oscillation resonance, 210e212, 211t Duffing oscillator, 253e254 E Electromagnetic and piezoelectric vibration energy harvesters damping coefficient, 118e119 environmental sustainability, 107 interface circuits duality, 115e118 289Electromagnetic and piezoelectric vibration energy harvesters (Continued) series and parallel SSHI, 115, 117t standard and SECE, 115, 116t types, 115 mechanical damping, 119 single load resistance, 112t dimensionless harvested resonant power vs. dimensionless force factor, 112, 113f dimensionless harvested resonant power vs. dimensionless resistance, 112, 114f dimensionless harvested resonant power vs. dimensionless resistance and force factors, 110, 111f dimensionless resonant energy harvesting efficiency vs. dimensionless resistance and force factors, 109e110, 109f efficiency, 110 electric circuit oscillation resonance, 108e109 mechanical damping, 111e112 nonzero constant, 110e111 piezoelectric/magnetic and coil materials, 112e115 radial resonant frequency, 108e109 resonant energy harvesting efficiency, 108 Electromagnetic vibration energy harvester (EMVEH) damping coupling, 80 dimensionless analysis, 70, 71f dimensionless frequency response analysis, 78e79 dimensionless harvested resonant power vs. dimensionless resistance and equivalent force factors, 81e82, 82f efficiency vs. dimensionless resistance and force factor, 101 electric energy extraction, 69e70 electric power generation efficiency, 80 electromechanical coupling, 99 energy extraction and storage circuits, types, 84 equivalent force factor, 71 Fourier transform, 71 frequency response method amplitude vs. frequency, 76 base acceleration and relative oscillator displacement, 77 definition, 75e76 displacement and voltage frequency response functions, 76 Matlab codes, 76, 77f open and short circuit stiffness, 77e78 root mean squared (RMS) ratio, 78 harmonic excitation, 212e217 inductor circuit parallel synchronous switch harvesting. See Parallel synchronous switch harvesting series synchronous switch harvesting, 95e99, 96f Laplace transform, 71 Matlab Simulink, 72, 72f mechanical damping, 83 nonzero constant, 101e102 optimized power generation, 70 reference power amplitude, 83 resonant energy harvesting efficiency vs. dimensionless resistance and equivalent force factors, 80e81, 81f SDOF electromagnetic harvester, 101e102, 101f SDOF electromagneticemechanical system, 79 sinusoidal displacement excitation, 71 standard interface circuit dimensionless harvested power, 87 efficiency, 87e88 external load resistance, 86 mechanical vibration period, 85 resonant frequency, 86 system energy equilibrium equation, 85 state space method, 74e75 storage interface circuits, 69e70 synchronous electric charge extraction (SECE) circuit, 88e91 system optimization analyses, 99e101, 100t time domain integration method, 72e73, 73f transfer function method, 71 variable range limits, 84 Energy harvesting efficiency at circuit oscillation resonance, 210e212, 211t without the circuit oscillation resonance, 210e212, 211t F Fourier transform correlation and frequency response analyses, 231e232 electromagnetic vibration energy harvester (EMVEH), 71 piezoelectric vibration energy harvester (PVEH) system, 26 spring-mass-dashpot system, 1e2, 4 Frequency response analysis method 290 Indexelectromagnetic vibration energy harvester (EMVEH). See Electromagnetic vibration energy harvester (EMVEH) piezoelectric vibration energy harvester (PVEH) system. See Piezoelectric vibration energy harvester (PVEH) system spring-mass-dashpot system, 8 H Harvested resonant power, 25e26 piezoelectric vibration energy harvester (PVEH) system vs. base excitation acceleration amplitude, 25e26, 27f vs. electrical load resistance, 25e26, 29f vs. force factor, 25e26, 29f, 62, 63f vs. mechanical damping, 25e26, 28f, 60e61, 62f I Inductor circuit electromagnetic vibration energy harvester (EMVEH) parallel synchronous switch harvesting. See Parallel synchronous switch harvesting series synchronous switch harvesting, 95e99, 96f piezoelectric vibration energy harvester (PVEH) system parallel synchronous switch harvesting. See Parallel synchronous switch harvesting series synchronous switch harvesting. See Series synchronous switch harvesting Inverse Laplace transform, 4 L Laplace transform correlation and frequency response analyses, 231e232 2DOF piezoelectric vibration energy harvester, 126 electromagnetic vibration energy harvester (EMVEH), 71 piezoelectric vibration energy harvester (PVEH) system, 19 spring-mass-dashpot system, 1e2 Linear vibration energy harvesting systems, 231e232 vs. coupling analysis coupling loss factor, 222 critical coupling strength, 221e222 damping loss factor, 204e205 electromagnetic vibration energy harvesting system, 212e217 electromechanical coupling factor, 204e205 environment vibrations/excitations, 203e204 force factor, 221e222 linear SDOF piezoelectric VEH, 222 moderate coupling, 223e224 normalized dimensionless force factor, 222 oscillation circuit resonance, 223 piezoelectric vibration energy harvesting system. See Piezoelectric vibration energy harvesting (PVEH) system random excitations, 217e221, 218f resonant energy harvesting efficiency, 224, 224f statistical energy analysis (SEA), 203e204 weak and non-weak coupling, 223e224, 223f M Multiple degree of freedom piezoelectric vibration energy harvester ambient excitation frequency, 155e156 dynamic magnifier, 156 generalized model, 173e177, 177f, 178t, 179f modal analysis, 178e180 multifrequency arrays, 156 piezoelectric patch elements advantage, 161 configuration, 156e157, 157f dimensionless harvested power, 166, 166fe167f dimensionless resonant frequencies vs. mass ratio and dimensionless frequency ratio, 156e157, 162f dimensionless voltages, 158e160 energy harvesting efficiency, 166e167, 168fe169f frequency difference, 163e164, 163f governing equations, 157e158 harvesting efficiencies, 160 Laplace transform, 157e158 mass ratio, 164e165, 164f output voltage signals, 160e161 stiffness ratio, 165e166, 165f system parameters, 160, 161t power density, 180e181 three degree of freedom piezoelectric vibration energy harvester dimensionless analysis, 171 dimensionless damping coefficient, 172, 174fe175f Index 291Multiple degree of freedom piezoelectric vibration energy harvester (Continued) governing equations, 168 harvested efficiency, 172e173, 175f Laplace transform, 169e170 mass ratio, 171, 172f stiffness ratio, 171e172, 173f total oscillator mass, 167, 170f Multiple degrees of freedom electromagnetic vibration energy harvesters civil structures, 283e284 classification, 281 N degree of freedom electromagnetic vibration energy harvester oscillator system, 279e280, 280f four degree of freedom electromagnetic vibration energy harvester oscillator system, 275f, 276e278 iron cores, 281 large vibration amplitudes, 283 regenerative shock absorbers, 284f ballescrew mechanism, 281, 283f linear generator, 281, 282f rack and pinion mechanism, 281, 282f rotational generator, 282 two degrees of freedom electromagnetic vibration energy harvester (2DOF EMVEH) oscillator system. See Two degrees of freedom electromagnetic vibration energy harvester (2DOF EMVEH) oscillator system N Nonlinearization methods, 253 Normalized dimensionless harvested resonant power, 50e51 O Ocean wave energy conversion analysis buoy-type wave energy harvester, 252e254 dual mass system, 253 Duffing oscillator, 253e254 Halbach array, 252 nonlinearization methods, 253 Pelamis, hydraulic actuation type, 250, 250f renewable energy sources, 249 requirements, 251e252 rotational turbine type, 250, 251f single degree of freedom nonlinear oscillator. See Single degree of freedom nonlinear oscillator, cylindrical tube generator wave energy converters (WEC), 250 types, 250, 251f P Parallel synchronous switch harvesting dimensionless force factor, 52, 95 dimensionless input resonant power, 92 electric charge, 49 electromagnetic coil outgoing current, 91 harmonic base excitation, 50 load voltage, 92 mean harvested power, 50 mean harvested resonant power, 92 mean input resonant power, 50 nonzero finite number, 94 normalized dimensionless harvested resonant power, 50e51 parallel SSHI circuit, 93 efficiency, 51 piezoelectric outgoing current, 49 Piezoelectric vibration energy harvester (PVEH) system base acceleration, 64, 64t excitation, 19 dimensionless frequency response analysis applied force/excitation acceleration, 37 dimensionless harvested resonant power vs. dimensionless resistance and force factors, 36, 36f electrical load resistance, 39 mechanical damping, 39e40 RC oscillation circuit, 37e38 resonant energy harvesting efficiency vs. dimensionless resistance and force factors, 37e38, 38f dimensionless harvested resonant power, 54e56, 55t vs. dimensionless resistance and dimensionless force factor, 56e57, 57f electric energy extraction, 43, 44f, 64e65 energy harvesting efficiency, 54e56, 55t force factor variation, 59e60, 59t frequency bandwidth, 16e17 frequency response analysis method base excitation acceleration amplitude, 32e33 displacement frequency response function, 31, 31f Fourier transform, 26 harvested resonant power and output voltage vs. frequency, 34, 34f input excitation acceleration, 30e31, 30f optimized electrical load resistance, 33 RungeeKuta method, 33 voltage frequency response function, 31, 32f 292 Indexharmonic excitation blocking capacitance, 205 circuit oscillation resonance, 207e208 dimensionless mean harvested resonant power, 210e212, 210te211t eigen equation, 206 electrical load resistance and force factor, 206 electromechanical coupling, 208e209 energy harvesting efficiency, 210e212, 211t force factor, 205 frequency ratios, 208e209 governing equation, 205 harvested voltage ratio, 207 harvested voltage frequency response function, 206 linear SDOF system, 208, 210 mechanical damping loss factor, 209 mechanical subsystem, 205 open circuit and close circuit damping coefficient, 208 resonance frequency, 208 resonant energy harvesting efficiency, 206 transfer function, 209 zero coupling, 207 harvested electric energy, 17e18 harvested resonant power vs. force factor, 62, 63f vs. mechanical damping, 60e61, 62f inductor circuit parallel synchronous switch harvesting. See Parallel synchronous switch harvesting series synchronous switch harvesting. See Series synchronous switch harvesting Laplace transform, 19 Matlab Simulink, 20e21, 20f mechanical damping, 57 mechanicaleelectrical system, 17 optimization design methods, 16e17 output voltage and harvested resonant power vs. electric resistance, 61e62, 63f vs. input excitation acceleration, 60, 61f parameters, 20e21, 20t, 60, 60t relative oscillator displacement, 19 resistance variation, 58e59, 58t resonant energy harvesting efficiency vs. dimensionless resistance and force factors, 56, 56f single electric load resistor, 15e16, 16f single load resistance, 18 spring-mass oscillating system, 15e16 standard interface circuit base excitation acceleration, 45e46 bridge rectification circuit, 44e45, 45f dimensionless force factor, 47 mechanical vibration period, 44e45 peak efficiency, 46e47 resonant power, 46 system energy equilibrium equation, 45e46 state space method harvested resonant power, 25e26. See also Harvested resonant power output sinusoidal voltage signal, 24e25, 25f output voltage amplitude vs. electrical load resistance, 25e26, 28f output voltage amplitude vs. mechanical damping, 25e26, 27f output voltage frequency response, 24e25 output voltage vs. base excitation acceleration amplitude, 25e26, 26f Simulink codes, 24e25, 24f state space equations, 23 storage interface circuits, 43, 44f, 64e65 synchronous electric charge extraction (SECE) circuit. See Synchronous electric charge extraction (SECE) circuit time domain integration method. See Time domain integration method transfer function method, 19 weak electromechanical coupling, 54e56 Polytec Laser Doppler Vibrometer System display, 190, 190f Q Quarter vehicle suspension model analytical frequency response analysis methods, 138e139 bouncing resonant voltage magnitude, 144e146 coupling system, 146 dimensionless harvested resonant power vs. damping ratio, 147, 149f vs. stiffness ratio, 146, 147f displacement amplitude ratios, 134e136, 137f Laplace/Fourier transform, 142 Matlab program code, 143 Matlab Simulink, 134 output power, 134, 136f output voltage, 134, 135f vs. frequency for various quarter vehicle masses, 144, 144f vs. frequency for various suspension damping coefficients, 147, 148f vs. frequency for various suspension stiffness, 145e146, 145f Index 293Quarter vehicle suspension model (Continued) vs. frequency for various wheeletire damping coefficients, 147, 148f vs. frequency for various wheeletire mass, 143e144, 143f vs. frequency for various wheeletire stiffness, 145e146, 145f output voltage and harvested resonant power vs. electric load resistance, 139e140, 139f vs. force factor, 141e142, 142f vs. frequency, 138e139, 138f vs. input excitation acceleration amplitude, 137f, 138 vs. suspension damping, 140e141, 141f vs. wheeletire damping, 140, 140f parameters, 132, 134t and simulation, 132, 133f sine wave base excitation input and sinusoidal voltage output, 134, 135f time domain simulation, 138e139 R Regenerative shock absorbers, 284f ballescrew mechanism, 281, 283f linear generator, 281, 282f rack and pinion mechanism, 281, 282f RungeeKuta method, 33 S Series synchronous switch harvesting, 95e99, 96f dimensionless force factor, 54 electrical quality factor, 52 equivalent load resistance, 52 mean harvested power, 53 mechanical vibration period, 52 rectified voltage, 52 SSHI circuit efficiency, 53e54 Single degree of freedom (SDOF) piezoelectric vibration energy harvester (PVEH) system. See Piezoelectric vibration energy harvester (PVEH) system spring-mass-dashpot system displacement frequency response function, 4 Fourier transform, 1e2, 4 frequency response method, 8 inertial force, 3, 3f inverse Laplace transform, 4 Laplace transform, 1e2 Matlab Simulink transfer function method, 4e5, 5f normalized conversion, 2 parameters, 8, 9t “shunt” damping, 1e2 state space method, 6e7, 8f time domain integration method. See Time domain integration method transducer mechanism, 2 weak coupling, 2 Single degree of freedom nonlinear oscillator, cylindrical tube generator harmonic balance method, 258e261 perturbation method, 262e265 time domain integration method, 256f excitation frequencies and amplitude, 257 Gaussian meter, 257 Matlab Simulink, 256e257, 257f Newton’s second law, 255e256 nonlinearity, types, 254e255 state space analysis methods, 258 transfer function, 258 State space method electromagnetic vibration energy harvester (EMVEH), 74e75 piezoelectric vibration energy harvester (PVEH) system. See Piezoelectric vibration energy harvester (PVEH) system spring-mass-dashpot system, 6e7, 8f Statistical energy analysis (SEA) model, 203e204, 218, 218f Synchronous electric charge extraction (SECE) circuit, 88e91 dimensionless force factor, 49 dimensionless harvested resonant power, 48 efficiency, 48 mean harvested power, 47 mechanical velocity, 47 T Time domain integration method, 72e73, 73f excitation acceleration, 21e22 Matlab Simulink program codes, 11f frequency response analysis, 10e12, 11fe12f sine wave excitation acceleration, 9e10, 10f sine wave excitation signal, 9e10, 9f relative acceleration, 21e22, 22f relative displacement frequency response, 5, 6f single degree of freedom nonlinear oscillator, 256f excitation frequencies and amplitude, 257 Gaussian meter, 257 Matlab Simulink, 256e257, 257f Newton’s second law, 255e256 nonlinearity, types, 254e255 294 Indexstate space analysis methods, 258 transfer function, 258 Two degree of freedom piezoelectric vibration energy harvester (2DOF PVEH), 197e198, 198f advantages, 123e124 electrical system governing equation, 125e126 force factor and blocking capacitance, 126 frequency response analysis method, 149e150 harmonic excitation, 127 Laplace transform, 126 mechanical system governing equations, 125, 125f microelectromechanical system (MEMS) applications, 124 output voltage magnitude, 127 piezoelectric materials, 123e124 quarter vehicle suspension model, 125. See also Quarter vehicle suspension model resonant frequency tuning structure, 124 transfer function equations, 126 tuned auxiliary structure, 124 weakly coupled model, 128e131, 132fe133f Two degrees of freedom electromagnetic vibration energy harvester (2DOF EMVEH) oscillator system coil oscillators, 271e273, 272f damping coefficients, 271e273 electromechanical system governing equations, 271e273 Laplace transform, 273 Matlab program codes, 273 relative displacement responses and output voltages, 274f, 275 W Wave energy converters (WEC), 250, 251f #ماتلاب,#متلاب,#Matlab,
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