Mechanical Damping Ratio Research Articles

Moment analysis of galloping energy harvesters with a parallel circuit under stochastic excitation

In this paper, the stochastic response performance of a galloping energy harvester with a parallel circuit under Gaussian white noise excitation is analyzed. To overcome the complexity of the distributed parameter model and the two-degree-of-freedom characteristics of the harvester, firstly, the exact closed expressions for the mean square displacement, mean square voltage and mean output power are derived by the equivalent linearization method. The results obtained by the direct Monte Carlo simulations verify the accuracy of the theoretical analysis. Then, the conditions for the occurrence of the galloping phenomenon can be identified. The influences of dimensionless parameters and noise intensity on the statistical moments are discussed. The results show that most of the parameters have the same tendency to affect the mean square displacement and the mean square voltage. However, the electromechanical coupling term has the opposite effect on both of them. It is also shown that the mean output power decreases as the coefficients of the electromechanical coupling term and mechanical damping ratio increase, however, it increases dramatically with increasing the wind speed and the linear galloping force coefficient. In addition, utilizing the stationary probability density of the stochastic harvester, the influences of important parameters and noise intensity on the harvester’s performance are studied. It is observed that a larger level of noise intensity results in the greater peak value of the stationary probability density.

Investigation of nonlinear and transitional characteristics of flutter varying with wind angles of attack for some typical sections with different side ratios

This study conducted a comprehensive investigation into the flutter responses of three closed sections, namely two streamlined box girders and a 5:1 rectangular cylinder, which possess different side ratios. Different types of flutter were measured during the tests. The nonlinear and transitional characteristics of the flutter were thoroughly discussed. The effects of the wind angle of attack and side ratio on the reduced critical wind speed, limit cycle oscillation amplitude and the transition behaviors of the flutter were investigated. The complex modal motion information of the three systems was discussed. The main factors contributing to the transition of flutter, and the dynamic mechanisms of the occurrence of different flutter manifestations, were investigated. This was achieved by examining the evolution of their modal damping ratios with the vibration amplitude and wind speed. Finally, an examination was conducted to evaluate the impact of the mechanical damping ratio on the transition behavior of flutter as well as its effectiveness in mitigating flutter responses for the considered sections. Results showed that the various manifestations of the flutter are mainly ascribed to the gradually evolved modal damping ratio, where aerodynamic damping plays the most important role in producing different types of limit cycle oscillations and transitional behaviors of flutter. Moreover, the modal damping ratio slowly transitions with the wind angle of attack, wind speed, or the side ratio of a section, instead of experiencing any mutation, despite the observation of some distinct types of flutter during the tests.

Motion-amplitude-dependent nonlinear VIV model and maximum response over a full-bridge span

Nonlinear motion-amplitude $$(y_{T} )$$ -dependent energy-trapping properties of a bridge model undergoing vortex-induced vibration (VIV) are investigated. Energy-trapping properties of the model undergoing a full-process from still to a limit cycle oscillation (LCO) state are identified. A van der Pol-type model is adapted to describe the amplitude-dependent aerodynamic properties. Nonlinear parameter-amplitude relations, $$\varepsilon {-}y_{T}$$ and $$\xi_{\varepsilon } {-}y_{T}$$ , are established. Nonlinear aerodynamic damping is separated into two parts: the initial damping which varies with the reduced wind speed, and the $$\varepsilon $$ -related part which varies with both the reduced wind speed and the motion amplitude. The initial aerodynamic damping determines the threshold of VIV, while the $$\varepsilon $$ -related part dominates the evolution process and the LCO. The identified nonlinear analytical model is capable of predicting VIV responses at higher mechanical damping ratios. The energy-trapping properties of a section model in time are transformed into nonlinear properties distributed in space along an elongated 3-D elastic bridge span. According to this “time-space” transformation, the convection coefficient, which links the maximum response of a 3-D structure with that of a 2-D (1-DOF) sectional model, can be determined. Compared with a constant-parameter analytical model, an adapted nonlinear one brings to light significantly larger convection coefficients. Finally, parameter overflowing phenomena are revealed and discussed.

Parametric Analysis of Nonlinear Bi-Stable Piezoelectric Energy Harvester Based on Multi-Scale Method

Given the geometric nonlinearity of the piezoelectric cantilever beam, this study establishes a distributed parameter model of the nonlinear bi-stable cantilever piezoelectric energy harvester, following the generalized Hamilton variational principle. The analytical expressions of the dynamic response were obtained for the energy harvesting system using Galerkin modal decomposition and the multi-scale method. The investigation focuses on how the performance of the energy harvesting system is influenced by the gap distance between magnets, external excited amplitude, mechanical damping ratio and external load resistance. The calculation results were compared with those obtained neglecting the geometric nonlinearity of the beam. The results show that the system responses contain jump and multiple solutions. The consideration of the geometrical nonlinearity significantly amplified the peak displacement and peak output power of the intra-well and inter-well motions. There is an evident hardening effect of the inter-well motion frequency response curve. By reasonable adjusting the parameters, it is possible to improve the output power of the piezoelectric energy harvesting system and broaden the operating frequency of the system.

Geometrically nonlinear model of piezoelectric wind energy harvesters based on vortex-induced vibration and galloping

The interaction between vortex-induced vibration (VIV) and galloping could enhance the performance of wind energy harvesters (WEHs). Though VIV-galloping interaction may cause large amplitude wind-induced vibrations, the effects of geometrical nonlinearity were not considered in the modeling of VIV-galloping interactive piezoelectric WEHs (PWEHs). In this work, based on the extended Hamilton’s principle, a geometrically nonlinear model (GNM) of cantilevered PWEHs with VIV-galloping interaction was derived. The model includes both the transverse and axial aerodynamic forces, and considers the effect of the rotation of the bluff body on the aerodynamic forces. The aerodynamic coefficients were extracted by a piecewise polynomial fitting in a relatively large range of angle of attack for the square cross-sectional bluff body. Two flexible PWEH prototypes were fabricated and tested in a small wind tunnel to verify the proposed model. After the mechanical damping ratio of the low-coupling piezoelectric energy harvester prototypes were identified based on purely electrical measurements, the steady-state root mean square voltages of the prototypes with increasing wind speed were worked out using geometrically linear model (GLM) and the proposed GNM, respectively, and then compared with experiments. Both models can accurately predict the VIV-galloping interaction, but GNM is much more accurate than GLM at a relatively high wind speed. The proposed GNM provides a powerful tool to develop VIV-galloping interactive PWEHs.

Influence of effective electrode coverage on the energy harvesting performance of piezoelectric cantilevers

Cantilevers are widely used for piezoelectric vibration energy harvesters (PEHs), thanks to the low resonance frequency and large mechanical strain under weak excitations. Electrode coverage optimization is essential for maximizing output power. In this study, bimorph cantilevers with segmented electrodes are developed to investigate the correlation between effective electrode coverage (EEC) and the harvesting behavior of piezoelectric cantilevers. A nonlinear electromechanical model is established using Hamilton's principle, numerically solved with the method of harmonic balance, and further validated by experimental measurements. Equivalent linear stiffness and equivalent mechanical damping ratio are firstly introduced for evaluating the equivalent effective electromechanical coupling coefficients and identifying strong and weak coupling harvesters. Both harvesters fabricated with and without proof mass behave mainly as strongly coupled systems and can obtain the maximum possible power at a very low EEC. Variation of EEC in higher range acquires almost constant maximum power on dramatically changing resistance. A comprehensive EEC design principle for acquiring the maximum output power is obtained for cantilevered PEHs. An optimal EEC obtaining the maximum effective electromechanical coupling coefficient is needed for acquiring the unique maximum power pick for weakly coupled harvesters, whereas a wide range of EEC can be used to maximize output power for harvesters those mainly behave as strongly coupled systems.

Experimental investigation on post-flutter characteristics of a typical steel-truss suspension bridge deck

A testing device that was specially developed for large torsional amplitude vibrations was adopted to investigate the nonlinear post-flutter behaviors of a typical steel-truss girder bridge deck. Firstly, a series of free decay vibration tests of a narrow rigid rod model and a section model of the bridge were conducted in still-air to obtain mechanical properties and still-air induced aerodynamic properties. Then, post-flutter experiments of the bridge section model under initial attack angles of −3°, −0°, +3°, +5°, +7°and with different mechanical damping ratios were carried out. Finally, nonlinear and coupling behaviors of the bridge section model during post-flutter were described and analyzed in detail from several different aspects such as aeroelastic center, vibration mode, and wind-induced aerodynamic damping. The results showed that the truss girder section exhibited significant vertical-torsional coupled soft flutter behavior at various attack angles and there was a stable limit cycle oscillation (LCO) and an unstable LCO under a given wind speed near the critical speed. Moreover, the added aerodynamic damping, the coupling vertical deformation, the aerodynamic torsional center, the wind-induced aerodynamic damping, and the soft flutter mode in the post-critical state all exhibited the characteristic of amplitude dependence. Meanwhile, it was found that the coupling vertical deformation was strongly related to the torsional amplitude and wind attack angle. Finally, the effects of mechanical damping on the post-flutter performance were discussed and the results showed that some nonlinear dampers whose damping increased with the increasing amplitude can be adopted to improve the critical flutter speed without deteriorating or even enhancing the post-flutter performance.

Piezoelectric energy harvesting from vortex-induced vibration of a circular cylinder: Effect of Reynolds number

This study investigates the effects of the Reynolds number on the piezoelectric energy harvesting from vortex-induced vibrations (VIV) of a circular cylinder. The VIV-based piezoelectric energy harvester is simulated by a representative electro-aero-mechanical model, in which the aerodynamic force is represented by a model with amplitude-dependent aerodynamic parameters and an aerodynamic damping envelope model. The amplitude-dependent aerodynamic parameters at two typical Reynolds numbers are identified based on free vibration experiments and forced vibration numerical simulations, respectively. The aerodynamic damping envelope over a Reynolds number range of 500–33000 are identified based on the Reynolds number-dependent Griffin plot. Wind tunnel experiments are carried out to verify the accuracy of the electro-aero-mechanical model for the VIV-based piezoelectric energy harvester. A parametric study is then performed to investigate the Reynolds number effects on the bifurcation diagrams and maximum power output of a VIV-based piezoelectric energy harvester. It is demonstrated that the Reynolds number affects not only the levels of the harvested power, but also the global branches of the bifurcation diagram. The results also show that the performance of a VIV-based piezoelectric energy harvester is more sensitive to the Reynolds number for a case with low mechanical damping ratio.

A novel configuration for high power-output and highly efficient vibration energy harvesting

The most commonly used vibration energy harvesting system is the single-degree-of-freedom (SDOF) system. However, the power output of the SDOF system is limited by the mechanical damping ratio and the efficiency is only 50% at the maximal power output. In this paper, we modified the SDOF configuration by adding a spring in series with the energy transducer, which results in a configuration where a parallel energy-harvesting damper and inerter are in series with the spring, similar to the Maxwell model. The parameters of the proposed configuration are analyzed and optimized. The optimization results show the new configuration can produce many times more power output compared with the conventional SDOF system if the primary spring is relatively soft and the added spring is properly selected. In addition, the energy-harvesting efficiency can be significantly increased at the maximal power output. A lab experiment of backpack setup demonstrated 82% power increase and an efficiency of 83%. Lab test using a bridge vibration induced by freight train finds out that the proposed configuration can double the power output compared with the conventional SDOF system. With the augment of a simple spring, the proposed configuration disrupts the fundamental limit on the power and efficiency of the conventional SDOF energy-harvesting system, which opens a new perspective in the energy-harvesting field.

Post flutter energy map for H-shaped bridge girder

In this study, with the help of the three-degree-of-freedom random forced vibration device developed independently by Tongji University, high-precision measurement of bridge aerodynamic force under large torsional amplitude was realized, and a potential energy spectrum was established in consideration of the energy evolution and amplitude effect that varies with reduced velocity. As a result, a new research strategy for nonlinear flutter of bridge girders is proposed. Taking the Old Tacoma Narrows Bridge as an example, the mechanism of nonlinear vibration phenomena at post-flutter state was analyzed, and the wind-induced instability path of the bridge and its potential development path were deduced. The results show that flutter response can be divided into two components according to the energy spectrum. One component is related to the aerodynamic effect, and the other component is related to the mechanical damping ratio effect, which is helpful for studying the relationships between the mechanical damping ratio and bifurcation phenomena of bridge flutter. At the collapse of the Old Tacoma Narrows Bridge, both the aerodynamic nonlinearity and variations of the mechanical damping ratio with vibration amplitude have important contributions to the post-flutter behavior of the bridge in terms of energy effects. Variations of the mechanical damping ratio in torsion with amplitude greatly improve the flutter performance of the bridge. Thus, the development path of the structural displacement response depends on the competitive relationships among the structural torsional damping ratio, structural rigidity, and oncoming wind velocity.

Eksperimentalno istraživanje uticaja paralelnih električnih opterećenja na performanse elektromagnetnog sakupljača energije baziranog na vibracijama

At present, the researchers are grappling with the problems of maximizing the output power from a vibration-based electromagnetic energy harvester (VBEH). The parameters affecting the VBEH output power are: electrical damping ratio (ze), mechanical damping ratio (zm) and load impedances of shunted electrical load. Therefore, in this work, the experimental studies are carried out to study the effect of shunted electrical load on: i) the power output of VBEH and, ii) the determination of ze which maximizes the output power. For this purpose, a VBEH is designed and developed to obtain high open-circuit voltage. The effect of resistive, inductive, and capacitive loads on output power of VBEH is investigated using experimental setup developed exclusively for the same. The experimental results reveal that the output power of VBEH is maximum: i) at the resonant frequency ii) when equivalent resistive load impedance equals the internal resistance of electromagnetic coil and iii) value of ze is very small when compared to zm.

Assessment of wind-induced nonlinear post-critical performance of bridge decks

A quantitative analysis of the wind-induced post-critical performance of bridge decks is presented in order to highlight the underlying wind-resistant capacity after the critical wind speeds. Two simple indices, i.e., the wind speed extension (after the critical state) with acceptable post-critical vibrations and the relative post-critical capacity (i.e., the ratio between the aforementioned wind speed extension and the critical speed value), are utilized to quantify the post-critical performance of bridge decks. A short review on available limit states of vibration in literatures and design codes is presented in order to define a suitable threshold for acceptable post-critical vibrations. The post-critical performances of three typical cross-sections are quantitatively analyzed with the proposed indices, in which the limit states of vibration are reasonable defined considering either structural normal operation or structural safety. The effects of various mechanical and aerodynamic properties (e.g., mechanical damping ratio, natural frequency, initial angle of attack, and aerodynamic derivatives) on the post-critical behaviors of the selected cross-sections are highlighted. The quantitative analysis of the post-critical performance in the present work may deepen the understanding of the wind-resistant performance of bridges and potentially serve as a building block for a more economic wind-resistant design framework for future long-span bridges.

Equivalent impedance and power analysis of monostable piezoelectric energy harvesters

Nonlinearity has been successfully introduced into piezoelectric energy harvesting for power performance enhancement and bandwidth enlargement. While a great deal of emphasis has been placed by researchers on the structural design and broadband effect, this article is motivated to investigate the maximum power of a representative type of nonlinear piezoelectric energy harvesters, that is, monostable piezoelectric energy harvester. An equivalent circuit is proposed to analytically study and explain system behaviors. The effect of nonlinearity is modeled as a nonlinear stiffness element mechanically and a nonlinear capacitive element electrically. Facilitated by the equivalent circuit, closed-form solutions of power limit and critical electromechanical coupling, that is, minimum coupling to reach the power limit, of monostable piezoelectric energy harvesters are obtained, which are used for a clear explanation of the system behavior. Several important conclusions have been drawn from the analytical analysis and validated by numerical simulations. First, given the same level of external excitation, the monostable piezoelectric energy harvester and its linear counterpart are subjected to the same power limit. Second, while the critical coupling of linear piezoelectric energy harvesters depends on the mechanical damping ratio only, it also depends on the vibration excitation and magnetic field for monostable piezoelectric energy harvesters, which can be used to adjust the power performance of the system.

Effect of Electrical Damping Ratio on Maximum Average Power of an Electromagnetic Vibration Harvester at Resonance and off-Resonance Conditions

Energy harvesting has opened up new ways to power low energy electronics without any need for batteries. The most advanced types of harvesters are piezoelectric, electrostatic and electromagnetic which use kinetic energy in form of vibrations. Electromagnetic harvesters (EHs) are efficient, less costly and offer huge design flexibility. As such, in this paper some analytical and experimental studies on maximum average harvestable power (Pave) from typical design of EH using the popular coupling architecture (a cylindrical magnet oscillating in a copper coil) have been carried out. The effect of frequency of excitation and electrical (ζe ) and mechanical damping ratios (ζm) on Pave obtained from an EH has been investigated. For this purpose, an EH using NdFeB (neodymium magnet) magnet oscillating in a copper coil has been designed and developed. An experimental test set up has developed which comprises of (i) a cam-follower type drive mechanism to simulate a vibration source with variable amplitude and frequency of base excitation, (ii) a spring-mass system with an EH system in parallel with the spring to provide the relative motion between the magnet and the copper coil, (iii) the attendant instrumentation comprising of ultrasonic sensor to sense the displacements of base and mass, motor drive speed measurement and an open source platform (ARDUINO) with programmable circuit board and software and data acquisition system. Using the developed experimental set up, (i)frequency response curve for vibrating system with EH has been obtained from which equivalent electrical damping ζe has been determined using half power point method, and (ii)the values of relative displacements of magnet and coil, at various excitation frequencies have been recorded. Using the analytical expression for the Pave of SDOF EH has been determined both at resonance and off resonance conditions, for three different values of ζe namely ζe = ζm, ζe = 5ζm, and ζe = 10ζm. It is observed that the maximum Pave is obtained for condition ζe = ζm. As the value of ζe increases, the Pave increases, at resonant frequency and considerable power is made available at off-resonance frequencies. It is concluded that ζe should be increased to enhance the maximum average harvestable power from EH and vibrating system should be so designed to have ζm, as small as possible.

Added mass and damping effects on vibrating bridge decks in still air

Due to the windless-air-induced added mass and damping effects, the mechanical frequency and damping ratio of a bridge deck model in wind tunnel test are usually under-estimated and over-estimated, respectively. In this study, the vertical and torsional added mass and damping and their effects on the mechanical frequency and damping ratio of an ideal plate and five typical bridge decks are thoroughly investigated based on single degree-of-freedom free vibration numerical simulations. The aerodynamic forces are linearly expressed as functions of vibrating accelerations, velocities, added mass coefficients, and added damping coefficients. The discrepancies between two-dimensional (2D) and three-dimensional (3D) simulations are found to be small. The numerical simulation accuracy is validated through comparison with the results from the Theodorsen expression and forced vibration simulations. It is found that the added mass coefficients are insensitive to the vibration amplitudes, while the added damping coefficients almost linearly increase with the amplitudes. Based on the added mass and damping coefficients, the vertical and torsional air-induced frequency reduction coefficients and aerodynamic damping ratios are calculated. Both the frequency reduction coefficients and aerodynamic damping ratios are closely related to deck configurations and model mass/mass moment of inertia, while independent of vibration frequencies. For the concerned six section models, the vertical and torsional frequency reduction coefficients are mainly located in the ranges of (1.0%, 2.5%) and (0.4%, 1.2%), respectively. The aerodynamic damping ratios are significantly dependent on the vibration amplitudes and may exceed the mechanical damping ratios for large amplitude cases. Based on numerical simulations, the mechanical frequency and damping ratio of a spring-suspended deck model can be identified by excluding the added mass and damping effects.

Model update and validation of a mistuned high-pressure compressor blisk

ABSTRACTIn order to prepare an advanced 4-stage high-pressure compressor rig test campaign, details regarding both accomplishment and analysis of preliminary experiments are provided in this paper. The superior objective of the research project is to contribute to a reliable but simultaneously less conservative design of future high pressure blade integrated disks (blisk). It is planned to achieve trend-setting advances based on a close combination of both numerical and experimental analyses. The analyses are focused on the second rotor of this research compressor, which is the only one being manufactured as blisk. The comprehensive test program is addressing both surge and forced response analyses e.g. caused by low engine order excitation. Among others the interaction of aeroelastics and blade mistuning is demanding attention in this regard. That is why structural models are needed, allowing for an accurate forced response prediction close to reality. Furthermore, these models are required to support the assessment of blade tip timing (BTT) data gathered in the rig tests and strain gauge (s/g) data as well. To gain the maximum information regarding the correlation between BTT data, s/g-data and pressure gauge data, every blade of the second stage rotor (28 blades) is applied with s/g. However, it is well known that s/g on blades can contribute additional mistuning that had to be considered upon updating structural models.Due to the relevance of mistuning, efforts are made for its accurate experimental determination. Blade-by-blade impact tests according to a patented approach are used for this purpose. From the research point of view, it is most interesting to determine both the effect s/g-instrumentation and assembling the compressor stages on blade frequency mistuning. That is why experimental mistuning tests carried out immediately after manufacturing the blisk are repeated twice, namely, after s/g instrumentation and after assembling. To complete the pre-test program, the pure mechanical damping and modal damping ratios dependent on the ambient pressure are experimentally determined inside a pressure vessel. Subsequently the mistuning data gained before is used for updating subset of nominal system mode (SNM) models. Aerodynamic influence coefficients (AICs) are implemented to take aeroelastic interaction into account for forced response analyses. Within a comparison of different models, it is shown for the fundamental flap mode (1F) that the s/g instrumentation significantly affects the forced response, whereas the impact of assembling the compressor plays a minor role.

Effect of elastic modulus of cantilever beam on the performance of unimorph type piezoelectric energy harvester

Piezoelectric energy harvesting is a technique that can utilize ambient vibration energy to generate useful electrical energy, which is promising for powering small-scale autonomous devices such as sensors for wearable, biomedical, and industrial applications. Typically, cantilever-type piezoelectric energy harvesters (PEHs) are operated under resonance condition to achieve the maximum output power at low frequency stimuli. Along with resonance matching, it is also necessary to optimize the PEH configuration with high electromechanical properties for the efficient energy conversion. The purpose of this study is to investigate the effect of the elastic modulus of the passive layer in the cantilever structured PEH on the electromechanical properties and thus harvesting performance. In this regard, two unimorph type PEHs having the identical geometry, piezoelectric properties, and proof mass but with different elastic modulus (55 GPa and 97 GPa) of Ti alloy-based passive layers were fabricated and their output performance was compared under the same acceleration amplitude excitation stimuli. The PEH with the smaller elastic modulus passive layer exhibited almost 53% improvement in the maximum power than that with the higher elastic modulus passive layer, which is attributed to a smaller mechanical damping ratio, higher quality factor, and larger vibration amplitude.

Characterization of Piezoelectric Energy Harvesters Using Frequency-Weighted Impedance Method

This paper presents a frequency-weighted impedance method for the characterization of piezoelectric bending beam energy harvesters. The method uses the frequency response of the beam to determine the beam model parameters. Existing methods often need to measure the beam tip deflection for the characterization of the parameters. This requires the accessibility of the tip and advanced measurement tools, such as a laser vibrometer. However, in the proposed method only the knowledge of the base acceleration is adequate to determine the electromechanical coupling coefficient of the beam. Furthermore, the rest of parameters including mechanical damping ratio can be determined based on a pure electrical test. The proposed method is useful for the characterization of all types of piezoelectric energy harvesters, particularly the vacuum-packed ones in which the tip of the beam is inaccessible. The validity of the method is experimentally verified using a test setup including a vacuum-packed piezoelectric energy harvester. The measured parameters of the beam are compared with the bulk material parameters showing that the proposed method can accurately estimate the model parameters without using the information of the tip deflection and with an error less than 4%.

On optimization of energy harvesting from base-excited vibration

This paper re-examines and clarifies the long-believed optimization conditions of electromagnetic and piezoelectric energy harvesting from base-excited vibration. In terms of electromagnetic energy harvesting, it is typically believed that the maximum power is achieved when the excitation frequency and electrical damping equal the natural frequency and mechanical damping of the mechanical system respectively. We will show that this optimization condition is only valid when the acceleration amplitude of base excitation is constant and an approximation for small mechanical damping when the excitation displacement amplitude is constant. To this end, a two-variable optimization analysis, involving the normalized excitation frequency and electrical damping ratio, is performed to derive the exact optimization condition of each case. When the excitation displacement amplitude is constant, we analytically show that, in contrast to the long-believed optimization condition, the optimal excitation frequency and electrical damping are always larger than the natural frequency and mechanical damping ratio respectively. In particular, when the mechanical damping ratio exceeds a critical value, the optimization condition is no longer valid. Instead, the average power generally increases as the excitation frequency and electrical damping ratio increase. Furthermore, the optimization analysis is extended to consider parasitic electrical losses, which also shows different results when compared with existing literature. When the excitation acceleration amplitude is constant, on the other hand, the exact optimization condition is identical to the long-believed one. In terms of piezoelectric energy harvesting, it is commonly believed that the optimal power efficiency is achieved when the excitation and the short or open circuit frequency of the harvester are equal. Via a similar two-variable optimization analysis, we analytically show that the optimal excitation frequency depends on the mechanical damping ratio and does not equal the short or open circuit frequency. Finally, the optimal excitation frequencies and resistive loads are derived in closed-form.

Parametric Analysis and Experimental Verification of a Hybrid Vibration Energy Harvester Combining Piezoelectric and Electromagnetic Mechanisms

Considering coil inductance and the spatial distribution of the magnetic field, this paper developed an approximate distributed-parameter model of a hybrid energy harvester (HEH). The analytical solutions were compared with numerical solutions. The effects of load resistances, electromechanical coupling factors, mechanical damping ratio, coil parameters and size scale on performance were investigated. A meso-scale HEH prototype was fabricated, tested and compared with a stand-alone piezoelectric energy harvester (PEH) and a stand-alone electromagnetic energy harvester (EMEH). The peak output power is 2.93% and 142.18% higher than that of the stand-alone PEH and EMEH, respectively. Moreover, its bandwidth is 108%- and 122.7%-times that of the stand-alone PEH and EMEH, respectively. The experimental results agreed well with the theoretical values. It is indicated that the linearized electromagnetic coupling coefficient is more suitable for low-level excitation acceleration. Hybrid energy harvesting contributes to widening the frequency bandwidth and improving energy conversion efficiency. However, only when the piezoelectric coupling effect is weak or medium can the HEH generate more power than the single-mechanism energy harvester. Hybrid energy harvesting can improve output power even at the microelectromechanical systems (MEMS) scale. This study presents a more effective model for the performance evaluation and structure optimization of the HEH.