Bimorph Cantilever Research Articles

Electromechanical modeling and analytical investigation of nonlinearities in energy harvesting piezoelectric beams

Piezoelectric materials are extensively applied for vibrational energy harvesting especially in micro-scale devices where other energy conversion mechanisms such as electromagnetic and electrostatic methods encounter fabrication limitations. A cantilevered piezoelectric bimorph beam with an attached proof (tip) mass for the sake of resonance frequency reduction is the most common structure in vibrational harvesters. According to the amplitude and frequency of applied excitations and physical parameters of the harvester, the system may be pushed into a nonlinear regime which arises from material or geometric nonlinearities. In this study nonlinear dynamics of a piezoelectric bimorph harvester implementing constitutive relations of nonlinear piezoelectricity together with nonlinear curvature and shortening effect relations, is investigated. To achieve this goal first of all a comprehensive fully-coupled electromechanical nonlinear model is presented through a variational approach. The governing nonlinear partial differential equations of the proposed model are order reduced and solved by means of the perturbation method of multiple scales. Results are presented for a PZT/Silicon/PZT laminated beam as a case study. Findings indicate that material nonlinearities of the PZT layer has the dominant effect leading to softening behavior of the frequency response. At the primary resonance, different frequency responses of the extracted power can be distinguished according to the excitation amplitude, which is due to harmonic generation as a result of piezoelectric nonlinearity. The extracted power is analytically computed and validated with a good agreement by a numerical solution.

A Self-Aligned 45°-Tilted Two-Axis Scanning Micromirror for Side-View Imaging

This paper presents a two-axis electrothermal single-crystal-silicon micromirror that is tilted 45° out of plane on a silicon optical bench (SiOB). The SiOB provides mechanical support and electrical wiring to the tilted two-axis scanning mirror, as well as aligned trenches for assembling other optical components, such as optical fibers and GRIN lens. The pre-set tilting of the mirror plate is achieved via the bending of a set of stressed bimorph cantilevers. A stopper connected on the bending bimorphs reaches and is stopped by the adjacent silicon sidewall. The tilt angle can be precisely controlled by properly choosing the distance from the stopper to the silicon sidewall and the flexure bimorph length. The fabricated mirror plate is 0.72 mm $\times $ 0.72 mm and the footprint of the entire MEMS device is 2.22 mm $\times$ 1.25 mm. The measured maximum total optical scan angle of the mirror is approximately 40° in both $x$ -axis and $y$ -axis at only $5.5~V_{\mathrm{ dc}}$ . This technology will enable a new class of more compact microendoscopic optical imaging probes for in vivo early cancer detection. [2015-0315]

A broadband quad-stable energy harvester and its advantages over bi-stable harvester: Simulation and experiment verification

This paper presents a novel quad-stable energy harvester (QEH) to harvest vibration energy. The QEH is composed of a piezoelectric bimorph cantilever beam with a tip magnet and three external fixed magnets. The potential energy of the QEH is derived, which has four potential wells and three low barriers. This implies that the QEH needs relatively lower excitation energy to cross the barriers than the bi-stable energy harvester (BEH). Simulations are carried out for the BEH and QEH at different levels of excitations. Results show that the QEH owns a smaller threshold and a wider range of frequencies for occurrence of snap-through than the BEH. Thus the QEH can give out a large output voltage. Corresponding experiments were performed for validation. The experimental results are in agreement with the simulations, which means that the QEH can improve the harvesting efficiency considerably.

Design and characterisation of a piezoelectric knee-joint energy harvester with frequency up-conversion through magnetic plucking

Piezoelectric energy harvesting from human motion is challenging because of the low energy conversion efficiency at a low-frequency excitation. Previous studies by the present authors showed that mechanical plucking of a piezoelectric bimorph cantilever was able to provide frequency up-conversion from a few hertz to the resonance frequency of the cantilever, and that a piezoelectric knee-joint energy harvester (KEH) based on this mechanism was able to generate sufficient energy to power a wireless sensor node. However, the direct contact between the bimorph and the plectra leads to reduced longevity and considerable noise. To address these limitations, this paper introduces a magnetic plucking mechanism to replace the mechanical plucking in the KEH, where primary magnets (PM) actuated by knee-joint motion excite the bimorphs through a secondary magnet (SM) fixed on the bimorphs tip and so achieve frequency up-conversion. The key parameters of the new KEH that affect the energy output of a plucked bimorph were investigated. It was found that the bimorph plucked by a repulsive magnetic force produced a higher energy output than an attractive force. The energy output peaked at 32 PMs and increased with a decreasing gap between PM and SM as well as an increasing rotation speed of the PMs. Based on these investigations, a KEH with high energy output was prototyped, which featured 8 piezoelectric bimorphs plucked by 32 PMs through repulsive magnetic forces. The gap between PM and SM was set to 1.5 mm with a consideration on both the energy output and longevity of the bimorphs. When actuated by knee-joint motion of 0.9 Hz, the KEH produced an average power output of 5.8 mW with a life time >7.3 h (about 3.8 × 105 plucking excitations).

Numerical modeling of shape and topology optimisation of a piezoelectric cantilever beam in an energy-harvesting sensor

Piezoelectric materials are excellent transducers for converting mechanical energy from the environment for use as electrical energy. The conversion of mechanical energy to electrical energy is a key component in the development of self-powered devices, especially enabling technology for wireless sensor networks. This paper proposes an alternative method for predicting the power output of a bimorph cantilever beam using a finite-element method for both static and dynamic frequency analyses. A novel approach is presented for optimising the cantilever beam, by which the power density is maximised and the structural volume is minimised simultaneously. A two-stage optimisation is performed, i.e., a shape optimisation and then a topology hole opening optimisation.

Thermal Management Using MEMS Bimorph Cantilever Beams

This paper examines a passive cooling technique using microelectromechanical systems (MEMS) for localized thermal management of electronic devices. The prototype was designed using analytic equations, simulated using finite element methods (FEM), and fabricated using the commercial PolyMUMPs™ process. The system consisted of an electronic device simulator (EDS) and MEMS bimorph cantilever beams (MBCB) array with beams lengths of 200, 250, and 300 μm that were tested to characterize deflection and thermal behavior. The specific beam lengths were chosen to actuate in response to heating associated with the EDS (i.e. the longest beams actuated first corresponding to the hottest portion of the EDS). The results show that the beams deflected as designed when thermally actuated and effectively transferred heat away via thermal conduction. The temperature when the beams reached “net-zero” deflection (i.e. uncurled and flat) was related to the initial deflection distance while the contact deflection temperature and rate of actuation was related to beam length. Initial beam deflections, after release, and contact temperatures, when fully actuated, were approximately 5.05, 9.45, 14.05 μm, and 231, 222, 216 °C, respectively with the longer beams making contact first. This innovative passive thermal management system enables selective device cooling without requiring active control or forced convection to maintain steady-state operating temperatures for sensitive microelectronic devices.

Improve efficiency of harvesting random energy by snap-through in a quad-stable harvester

In response to the defects of the bi-stable energy harvester (BEH), we develop a novel quad-stable energy harvester (QEH) to improve the harvesting efficiency. The device is made up of a bimorph cantilever beam having a tip magnet and three external fixed magnets. By introducing the repulsion forces between magnets, the function of potential energy of the QEH is given. It owns four potential wells. It is proved that the quad-stable harvester can cross the barriers and realize snap-through easier. Validation experiments were performed by frequency sweeping and random excitations. Results show that compared to the BEH the frequency bandwidth of snap-through of the novel device is much wider. The QEH can make a dense snap-through in response under random excitation, and give out a large output voltage. This shows that the proposed QEH is more effective in energy harvesting applications.

Energy harvesting in a quad-stable harvester subjected to random excitation

In response to the defects of bi-stable energy harvester (BEH), we develop a novel quad-stable energy harvester (QEH) to improve harvesting efficiency. The device is made up of a bimorph cantilever beam having a tip magnet and three external fixed magnets. By adjusting the positions of the fixed magnets and the distances between the tip magnet and the fixed ones, the quad-stable equilibrium positions can emerge. The potential energy shows that the barriers of the QEH are lower than those of the BEH for the same separation distance. Experiment results reveal that the QEH can realize snap-through easier and make a dense snap-through in response under random excitation. Moreover, its strain and voltage both become large for snap-through between the nonadjacent stable positions. There exists an optimal separation distance for different excitation intensities.

Dynamical Model for an Interharmonic Property of a Piezoelectric Bimorph Cantilever Beam with Self-Sensing Function

A piezoelectric bimorph cantilevered beam is analyzed dynamically by a longitudinal and transverse coupling theory. When a sinusoidal voltage is applied on the actuating layer of the bimorph, the output voltage of the sensing layer appears as interharmonic component signal. The interharmonic frequency is noninteger harmonic frequency of the applied voltage. A dynamic model is proposed to describe the interharmonic property of the piezoelectric bimorph beam. Through some simulations and experiments, the theoretical model is verified effectively to express the nonlinear characteristic. Furthermore, when the piezoelectric bimorph resonance happens, some interharmonic response at low frequency will modulate with the resonance response.

A flexoelectric microelectromechanical system on silicon.

Flexoelectricity allows a dielectric material to polarize in response to a mechanical bending moment and, conversely, to bend in response to an electric field. Compared with piezoelectricity, flexoelectricity is a weak effect of little practical significance in bulk materials. However, the roles can be reversed at the nanoscale. Here, we demonstrate that flexoelectricity is a viable route to lead-free microelectromechanical and nanoelectromechanical systems. Specifically, we have fabricated a silicon-compatible thin-film cantilever actuator with a single flexoelectrically active layer of strontium titanate with a figure of merit (curvature divided by electric field) of 3.33 MV(-1), comparable to that of state-of-the-art piezoelectric bimorph cantilevers.

Improvements in energy harvesting capabilities by using different shapes of piezoelectric bimorphs

The small amount of power needed by microelectronic devices opens up the possibility to convert part of the vibration energy present in the environment into electrical energy, using several methods. One such method is to use piezoelectric material as an additional layer in cantilever beams to harvest vibration energy for self-powered sensors. The geometry of a piezoelectric cantilever beam will greatly affect its vibration energy harvesting ability. Tapering and changing the configuration as ways to increase the generated output power of cantilever piezoelectric energy harvesters have gained popularity in recent years. In this work vibration energy harvesting via piezoelectric resonant bimorph cantilevers is studied and new designs for obtaining optimal piezoelectric energy harvesters are suggested. This paper deduces a very precise and simple analytical formula that can be used as a rule of thumb for calculating the resonant frequency of bimorph trapezoidal V-shaped cantilevers using the Rayleigh–Ritz method. This analytical formula is then analyzed using MATLAB as well as finite element methods and validated by ABAQUS simulation. Also, mathematical derivations for the output voltage of bimorph piezoelectric energy harvesters are presented and validated by simulation and experimental results. These formulas provide a new perspective that, among all the bimorph trapezoidal V-shaped cantilever beams with uniform thickness, the bimorph triangular tapered cantilever can lead to the highest resonant frequency and therefore maximum sensitivity, and by increasing the ratio of the trapezoidal bases the sensitivity decreases. Also, the output voltage and strain distribution show that the triangular cantilever has the highest efficiency and power density.

Nonlinear dynamic analyses on a magnetopiezoelastic energy harvester with reversible hysteresis

This paper presents a systematic approach through a series of nonlinear analyses to predict and design the nonlinear resonance characteristics and energy-harvesting performance of a magnetopiezoelastic vibration energy harvester with reversible hysteresis. To this end, a mathematical model of the energy harvester system, composed of a bimorph cantilever beam along with three permanent magnets, is first derived. With this model, frequency response analyses are conducted using the methods of multiple scales and harmonic balance. In the case of weak excitation, the system’s stationary forced response is obtained through multiple-scale analysis (MSA). With this MSA solution, analytical design criteria in terms of the position parameters of the magnets are derived to determine the hysteresis type of the nonlinear resonance (stiffness hardening or stiffness softening). However, in the case of a relatively strong excitation, the high-dimensional harmonic balance analysis (HDHBA) shows that the stiffness-softening effect tends to strengthen for the oscillation amplitude in a specific condition, and this effect can even lead to hysteresis transition or potential well escape. Using the HDHBA, design criteria are set up and evaluated in terms of source vibration strength to detect the hysteresis transition or the condition required to determine the potential well escape. Finally, based on the present systematic analyses, the complicated resonant responses and energy-harvesting performance of the present system are examined with respect to several design parameters, and the results are discussed in detail.

A finger-like hardness tester based on the contact electromechanical impedance of a piezoelectric bimorph cantilever.

We proposed a finger-like hardness tester based on the electromechanical impedance of a piezoelectric bimorph cantilever. A Vickers indenter was fabricated to the free end of the bimorph to contact the sample. The contact force was monitored by a strain gauge and the contact area was obtained by tracking the bimorph's resonance frequency. The bimorph-sample contact system was modeled by the electromechanical equivalent circuit method. Verification experiments on standard hardness samples were conducted and the measured hardness values agreed well with those given by a conventional Vickers hardness tester. Further hardness measurement on a gear wheel showed that the proposed hardness tester is very adaptive and can be used for inner surface testing or in situ testing, where other hardness testers may not be applicable. The proposed hardness tester can be regarded as an improved ultrasonic hardness tester.

A forefinger-like tactile sensor for elasticity sensing based on piezoelectric cantilevers

Current researches on tactile sensors are mainly focused on force sensing, but studies on elasticity sensing are very lacking. In this work, a forefinger-like tactile sensor is proposed with the functions of force sensing, contact prediction, and elasticity sensing. The sensor is made of a piezoelectric bimorph cantilever with a strain gauge for deformation monitoring. A cone-shaped tip is fabricated on the cantilever’s free end to contact the testing sample. When the sensor approaches to the sample’s surface, the bimorph cantilever is excited to vibrate in its flexure mode. Consequently, the vibration amplitude would shift regularly when the tip is close enough to the sample, leading to the contact prediction function. When the tip touches the sample, the sample’s elasticity can be derived by tracking the contact resonance frequency of the cantilever-sample system. The functions of the proposed sensor were carefully examined and excellent performances were achieved. The proposed sensor is adaptive and may hold potentials in sample characterization in industry.

Fabrication of multilayer Pb(Zr,Ti)O3 thin film by sputtering deposition for MEMS actuator applications

Multimorph structures composed of multiple thin-film piezoelectric layers and electrode layers were realized by sputtering deposition. Cantilevers with four layers of Pb(Zr,Ti)O3 (PZT) thin film are fabricated and evaluated. The electrical and piezoelectric characteristics of each PZT layer are very similar and are comparable to those of conventional single-layer PZT thin film. The piezoelectric constant d31 is estimated to be −38.3 pm/V from the relationship between tip displacement and applied voltage under the condition of driving four all piezoelectric layers. The generative force is also estimated from the tip displacement. It is confirmed that multimorph cantilevers have larger generative force than those of conventional unimorph and bimorph cantilevers for identical driving voltage.

Preloaded freeplay wide-bandwidth low-frequency piezoelectric harvesters

We propose a technique for increasing the bandwidth of resonant low-frequency (<100 Hz) piezoelectric energy harvesters based on the modification of the clamped boundary condition of cantilevers, termed here as preloaded freeplay boundary condition. The effects of the preloaded freeplay boundary condition are quantified in terms of the fundamental frequency, frequency response, and power output for two beam configurations, namely, classical cantilevered bimorph piezoelectric energy harvester and zigzag unimorph piezoelectric energy harvester. A comparative analysis was performed between both the harvesters to empirically establish the advantages of the preloaded freeplay boundary condition. Using this approach, we demonstrate that the coupled degree-of-freedom dynamics results in an approximate 4–7 times increase in half-power bandwidth over the fixed boundary condition case.

Vibration-based piezoelectric micropower generator for power plant wireless monitoring application

This paper investigates the factors affecting the performance of a vibration based micropower generator for a power plant wireless monitoring application. A bimorph bender, one of micropower generator methods, was used in this work. The ANSYS program was used to study the distribution of stress–strain in each model design, and MATLAB was used to simulate and investigate the extracted power. Triangular, rectangular and trapezoidal cantilevers were chosen to analyse the effect of the bender shape on the power produced. The tests were conducted using the same input excitation conditions (10m/s2), frequency range of (50–150Hz). The effect of the configuration arrangement (horizontal and vertical) of the PZT bimorph cantilever on the power generation (poling series) was also investigated. The simulation result showed that the maximum stress–strain value was produced in the triangular bender shape with equal distribution on all the surface areas. The measurements’ results showed that the triangular cantilever produced maximum power compared to other bender shapes for both horizontal and vertical directions and single test. The horizontal and vertical arrangements of the cantilevers showed that the vertical arrangement could produce more power than the horizontal arrangement. The experimental results showed concurrence with theoretical models.

Quantitative electromechanical impedance method for nondestructive testing based on a piezoelectric bimorph cantilever

The electromechanical impedance (EMI) method, which holds great promise in structural health monitoring (SHM), is usually treated as a qualitative method. In this work, we proposed a quantitative EMI method based on a piezoelectric bimorph cantilever using the sample’s local contact stiffness (LCS) as the identification parameter for nondestructive testing (NDT). Firstly, the equivalent circuit of the contact vibration system was established and the analytical relationship between the cantilever’s contact resonance frequency and the LCS was obtained. As the LCS is sensitive to typical defects such as voids and delamination, the proposed EMI method can then be used for NDT. To verify the equivalent circuit model, two piezoelectric bimorph cantilevers were fabricated and their free resonance frequencies were measured and compared with theoretical predictions. It was found that the stiff cantilever’s EMI can be well predicted by the equivalent circuit model while the soft cantilever’s cannot. Then, both cantilevers were assembled into a homemade NDT system using a three-axis motorized stage for LCS scanning. Testing results on a specimen with a prefabricated defect showed that the defect could be clearly reproduced in the LCS image, indicating the validity of the quantitative EMI method for NDT. It was found that the single-frequency mode of the EMI method can also be used for NDT, which is faster but not quantitative. Finally, several issues relating to the practical application of the NDT method were discussed. The proposed EMI-based NDT method offers a simple and rapid solution for damage evaluation in engineering structures and may also shed some light on EMI-based SHM.

Enhanced performance of magnetoelectric energy harvester based on compound magnetic coupling effect

We have theoretically and experimentally demonstrated the greatly enhanced energy harvesting property of the specific magnetoelectric (ME) device, comprising a piezoelectric bimorph cantilever with a permanent magnet tip mass based on a compound interaction between the remanent magnetic moment of the magnet and a nonuniform alternating magnetic field. With appropriate positioning of the device, the coexistence of torque-mode and force-mode excitations leads to reinforced magneto-mechanical coupling, which subsequently yields improvements in both ME response and power conversion. In the experiments, a piezoelectric bimorph/magnet energy harvester was placed at a distance of 10 mm from a power line that was conducting a 50 Hz, 10 A current, and a maximum power of 2.136 mW was achieved via the optimal cooperative magnetic coupling mode. This output power is 7.8× larger than that produced using the conventional torque mode.

Analysis of Energy Harvesting Performance for <inline-formula><tex-math>$d_{15}$</tex-math> </inline-formula> Mode Piezoelectric Bimorph in Series Connection Based on Timoshenko Beam Model

Shear strain is exactly described by combining the piezoelectric constitutive equations of d 15 mode with the Timoshenko beam model. We provide an electromechanical model to evaluate the energy harvesting performance of shear mode cantilevered piezoelectric bimorph in series connection. The output peak voltage-frequency and the output peak power-frequency curves are simulated by using the proposed model at different load resistances. The results are in good agreement with the finite element simulation results, indicating that the proposed model is valid for evaluating the energy harvesting performances of the cantilevered piezoelectric bimorph. The variations of the output peak voltage and power with load resistance change for excitation are also discussed. Moreover, the effect of proof mass on the energy harvesting performance is analyzed at certain load resistance. The proposed model can be used for the structure optimized design and performance improvement of shear mode piezoelectric energy harvesting.