Good morning! It is my pleasure to present our recent work on nonlinear dynamics inenergy harvesting. Mr. Tianchen Yuan is my Ph D student.

Firstof all, I’ll show you the outline of the talk.

Thetalk is divided into 5 parts. It begins with an introduction to explain the background and the motivation. Then the physical model is presented and the electromechanical coupling relation is analytically established. It follows the parameter identification from the experimental data. Then the experimental amplitude-frequency response curves are presented and compared with numerical results and harmonic-balance-analysis results. Finally, some concluding remarks end the talk.

Now turn to the Introduction.  

As we well know, energy harvesting is a significant issue. Especially, vibratory energy harvesting is to transform the kinetic energy of waste vibration intoelectricity.

Linear beams have been widely used to model and to analyze energy harvesters. However, there are some essential limitations of linear beam energy harvesters. As a resonator, a linear energy harvester works only in a narrow frequency range near the resonance. The electromechanical coupling is rather weak.

A plate is a promising structure for piezoelectric energy harvesting because of its strong coupling and its compact volume. Especially, a circular plate has an axial symmetric structure which is easy to be processed and thus attracts more and more attentions. Some significant works are list here without detailed reviews.

An isotropic piezoelectric plate is not suitable for energy harvesting due to its brittleness. A composite plate manufactured by the metal base plate can solve the problem thank to the toughness of metals. A center mass is necessary for reducing the natural frequency of the harvester in order to achieve the resonance with the environment. In addition, nonlinearities can be employed to broaden the response frequency bandwidth of the harvester.Therefore, it is important to investigate the nonlinear behaviors of a circular piezoelectric plate theoretically and experimentally. It has been found that a no-proof-mass circular isotropic piezoelectric plate with the von Karman geometric nonlinearity produces hardening effect associated with a jumping near resonance. There have been no investigations on nonlinear dynamics of circular composite piezoelectric plate with a proof mass.

If jumping can enhance energy harvesting, it is a natural idea that double-jumping, jumping in both sides, does the job better. Internal resonance leads to double-jumping. Our group proposed conceptually an energy harvester with internal resonance. Do softening and hardening nonlinearities coexist in responses of a composit eplate?

The objective of the workis to explore the nonlinear behaviors of a proof-mass circular composite plateenergy harvester. To do so, the harvester is modeled analytically and experimentally with the parameters identified from the data. Then the frequency response of the harvester is investigated experimentally, numerically, and analytically.

A thin composite piezoelectric plate is clamped between two steel support rings. Two steel ringsare employed to implement a clamped boundary condition. The composite plate is manufactured by a brass plate and two identical piezoelectric plates.Two piezoelectric plates are connected in series. The upper and the lower surfaces of the composite plate are the silver electrodes so that they have the opposite direction of polarization. A proof mass shaped as an inverted cone isset at the center of the composite plate. The harvester is modeled as asingle-degree-of-freedom oscillator with a nonlinear restoring force

Newton’s second law and Kirchhoff's second law yield the governing equations of the harvester. However, there are some parameters need to be determined based on the experimental data. In the following, the electrical parameters are both modeled and identified, while themechanical parameters are identified from experiment.

The following 3 slides present the outline of the derivation of the electromechanical coupling relation. The piezoelectric equations lead to the electric displacement. Then the charge on the electrode surface is given via the integration.

Express the deflection of the plate as the product of the center displacement and the shape function. The average strains are used in the piezoelectric constitutive equations.

The output current is differential of the charge with respect to the time. Some mathematical operations lead to the current equation, actually the electromechanical equation with the electromechanical coupling coefficient and the equivalentcapacitance.

Parameter identification begins with the description of the experimental system, presented in the figure. The shaker is driven by a power amplifier and controlled by a closed loop vibration controller. Base acceleration is feedback to the controller by an impedance head. These experiment equipments can provide the harmonic excitation at the specified frequency and amplitude, the up and down frequency sweeps at constant amplitude or the random excitement in the specified frequency range. The accelerations of the base and the proof massare measured by two accelerometers. The displacements of the base and the proofmass are measured by two absolute eddy current sensors. The output voltage ismeasured by a hall voltage sensor. Data is recorded by a data acquisitioninstrument.

The electrical parametersare identified based on the linear model under small excitations. Thus the frequencies of responses are the same as those of the excitations. Here is the relation between the amplitudes of the displacement and the output voltage.

Based on the experimentwith the excitation frequency, the load resistance, the mass acceleration amplitude, andthe output voltage amplitude, N times of experiments yield the parameters identified via the least square method

The table lists the theoretical and the identified plate coefficients as well as the theoretical beam electromechanical coupling coefficient. The beam is with larger area of piezoelectric bimorph and thicker piezoelectric layers than those of the plate. However, the electromechanical coupling coefficient of the beam is much smaller than that of the plate. Thus the circular plate has the much larger electromechanical coupling coefficient than that of a beam in similar sizes. The strong coupling is beneficial to energy harvesting.

Damping and stiffness coefficients can be identified from measured accelerations and its integrations filtered by the empirical mode decomposition. The restoring forcecan beknown at every sampling instant though the equation. The displacement and the velocity can be integrated from the acceleration data or measured directly.

In order to get the more wide distribution of the points in the phase plane, the band-limited random excitation is used. The velocities and the displacements are integrated based on the measured accelerations. The integration data are filtered by the empirical mode decomposition method in order to remove low frequency interferences. The figure shows the displacement of the center mass. Line 1 represents the original data measuredby the absolute eddy current sensor. Line 2 represents the integral data from the accelerometer sensor with the filter. Line 3 represents the integral data from the accelerometer sensor without the filter. The filtered integration results are consistent withthe measured displacement without any phase deviation.

The excitation should be large enough to highlight the geometrical nonlinearity. In this case the displacement exceeds the full scale of the absolute eddy current sensors. The integration and filter method mentioned before are used to obtain the velocities and displacements. The figure shows the restoring force surface. There storing force can be defined by the fifth polynomial with coefficients listed in the table. The figure shows the fitting results of the restoring force. Other parameters identified from theexperimental date are summarized in the table.

Both time-domain simulations and harmonic-balance-analysis solutions (1-term and 3-terms) are compared with experiment results. The system exhibits linear behavior in the 0.3 gexcitation case and the dash lines reflects the linear natural frequency (87Hz). The simulations and 1-term and 3-terms harmonic balance solutions agree well at such low excitation. Under the 1.5 g excitation, voltage response demonstrates softening nonlinearity in the experiment, the simulation and the analysis. The amplitude-frequency response curve bends to the left with a slight jump. The resonant frequent is smaller than that of the linear system.

Both the softening and the hardening nonlinearities are revealed in the 2 g excitation case. At the left side of the resonance peak, there is a slight jump resulted from the softening. At the right side of the resonance peak, the response curve displays a significant nonlinear behaviorresulted from the hardening. The resonant frequency is also smaller than thelinear one, which is different from the typical Duffing system. Overall, the solution with 3-terms exhibits much better agreement with the time-domain numerical solution than the solution with 1-term only.

Now a few comments are given about the working bandwidth of the harvester. The bandwidth is defined by the half power points for linear systems. The definition can be extended to nonlinear systems.

The definition can beused for jumping with both softening and hardening. The results show that it improves the performance of bandwidth.

Finally, I’ll end thepresentation with a few concluding remarks.

The investigation focuses on a circular composite piezoelectric plate energy harvester. The electromechanical model is established analytically. The parameters are identified from the experiment, which verifies the modeling results. The dynamics is examined numerically, analytically and experimentally. Softening nonlinearity is found under small excitation. Both softening and hardening nonlinearities are found under larger excitation.