Theoretical Study of a Two-Degree-of-Freedom Piezoelectric Energy Harvester under Concurrent Aeroelastic and Base Excitation

This paper presents a study of a two-degree-of-freedom (2DOF) piezoelectric energy harvester (PEH) under concurrent aeroelastic and base excitation. The governing equations of the theoretical model under the combined excitation are developed and solved analytically using the harmonic balance method. Based on the electro-mechanical analogies, an equivalent circuit model is established. The energy harvesting performance of the 2DOF PEH under different wind speeds but the same base excitation is investigated. Voltage amplitudes of various response components with different frequencies are predicted by the analytical method and verified by the circuit simulation. The root-mean-square (RMS) voltage is used to measure the actual performance of the 2DOF PEH. Around the resonance state, the 2DOF PEH has been found to produce a larger voltage output than the conventional SDOF PEH. Moreover, several interesting phenomena, such as the quasi-periodic oscillation and the peak-to-valley transition, have been observed in the circuit simulation and explained by the analytical solution. The developed methodology in this paper can be easily adapted to analyze other similar types of multiple-degree-of-freedom (MDOF) PEHs under concurrent aeroelastic and base excitation.

Rotational nonlinear double-beam energy harvesting

This paper presents an investigation of the performance of a coupled rotational double-beam energy harvester (DBEH) with magnetic nonlinearity. Two spring-connected cantilever beams are fixed on a rotating disc. Repelling magnets are attached to the frame and to the lower beam tip, and an equal-mass block is attached to the tip of the upper beam. To describe the dynamic response, a theoretical model related to the rotational motion of the coupled cantilever beam is derived from the Lagrange equations. In addition, the harmonic balance method, together with the arc-length continuation method, is applied to obtain the frequency response functions (FRFs). Parametric studies are then conducted to analyze the effect of varying the parameters on the energy harvesting performance, and numerical analysis is performed to validate the analytical solutions. Finally, the theoretical model is verified by forward- and reverse-frequency-sweeping experiments. The DBEH in rotational motion can perform effective energy harvesting over a wide range of rotational frequencies (10 to 35 rad/s). The upper beam is found to exhibit better energy harvesting efficiency than the lower beam around the resonant frequency. This study effectively broadens the energy harvesting bandwidth and provides a theoretical model for the design of nonlinear magnet-coupled double-beam structure in rotational energy harvesting.

An Effective Solution of the Cube-Root Truly Nonlinear Oscillator: Extended Iteration Procedure

The cube-root truly nonlinear oscillator and the inverse cube-root truly nonlinear oscillator are the most meaningful and classical nonlinear ordinary differential equations on behalf of its various applications in science and engineering. Especially, the oscillators are used widely in the study of elastic force, structural dynamics, and elliptic curve cryptography. In this paper, we have applied modified Mickens extended iteration method to solve the cube-root truly nonlinear oscillator, the inverse cube-root truly nonlinear oscillator, and the equation of pendulum. Comparison is made among iteration method, harmonic balance method, He’s amplitude-frequency formulation, He’s homotopy perturbation method, improved harmonic balance method, and homotopy perturbation method. After comparison, we analyze that modified Mickens extended iteration method is more accurate, effective, easy, and straightforward. Also, the comparison of the obtained analytical solutions with the numerical results represented an extraordinary accuracy. The percentage error for the fourth approximate frequency of cube-root truly nonlinear oscillator is 0.006 and the percentage error for the fourth approximate frequency of inverse cube-root truly nonlinear oscillator is 0.12.

Numerical and Experimental Investigations on a Friction Ring Damper for a Flywheel

A damping strategy using a friction ring damper for an industrial flywheel was numerically and experimentally investigated. The friction ring damper, located on the arms of the flywheel, was experimentally found to effectively reduce the vibration amplitude of the flywheel. The vibration energy is dissipated when relative motions occur at the friction contact interfaces. Nonlinear dynamic analysis based on a lumped-parameter model of a flywheel equipped with a friction ring damper was conducted. A dimensionless parameter, κ, defined as the ratio of the critical friction force to the amplitude of harmonic force, was used to evaluate the damping performance. For several values of κ, steady-state responses under harmonic excitation and nonlinear modes were obtained using the harmonic balance method (HBM) combined with the alternating frequency–time domain method (AFT). The forced response analysis proved the existence of an optimal value of κ, which can minimize the vibration amplitude of the flywheel. The nonlinear modal analysis showed that all the damping ratio–frequency curves are completely coincident even for different κ, and the frequency corresponding to the maximum damping ratio is equal to the frequency at the intersection of the forced response curves under the fully slip and the fully stick states of the friction contact interface. By analyzing the behaviors of the friction contact interface, it is shown that the friction contact interface provides damping in the combined stick–slip state. The forced response under random excitation was calculated using the Runge–Kutta method and the friction interface behaviors were analyzed. Finally, spectral testing was conducted to verify the numerical results.

Vibration suppression and dynamic behavior analysis of an axially loaded beam with NES and nonlinear elastic supports

Recently, dynamic analysis of a beam structure with nonlinear energy sink (NES) and various supports is attracting great attention. Most of the existing studies are about the beam structure with NES or nonlinear boundary supports with zero rotational restraint, respectively. However, there is little research accounting for such two types of complex factors simultaneously. In this work, the dynamic behavior of an axially loaded beam with both NES and general boundary supports is modeled and studied. The Galerkin truncated method (GTM) is employed to make the prediction of dynamic behavior of such a beam system, in which the mode functions of axially loaded Euler–Bernoulli beam with linear elastic boundary conditions are selected as the trail and weight functions. Then, the Galerkin condition is used to discretize the nonlinear governing equation of the beam system and establish the residual equations. The Runge–Kutta method is used to solve the residual matrix which consists of residual equations directly, and the harmonic balance method is also used to verify the results from the GTM. The influence of NES on vibration suppression and dynamic behavior of the beam structure is investigated and discussed. Results show that the vibration states of the beam structure can be transformed effectively through the change of NES parameters. On the other hand, the NES with suitable parameters has a beneficial effect on the vibration suppression at both ends of the beam structure.

Stability and bifurcation analysis of super- and sub-harmonic responses in a torsional system with piecewise-type nonlinearities

The nonlinear dynamic behaviors induced by piecewise-type nonlinearities generally reflect super- and sub-harmonic responses. Various inferences can be drawn from the stability conditions observed in nonlinear dynamic behaviors, especially when they are projected in physical motions. This study aimed to investigate nonlinear dynamic characteristics with respect to variational stability conditions. To this end, the harmonic balance method was first implemented by employing Hill’s method, and the time histories under stable and unstable conditions were examined using a numerical simulation. Second, the super- and sub-harmonic responses were investigated according to frequency upsweeping based on the arc-length continuation method. While the stability conditions vary along the arc length, the bifurcation phenomena also show various characteristics depending on their stable or unstable status. Thus, the study findings indicate that, to determine the various stability conditions along the direction of the arc length, it is fairly reasonable to determine nonlinear dynamic behaviors such as period-doubling, period-doubling cascade, and quasi-periodic (or chaotic) responses. Overall, this study suggests analytical and numerical methods to understand the super- and sub-harmonic responses by comparing the arc length of solutions with the variational stability conditions.