Nonlinear Two-to-One Internal Resonance for Broadband Energy Harvesting

2015 ◽  
Author(s):  
D. X. Cao ◽  
S. Leadenham ◽  
A. Erturk
Keyword(s):  
Energy Harvesting ◽  
Frequency Response ◽  
Internal Resonance ◽  
Numerical Solutions ◽  
Primary Resonance ◽  
Frame Structure ◽  
Response Curves ◽  
Current Effort ◽  
Bandwidth Enhancement ◽  
Piezoelectric Coupling

The transformation of waste vibration energy into low-power electricity has been heavily researched to enable self-sustained wireless electronic components. Monostable and bistable nonlinear oscillators have been explored by several researchers in an effort to enhance the frequency bandwidth of operation. Linear two degree of freedom (2-DOF) configurations as well as combination of a nonlinear single-DOF harvester with a linear oscillator to constitute a nonlinear 2-DOF harvester have also been explored to develop broadband energy harvesters. In the present work, the concept of nonlinear internal resonance in a continuous frame structure is explored for broadband energy harvesting. The L-shaped beam-mass structure with quadratic nonlinearity was formerly studied in the nonlinear dynamics literature to demonstrate modal energy exchange and the saturation phenomenon when carefully tuned for two-to-one internal resonance. In the current effort, piezoelectric coupling is introduced, and electromechanical equations of the L-shaped energy harvester are employed to explore the primary resonance behaviors around the first and the second linear natural frequencies for bandwidth enhancement. Simulations using approximate analytical frequency response equations as well as time-domain numerical solutions reveal that 2-DOF configuration with quadratic and two-to-one internal resonance could extend the bandwidth enhancement capability. Both electrical power and shunted vibration frequency response curves of steady-state solutions are explored in detail. Effects of various electromechanical system parameters, such as piezoelectric coupling and load resistance, on the overall dynamics of the internal resonance energy harvesting system are reported.

Effects of nonlinear interactions of flexural modes on the performance of a beam autoparametric vibration absorber

2019 ◽  
Vol 26 (7-8) ◽  
pp. 459-474
Author(s):  
Saeed Mahmoudkhani ◽  
Hodjat Soleymani Meymand
Keyword(s):  
Frequency Response ◽  
Internal Resonance ◽  
Continuation Method ◽  
Harmonic Balance Method ◽  
Vibration Absorber ◽  
Primary System ◽  
Lumped Mass ◽  
Response Curves ◽  
Internal Resonances ◽  
The One

The performance of the cantilever beam autoparametric vibration absorber with a lumped mass attached at an arbitrary point on the beam span is investigated. The absorber would have a distinct feature that in addition to the two-to-one internal resonance, the one-to-three and one-to-five internal resonances would also occur between flexural modes of the beam by tuning the mass and position of the lumped mass. Special attention is paid on studying the effect of these resonances on increasing the effectiveness and extending the range of excitation amplitudes at which the autoparametric vibration absorber remains effective. The problem is formulated based on the third-order nonlinear Euler–Bernoulli beam theory, where the assumed-mode method is used for deriving the discretized equations of motion. The numerical continuation method is then applied to obtain the frequency response curves and detect the bifurcation points. The harmonic balance method is also employed for detecting the type of internal resonances between flexural modes by inspecting the frequency response curves corresponding to different harmonics of the response. Parametric studies on the performance of the absorber are conducted by varying the position and mass of the lumped mass, while the frequency ratio of the primary system to the first mode of the beam is kept equal to two. Results indicated that the one-to-five internal resonance is especially responsible for the considerable enhancement of the performance.

scholarly journals Performance Analysis of an Electromagnetically Coupled Piezoelectric Energy Scavenger

Energies ◽  
2020 ◽  
Vol 13 (4) ◽  
pp. 845 ◽  
Author(s):  
Abdolreza Pasharavesh ◽  
Reza Moheimani ◽  
Hamid Dalir
Keyword(s):  
Energy Harvesting ◽  
Multiple Scales ◽  
Nonlinear Partial Differential Equations ◽  
Low Frequency ◽  
Primary Resonance ◽  
Excitation Amplitude ◽  
Load Resistance ◽  
Response Curves ◽  
Piezoelectric Energy ◽  
Resonance Frequencies

The deliberate introduction of nonlinearities is widely used as an effective technique for the bandwidth broadening of conventional linear energy harvesting devices. This approach not only results in a more uniform behavior of the output power within a wider frequency band through bending the resonance response, but also contributes to energy harvesting from low-frequency excitations by activation of superharmonic resonances. This article investigates the nonlinear dynamics of a monostable piezoelectric harvester under a self-powered electromagnetic actuation. To this end, the governing nonlinear partial differential equations of the proposed harvester are order-reduced and solved by means of the perturbation method of multiple scales. The results indicate that, according to the excitation amplitude and load resistance, different responses can be distinguished at the primary resonance. The system behavior may involve the traditional bending of response curves, Hopf bifurcations, and instability regions. Furthermore, an order-two superharmonic resonance is observed, which is activated at lower excitations in comparison to order-three conventional resonances of the Duffing-type resonator. This secondary resonance makes it possible to extract considerable amounts of power at fractions of natural frequency, which is very beneficial in micro-electro-mechanical systems (MEMS)-based harvesters with generally high resonance frequencies. The extracted power in both primary and superharmonic resonances are analytically calculated, then verified by a numerical solution where a good agreement is observed between the results.

Nonlinear Finite Element-Based Path Following of Periodic Solutions

2011 ◽  
Author(s):  
Andrea Arena ◽  
Giovanni Formica ◽  
Walter Lacarbonara ◽  
Harry Dankowicz
Keyword(s):  
Finite Element ◽  
Vector Field ◽  
Frequency Response ◽  
Periodic Solutions ◽  
Parametric Resonance ◽  
Primary Resonance ◽  
Path Following ◽  
State Variables ◽  
Response Curves ◽  
Space Dependence

A computational framework is proposed to path follow the periodic solutions of nonlinear spatially continuous systems and more general coupled multiphysics problems represented by systems of partial differential equations with time-dependent excitations. The set of PDEs is cast in first order differential form (in time) u˙ = f(u,s,t;c) where u(s,t) is the vector collecting all state variables including the velocities/time rates, s is a space coordinate (here, one-dimensional systems are considered without lack of generality for the space dependence) and t denotes time. The vector field f depends, in general, not only on the classical state variables (such as positions and velocities) but also on the space gradients of the leading unknowns. The space gradients are introduced as part of the state variables. This is justified by the mathematical and computational requirements on the continuity in space up to the proper differential order of the space gradients associated with the unknown position vector field. The path following procedure employs, for the computation of the periodic solutions, only the evaluation of the vector field f. This part of the path following procedure within the proposed combined scheme was formerly implemented by Dankowicz and coworkers in a MATLAB software package called COCO. The here proposed procedure seeks to discretize the space dependence of the variables using finite elements based on Lagrangian polynomials which leads to a discrete form of the vector field f. A concurrent bifurcation analysis is carried out by calculating the eigenvalues of the monodromy matrix. A hinged-hinged nonlinear beam subject to a primary-resonance harmonic transverse load or to a parametric-resonance horizontal end displacement is considered as a case study. Some primary-resonance frequency-response curves are calculated along with their stability to assess the convergence of the discretization scheme. The frequency-response curves are shown to be in close agreement with those calculated by direct integration of the PDEs through the FE software called COMSOL Multiphysics. Besides primary-resonance direct forcing conditions, also parametric forcing causing the principal parametric resonance of the lowest two bending modes is considered through construction of the associated transition curves. The proposed approach integrates algorithms from the finite element and bifurcation domains thus enabling an accurate and effective unfolding of the bifurcation and post-bifurcation scenarios of nonautonomous PDEs with the underlying structures.

scholarly journals Using passive control by a pendulum in a portal frame platform with piezoelectric energy harvesting

2017 ◽  
Vol 24 (16) ◽  
pp. 3684-3697 ◽  
Author(s):  
Rodrigo T Rocha ◽  
Jose M Balthazar ◽  
Angelo M Tusset ◽  
Vinicius Piccirillo
Keyword(s):  
Energy Harvesting ◽  
Periodic Orbit ◽  
Internal Resonance ◽  
Passive Control ◽  
Dynamical Behavior ◽  
Frame Structure ◽  
Nonlinear Dynamical ◽  
Piezoelectric Energy ◽  
Portal Frame ◽  
Portal Frame Structure

This work presents a passive control strategy using a pendulum on a simple portal frame structure, with two-to-one internal resonance, with a piezoelectric material coupling as a means of energy harvesting. In addition, the system is externally base-excited by an electro-dynamical shaker with harmonic output. Due to internal resonance the system may present the phenomenon of saturation, which provides some nonlinear dynamical behavior to the system. A pendulum is coupled to control nonlinear behaviors, leading to a periodic orbit, which is necessary to maintain energy harvesting. The results show that the system presents, most of the time, as being quasiperiodic. However, it does not present as being chaotic. With the pendulum, it was possible to control most of these quasiperiodic behaviors, leading to a periodic orbit. Moreover, it is possible to eliminate the need for an active or semi-active control, which are usually more complex. In addition, the control provides a way to detune the energy captured to the desired operating frequency.

Nonlinear Response of Short Squeeze Film Dampers

1980 ◽  
Vol 102 (1) ◽  
pp. 51-58 ◽  
Author(s):  
D. L. Taylor ◽  
B. R. K. Kumar
Keyword(s):  
Frequency Response ◽  
Phase Angle ◽  
Transient Response ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Polar Coordinates ◽  
Squeeze Film ◽  
Response Curves ◽  
Squeeze Film Damper ◽  
Fluid Forces

This paper considers the methodology of numerical integration for prediction of dynamic response of squeeze film damper systems. A planar rotor carried in a squeeze film damper with linear centering spring is considered. Governing differential equations are expressed in polar coordinates, and fluid forces are obtained from the Ocvirk short bearing integrals. The rotating unbalance response is presented as a function of speed, unbalance, and a bearing parameter. Runge Kutta integration techniques are used to obtain numerical solutions for transient response and frequency response. The 2π film approximation results in almost linear frequency response curves. However, the π film response is very nonlinear, demonstrating the well known multiple valued response and associated hardening jump/drop phenomenon. The π film transient response is analyzed within the speed range of bistable operation to determine the effects of initial conditions, the domains of convergence, and the relative strengths of stability of each solution. The transient response is found to be most sensitive to initial values of phase angle and phase angle velocity. Initial eccentricity and eccentric velocity are much less important. In general, of the two steady state solutions, the one with lower eccentricity appears to be more stable, with a larger domain of convergence. Examples show how premature termination of the integration can lead to erroneous conclusions.

Vibration Control of Horizontally Excited Structures Utilizing Internal Resonance of Liquid Sloshing in Nearly Square Tanks

2017 ◽  
Vol 139 (4) ◽  
Author(s):  
Takashi Ikeda ◽  
Yuji Harata
Keyword(s):  
Frequency Response ◽  
Internal Resonance ◽  
Passive Control ◽  
Harmonic Excitation ◽  
Liquid Sloshing ◽  
Maximum Efficiency ◽  
System Response ◽  
Response Curves ◽  
System Parameters ◽  
Theoretical Results

Passive control of vibrations in an elastic structure subjected to horizontal, harmonic excitation by utilizing a nearly square liquid tank is investigated. When the natural frequency ratio 1:1:1 is satisfied among the natural frequencies of the structure and the two predominant sloshing modes (1,0) and (0,1), the performance of a nearly square tank as a tuned liquid damper (TLD) is expected to be superior to rectangular TLDs due to internal resonance. In the theoretical analysis, Galerkin's method is used to determine the modal equations of motion for liquid sloshing considering the nonlinearity of sloshing. Then, van der Pol's method is used to obtain the expressions for the frequency response curves for the structure and sloshing modes. Frequency response curves and bifurcation set diagrams are shown to investigate the influences of the aspect ratio of the tank cross section and the tank installation angle on the system response. From the theoretical results, the optimal values of the system parameters can be determined in order to achieve maximum efficiency of vibration suppression for the structure. Hopf bifurcations occur and amplitude modulated motions (AMMs) may appear depending on the values of the system parameters. Experiments were also conducted, and the theoretical results agreed well with the experimental data.

scholarly journals Influence of the Motion of a Spring Pendulum on Energy-Harvesting Devices

2021 ◽  
Vol 11 (18) ◽  
pp. 8658
Author(s):  
Mohamed K. Abohamer ◽  
Jan Awrejcewicz ◽  
Roman Starosta ◽  
Tarek S. Amer ◽  
Mohamed A. Bek
Keyword(s):  
Energy Harvesting ◽  
Power Supply ◽  
Numerical Solutions ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Mechanical Vibration ◽  
Real Life ◽  
Nonlinear Stability Analysis ◽  
Response Curves ◽  
Stability And Instability

Energy harvesting is becoming more and more essential in the mechanical vibration application of many devices. Appropriate devices can convert the vibrations into electrical energy, which can be used as a power supply instead of ordinary ones. This study investigated a dynamical system that correlates with two devices, namely a piezoelectric device and an electromagnetic one, to produce two novel models. These devices are connected to a nonlinear damping spring pendulum with two degrees of freedom. The damping spring pendulum is supported by a point moving in a circular orbit. Lagrange’s equations of the second kind were utilized to obtain the equations of motion. The asymptotic solutions of these equations were acquired up to the third approximation using the approach of multiple scales. The comparison between the approximate and the numerical solutions reveals high consistency between them. The steady-state solutions were investigated, and their stabilities were checked. The influences of excitation amplitudes, damping coefficients, and the different frequencies on energy-harvesting device outputs are examined and discussed. Finally, the nonlinear stability analysis of the modulation equations is discussed through the stability and instability ranges of the frequency response curves. The work is significant due to its real-life applications, such as a power supply of sensors, charging electronic devices, and medical applications.

Subharmonic Excitation for Electrically Actuated Microbeams

2005 ◽  
Author(s):  
Ali H. Nayfeh ◽  
Mohammad I. Younis
Keyword(s):  
Frequency Response ◽  
Rf Mems ◽  
Electrostatic Force ◽  
Primary Resonance ◽  
Resonance Excitation ◽  
Global Dynamics ◽  
Response Curves ◽  
Electrically Actuated ◽  
Subharmonic Excitation ◽  
Order Of Magnitude

We present analysis of the global dynamics of electrically actuated microbeams under subharmonic excitation. The microbeams are excited by a DC electrostatic force and an AC harmonic force with a frequency tuned near twice their fundamental natural frequencies. We show that the dynamic pull-in instability can occur in this case for an electric load much lower than that predicted with static analysis and the same order-of-magnitude as that predicted in the case of primary-resonance excitation. We show that, once the subharmonic resonance is activated, all frequency-response curves reach pull-in, regardless of the magnitude of the AC forcing. Our results show a limited influence of the quality factor on the frequency response. This result and the fact that the frequency-response curves have very steep passband-to-stopband transitions make the combination of a DC voltage and a subhormonic of order one-half a promising candidate for designing improved high-sensitive RF MEMS filters.

Enhanced Energy Harvesting of a Nonlinear Energy Sink by Internal Resonance

2019 ◽  
Vol 11 (10) ◽  
pp. 1950100 ◽  
Author(s):  
Shuai Hou ◽  
Ying-Yuan Teng ◽  
Ye-Wei Zhang ◽  
Jian Zang
Keyword(s):  
Energy Harvesting ◽  
Internal Resonance ◽  
Numerical Solutions ◽  
Nonlinear Energy Sink ◽  
Vibration Energy ◽  
Resonance System ◽  
Vibration Energy Harvesting ◽  
Energy Sink ◽  
The Relationship ◽  
Nonlinear Energy

Given its essential nonlinearity, nonlinear energy sink (NES) has been extensively studied as a promising vibration energy harvesting device. Internal resonance, which is due to strong energy exchange between modes, also provides a valuable idea for vibration energy harvesting. Combining these two advantages, we put forward a 3:1 internal resonance system, which consists of an NES and a coupled linear oscillator, as an enhanced method for vibration energy harvesting. The multiscale method is applied to derive the relationship between amplitude and frequency response. Simulations are carried out to evaluate the performance of the proposed method. Results show that the internal resonance system can remarkably improve the vibration energy harvesting performance. The numerical solutions verify the accuracy of the analytical solutions. The results demonstrate that the internal resonance system with NES for energy harvesting has better output power and bandwidth compared with noninternal resonance system. Overall, the comprehensive design improves the performance of NES for vibration energy harvesting.

Internal Resonance Energy Harvesting

2015 ◽  
Vol 82 (3) ◽  
Author(s):  
Li-Qun Chen ◽  
Wen-An Jiang
Keyword(s):  
Energy Harvesting ◽  
White Noise ◽  
Frequency Response ◽  
Internal Resonance ◽  
Energy Harvester ◽  
Multiple Scales ◽  
Gaussian White Noise ◽  
Resonance Energy ◽  
Correlated Noise ◽  
Amplitude Frequency Response

Internal resonance is explored as a possible mechanism to enhance vibration-based energy harvesting. An electromagnetic device with snap-through nonlinearity is proposed as an archetype of an internal resonance energy harvester. Based on the equations governing the vibration measured from a stable equilibrium position, the method of multiple scales is applied to derive the amplitude–frequency response relationships of the displacement and the power in the first primary resonances with the two-to-one internal resonance. The amplitude–frequency response curves have two peaks bending to the left and the right, respectively. The numerical simulations support the analytical results. Then the averaged power is calculated under the Gaussian white noise, the narrow-band noise, the colored noise defined by a second-order filter, and the exponentially correlated noise. The results demonstrate numerically that the internal resonance design produces more power than other designs under the Gaussian white noise and the exponentially correlated noise. Besides, the internal resonance energy harvester can outperform the linear energy harvesters with the same natural frequencies and in the same size under Gaussian white noise.

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