Nonlinear energy harvester with coupled Duffing oscillators

Abstract Structural vibrations are very common in aerospace and mechanical engineering systems, where dynamic analysis of modern aerospace structures and industrial machines has become an indispensable step in their design. Suppression of unwanted vibrations and their exploitation for energy harvesting at the same time would be the most desirable scenario. The dynamical system presented in this communication is based on a discrete model of energy harvesting device realized in such a manner as to achieve both vibration suppression and harvesting of vibration energy by introducing the nonlinear energy sink concept. The mechanical model is formed as a two-degree of freedom nonlinear oscillator with an oscillating magnet and harmonic base excitation. The corresponding mathematical model is based on the system of nonlinear nonhomogeneous Duffing type differential equations. To explore complex dynamical behaviour of the presented model, periodic solutions and their bifurcations are found by using the incremental harmonic balance (IHB) and continuation methods. For the detection of unstable periodic orbits, the Floquet theory is applied and an interesting harmonic response of the presented nonlinear dynamical model is detected. The main advantage of the presented approach is its ability to obtain approximated periodic responses in terms of Fourier series and estimate the voltage output of an energy harvester for a system with strong nonlinearity. The accuracy of the presented methodology is verified by comparing the results obtained in this work with those obtained by a standard numerical integration method and results from the literature. Numerical examples show the effects of different physical parameters on amplitude-frequency, response amplitude - base amplitude and time response curves, where a qualitative change is explored and studied in detail. Presented theoretical results demonstrate that the proposed system has advanced performance in both system requirements - vibration suppression, and energy harvesting.

[1]  S. E. Khadem,et al.  Nonlinear vibration control and energy harvesting of a beam using a nonlinear energy sink and a piezoelectric device , 2014 .

[2]  Alexey S. Matveev,et al.  Stability Analysis via Averaging Functions , 2016, IEEE Transactions on Automatic Control.

[3]  Ali H. Nayfeh,et al.  Effectiveness of a nonlinear energy sink in the control of an aeroelastic system , 2016 .

[4]  A. Nayfeh,et al.  Applied nonlinear dynamics : analytical, computational, and experimental methods , 1995 .

[5]  Alexander Yu. Pogromsky,et al.  Frequency Domain Performance Analysis of Lur'e Systems , 2014, IEEE Transactions on Control Systems Technology.

[6]  R. Seydel.  Practical Bifurcation and Stability Analysis , 1994 .

[7]  G. Kerschen,et al.  Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems , 2008 .

[8]  Xiang Li,et al.  Integration of a nonlinear energy sink and a giant magnetostrictive energy harvester , 2017 .

[9]  K. Y. Sze,et al.  Bifurcation and route-to-chaos analyses for Mathieu–Duffing oscillator by the incremental harmonic balance method , 2008 .

[10]  B. Wen,et al.  Nonlinear analysis of a parametrically excited beam with intermediate support by using Multi-dimensional incremental harmonic balance method , 2016 .

[11]  Saad A. Ragab,et al.  Passive control of transonic flutter with a nonlinear energy sink , 2017 .

[12]  Shaopu Yang,et al.  Nonlinear dynamics of a spur gear pair with time-varying stiffness and backlash based on incremental harmonic balance method , 2006 .

[13]  Ray Kai Leung Su,et al.  Nonlinear vibration of a curved beam under uniform base harmonic excitation with quadratic and cubic nonlinearities , 2011 .

[14]  Alexander F. Vakakis,et al.  Enhancing the Robustness of Aeroelastic Instability Suppression Using Multi-Degree-of-Freedom Nonlinear Energy Sinks , 2008 .

[15]  Kefu Liu,et al.  A nonlinear energy sink with an energy harvester: Harmonically forced responses , 2017 .

[16]  Oleg Gendelman,et al.  Bifurcations of Nonlinear Normal Modes of Linear Oscillator with Strongly Nonlinear Damped Attachment , 2004 .

[17]  Weidong Zhu,et al.  Nonlinear Dynamics of a High-Dimensional Model of a Rotating Euler–Bernoulli Beam Under the Gravity Load , 2014 .

[18]  Alexander F. Vakakis,et al.  Inducing Passive Nonlinear Energy Sinks in Vibrating Systems , 2001 .

[19]  Daniel J. Inman,et al.  Piezoelectric energy harvesting from broadband random vibrations , 2009 .

[20]  Daniel J. Inman,et al.  Piezoelectric Energy Harvesting , 2011 .

[21]  Li-Qun Chen,et al.  Complexification-averaging analysis on a giant magnetostrictive harvester integrated with a nonlinear energy sink , 2018 .

[22]  Kefu Liu,et al.  A nonlinear energy sink with an energy harvester: Transient responses , 2014 .

[23]  Ray Kai Leung Su,et al.  Precise Hsu's method for analyzing the stability of periodic solutions of multi-degrees-of-freedom systems with cubic nonlinearity , 2009 .

[24]  Alexander F. Vakakis,et al.  Vortex-induced vibration of a linearly sprung cylinder with an internal rotational nonlinear energy sink in turbulent flow , 2020, Nonlinear Dynamics.

[25]  Xianren Kong,et al.  Dynamics of 1-dof and 2-dof energy sink with geometrically nonlinear damping: application to vibration suppression , 2017 .

[26]  Xilin Lu,et al.  Nonlinear dissipative devices in structural vibration control: A review , 2018, Journal of Sound and Vibration.

[27]  Anil K. Bajaj,et al.  Periodic response predictions of beams on nonlinear and viscoelastic unilateral foundations using incremental harmonic balance method , 2016 .

[28]  Oleg Gendelman,et al.  Response regimes of linear oscillator coupled to nonlinear energy sink with harmonic forcing and frequency detuning , 2008 .

[29]  Ali H. Nayfeh,et al.  Effects of a non-linear energy sink (NES) on vortex-induced vibrations of a circular cylinder , 2014 .

[30]  Kefu Liu,et al.  Broadband piezoelectric vibration energy harvesting using a nonlinear energy sink , 2018 .

[31]  Lin Hua,et al.  Nonlinear vibrations of a piecewise-linear quarter-car truck model by incremental harmonic balance method , 2018 .

[32]  Willy Govaerts,et al.  MATCONT: A MATLAB package for numerical bifurcation analysis of ODEs , 2003, TOMS.

[33]  Kefu Liu,et al.  Piezoelectric energy harvesting with a nonlinear energy sink , 2015, Smart Structures.

[34]  A. Vakakis.  Passive nonlinear targeted energy transfer , 2018, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[35]  Yi Huang,et al.  Nonlinear dynamics of heave motion of the sandglass-type floating body with piecewise-nonlinear, time-varying stiffness , 2018, Marine Structures.

[36]  Ali H. Nayfeh,et al.  Steady-State dynamics of a linear structure weakly coupled to an essentially nonlinear oscillator , 2006 .

[37]  Alexander F. Vakakis,et al.  Steady State Passive Nonlinear Energy Pumping in Coupled Oscillators: Theoretical and Experimental Results , 2003 .

[38]  Grzegorz Litak,et al.  Magnetopiezoelastic energy harvesting driven by random excitations , 2010 .

[39]  O. Gendelman.  Transition of Energy to a Nonlinear Localized Mode in a Highly Asymmetric System of Two Oscillators , 2001 .

[40]  A. Leung,et al.  Non-linear vibration of coupled duffing oscillators by an improved incremental harmonic balance method , 1995 .

[41]  Lin Hua,et al.  Applications of incremental harmonic balance method combined with equivalent piecewise linearization on vibrations of nonlinear stiffness systems , 2019, Journal of Sound and Vibration.