Study of wave propagation in strongly nonlinear periodic lattices using a harmonic balance approach

Wave Motion ◽  
2012 ◽  
Vol 49 (2) ◽  
pp. 394-410 ◽  
Author(s):  
Raj K. Narisetti ◽  
Massimo Ruzzene ◽  
Michael J. Leamy
Keyword(s):  
Wave Propagation ◽  
Harmonic Balance ◽  
Strongly Nonlinear ◽  
Balance Approach

scholarly journals Dynamics and Wave Propagation in Nonlinear Piezoelectric Metastructures

2021 ◽  
Author(s):  
Jaime Alberto Mosquera-Sánchez ◽  
Carlos De Marqui
Keyword(s):  
Wave Propagation ◽  
Vibration Control ◽  
Harmonic Balance ◽  
Electrical Circuit ◽  
Dispersion Relations ◽  
Dynamical Response ◽  
Negative Capacitance ◽  
Strongly Nonlinear ◽  
Time Domains ◽  
Nonlinear Regimes

Abstract This paper reports dynamical effects in onedimensional locally resonant piezoelectric metastructures that can be leveraged by nonlinear electrical attachments featuring either combined quadratic and quartic, or essentially quartic potentials. The nonlinear electromechanical unit cell is built upon a linear host oscillator coupled to a nonlinear electrical circuit via piezoelectricity. Its dynamical response to prescribed longitudinal harmonic displacements is approached in the frequency and time domains. Semi-analytical harmonic balance (HB)-based dispersion relations are derived to predict the location and edges of the nonlinear attenuation band. Numerical responses show that weakly and moderately nonlinear piezoelectric metastructures (NPMSs) promote a class of nonlinear attenuation band where a bandgap and a wave supratransmission band coexist, while also imparting nonlinear attenuation at the resonances around the underlying linear bandgap. Besides, strongly nonlinear regimes are shown to elicit broadband chaotic attenuation. Negative capacitance (NC)-based essentially cubic piezoelectric attachments are found to potentiate the aforementioned effects over a broader bandwidth. Excellent agreement is found between the predictions of the HB based dispersion relations and the nonlinear transmissibility functions of undamped and weakly damped NPMSs at weakly and moderately nonlinear regimes, even in the presence of NC circuits. This research is expected to pave the way towards fully tunable smart periodic metastructures for vibration control via nonlinear piezoelectric attachments.PACS 05.45.-a . 62.30.+d . 62.65.+kMathematics Subject Classification (2010) 37N15 . 74H45 . 74H65 . 74J30

On Steady-State Solutions of a Wave Equation by Solving a Delay Differential Equation With an Incremental Harmonic Balance Method

2018 ◽  
Author(s):  
Xuefeng Wang ◽  
Mao Liu ◽  
Weidong Zhu
Keyword(s):  
Differential Equation ◽  
Wave Propagation ◽  
Wave Equation ◽  
Steady State ◽  
Delay Differential Equation ◽  
Harmonic Balance ◽  
Incremental Harmonic Balance ◽  
Delay Differential ◽  
Ihb Method ◽  
Steady State Solutions

For wave propagation in periodic media with strong nonlinearity, steady-state solutions can be obtained by solving a corresponding nonlinear delay differential equation (DDE). Based on the periodicity, the steady-state response of a repeated particle or segment in the media contains the complete information of solutions for the wave equation. Considering the motion of the selected particle or segment as a variable, motions of its adjacent particles or segments can be described by the same variable function with different phases, which are delayed variables. Thus, the governing equation for wave propagation can be converted to a nonlinear DDE with multiple delays. A modified incremental harmonic balance (IHB) method is presented here to solve the nonlinear DDE by introducing a delay matrix operator, where a direct approach is used to efficiently and automatically construct the Jacobian matrix for the nonlinear residual in the IHB method. This method is presented by solving an example of a one-dimensional monatomic chain under a nonlinear Hertzian contact law. Results are well matched with those in previous work, while calculation time and derivation effort are significantly reduced. Also there is no additional derivation required to solve new wave systems with different governing equations.

scholarly journals The Spreading Residue Harmonic Balance Method for Strongly Nonlinear Vibrations of a Restrained Cantilever Beam

2017 ◽  
Vol 2017 ◽  
pp. 1-8 ◽  
Author(s):  
Y. H. Qian ◽  
J. L. Pan ◽  
S. P. Chen ◽  
M. H. Yao
Keyword(s):  
Exact Solutions ◽  
Nonlinear Vibration ◽  
Harmonic Balance ◽  
Nonlinear Dynamical Systems ◽  
Harmonic Balance Method ◽  
Time History ◽  
Approximate Solutions ◽  
Balance Method ◽  
Lumped Mass ◽  
Strongly Nonlinear

The exact solutions of the nonlinear vibration systems are extremely complicated to be received, so it is crucial to analyze their approximate solutions. This paper employs the spreading residue harmonic balance method (SRHBM) to derive analytical approximate solutions for the fifth-order nonlinear problem, which corresponds to the strongly nonlinear vibration of an elastically restrained beam with a lumped mass. When the SRHBM is used, the residual terms are added to improve the accuracy of approximate solutions. Illustrative examples are provided along with verifying the accuracy of the present method and are compared with the HAM solutions, the EBM solutions, and exact solutions in tables. At the same time, the phase diagrams and time history curves are drawn by the mathematical software. Through analysis and discussion, the results obtained here demonstrate that the SRHBM is an effective and robust technique for nonlinear dynamical systems. In addition, the SRHBM can be widely applied to a variety of nonlinear dynamic systems.

Harmonic Balance Approach for an Airfoil with a Freeplay Control Surface

AIAA Journal ◽  
2005 ◽  
Vol 43 (4) ◽  
pp. 802-815 ◽  
Author(s):  
Liping Liu ◽  
Earl H. Dowell
Keyword(s):  
Harmonic Balance ◽  
Control Surface ◽  
Balance Approach
Sign in / Sign up

Export Citation Format

Share Document