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.
- Dynamic Systems and Control Division
On Steady-State Solutions of a Wave Equation by Solving a Delay Differential Equation With an Incremental Harmonic Balance Method
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Wang, X, Liu, M, & Zhu, W. "On Steady-State Solutions of a Wave Equation by Solving a Delay Differential Equation With an Incremental Harmonic Balance Method." Proceedings of the ASME 2018 Dynamic Systems and Control Conference. Volume 3: Modeling and Validation; Multi-Agent and Networked Systems; Path Planning and Motion Control; Tracking Control Systems; Unmanned Aerial Vehicles (UAVs) and Application; Unmanned Ground and Aerial Vehicles; Vibration in Mechanical Systems; Vibrations and Control of Systems; Vibrations: Modeling, Analysis, and Control. Atlanta, Georgia, USA. September 30–October 3, 2018. V003T39A002. ASME. https://doi.org/10.1115/DSCC2018-8933
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