Similar articles

Abstract: This is a supplementary study of the solution method previously proposed by the author. The proposed method is used for solving three-coupled integrable dispersionless equations with disturbance terms. It has only been adopted for solving (1) a nonlinear beam problem, and (2) a nonlinear vibro-acoustic problem. In the solution process, the three-coupled nonlinear equations can be transformed into only one Duffing equation. The higher-level nonlinear solutions, which were ignored in the previous method, can be generated using the proposed approach. Hence, in each step in the solution, only one independent nonlinear algebraic equation need be solved. As in the previous method, the proposed method has the advantage that the periodic solutions are represented by Fourier functions rather than the tedious implicit functions. The solutions from the proposed method agree reasonably well with those obtained from the classical harmonic balance method.

This is a continuation of the author's study on the solutions of nonlinear coupled integrable dispersionless equations. The effect of disturbance terms is taken into consideration. Also, the proposed method can lead to higher-order nonlinear solution that is ignored in the old method.

一种针对有扰动项的耦合可积非色散方程的修正残差谐波平衡求解方法

[1]Guner O, Bekir A, 2018. Solving nonlinear space-time fractional differential equations via ansatz method. Computational Methods for Differential Equations, 6(1):1-11.

[2]Hasan ASMZ, Lee YY, Leung AYT, 2016. The multi-level residue harmonic balance solutions of multi-mode nonlinearly vibrating beams on an elastic foundation. Journal of Vibration and Control, 22(14):3218-3235.

[3]Hashemi MS, Akgul A, 2018. Solitary wave solutions of time-space nonlinear fractional Schrödinger’s equation: two analytical approaches. Journal of Computational and Applied Mathematics, 339:147-160.

[4]Hu YF, Wang B, 2008. Solution of two-dimensional scattering problem in piezoelectric/piezomagnetic media using a polarization method. Applied Mathematics and Mechanics-English Edition, 29(12):1535-1552.

[5]Huang JB, Xiao ZX, Liu J, et al., 2012. Simulation of shock wave buffet and its suppression on an OAT15A supercritical airfoil by IDDES. Science China Physics, Mechanics and Astronomy, 55(2):260-271.

[6]Huang JL, Zhu WD, 2017. An incremental harmonic balance method with two timescales for quasiperiodic motion of nonlinear systems whose spectrum contains uniformly spaced sideband frequencies. Nonlinear Dynamics, 90(2):1015-1033.

[7]Huang JL, Chen SH, Su RKL, et al., 2011. Nonlinear analysis of forced responses of an axially moving beam by incremental harmonic balance method. Mechanics of Advanced Materials and Structures, 18(8):611-616.

[8]Huang JL, Su KLR, Lee YYR, et al., 2018. Various bifurcation phenomena in a nonlinear curved beam subjected to base harmonic excitation. International Journal of Bifurcation and Chaos, 28(7):1830023.

[9]Lee YY, 2002. Structural-acoustic coupling effect on the nonlinear natural frequency of a rectangular box with one flexible plate. Applied Acoustics, 63(11):1157-1175.

[10]Lee YY, 2017. Large amplitude free vibration of a flexible panel coupled with a leaking cavity. Journal of Zhejiang University-SCIENCE A (Applied Physics & Engineering), 18(1):75-82.

[11]Lee YY, 2018. Nonlinear structure-extended cavity interaction simulation using a new version of harmonic balance method. PLoS One, 13(7):e0199159.

[12]Leung AYT, Yang HX, Guo ZJ, 2012. Periodic wave solutions of coupled integrable dispersionless equations by residue harmonic balance. Communications in Nonlinear Science and Numerical Simulation, 17(11):4508-4514.

[13]Li W, Yang Y, Sheng DR, et al., 2011. Nonlinear dynamic analysis of a rotor/bearing/seal system. Journal of Zhejiang University-SCIENCE A (Applied Physics & Engineering), 12(1):46-55.

[14]Lu QQ, Shao W, Wu YF, et al., 2018. Vibration analysis of an axially moving plate based on sound time-frequency analysis. International Journal of Acoustics and Vibration, 23(2):226-233.

[15]Mohammadian M, Shariati M, 2017. Approximate analytical solutions to a conservative oscillator using global residue harmonic balance method. Chinese Journal of Physics, 55(1):47-58.

[16]Qian YH, Pan JL, Chen SP, et al., 2017. The spreading residue harmonic balance method for strongly nonlinear vibrations of a restrained cantilever beam. Advances in Mathematical Physics, 2017:5214616.

[17]Rahman MS, Lee YY, 2017. New modified multi-level residue harmonic balance method for solving nonlinearly vibrating double-beam problem. Journal of Sound and Vibration, 406:295-327.

[18]Tasbozan O, Şenol M, Kurt A, et al., 2018. New solutions of fractional Drinfeld-Sokolov-Wilson system in shallow water waves. Ocean Engineering, 161:62-68.

[19]Wang B, Guo JF, Feng JG, et al., 2016. Nonlinear dynamics and coupling effect of libration and vibration of tethered space robot in deorbiting process. Journal of Central South University, 23(5):1095-1105.

[20]Yin XL, Ma J, Wang XG, et al., 2012. Spin squeezing under non-Markovian channels by the hierarchy equation method. Physical Review A, 86(1):012308.

[21]Zhang CL, Wang XY, Chen WQ, et al., 2016. Propagation of extensional waves in a piezoelectric semiconductor rod. AIP Advances, 6(4):045301.

[22]Zheng XH, Zhang BF, Jiao ZX, et al., 2016. Tunable, continuous-wave single-resonant optical parametric oscillator with output coupling for resonant wave. Chinese Physics B, 25(1):014208.

[23]Zhong W, Ma J, Liu J, et al., 2014. Derivation of quantum Chernoff metric with perturbation expansion method. Chinese Physics B, 23(9):090305.

[24]Zhou WJ, Wei XS, Wei XZ, et al., 2014. Numerical analysis of a nonlinear double disc rotor-seal system. Journal of Zhejiang University-SCIENCE A (Applied Physics & Engineering), 15(1):39-52.

[25]Zhu ZQ, Jin XL, Hu SD, et al., 2013. The approximate response of real order nonlinear oscillator using Homotopy analysis method. Advances in Vibration Engineering, 12(2):123-134.