Full Text:   <1122>

Suppl. Mater.: 

CLC number: 

On-line Access: 2019-04-02

Received: 2019-01-22

Revision Accepted: 2019-03-06

Crosschecked: 2019-03-19

Cited: 0

Clicked: 3592

Citations:  Bibtex RefMan EndNote GB/T7714


Yiu-yin Lee


-   Go to

Article info.
Open peer comments

Journal of Zhejiang University SCIENCE A 2019 Vol.20 No.4 P.300-304


Modified residue harmonic balance solution for coupled integrable dispersionless equations with disturbance terms

Author(s):  Yiu-yin Lee

Affiliation(s):  Department of Architecture and Civil Engineering, City University of Hong Kong, Kowloon Tong, Kowloon, Hong Kong, China

Corresponding email(s):   bcraylee@cityu.edu.hk

Key Words:  Coupled integrable dispersionless equations, Residue harmonic balance, Periodic solution

Yiu-yin Lee. Modified residue harmonic balance solution for coupled integrable dispersionless equations with disturbance terms[J]. Journal of Zhejiang University Science A, 2019, 20(4): 300-304.

@article{title="Modified residue harmonic balance solution for coupled integrable dispersionless equations with disturbance terms",
author="Yiu-yin Lee",
journal="Journal of Zhejiang University Science A",
publisher="Zhejiang University Press & Springer",

%0 Journal Article
%T Modified residue harmonic balance solution for coupled integrable dispersionless equations with disturbance terms
%A Yiu-yin Lee
%J Journal of Zhejiang University SCIENCE A
%V 20
%N 4
%P 300-304
%@ 1673-565X
%D 2019
%I Zhejiang University Press & Springer
%DOI 10.1631/jzus.A1900023

T1 - Modified residue harmonic balance solution for coupled integrable dispersionless equations with disturbance terms
A1 - Yiu-yin Lee
J0 - Journal of Zhejiang University Science A
VL - 20
IS - 4
SP - 300
EP - 304
%@ 1673-565X
Y1 - 2019
PB - Zhejiang University Press & Springer
ER -
DOI - 10.1631/jzus.A1900023

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.


创新点:1. 在取得每一阶段破解方案的过程中, 只需处理一条非线性代数方程式及一组线性代数方程式; 2. 能找出旧方法不能找出的非线性答案.
方法:1. 使用理论推导、方程式替换及残余谐波平衡方法; 2. 通过仿真模拟,推导震动位移与频率之间的关系(图1)以及位移与速度之间的关系 (图2).
结论:1. 成功将改进残余谐波平衡方法应用于有扰动项的耦合可积非色散方程; 2. 通过与其他方法产生的数据进行比较,验证了所提方法的可行性和有效性(表1-3).

关键词:大幅自主震动; 残余谐波平衡; 有扰动项的耦合可积非色散方程

Darkslateblue:Affiliate; Royal Blue:Author; Turquoise:Article


[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.

Open peer comments: Debate/Discuss/Question/Opinion


Please provide your name, email address and a comment

Journal of Zhejiang University-SCIENCE, 38 Zheda Road, Hangzhou 310027, China
Tel: +86-571-87952783; E-mail: cjzhang@zju.edu.cn
Copyright © 2000 - 2022 Journal of Zhejiang University-SCIENCE