A new numerical scheme with Wavelet–Galerkin followed by spectral deferred correction for solving string vibration problems

• Published in: Mechanism and machine theory 2019-12, Vol.142, p.103623, Article 103623
• Main Authors: Ma, Xiaolong, Wu, Bo, Zhang, Jiaohua, Shi, Xi
• Format: Article
• Language: English
• Subjects: Coupled; Nonlinear; SDC; String vibrations; Wavelet–Galerkin
• ISSN: 0094-114X; 1873-3999
• DOI: 10.1016/j.mechmachtheory.2019.103623

•The Wavelet–Galerkin method using spectral deferred correction is proposed.•The validity of the proposed method is demonstrated by comparing with experiment.•The proposed method has high accuracy and efficiency.•The proposed method can solve coupled nonlinear string vibrations problems. In this paper, the nonlinear governing equations for longitudinal and transverse vibrations of axially moving strings are presented first and a new numerical scheme with Wavelet–Galerkin method followed by spectral deferred correction (SDC) is developed to solve such governing equations. A string vibration problem with exact solution is used to demonstrate the accuracy and efficiency of the proposed method by comparing it to the traditional methods with a classic finite element method (FEM) followed by Runge-Kutta or SDC, Wavelet–Galerkin followed by Runge–Kutta. Moreover, the proposed method is applied in solving for the coupled nonlinear string vibration of the main drive chain system of escalator and the results show a good agreement with experimental and multi-body dynamic (MBD) simulation results, which demonstrates the validity of the proposed method for solving coupled nonlinear string vibration problems.