{"title":"Coupled-mode subharmonic resonance of a piecewise-linear gear transmission system with 1:2 internal resonance","authors":"Y.L. Li , J.L. Huang , F.L. Liao , W.D. Zhu","doi":"10.1016/j.ymssp.2025.113383","DOIUrl":null,"url":null,"abstract":"<div><div>The coupled-mode subharmonic resonance of a piecewise-linear gear transmission system with 1:2 internal resonance is investigated. When the system is excited at a frequency close to the sum of its first and third natural frequencies, both modes are simultaneously activated, giving rise to a period-3 coupled-mode response. An efficient and robust incremental harmonic balance (ER-IHB) method with significantly improved computational efficiency and convergence is developed for computing periodic responses of the piecewise-linear gear transmission system, and its two-time-scale variant, termed the two-time-scale efficient and robust incremental harmonic balance (TER-IHB) method, is formulated for quasi-periodic (QP) responses. Fast Fourier transform (FFT) and two-dimensional FFT are employed to efficiently evaluate the residual vectors and Jacobian matrices for periodic and QP solutions, respectively, and the Levenberg–Marquardt algorithm is utilized to enhance the convergence. Additionally, a path-following continuation technique is integrated to calculate response curves. The amplitude–frequency response curves of the gear transmission system are traced using the proposed ER-IHB and TER-IHB methods. The proposed methods are approximately two and four orders of magnitude more efficient than the conventional IHB method for periodic and QP solutions, respectively, and their accuracy is verified against the fourth-order Runge–Kutta (RK) method. Floquet and extended Floquet theories are employed to assess the stability of periodic and QP solutions, respectively. Multiple saddle–node and secondary Hopf bifurcations are identified, leading to transitions among periodic (period-1, -3, -15, and -63), QP, and chaotic responses.</div></div>","PeriodicalId":51124,"journal":{"name":"Mechanical Systems and Signal Processing","volume":"240 ","pages":"Article 113383"},"PeriodicalIF":8.9000,"publicationDate":"2025-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mechanical Systems and Signal Processing","FirstCategoryId":"5","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0888327025010842","RegionNum":1,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"ENGINEERING, MECHANICAL","Score":null,"Total":0}
引用次数: 0
Abstract
The coupled-mode subharmonic resonance of a piecewise-linear gear transmission system with 1:2 internal resonance is investigated. When the system is excited at a frequency close to the sum of its first and third natural frequencies, both modes are simultaneously activated, giving rise to a period-3 coupled-mode response. An efficient and robust incremental harmonic balance (ER-IHB) method with significantly improved computational efficiency and convergence is developed for computing periodic responses of the piecewise-linear gear transmission system, and its two-time-scale variant, termed the two-time-scale efficient and robust incremental harmonic balance (TER-IHB) method, is formulated for quasi-periodic (QP) responses. Fast Fourier transform (FFT) and two-dimensional FFT are employed to efficiently evaluate the residual vectors and Jacobian matrices for periodic and QP solutions, respectively, and the Levenberg–Marquardt algorithm is utilized to enhance the convergence. Additionally, a path-following continuation technique is integrated to calculate response curves. The amplitude–frequency response curves of the gear transmission system are traced using the proposed ER-IHB and TER-IHB methods. The proposed methods are approximately two and four orders of magnitude more efficient than the conventional IHB method for periodic and QP solutions, respectively, and their accuracy is verified against the fourth-order Runge–Kutta (RK) method. Floquet and extended Floquet theories are employed to assess the stability of periodic and QP solutions, respectively. Multiple saddle–node and secondary Hopf bifurcations are identified, leading to transitions among periodic (period-1, -3, -15, and -63), QP, and chaotic responses.
期刊介绍:
Journal Name: Mechanical Systems and Signal Processing (MSSP)
Interdisciplinary Focus:
Mechanical, Aerospace, and Civil Engineering
Purpose:Reporting scientific advancements of the highest quality
Arising from new techniques in sensing, instrumentation, signal processing, modelling, and control of dynamic systems