Title: Exact Analytical Periodic Solutions in Special Cases and Numerical Analysis of a Half-Undamped 1-DOF Piecewise-Linear Vibratory System
Abstract: A lot of realistic systems are better described by piecewise-linear models than by continuous models, including systems with dry friction or impact as well as some controlled systems. Such models are usually strongly nonlinear and do not have general exact analytical solutions. This paper analyzes a harmonically excited 1-DOF piecewise-linear vibratory system that has a damped domain and an undamped domain. The purposes are to find some exact analytical solutions of the system and find the connection between the analytical solutions and the numerical ones. Though the system is linear and can be analytically analyzed in each domain, no general closed-form solution for its motion has been found because the switching times between the two domains are solutions of transcendental equations. To solve the problem, a special method is proposed: the initial conditions and the parameters are adjusted so that the homogeneous solution in the damped domain is cancelled, and the excitation is subharmonic in the undamped domain. As a result, the transcendental equations are simplified and exact analytical expressions for periodic motions are found for special cases of the parameters. The obtained motions are single-penetration multi-periodic, which is a typical behavior of the considered bilinear system as seen in the bifurcation diagram. Thus, the proposed method may be used to investigate dynamic behaviors of more complicated piecewise-linear systems.
Publication Year: 2021
Publication Date: 2021-12-14
Language: en
Type: book-chapter
Indexed In: ['crossref']
Access and Citation
AI Researcher Chatbot
Get quick answers to your questions about the article from our AI researcher chatbot