Finite Difference Analysis of a Nonlinearly Moving String with Time-Dependent Length

含時變長度之非線性移動索的有限差分分析

陳狄成(Dyi-Cheng Chen) ； 陳本明(Beng-Ming Chen) ； 馮榮豐(Rong-Fong Fung)

《中華民國振動與噪音工程學會論文集》 1996 (1996/04) Pp. 13-24

https://doi.org/10.30028/CSSVANNUAL.199604.0013

摘要

本文探討含時變長度繩索的非線性振動分析，繩索一端掛有重量。系統方程式的導證和解法分別使用Hamilton原理、有限差分法和Rung-Kutta方法。文中，比較了系統的線性和非線性效應，末了，將軸向速度對繩索振幅的影響作一結論。
解決本題之技巧必須不同於傳統固定空間範圍的方法。當索長度變化時，自然頻率與時問有關連，而振動模態之獨立性便消失。因此，振動模態和頻率之概念便毫無意義。
本研究以有限差分方法發展一新的技巧來解決該時變問題，分析則分別以拋物線及正弦波之索軸向運動來進行。

關鍵字

Hamilton原理 ； Runge-Kutta方法 ； 有限差分法 ； 移動索

並列摘要

This paper presents the nonlinear vibration analysis of a string with time-dependent length and a weight at one end. The system equations are derived and solved using Hamilton's principle, and finite difference and the Runge-Kutta methods, respectively. The linear and nonlinear effects of the system are compared. As a result, the axial velocity effect on the amplitude of the string is summarized.
To solve the problem in question, technique has to be different from the classical methods utilized to solve the fixed spatial domain problems. For instance, when the length of the string varies, the natural frequencies are time-dependent and the independence of the natural modes of oscillation is lost. Therefore, the concepts of the natural modes and frequencies become meaningless.
In this study, a new technique of the finite difference method is developed to solve this time-dependent problem. Numerical simulations are conducted for the string having axially parabolic and sinusoidal motion.

並列關鍵字

Hamilton's principle ； Runge-Kutta method ； Finite difference method ； Moving string