Free vibration analysis of a finite bar with an external spring

Abstract

In this paper, we study the free vibration of a finite bar with an external spring. Several methods, including the method of separation variables, the finite element method (FEM) and the approach of diamond rule, are employed to solve this problem. Two analytical solutions are derived by using the approach of diamond rule and the method of separation variables. A numerical solution by using the FEM is also provided for comparison. Advantages and disadvantages of each method are addressed. It is found that the approach of diamond rule can satisfy the causality condition, while the other two disobey. Besides, the Gibbs phenomenon occurs in the two series-based solutions obtained by the method of separation variables and mode superposition using the FEM. 本文使用分離變數法、有限元素法與鑽石法則來求解有限桿加邊彈簧的自由振動經典問題。文中分別以鑽石法則以及分離變數法推導出解析解，其結果與有限元素法的數值結果做比較，並討論各方法間的優劣性。我們發現鑽石法則滿足因果律，其它兩方法會違背此規律；此外，由於使用有限元素法中的模態疊加與分離變數法，可以從結果觀察到吉布斯現象。