Support motion of a finite bar with an external spring

Abstract

In this paper, we gave the vibration analysis of a finite bar with an external spring on one side and the support motion on the other side. Two analytical methods, the mode superposition method in conjunction with the quasi-static decomposition method and the method of characteristics using the diamond rule, were employed to solve this problem. Both advantages and disadvantages of two methods were discussed. It is interesting to find that the mode superposition method can capture the silent area in terms of sum of an infinite series while the method of characteristics using the diamond rule can exactly derive the dead zone. Besides, it is found that discontinuities always occur at the location on the characteristic lines. Discussions of direct and inverse problems are also addressed.