4.1 INTRODUCTION

When a steel bar is hit by a hammer, a clear sound can be heard because the steel bar vibrates at its resonant frequency. If the bar is oscillated at this resonant frequency, it will be found that the vibration amplitude of the bar becomes very large. Therefore, when a machine is designed, it is important to know the resonant frequency of the machine. The analysis to obtain the resonant frequency and the vibration mode of an elastic body is called "mode analysis."

It is said that there are two methods for mode analysis. One is the theoretical analysis and the other is the finite-element method (FEM). Theoretical analysis is usually used for a simple shape of an elastic body, such as a flat plate and a straight bar, but theoretical analysis cannot give us the vibration mode for the complex shape of an elastic body. FEM analysis can obtain the vibration mode for it.

In this chapter, three examples are presented, which use beam elements, shell elements, and solid elements, respectively, as shown in Figure 4.1

Figure 4.1: FEM element types: (a) beam element; (b) shell element; and (c) solid element

4.2 MODE ANALYSIS OF A STRAIGHT BAR

4.2.1 PROBLEM DESCRIPTION

Obtain the lowest three vibration modes and resonant frequencies in the y direction of the straight steel bar shown in Figure 4.2

Figure 4.2: Cantilever beam for mode analysis

Thickness of the bar is 0.005 m, width is 0.01 m, and the length is 0.09...