ABSTRACT
Before we discuss longitudinal vibration in bars, it would be useful to look at the vibration of multidegree-of-freedom (MDOF) discrete systems first. The governing equations for their motion can be easily formed using Newton’s second law and can be analyzed by casting them into the eigenvalue form. In addition, the concept of mode shapes can be easily explained for an MDOF discrete system; the mode shapes presented in the form of phasors can be easily visualized. The mode shapes of a discrete system can be used to obtain a system of uncoupled equations that can be solved to obtain their response. This is known as the modal approach and a similar approach can also be used to obtain the response of continuous systems. Once the physical concepts of natural frequencies and mode shapes for a MDOF discrete system are clearly understood, the same can be easily applied to continuous systems. However, the governing equations for bars are expressed in the form of partial differential equations that are solved in a different way than an MDOF that is represented by a system of ordinary differential equations.
