3 Multi-Degree of Freedom Systems The methods of vibration analysis of single degree of freedom systems can be generalized and extended to study systems with an arbitrary finite number of degrees of freedom. Mechanical systems in general consist of structural ele- ments which have distributed mass and elasticity. In many cases, these systems can be represented by equivalent systems which consist of some elements which are bulky solids which can be treated as rigid elements with specified inertia properties while the other elements are elastic elements which have negligible inertia effects. In fact, the single degree of freedom systems discussed in the preceding chapters are examples of these equivalent models which are called lumped mass systems. We have shown in the preceding chapters that a single degree of freedom system exhibits motion governed by one second-order ordinary differential equation while a two degree of freedom system exhibits motion governed by two second-order ordinary differential equations. It is expected, therefore, that a system having n degrees of freedom exhibits motion which is governed by a set of n simultaneous second-order differential equations. An example of these systems is shown in Fig. 1. In Section 1 of this chapter, the general form of the second-order ordinary differential equations of motion that govern the vibration of multi-degree of freedom systems is presented, and the use of these equations is demonstrated by several applications. In Section 2, the undamped free vibration of multi- degree of freedom systems is discussed and it is shown that a system with n degrees of freedom has n natural frequencies. Methods for determining the mode shapes of the undamped systems are presented and the orthogonality of these mode shapes is discussed in Section 3. In this section, we also discuss the use of the modal transformation to obtain n uncoupled second-order ordinary differential equations of motion in terms of the modal coordinates. Sections 4 and 5 are devoted, respectively, to the analysis of semidefinite systems, and the conservation of energy in the case of undamped free vibra- tion. In Section 6, the forced vibration of the undamped multi-degree of freedom systems is discussed. The solution of the equations of motion of 98 A. A. Shabana, Vibration of Discrete and Continuous Systems © Springer-Verlag New York, Inc. 1997