Introduction

In Chapter 2, the dynamic equation of motion was established for an SDOF system, and the solutions to different deterministic load conditions were described. It was also shown that the SDOF system could be used to model important features of rigid-body and flexible motions of various structures. The SDOF can be translational or rotational motion. The number of degrees of freedom is equal to the number of variables found necessary to describe the dynamic behavior of a structure with the required accuracy. This chapter deals with formulation and solution of dynamic problems which include both rigid-body motions and structural deformations, and are based on multiple degrees of freedom.

In Example 2.3.3, the concept of a generalized DOF is introduced, and it is illustrated how this could be used to effectively reduce a large number of DOFs to just one generalized DOF by combining this with the appropriate shape function. Thus, the instantaneous position of all mass points in the system were specified without ambiguity and without redundancy. The simplest example of this is a mass point moving in a plane, which is described by two coordinates. Of course, the concept of generalized coordinates is not needed in this case. However, a rigid disc moving in a plane needs three generalized coordinates. A rigid body moving in the paper plane, see Fig. 3.1, may be described by uG(t), vG(t), and ϕG(t) of the center of gravity. Clearly, using the three quantities with reference to one of the ends of the body also represents a (unique) set of generalized coordinates. If considered as a rigid body, the articulated loading tower in Example 2.2.8 can be described by a single generalized coordinate (DOF) when a plane motion is considered.