TABLE OF CONTENTS Single degree of freedom (single-DOF) theory, as outlined in this and in the next two chapters, enables a surprisingly large proportion of day-to-day structural dynamics problems to be solved. This chapter first describes how the equations of motion of single-DOF systems can be set up, using a variety of methods. It will be seen that most systems can be reduced to one of the two basic configurations:
Systems excited by an external force;
Systems excited by the motion of the base or supports.
Having set up the differential equations of motion, their direct solution by 'classical' methods is then discussed. These methods are somewhat tedious in use, and more practical methods of solution are discussed in Chapters 3 and 4. However, they are fundamental to an understanding of vital concepts such as damped and undamped natural frequencies, critical damping, etc., and cannot be omitted from any study of structural dynamics.
Several methods can be used to set up the differential equation of motion. The three methods introduced here are the following:
By inspection of the forces involved, using Newton's second law, with D'Alembert's principle, and the fundamental properties of mass, stiffness and damping. This is only suitable for very simple systems, such as that shown in Fig. 2.1.
For single-DOF systems with multiple mass, spring or damper elements connected together by levers or gears, the principle of virtual work, introduced in Chapter 1, provides a simple approach.
Using Lagrange's equations,
