# Dynamics of Structures (4th Edition)

## Book Preface

In this opening chapter, the structural dynamics problem is formulated for simple structures that can be idealized as a system with a lumped mass and a massless supporting structure. Linearly elastic structures as well as inelastic structures subjected to applied dynamic force or earthquake-induced ground motion are considered. Then four methods for solving the differential equation governing the motion of the structure are reviewed briefly. The chapter ends with an overview of how our study of the dynamic response of single-degree-of-freedom systems is organized in the chapters to follow.

**SIMPLE STRUCTURES**

We begin our study of structural dynamics with simple structures, such as the pergola shown in Fig. 1.1.1 and the elevated water tank of Fig. 1.1.2. We are interested in understanding the vibration of these structures when subjected to a lateral (or horizontal) force at the top or horizontal ground motion due to an earthquake.

We call these structures simple because they can be idealized as a concentrated or lumped mass m supported by a massless structure with stiffness k in the lateral direction. Such an idealization is appropriate for this pergola with a heavy concrete roof supported by light-steel-pipe columns, which can be assumed as massless. The concrete roof is very stiff and the flexibility of the structure in lateral (or horizontal) motion is provided entirely by the columns. The idealized system is shown in Fig. 1.1.3a with a pair of columns supporting the tributary length of the concrete roof. This system has a lumped mass m equal to the mass of the roof shown, and its lateral stiffness k is equal to the sum of the stiffnesses of individual pipe columns. A similar idealization, shown in Fig. 1.1.3b, is appropriate for the tank when it is full of water. With sloshing of water not possible in a full tank, it is a lumped mass m supported by a relatively light tower that can be assumed as massless. The cantilever tower supporting the water tank provides lateral stiffness k to the structure. For the moment we will assume that the lateral motion of these structures is small in the sense that the supporting structures deform within their linear elastic limit. We shall see later in this chapter that the differential equation governing the lateral displacement u(t) of these idealized structures without any external excitatio —applied force or ground motion—is

m ¨ u + ku = 0 (1.1.1)

where an overdot denotes differentiation with respect to time; thus ˙ u denotes the velocity of the mass and ¨ u its acceleration. The solution of this equation, presented in Chapter 2, will show that if the mass of the idealized systems of Fig. 1.1.3 is displaced through some initial displacement u(0), then released and permitted to vibrate freely, the structure will oscillate or vibrate back and forth about its initial equilibrium position. As shown in Fig. 1.1.3c, the same maximum displacement occurs oscillation after oscillation; these oscillations continue forever and these idealized systems would never come to rest. This is unrealistic, of course. Intuition suggests that if the roof of the pergola or the top of the water tank were pulled laterally by a rope and the rope were suddenly cut, the structure would oscillate with

ever-decreasing amplitude and eventually come to rest. Such experiments were performed on laboratory models of one-story frames, and measured records of their free vibration response are presented in Fig. 1.1.4. As expected, the motion of these model structures decays with time, with the decay being more rapid for the plexiglass model relative to the aluminum frame.

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