CHAPTER 2

BASIC VIBRATION THEORY

Ralph E. Blake

INTRODUCTION

This chapter presents the theory of free and forced steady-state vibration of single degree-of-freedom systems. Undamped systems and systems having viscous damping and structural damping are included. Multiple degree-of-freedom systems are discussed, including the normal-mode theory of linear elastic structures and

Lagrange’s equations.

ELEMENTARY PARTS OF VIBRATORY SYSTEMS

Vibratory systems comprise means for storing potential energy (spring), means for storing kinetic energy (mass or inertia), and means by which the energy is gradually lost (damper). The vibration of a system involves the alternating transfer

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2.4. The relations of Eq. (2.8) are shown by the solid lines in Fig. 2.5.

FIGURE 2.5 Natural frequency relations for a single degree-of-freedom system. Relation of natural frequency to weight of supported body and stiffness of spring [Eq. (2.8)] is shown by solid lines. Relation of natural frequency to static deflection [Eq. (2.10)] is shown by diagonal-dashed line. Example: To find natural frequency of system with W = 100 lb and k = 1000 lb/in., enter at

W = 100 on left ordinate scale; follow the dashed line horizontally to solid line k = 1000, then vertically down to diagonal-dashed line, and finally horizontally to read fn = 10 Hz from right ordinate scale.

Initial Conditions. In Eq. (2.5), B is the value of x at time t = 0, and the value of A is equal to x/ωn at time t = 0. Thus, the conditions of displacement and velocity which

˙

exist at zero time determine the subsequent oscillation completely.

Phase Angle. Equation (2.5) for the displacement in oscillatory motion can be written, introducing the frequency relation of Eq. (2.6), x = A sin ωnt + B cos ωnt = C sin (ωnt + θ) where C = (A + B )

2

2 1/2

(2.9)

−1

and θ = tan (B/A). The angle θ is