# Vibration theory

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 of energy between its potential and kinetic forms. In a damped system, some energy is dissipated at each cycle of vibration and must be replaced from an external source if a steady vibration is to be maintained. Although a single physical structure may store both kinetic and potential energy, and may dissipate energy, this chapter considers only lumped parameter systems composed of ideal springs, masses, and dampers wherein each element has only a single function. In translational motion, displacements are defined as linear distances; in rotational motion, displacements are defined as angular motions.

TRANSLATIONAL MOTION

Spring. In the linear spring shown in Fig. 2.1, the

change in the length of the spring is proportional

to the force acting along its length:

F = k(x − u)

(2.1)

FIGURE 2.1 Linear spring.

The ideal spring is considered to have no mass;

thus, the force acting on one end is equal and

2.1

8434_Harris_02_b.qxd 09/20/2001 11:37 AM Page 2.2

2.2

CHAPTER TWO

opposite to the force acting on the other end.The constant of proportionality k is the spring constant or stiffness.

Mass. A mass is a rigid body (Fig. 2.2) whose

acceleration x according to Newton’s second law is

¨

proportional to the resultant F of all forces acting on

the mass:*

F = mx

¨

FIGURE 2.2 Rigid mass.

Damper. In the viscous damper shown in Fig. 2.3,

the applied force is proportional to the relative

velocity of its connection points:

F = c(˙ − u)

x ˙

FIGURE 2.3 Viscous damper.

(2.2)

(2.3)

The constant c is the damping coefficient, the characteristic parameter of the damper. The ideal damper is considered to have no mass; thus the force at one

end is equal and opposite to the force at the other

end. Structural damping is considered below and

several other types of damping are considered in

Chap. 30.

ROTATIONAL MOTION

The elements of a mechanical system which moves with pure rotation of the parts are wholly analogous to the elements of a system that moves with pure translation. The property of a rotational system which stores kinetic energy is inertia; stiffness and damping coefficients are defined with reference to angular displacement and angular velocity, respectively. The analogous quantities and equations are listed in Table 2.1.

TABLE 2.1 Analogous Quantities in Translational

and Rotational Vibrating Systems

Translational quantity

Rotational quantity

Linear displacement x

Force F

Spring constant k

Damping constant c

Mass m

Spring law F = k(x1 − x2)

Damping law F = c(˙ 1 − x2)

x ˙

Inertia law F = m¨

x

Angular displacement α

Torque M

Spring constant kr

Damping constant cr

Moment of inertia I

Spring law M = kr(α1 − α2)

Damping law M = cr(¨ 1 − α2)

α ˙

¨

Inertia law M = Iα

* It is common to use the word mass in a general sense to designate a rigid body. Mathematically, the mass of the rigid body is defined by m in Eq. (2.2).

8434_Harris_02_b.qxd 09/20/2001 11:37 AM Page 2.3

2.3

BASIC VIBRATION THEORY

Inasmuch as the mathematical equations for a rotational system can be written by analogy from the equations for a translational system, only the latter are discussed in detail.Whenever translational systems are...

Please join StudyMode to read the full document