Vibration theory

Topics: Damping, Force, Harmonic oscillator Pages: 32 (8772 words) Published: April 21, 2014
8434_Harris_02_b.qxd 09/20/2001 11:37 AM Page 2.1


Ralph E. Blake

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.

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.

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)


FIGURE 2.1 Linear spring.

The ideal spring is considered to have no mass;
thus, the force acting on one end is equal and

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



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.



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.

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¨

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).

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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...
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