Vibration Theory

1. Introduction 2. Harmonic Motion

3. Vibration Model

4. Equation of Motion: Natural Frequency

5. Viscously Damped Free Vibration

6. Forced Harmonic Vibration

7. Rotating Unbalance

Introduction
There are two general classes of vibrations – free and forced. Free vibration takes place when a system oscillates under the action of forces inherent in the system itself, and when external impressed forces are absent. The system under free vibration will vibrate at one or more of its natural frequencies, which are properties of the dynamic system established by its mass and stiffness distribution.

Vibration that takes place under the excitation of external forces is called forced vibration. When the excitation is oscillatory, the system is forced to vibrate at the excitation frequency. If the frequency of excitation coincides with one of the natural frequencies of the system, a condition of resonance is encountered, and dangerously large oscillations may result. The failure of major structures such as bridges, buildings, or airplane wings is an awesome possibility under resonance. Thus, the calculation of the natural frequencies of major importance in the study of vibrations.

Vibrating systems are all subject to damping to some degree because energy is dissipated by friction and other resistances. If the damping is small, it has very little influence on the natural frequencies of the system, and hence the calculation for the natural frequencies are generally made on the basis of no damping. On the other hand, damping is of great importance in limiting the amplitude of oscillation at resonance.

The number of independent coordinates required to describe the motion of a system is called degrees of freedom of the system. Thus, a free particle undergoing general motion in space will have three degrees of freedom, and a rigid body will have six degrees of freedom, i.e., three components of position and three angles defining its orientation. Furthermore, a continuous elastic body will require an infinite number of coordinates (three for each point on the body) to describe its motion; hence, its degrees of freedom must be infinite. However, in many cases, parts of such bodies may be assumed to be rigid, and the system may be considered to be dynamically equivalent to one having finite degrees of freedom. In fact, a surprisingly large number of vibration problems can be treated with sufficient accuracy by reducing the system to one having a few degrees of freedom.

Harmonic Motion

Oscillatory motion may repeat itself regularly, as in the balance wheel of a watch, or display considerable irregularity, as in earthquakes. When the motion is repeated in equal intervals of time T, it is called period motion. The repetition time t is called the period of the oscillation, and its reciprocal, ,is called the frequency. If the motion is designated by the time function x(t), then any periodic motion must satisfy the relationship .

Harmonic motion is often represented as the projection on a straight line of a point that is moving on a circle at constant speed, as shown in Fig. 1. With the angular speed of the line o-p designated by w , the displacement x can be written as

(1)

Figure 1 Harmonic Motion as a Projection of a Point Moving on a Circle

The quantity w is generally measured in radians per second, and is referred to as the angular frequency. Because the motion repeats itself in 2p radians, we have the relationship

(2)

where t and f are the period and frequency of the harmonic motion, usually measured in seconds and cycles per second, respectively.

The velocity and acceleration of harmonic motion can be simply determined by differentiation of Eq. 1. Using the dot notation for the derivative, we obtain

(3)

(4)

Vibration Model

The basic vibration model of a simple oscillatory system consists of a mass, a massless spring, and a damper. The spring supporting the mass is assumed to be of negligible mass. Its force-deflection relationship is considered to be linear, following Hooke’s law, , where the stiffness k is measured in newtons/meter.

The viscous damping, generally represented by a dashpot, is described by a force proportional to the velocity, or . The damping coefficient c is measured in newtons/meter/second.

Rotating Unbalance

Unbalance in rotating machines is a common source of vibration excitation. We consider here a spring-mass system constrained to move in the vertical direction and excited by a rotating machine that is unbalanced, as shown in Fig. 10. The unbalance is represented by an eccentric mass m with eccentricity e that is rotating with angular velocity w . By letting x be the displacement of the non rotating mass (M – m) from the static equilibrium position, the displacement of m is :

Figure 10 Harmonic Disturbing Force Resulting from Rotating Unbalance

The equation of motion is then :

which can be rearranged to :

(37)

It is evident that this equation is identical to Eq. (29), where is replaced by , and hence the steady-state solution of the previous section can be replaced by :

(38)

and

(39)

These can be further reduced to non dimensional form :

(40)

and

(41)

Example

A counter rotating eccentric weight exciter is used to produce the forced oscillation of a spring-supported mass as shown in Fig. 11. By varying the speed of rotation, a resonant amplitude of 0.60 cm was recorded. When the speed of rotation was increase considerably beyond the resonant frequency, the amplitude appeared to approach a fixed value of 0.08 cm. Determine the damping factor of the system.