Theory of Vibration

The response of a machine foundation is generally analyzed by Lumped Parameter Approach. In this approach the foundation is represented quite well by a mass, spring and a dash-pot and the problems are usually studied under following headings.

The properties of Simple Harmonic Motion are to be understood clearly to make the above studies.

Simple Harmonic Motion

In simple harmonic motion acceleration is proportional to displacement from some fixed point. A weight suspended to a spring as shown Fig.1 is a simple example of such motion. The spring in Fig.1 is assumed to have a spring constant, k. A weight, W, is suspended at the bottom end of the spring. The weight W produces a static deflection of z_st and a static position of equilibrium is achieved in position OO. The system is then set into vertical motion by pulling down from the position static equilibrium OO, and then released. As a result, the weight W oscillates about OO, undamped. The maximum displacement z_max from the position OO is called the amplitude A. The total maximum displacement is 2A and is called double amplitude. This phenomenon is illustrated in Fig.1 (a) to (e). At any time t the displacement achieved is z. This phenomenon of vertical oscillation can be illustrated graphically. Consider a circle with center at O and of radius z_max [Fig 1 (g)].

A particle is allowed to move around the circle with an angular velocity Ï‰. At any time’t’ the position of the particle is at ‘a’. If a projection of particle at different time in a vertical diameter of the circle is made, the particle will be oscillating from its equilibrium position O to ‘a’, ‘a’ to O, O to ‘b’ and ‘b’ to O in one complete revolution. A time displacement plot of such vibration will as shown in Fig.1 (g). From the figure,

$\dfrac{Po}{oa} = Sin\omega t$

Or, $\dfrac{z}{z_max} = Sin\omega t$

Or, $z = z_{max} \times sin\omega t = A \times sin\omega t$ ———- (1)

The above equation represents equation of motion in terms of a sine function. Since the particle covers an angular displacement of $2\pi$ in on complete cycle, we can write

$\omega \times t = 2\pi$ ———- (2)

Thus, $t=T=\dfrac{2\pi}{\omega}$ ———- (3)

Where, T is time period.

The frequency of Oscillation is given by,

$f=\dfrac{1}{T}=\dfrac{\omega}{2\pi}$ ———- (4)

Vectorial Representation

The mechanism of vibration can also be analyzed by making use of rotating vectors. The displacement equation is:

$z = A\times sin\omega t$

Velocity is, $\dfrac{dz}{dt} = \omega\times A\times Cos\omega t = \omega \times A\times Sin(\omega t + \dfrac{\pi}{2})$ ———- (5)

and acceleration is $\dfrac{d^2Z}{dt^2}=-\omega^2 \times A \times Sin\omega t = \omega^2 \times A \times Sin(\omega t + \pi)$ ———- (6)

The above equation clearly shows that velocity leads displacement by 90 ° and acceleration leads displacement by 180 °. The vectorial representation of displacement, velocity and acceleration are shown in Fig.2.