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# Vibrations

Introduction

Vibration is an oscillation wherein the quantity is a parameter defining the motion of a mechanical system.

Oscillation is the variation, usually with time, of the magnitude of a quantity with respect to a specified reference when the magnitude is alternately greater and smaller than the reference.

Vibration is mechanical oscillation about a reference position. Vibration is an everyday phenomenon, we meet it in our homes, during transport and at work. Vibration is often a destructive and annoying side effect of a useful process, but is sometimes generated intentionally to perform a task.

Free vibration

When a free undamped mass-spring system is set into oscillation the added energy is constant, but changes form from kinetic to potential during the motion.

At maximum displacement the velocity and therefore also the kinetic energy is zero, while the potential energy is 1/2kD˛. At the equilibrium position the potential energy is zero and the kinetic energy is maximum at 1/2mV˛.

For the sinusoidal motion

• d = D sin wn t

we can also find the velocity:

• V = 2*PI*fnD.

Using energy conservation laws we then get the natural resonance frequency:

• fn=squareroot(K/m) / (2PI)

Forced vibration

If an external sinusoidal force is applied to the system, the system will follow the force, which means that the movement of the system will have the same frequency as the external force. There might, however, be a difference in amplitude (and phase).

For frequencies below its natural frequency, the amplitude of the vibrating system will increase as the frequency is increased, a maximum being reached at the natural frequency. If there was no damping in the system (c = 0), the amplitude would approach infinity.

If the frequency of the external force is increased the frequency of the spring/mass/damper system will increase to the same value, but the amplitude (and the phase) will change in accordance.