Vibration

Vibration:

Types of Vibrations

There are three types of vibrations:

1. Free or normal vibrations

2. Damped vibrations

3. Forced vibrations

When a body which is held in position by elastic constraints is displaced from its equilibrium position by the application of an external force and then released, the body commences to vibrate assuming that there are no external or internal resistances to prevent the motion and the material of constraints is perfectly elastic, the body will continue vibrating indefinitely. In that case at the extreme positions of oscillations; the energy imparted to the body by the external force is entirely stored in the elastic constraint as internal or elastic or strain energy. When the body falls back to its original equilibrium position, whole strain energy is converted into the kinetic energy which further takes the body to the other extreme position, when again the energy is stored in the elastic constraint; at the expense of which the body again moves towards its initial equilibrium position; and this cycle continues repeating indefinitely. This is how the body oscillates between two extreme positions. A vibration of this kind in which, after initial displacement, no external forces act and the motion is maintained by the internal elastic forces are termed as natural vibrations.

Free Vibrations

Consider a bar of length l, diameter d, the upper end of which is held by the elastic constraints and at the lower end, it carries a heavy disc of mass m.

The system may have one of the three simple modes of free vibrations given below:

a. Longitudinal vibrations

b. Transverse vibrations

c. Torsional vibrations

a. Longitudinal Vibrations

When the particles of the shaft or disc move parallel to the axis of the shaft as shown in fig. Than the vibrations are known as longitudinal vibrations.

b. Transverse Vibrations

When the particles of the shaft or disc move approximately perpendicular to the axis of the shaft shown in fig. Then the vibrations are known as transverse vibrations.

c. Torsional Vibrations

When the particles of the shaft or disc move in a circle about the axis of the shaft, thenthe vibrations are known as torsional vibrations. Before studying frequencies of general vibrations we must understand degree of freedom.

Natural Frequency of Free Longitudinal Vibrations

The natural frequency of the free longitudinal vibrations may be determined by the following three methods.

1. Equilibrium Method

2. Energy Method

3. Rayleigh’s Method

Damping Factor or Damping Ratio

The ratio of damping coefficient C to the critical damping coefficient Cc is known as damping factor or damping ratio. Mathematically,

Damping factor = C/ Cc = C/2mwn (Cc =2mwn)

The damping factor is the measure of the relative amount of damping in the existing system with that necessary for the critical damped systems.

TRANSVERSE & TORSIONAL VIBRATIONS

Generally when the particles of the shaft or disc move in a circle about the axis of the shaft as already discussed in previous chapter, then the vibrations are known as torsional vibrations. In this case, the shaft is twisted and alternately and the torsional shear stresses are induced in the shaft.

When the particles of the shaft or disc move in a circle about the axis of the shaft as shown in fig as already explained in previous chapter , then the vibrations are known as known as transverse vibrations. Natural Frequency of Free Transverse Vibrations Due to Point Load Acting Over a Simple Supported Shaft

Natural Frequency of Free Transverse Vibrations of a Shaft Fixed at Both Ends Carrying a Uniformly Distributed Load

Natural Frequency of Free Transverse Vibrations for a Shaft Subjected to a Number of Point Loads Critical or Whirling Speed of a Shaft In general, a rotating shaft carries different mountings and accessories in the form of gears, pulleys, etc. When the gears or pulleys are out on the shaft, the centre of gravity of the pulley of gear does not coincide with the centre of the bearings or with the axis of the shaft, when the shaft is stationary, This means that the centre of gravity of the pulley of gear is at a certain distance from the axis of rotation and due to this, the shaft is subjected to centrifugal force. This force will bend the shaft, which will further increase the distance of centre of gravity of the pulley or gear from the axis of rotation. This correspondingly increases the value of centrifugal force, which further increases the distance of centre of gravity from the axis rotation. This effect is cumulative and ultimately the shaft fails. The bending of shaft not only depends upon the value of eccentricity (distance between centre of gravity of the pulley and the axis of rotation)But also depends upon the speed at which the shaft rotates. The speed, at which the shaft runs so that the additional deflection of the shaft from the axis of rotation becomes infinite, is known as critical or whirling speed.

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