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Vibrations - Mechanical Engineering

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A Brief about
A body is said to vibrate if it has a to-and-fro motion about a fixed position. A pendulum swinging on either side of a mean position does so under the influence of gravity. Usually, vibrations are due to elastic forces. Whenever a body is displaced from its equilibrium position, work is done on the elastic constraints of the forces on the body and is stored as strain energy. Now, if the body is released, the internal forces cause the body to move towards its equilibrium position. If the motion is frictionless, the strain energy stored in the body is converted into kinetic energy. The body passes through the mean position, the kinetic energy is utilized to overcome the elastic forces and is stored in the form of strain energy, and so on.

The vibration of a body is characterized by the following parameters:
1.  Period- It is the time taken by a motion to repeat itself and has a unit of seconds.
2.  Frequency- Frequency is the number of cycles of motion completed in one second. It is expressed in hertz(Hz) and is equal to one cycle per second.
3.  Resonance- When the frequency of the external force is the same as that of the natural frequency of the system, a state of resonance is said to have been reached.

Types of Resonance
1.  Based on the number of degrees of freedom
It can be Single degree of freedom or Multiple degrees of freedom.
2.  Based on the nature of vibration
It can be classified as Free(Natural) vibrations, Damped vibrations, and Forced Vibration. The force can be either periodic or harmonic.
3.  Based on the direction of vibration
It can be Longitudinal vibration, Transverse vibration, and Torsional vibration.
Vibrations are everywhere around us. The most common example being cars and motorcycles. Sometimes these vibrations can be good for the functioning of a machine but many times vibrations are harmful and therefore undesirable. If the vibration is natural and free from any effort to oppose it, the part will continue to vibrate without any reduction in the amplitude of vibration. But there are many ways by which the vibrations can be damped or the frequency of vibration can be reduced.

Free Vibration
The equation of motion of a simple spring-mass system undergoing free longitudinal vibrations is given by-
Where m is the mass of the body undergoing vibration, y is the position of the body,  is the acceleration of the body and k is the stiffness of the spring.
The frequency of vibration of this natural system is given by,

Damped Vibration
When an elastic body is set in vibratory motion, the vibrations die out after sometime due to the internal molecular friction of the mass of the body and the friction of the medium in which it vibrates. The diminishing vibrations are called damped vibrations. External damping can be increased by using dashpots or dampers. A dashpot has a piston which moves in a cylinder filled with some fluid. Shock absorbers, fitted in the suspension system of a motor vehicle reduce the movement of the springs when there are sudden shocks, thus damping out the bouncing which could have occurred otherwise. The equation of motion of a body undergoing damped vibration is given by-
Here  is the damping coefficient of the damper which is basically damping force per unit velocity and ẏ is the velocity of the body.
The ratio of  represents the degree of dampness provided in the system and its square root is known as damping factor or damping ratio ,i.e.
When , the damping is known as critical damping.
When , the system is overdamped.
When , the system is underdamped.

Forced Vibration
The forcing input may be step-input, harmonic or periodic. The equation of motion being given by-
For a step-input force F is constant and for a harmonic forcing, F can be either a sine or a cosine function.
The equation of motion of a forced damped vibration is given by-
The solution of this equation contains a complementary function and a principal integral and can be solved using the methods for an ordinary differential equation.
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