Unleashing the Power of Vibration and Benefits of Oscillation [PDF]

Definition:

Vibration is defined as when an elastic body such as a spring, a beam, and a shaft is displaced from the equilibrium position by the application of external forces and then released they execute a vibratory motion. When body particles are displaced by the application of external force, internal forces are present in the body in the form of elastic energy. It tries to bring the body to its original position. At equilibrium position, the elastic energy is converted into kinetic energy and the body continues to move in the opposite direction.

The kinetic energy is again converted into elastic or strain energy due to which the body again returns to its equilibrium position. In this way, vibratory motion is repeated infinite times and exchange of energy takes place. Thus we can say that any motion which repeats itself after regular intervals of time is called Vibration or Oscillation.

Vibration Types:

The following are the types of Vibration:

1. Free or Natural
2. Forced
3. Damped Vibration

1. Free or Natural Vibration:

When no external force acts on the body, after giving it an initial displacement, then the body is said to be under free or natural vibration. The frequency of free or natural vibration is called free or natural frequency.

Types of Free Vibration:

1. Longitudinal
2. Transverse ad 
3. Torsional Vibration.

1. Longitudinal Vibrations: 

The particles of the shaft or disc move parallel to the axis of the shaft as shown in the above diagram. The shaft is elongated and shortened alternately thus applying the tensile and compressive stresses alternately on the shaft.

2. Transverse vibrations: 

The particles of the shaft or disc move perpendicular to the axis of the shaft as shown in the above diagram. Here the shaft is straight and bent alternatively. Hence bending stresses are induced in the shaft.

3. Torsional Vibrations: 

The particles of the shaft or disc move in a circle about the axis of the shaft as shown in the above diagram. Here the shaft is twisted and untwisted alternatively. Hence torsional shear stress is induced in the shaft.

2. Forced Vibrations:

When the body vibrates under the influence of an external force, the body is said to be under forced vibrations. An external force is applied to the body periodically distributing force created by unbalance. This has the same frequency as the applied force. The amplitude remains constant with respect to time.

3. Damped Vibration:

When there is a reduction in amplitude over every cycle of vibration, the motion is said to be damped vibration. It is observed that the amplitude reduces abruptly with respect to time as shown in the above diagram.

Equilibrium Method:

It is based on the principle that whenever a vibratory system is in equilibrium, the sum of all forces and moments acting on it is zero. According to D’Alembert’s Principle, the sum of inertia forces and external forces on a body in equilibrium must be zero.

Let, Δ = static deflection

k = Stiffness of the spring

Inertial force = ma ( upwards, a = acceleration)

Spring force = kx ( upwards)

So the equation becomes

ma + kx = 0

⇒ω_n = √(k/m)

Linear frequency f_n = (1/2π)√(k/m)

Time period T = 1/f_n  = 2π√(m/k)

Energy method:

In a conservative system (system with no damping). The total mechanical energy i.e. the sum of the kinetic and the potential energies remains constant.

d/dt (K.E+ P.E.) = 0

Rayleigh’s Method:

In this method, the maximum kinetic energy at the mean position is made equal to the maximum potential energy( or strain energy) of the extreme position. The displacement of the mass ‘m’ from the mean position at any instant is given by

a+ω_n² x = 0

x = A sinω_n t + B Cosω_n t

Let A = X cos φ ; B = X Sin φ

x = X sin(ω_n t +φ)

Velocity, V = Xω_n Sin [π/2 + (ω_n t +φ)]

Acceleration = Xω_n² Sin[ π + (ω_n t +φ)]

These relationships indicate that

• the velocity vector leads the displacement vector by  π/2.
• acceleration vector leads the displacement vector by π.

Consider, ‘m’ = mass of the spring wire per unit length

l = total length of the spring wire m₁ = m’l

KE of the spring = 1/3 * KE of a mass equal to that of the spring moving with the same velocity as the free end.

f_n = (1/2π) √ (s/(m+(m₁/m)))

f_n = (1/2π) √g/Δ

Damped Vibrations:

When an elastic body is set in vibratory motion, the vibrations die out after some time due to the internal molecular friction of the mass of the body and the friction of the medium in which it vibrates. The reduction of vibrations with time is called damping. Shock absorbers, fitted in the suspension system of a motor vehicle, reduce the movement of the springs. when there is a sudden shock.

It is usual to assume that the damping force is proportional to the velocity of vibration at lower values of speed and proportional to the square of velocity at high speeds.

F∝ V at a lower speed

F∝ V² at a higher speed

C = damping coefficient (damping force per unit velocity)

ω_n = frequency of natural undamped vibrations

a + (c/m)v + (k/m)x = 0

α_1,2 = -(c/2m) ± √[(c/2m)²-(k/m)]

Degree of dampness = (c/2m)²/(k/m)

Damping factor (ξ) = c/(2√km)

Damping coefficient (c) = 2ξ√km = 2ξmω_n = 2ξk/ω_n

When ξ = 1, damping is critical, thus under critical damping conditions

ξ = 2√km = 2mω_n = 2k/ω_n

ξ = c/c_c = Actual damping coefficient / Critical damping coefficient

• ξ > 1 ; the system is over damped
• ξ < 1 ; the system is under damped
• ω_d = ω_n√(1-ξ²)

In a critically damped system, the displaced mass return to the position of rest in the shortest possible time without oscillation. Due to this reason large guns are critically damped so that they return to their original positions in minimum possible time. An undamped system (ξ = 0) vibrates at its natural frequency which depends upon the static deflection under the weight of its mass. At critical damping (ξ = 1); ω_d = 0 and T_d = ∞. The system does not vibrate and the mass ‘m’ moves back slowly to the equilibrium position. For overdamped system (ξ > 1) the system behaves in the same manner as for critical damping

FAQ’s

Conclusion

Hence we can conclude by saying that vibration is a common phenomenon that occurs in structures and machines. Excessive vibration can cause damage and reduce performance. Understanding the mechanism of vibration is crucial to identify potential problems and solutions to correct them.