**Vibrations are due to elastic forces**

## Types of Vibrations:

**Free Vibration:**Elastic vibration in which there are no friction and external forces after the initial release of the body is known as free or natural vibration.

Damped Vibration: When the energy of a vibrating system is gradually dissipated by friction and other resistances the vibration are said to be damped.

**Forced Vibrations:**When a repeated force continuously acts on a system, the vibrations is that not applied force and is independent of their own natural frequency of vibrations.

**Resonance:**When the frequency of external force is the same as that of the natural frequency of the system, a state of resonance is said to be have been reached.

**Longitudinal Vibrations:**If a body is elongated and shortened so that the mass moves up and down resulting in tensile and compressive stresses in the shaft, the vibrations are said to be longitudinal.

**Transverse vibrations:**When a body is bent alternatively and tensile and compressive stresses due to bending result, the vibrations are said to be transverse. The particles of the body move-approximately perpendicular to its axis.

**Torsional Vibrations:**When a body is twisted and untwisted alternatively and torsional shear stresses are induced, the vibrations are known as torsional vibrations.

**Free Longitudinal vibrations**

Equilibrium Method:

It is based on the principle that whenever a vibratory system is in equilibrium, the algebraic sum of forces and moments acting on it is zero recordings to D’Alembert’s Principle that 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_{n}^{2} 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,f = Xω_{n}^{2} 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’ = man of the spring wire per unit length

l = total length of the spring wire m_{1} = 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+(m_{1}/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 diminishing of the 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∝ V^{2} 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 damoing coefficient

- ξ > 1 ; the system is over damped
- ξ < 1 ; the system is under damped
**ω**_{d}= ω_{n}√(1-ξ^{2})

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

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