A body is said to be vibrate if it has a to-and-fro motion.

Most vibrations in machines and structures are undesirable due to increased stresses and energy losses.

If body has harmonic motion it will also be vibrate.

Longitudinal vibration- The different particles of the body move parallel to the axis of the body.

Transverse vibration- When the shaft is bent alternately and tensile and compressive stresses due to bending result, the vibrations are said to be transverse.

Torsional vibration- Because of twisting of shaft alternately torsional vibrations are created. The particles of the body move in a circle about the axis of the shaft.

Free (Natural) Vibration- No friction and no external forces after the initial release of the body.

Damped Vibration- In this vibrating system energy is dissipated by friction and other resistances.

Forced Vibration- Repeated force continuously acts on a system, vibration are said to be forced.

It states that the inertia forces and

couples, and the external forces

and torques on a body together

give statically equilibrium.

i. Kinetic storing device (mass - m)

ii. Potential storing device (Stiffness – s)

iii. Kinetic Friction (never equals to zero).

iv. Vibrating causing harmonic force.

If a particle is displaced through a distance xm from its equilibrium position and released with no velocity, the particle will undergo simple harmonic motion, 0=+ kxxm

( )φω += txx nm sin

xv =xa =

• Spring-MassSystem in VerticalPosition

0=+ kxxm

=

m

knω

( )φω += txx nm sin

Vibrating systems can encounter damping in various ways like

Intermolecular frictionSliding frictionFluid resistance2. Damping estimation of any system is the most

difficult process in any vibration analysis.

Technical name of friction in vibration is known as damping.

High damping/ coulomb damping – Due to friction between dry surfaces.

Low damping/viscous damping – Friction because of fluid layer due to lubrication.

Damped SDF and free body diagramThe eqn of motion is given by

A particular solution is given byHence

The two roots of the above equation are given by

A general solution is given byTo make the solution more general the critical damping coefficient is defined as

which gives the criteria for various damping properties where ξ= C/c

So the solution now becomes in terms of as ξ

This gives rise to four cases as discussed next

Type of System depends on Damping Factor

Under damped ξ<1

Undamped ξ =0

Over damped ξ >1

Critically damped ξ=1

.

The general solution becomes

The solution will be non oscillatory and gradually comes to rest

The solution will be

Critical Damping response Is more fast than over damping.

Damping factor= actual damping coefficient/critical damping

coefficient

The ratio of the force transmitted to the foundation to the force applied.It is the measure of the effectiveness of the vibration isolating material.