VIBRATIONS

Vibration:Any motion that repeats itself in regular intervals of time.

Period of vibration or time period:It is the time interval after which the motion is repeated itself (seconds).

Cycle:It is the motion completed during one time period.

Frequency:It is the number of cycles described in one second. In S.I. units, the frequency is expressed in hertz

Free Vibration:No external force plays over the body after the initial displacement.

Forced Vibration:Body under vibration under the influence of external force.

Damped Vibration:Decreased amplitude of vibration for every cycle.

Terminology

Springs• A spring is an elastic object

used to store mechanical energy.

• Stiffness is the physical property of being inflexible and hard to bend. (or)

• Stiffness is the rigidity of an object — the extent to which it resists deformation in response to an applied force or load.

Combinations of Springs.

Do only SPRINGS have STIFFNESS

• “NO”………

• All the bodies resisting the force against deflection exhibit STIFFNESS

Stiffness in String

Check your skill of learning…..

Development of equation of motion

x

m

m

a b c

x

m

m

a b c

mg

k

mg

k kx

ma

mg k

0

k x mg ma

k x k mg mx

mx kx

• Consider 2nk m

2

0

0n

mx kx

x x

0mx kx

This is a second order linear homogeneous differential eqn.

Solution is: sin cosn nx A t B t

Woz dis fellow ??..

n “I’m Natural Frequency”

sin cosn nx A t B t

0 nA V 0B x0 0t 0 , and at x x x v

0 sin nx C t

After few modifications

x

m

m

a b c

tP tP

mg

k kx

ma

tP tP

k x mg P t ma

k x k mg P t mx

mx kx P t

x

m

m

a b c

tP tP

mg

k kx

ma

tP tP

k x mg P t ma

k x k mg P t mx

mx kx P t

mx kx P t This is a second order differential eqn.

Because even the load experiences vibrations along with the mass the load is also of form:

sinmP t P t

CF PIx x x

0 2sin sinmn

Px C t t

K m

sinmx kx P t

0 2sin sinmn

Px C t t

K m

Transition Part Steady State Part

Steady State Part exists as long as the force vanishes.

2

21

mm

mm

PxK mP K

xm K

2

2

11 .

1

1

m

m

n

xP K m K

MF

Magnification Factor or Dynamic Magnifier

• It is the ratio of maximum displacement of the forced vibration (x max ) to the deflection due to the static force F( x).

2

1

1n

MF