Download - Vibrational Analysis
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