VIBRATION ANALYSIS-2

Response to Periodic Forcing (in undamped systems)

x=xh+xp

Dividing numerator and denominator by k

Inıtial conditions: At t=0, x=0 and

Natural frequency Forcing frequency

General Solution

Transient Steady State

Excitation

Response

Special Cases

1)

Response will be equal to excitation itself. The motion is called as RIGID BODY MOTION. Mass will follow exciter’s motion. This case arises approximately when the spring is very stiff; and strictly for the unstrained systems.

2)

General solution may be written in the following form before substituting

n

Response to Harmonic Forcing (in damped systems)

Steady state Transient

Steady-State Response To Harmonic Forcing

Amplitude (Magnification) Ratio

Phase angle:

1) Amplitude is very large near the resonance for small damping ratios. 2) Peak amplitudes occur slightly before /n =1 3) Phase angle changes rapidly near the resonance.

21 2R n

Forcing Caused by Unbalance

t

1) For /n >>1 mX/mue 1 X mue/m Vibration amplitude can be

reduced by reducing the mass and eccentiricity of the unbalance.

2) Peak amplitudes occur slightly after /n =1

21 2

nR

Relative Motion

k(x-y) c

Vibration Isolation

Transmissibility:

1) For effective isolation, /n >2 2) For safe passage of resonance, a small damping should be present in the system.

2

2 2 2 20 0

1 (2 / )

(1 / ) (2 / )

ntr

n n

F XT

F y

Ftr/F0 : Force transmissibility X/y0 : Displacement transmissibility (Absolute motion)

Static Deflection and Natural Frequency W

y0

Rayleigh’s Principle

W

y0

Assuming harmonic motion,

Potential energy is maximum for

Kinetic energy is maximum when the velocity is maximum, for

Torsional Systems

Rigid body motion

Dunkerley’s Formula

• A formula for appromixate determination of first natural (fundamental) frequency of a beam (shaft) with more than one masses.

• : Fundamental frequency of the system

• : Natural frequency of the first mass

• : Natural frequency of the second mass

• : Natural frequency of the n th mass

2 2 2 2

1 11 22

1 1 1 1...

nn

1

11

22

nn

1