VIBRATION ANALYSIS-I

Classification of Vibrations

• Free Damped Steady state

• Forced Undamped Transient

• Axial Linear Deterministic

• Torsional Nonlinear Random

Differential Equations of Motion

• Forced, single-degree-of-freedom system. This simple system is composed of a mass, spring and damper.

Equation of motion

Natural frequency of this 1 DOF system

A linear, second order, ordinary diff. eq. with constant coefficients

Torsional System

Simulation of this torsional system:

Obtaining Equation of Motion and Natural Frequency of a 1-DOF System

Assume x> xo

m

A Vertical Model

Static equilibrium position

0

Static Equilibrium:

Solution of the Differential Equation of Motion

is another solution

General solution

2

cos

sin

cos

x B bt

x bB bt

x b B bt

2 cos cos

kb B bt B bt

m

sinx A bt

Initial Conditions and Different Solutions

1)

2)

3)

Phase-Plane Diagram Time-Displacement Diagram

Each vector is the phasor of the related phase plane.

OR

Step-Input Forcing

Phase-Plane Representation

Phase-Plane Analysis

From the new origin

1

1

cos(0 )

cos

o

o

x X

xX

0 sin(0 )

0 sin 0

n o

n o

X

X

From the old origin

Transient Disturbances

Free Vibration with Viscous Damping

Solution of Equation of Motion for Damped Free Vibration

1 2p t p tx Ae Be

2

cn

c k

m m

If >1 Overdamping

If =1 Critical damping

If 0<<1 Under damping

(Damped Vibration)

or 2c nc m

2c n

c c

c m

Damping

ratio

2( 1) np

Displacement in Damped Free Vibration

For <1

Damping Obtained by Experiment

For small

d

tn tN=tn+Nd

Nd