Free Vibrations
Topics:
• SDF System - Free Vibrations
• Torsional Vibration
• Effective Stiffness
• Equivalent Systems
• Determining the Frequency of the system
• Energy Method
• Conservation of Energy Principle
• Virtual Work Principle
• Response due to Initial Disturbance
• Response due to General Disturbance
• Duhamel’s Integral

Single Degree of Freedom System - Free vibrations
•This is the basic and simple model for vibrations
•Free vibrations are the vibrations at the natural frequency of the system
•Any system can be approximated as a SDOF system for very simple and fast analysis.
•Let us go through the examples as to how to model the systems.

Single Degree of Freedom System - Free vibrations
A big space structure can be crudely molded as a SDOF cantilever beam system as shown…

Single Degree of Freedom System - Free vibrations
A car or bike can be modeled as a SDOF system with the stiffness corresponding to the equivalent stiffness of the suspension and tries

Single Degree of Freedom System - Free vibrations
The figure shows the mathematical model for single degree of freedom spring mass system
Free body diagram
The equation of motion for this system is
..
 
0

Single Degree of Freedom System - Free vibrations
Where k – spring stiffness m – suspended mass
Natural frequency p
 k m
The natural frequency of the system is the frequency at which it vibrates when excited and left to itself x
Assumptions
 X – Downwards +ve
 Small Displacements
 Static Deflection


Single Degree of Freedom System - Free vibrations
Newton's laws of motion gives x
 k m x

0
It should be observed that solving this equation of motion yields the displacement of the system
PHYSICAL
DYNAMIC
SYSTEM
MATHEMATICAL
MODEL
GOVERNING
EQUATIONS
SOLVE
GOVERNING
EQUATIONS
RESPONSE

Single Degree of Freedom System - Free vibrations
From the equation for p p
 k m
Hence we have
Static equilibrium of spring
From the above figure we have hence
Which is approximated as

Free Vibrations
Topics:
• SDF System - Free Vibrations
• Torsional Vibration
• Effective Stiffness
• Equivalent Systems
• Determining the Frequency of the system
• Energy Method
• Conservation of Energy Principle
• Virtual Work Principle
• Response due to Initial Disturbance
• Response due to General Disturbance
• Duhamel’s Integral

Torsional System - Free vibrations
Many times we encounter the Torsional system like the shaft and gear systems .
Equilibrium equation of motion is
Hence natural frequency is
The stiffness is the elasticity of the shaft material

Torsional System - Free vibrations
Example for a torsional system

Free Vibrations
Topics:
• SDF System - Free Vibrations
• Torsional Vibration
• Effective Stiffness
• Equivalent Systems
• Determining the Frequency of the system
• Energy Method
• Conservation of Energy Principle
• Virtual Work Principle
• Response due to Initial Disturbance
• Response due to General Disturbance
• Duhamel’s Integral

Effective Stiffness
The bike suspension system has inclined springs
Effective spring stiffness of the inclined spring at an angle α is
K eq
 k cos
2

 of that in the direction of motion gives Equivalent K

Free Vibrations
Topics:
• SDF System - Free Vibrations
• Torsional Vibration
• Effective Stiffness
• Equivalent Systems
• Determining the Frequency of the system
• Energy Method
• Conservation of Energy Principle
• Virtual Work Principle
• Response due to Initial Disturbance
• Response due to General Disturbance
• Duhamel’s Integral

Equivalent Systems
All the systems can be reduced to Equivalent Systems for the sake of analysis and simplicity
Example
System can be modeled as

Equivalent Systems
Geared Torsional System
There are many practical application of geared Torsional systems
These can be equivalently modeled as
Motor pump gear system
The model equivalents are given by

Free Vibrations
Topics:
• SDF System - Free Vibrations
• Torsional Vibration
• Effective Stiffness
• Equivalent Systems
• Determining the Frequency of the system
• Energy Method
• Conservation of Energy Principle
• Virtual Work Principle
• Response due to Initial Disturbance
• Response due to General Disturbance
• Duhamel’s Integral

Determining the Frequency of the system
Energy Method
Conservation of Energy Principle
Virtual Work Principle
These all are various methods of applying Newton’s laws

Energy Method
Developed by Rayleigh and applicable for conservative systems
Consider a system having displacement in the form of x = a*cos(pt)
Using average energy principle we have T max
 k max
Maximum KE = Maximum PE
( at extreme positions )
For mechanical systems

1
2 2
T ma p
4 and 
1
V ka
2
2
Equating these two we get p
 k m
(max values)

Conservation of Energy Principle
“Energy can neither be created nor be destroyed”
Mathematically sum of total energy in the system is constant at any point of time hence d dt
( T

V )

0 or T+V= constant
Which again leads to p
 k m

Principle of Virtual Work
Principle of conservation of energy is not applicable for system where additional work input is required for the systems.
For virtual work
Assume a virtual displacement for the system from the mean position
Assuming the displacement changes the path of system
Virtual displacement

Principle of Virtual Work
Hamilton principle
Kinetic Energy
Potential Energy

Principle of Virtual Work
Hamilton principle
Lagrange’s equation
This considers the PE , KE variation for small displacements

Free Vibrations
Topics:
• SDF System - Free Vibrations
• Torsional Vibration
• Effective Stiffness
• Equivalent Systems
• Determining the Frequency of the system
• Energy Method
• Conservation of Energy Principle
• Virtual Work Principle
• Response due to Initial Disturbance
• Response due to General Disturbance
• Duhamel’s Integral

Response due to Initial Disturbance
System Natural Frequency
The Solution for mx kx

0 to an initial disturbances of
X(0)= A
V(0)=pB A & B are Constants of Integration
Solution is x=A*cos(pt)+B*sin(pt)
Can also be written as where X

A
2 
B
2 x
 x
0
pt
 v
0 p
pt x

X
pt
 
Velocity and acceleration are given by and v= a=
  
1
B
A

Xp
pt
 
 2
 

Free Vibrations
Topics:
• SDF System - Free Vibrations
• Torsional Vibration
• Effective Stiffness
• Equivalent Systems
• Determining the Frequency of the system
• Energy Method
• Conservation of Energy Principle
• Virtual Work Principle
• Response due to Initial Disturbance
• Response due to General Disturbance
• Duhamel’s Integral

Response due to General disturbance
Road profile….
For any general disturbance like shown in the figure..
• It can be divided into small blocks as shown
• Calculations are made for each block and finally they all are summed up to get the final output
This is done by DUHAMELS INTEGRAL

Duhamel’s Integral
The impulse signal due to applied force is given by
According to Newton’s law impulse will cause change in velocity
If initial conditions are
Then the solution is
For generalized forcing function x

mp
 t
0

Solving this integral gives response at any time of the action

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