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Damped Vibration
Presented by-
Name-Saurav Roy
Stream-Electrical
Section-1(B)
What is Damped Vibration?
• When a body or a particle vibrates it faces a number of frictional
forces(Internal or external).This forces decreases the amplitude
of vibration of time and the motion of the body ultimately stops
altogether.Such vibration of decaying amplitude is called Damped
Vibration.
Differential Equation of Damped Harmonic Vibration
The Newton's 2nd Law motion equation is:
This is in the form of a homogeneous second order differential equation and has
a solution of the form:
Substituting this form gives an auxiliary equation for λ.
The roots of the quadratic auxiliary equation are
Damping Coefficient
When a damped oscillator is subject to a damping force which is
linearly dependent upon the velocity, such as viscous damping, the
oscillation will have exponential decay terms which depend upon a
damping coefficient. If the damping force is of the form:
then the damping coefficient is given by:
DAMPED FREE VIBRATION SDOF
• If viscous damping is assumed, the equation of motion
becomes:
• There are 3 types of solution to this, defined as:
• Critically Damped
• Overdamped
• Underdamped
• A swing door with a dashpot closing mechanism is a
good analogy
• If the door oscillates through the closed position it is
underdamped
• If it creeps slowly to the closed position it is overdamped.
• If it closes in the minimum possible time, with no overswing, it
is critically damped.
0
)
(
)
(
)
( 

 t
ku
t
u
b
t
u
m 


Frequency and
period as before
Amplitude is a
function of damping
2% Damping
5% Damping
DAMPED VIBRATION
Time
Disp.
DAMPING WITH FORCED VIBRATION
 Apply a harmonic forcing function:
 note that is the DRIVING or INPUT frequency
 The equation of motion becomes
 The solution consists of two terms:
 The initial response, due to initial conditions which decays rapidly in the presence of
damping
 The steady-state response as shown:
 This equation is described on the next page
t
p 
sin
t
p
t
ku
t
u
b
t
u
m 
sin
)
(
)
(
)
( 

 


2
2
2
2
)
/
2
(
)
1
(
)
sin(
/
)
(
n
n
t
k
p
t
u











DAMPING WITH FORCED VIBRATION
 This equation deserves inspection as it shows several important
dynamic characteristics:
At  = n this term = (2 )^2 and controls
the scaling of the response
From this is derived the Dynamic
Magnification Factor 1/2
2
2
2
2
)
/
2
(
)
1
(
)
sin(
/
)
(
n
n
t
k
p
t
u










This is the static loading
and dominates as  tends
to 0.0
At  = n this term = 0.0
With no damping present this
results in an infinite response
Phase lead of the response relative to the input
(see next page)
At  >> n both terms drive the
response to 0.0


 Summary:
 For
 Magnification factor 1 (static solution)
 Phase angle 360º (response is in phase with the force)
 For
 Magnification factor 0 (no response)
 Phase angle 180º (response has opposite sign of force)
 For
 Magnification factor 1/2
 Phase angle 270º
1
ω
ω
n

1
ω
ω
n

1
ω
ω
n

DAMPING WITH FORCED VIBRATION
(Cont.)
THANK YOU

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dampedvibration02-160513044006.pdf

  • 1. Damped Vibration Presented by- Name-Saurav Roy Stream-Electrical Section-1(B)
  • 2.
  • 3.
  • 4. What is Damped Vibration? • When a body or a particle vibrates it faces a number of frictional forces(Internal or external).This forces decreases the amplitude of vibration of time and the motion of the body ultimately stops altogether.Such vibration of decaying amplitude is called Damped Vibration.
  • 5. Differential Equation of Damped Harmonic Vibration The Newton's 2nd Law motion equation is: This is in the form of a homogeneous second order differential equation and has a solution of the form: Substituting this form gives an auxiliary equation for λ. The roots of the quadratic auxiliary equation are
  • 6. Damping Coefficient When a damped oscillator is subject to a damping force which is linearly dependent upon the velocity, such as viscous damping, the oscillation will have exponential decay terms which depend upon a damping coefficient. If the damping force is of the form: then the damping coefficient is given by:
  • 7. DAMPED FREE VIBRATION SDOF • If viscous damping is assumed, the equation of motion becomes: • There are 3 types of solution to this, defined as: • Critically Damped • Overdamped • Underdamped • A swing door with a dashpot closing mechanism is a good analogy • If the door oscillates through the closed position it is underdamped • If it creeps slowly to the closed position it is overdamped. • If it closes in the minimum possible time, with no overswing, it is critically damped. 0 ) ( ) ( ) (    t ku t u b t u m   
  • 8. Frequency and period as before Amplitude is a function of damping 2% Damping 5% Damping DAMPED VIBRATION Time Disp.
  • 9. DAMPING WITH FORCED VIBRATION  Apply a harmonic forcing function:  note that is the DRIVING or INPUT frequency  The equation of motion becomes  The solution consists of two terms:  The initial response, due to initial conditions which decays rapidly in the presence of damping  The steady-state response as shown:  This equation is described on the next page t p  sin t p t ku t u b t u m  sin ) ( ) ( ) (       2 2 2 2 ) / 2 ( ) 1 ( ) sin( / ) ( n n t k p t u           
  • 10. DAMPING WITH FORCED VIBRATION  This equation deserves inspection as it shows several important dynamic characteristics: At  = n this term = (2 )^2 and controls the scaling of the response From this is derived the Dynamic Magnification Factor 1/2 2 2 2 2 ) / 2 ( ) 1 ( ) sin( / ) ( n n t k p t u           This is the static loading and dominates as  tends to 0.0 At  = n this term = 0.0 With no damping present this results in an infinite response Phase lead of the response relative to the input (see next page) At  >> n both terms drive the response to 0.0  
  • 11.  Summary:  For  Magnification factor 1 (static solution)  Phase angle 360º (response is in phase with the force)  For  Magnification factor 0 (no response)  Phase angle 180º (response has opposite sign of force)  For  Magnification factor 1/2  Phase angle 270º 1 ω ω n  1 ω ω n  1 ω ω n  DAMPING WITH FORCED VIBRATION (Cont.)