Sharanabasava C Pilli Principal, KLE Societys

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ME65 MECHANICAL VIBRATIONS
Sharanabasava C PilliPrincipal, KLE Societys
College of Engineering and Technology,
Udyambag, Belgaum-590008Email
scpilli_at_yahoo.co.in
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CHAPTER 3 DAMPED FREE VIBRATIONS

• Single degree of freedom systems
• Different types of damping
• Concept of critical damping and its importance
• Study of response of viscous damped systems
• under damped
• critically damped
• over damped system
• Logorthimic decrement

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MODELING SPRING MASS DAMPER SYSTEM
Imagine, at any instant, the system to be
displaced through a distance x from the
equilibrium position
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SOLUTION
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COMPARISON
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(No Transcript)
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LOGARITHMIC DECREMENT
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PROBLEM 1
The spring mass damper system is given an initial
velocity from the equilibrium position of x0?n,
where, ?n is the undamped natural frequency of
the system.

• Find the equation of the motion for the system
  when ? 0.2, 0.1 and 2.0.
• Plot the displacement curve if m 200 kg and k
  2400 N/m and x0 0.1 m.
• Compare the response for case 2, when
  ? 0.1, 0.25, 0.5, 0.75, and 0.95
  

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PROBLEM 1 contd
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PROBLEM 1 contd
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PROBLEM 1 contd
Under damped system
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PROBLEM 1 contd
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PROBLEM 1 contd
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PROBLEM 1 contd
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PROBLEM 1 contd

• As ? increases
• the response becomes sluggish
• the time period is increases
• the damped natural frequency reduces
• the peak over shoot decreases.

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Problem 2

• If the mass of the rod is negligible for small
  oscillations, find the condition for the system
  to be
• Over damped
• Critically damped
• Under damped and
• Also, find
• Frequency of damped oscillations.

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Problem 2 contd
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Problem 2 contd
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Problem 2 contd
Frequency of damped oscillations
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Problem 2 contd
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Problem 3

• Determine
• Natural frequency
• Damped natural frequency
• Logarithmic decrement
• If the mass is initially at rest and is given a
  velocity of 0.1 m / s, then calculate the
  amplitude after 5 oscillations.

For the system shown
k 10,000 N m 20 kg c 150 N-s / m
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Problem 3 contd
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Problem 3 contd
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Problem 3 contd
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Problem 3 contd
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PROBLEM 4
For what value of damping coefficient the damping
ratio is 1.25?
r1 0.1m r2 0.3m J 1.1 kg-m2 m1 10 kg m2
25 kg k1 1 x 104 N/m k21x105 N/m
Equivalent systems method is used for writing the
governing differential equation.
Let ? be the angular displacement of the disk in
counter clockwise direction.
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PROBLEM 4 contd
Potential energy of the system is
r1 0.1 m r2 0.3 m k1 1x104 N-m/rad k2
1x105 N-m/rad
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PROBLEM 4 contd
Kinetic energy of the system is
r1 0.1 m r2 0.3 m J 1.1 kg-m2 m1 10 kg
m2 25 kg
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PROBLEM 4 contd
Work done by the damping force between the two
instants ?1and ?2 is
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PROBLEM 4 contd
Equation of motion is
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PROBLEM 4 contd
r1 0.1m r2 0.3m J 1.1 kg-m2 m1 10 kg m2
25 kg k1 1 x 104 N/m k21x105 N/m
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PROBLEM 4 contd
If block m1 is displaced 20 mm and released, How
many cycles will be executed before the amplitude
is reduced to 1 mm?
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PROBLEM 4 contd
Logarithmic decrement is
n must be integer
Hence, after
7 cycles the amplitude is less than 1 mm.
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RECAP

• Study of response of viscous damped systems
• under damped
• critically damped
• over damped system
• Logarithmic decrement
• Illustrations

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QUIZ
Critical damping coefficient is
Damping ratio is