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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VISCOUS DAMPING
ordered mechanical energy of the system is
dissipated as disordered thermal energy to the
environment in an irreversible manner
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ENERGY DISSIPATION IN VISCOUS DAMPER
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ENERGY DISSIPATION IN VISCOUS DAMPER contd
Energy dissipation per cycle is proportional to
the square of the amplitude of motion,
amount of damping and damped
natural frequency.
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ENERGY DISSIPATION IN VISCOUS DAMPER contd
Presence of a spring parallel to a damper does
not alter the energy dissipated per cycle.
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ENERGY DISSIPATION IN VISCOUS DAMPER contd
This is expected since the spring force will not
do any net work over a complete cycle.
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RESPONSE OF SPRING MASS DAMPER
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RESPONSE OF SPRING MASS DAMPER
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RESPONSE OF SPRING MASS DAMPER
The general solution can be rearranged in either
of the following form for ? gt1, 1, lt1.
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RESPONSE OF SPRING MASS DAMPER
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RESPONSE OF SPRING MASS DAMPER
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UNDER DAMPED RESPONSE
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LOGARTHMIC DECREMENT
represents the rate at which the amplitude of a
free damped vibration decreases.
The logarithmic decrement can be used to find the
amount of damping in a physical system.
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LOGARTHMIC DECREMENT
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LOGARITHMIC DECREMENT
Logarithmic decrement can also be obtained as
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PROBLEM 1
Free vibration records of a 1000 kg machine
mounted on an isolator is shown in the figure.
Identify isolator characteristics.
We are given the amplitudes of 4 cycles. n 4
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PROBLEM 1 contd
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PROBLEM 2

• The disk of a torsional pendulum has a
  moment of inertia of 0.06 kg-m2. The disk is
  attached to a shaft of diameter 0.1 m and 0.4 m
  long. When the pendulum is vibrating in a viscous
  media the amplitude measured on same side of rest
  position for successive cycles are 9? 6? and 4?
  respectively. Determine
• Logarithmic decrement
• Damping torque at unit velocity and
• Time period of vibration
• Assume G 4.4 x 1010 N/m2 for the shaft.

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PROBLEM 2contd
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PROBLEM 3
A shock absorber is to be designed for 10
overshoot of the displacement when released.
Determine the damping factor.
If damping is
reduced to half the value, what will be the
over shoot?
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PROBLEM 3 contd
When damping is reduced to half ? 0.344 / 2
0.172
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Problem 4
An under damped shock absorber is to be designed
for an automobile. It is required that initial
amplitude to be reduced to 1/16th in one cycle.
The mass of the automobile is 200 kg and damped
period of vibration is 1 sec. Find the necessary
stiffness and damping constants for the shock
absorbers.
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Problem 4 contd
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Problem 4 contd
The damping coefficient c 1107.75 N-s / m The
stiffness is k 9411.92 N / m
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PROBLEM 5
An under damped shock absorber is to be designed
for a motorcycle of mass 200 kg. When the shock
absorber is subjected to an initial vertical
velocity due to road bump,
The resulting displacement -time curve is as
shown.
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PROBLEM 5 contd
2.0 s
Find the stiffness and damping coefficients of
the shock absorber if the damped period of
vibration is to be 2 sec and the amplitude x1 is
to be reduced to one-fourth in one half cycle.
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PROBLEM 5 contd
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PROBLEM 5 contd
The damping coefficient c 554.4981 N-s / m The
stiffness is k 2358.2652 N / m
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RECAP

• Energy dissipated by viscous damper
• Response of spring mass damper system
• Logarithmic decrement
• Illustrations

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RECAP
initial conditions

• 0 undamped system free vibrations
• 0 lt ? lt 1 underdamped oscillatory behavior
• 1 critically damped aperiodic motion
• gt 1 over damped aperiodic motion more sluggish

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RECAP
Response
What is the Response of 1) under damped,
2)over damped and 3)critical damped
system?
Hint 1-C, 2- A, 3- B
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RECAP
Formule