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

S. C. Pilli, KLE Belgaum
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RECAP OF 13 March 07

• Introduction
• Damping
• Viscous damping
• Illustrations
• Single DOF system
• Damped free vibrations with viscous damping
• Modeling single DOF system
• Response of over damped system
• Critical damping and damping coefficient
• Significance of damping
• Illustrations

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CRITICALLY DAMPED SYSTEM
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CRITICAL DAMPING contd
Because of the repeated roots the solution of the
governing equation and Its dedrivative are
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CRITICAL DAMPING contd
The solution is
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CRITICAL DAMPING contd
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CRITICAL DAMPING contd
Comparison of solution for different initial
condition
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PROBLEM 1
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PROBLEM 2

• A gun barrel of mass 500 kg has a recoil
  springof stiffness 10,000 N/m. If the barrel
  recoils 0.4 meters on firing determine
• The critical damping coefficient of the dashpot.
• Initial recoil velocity of the barrel
• Time taken by the gun to return to a position
  0.1m from its initial position.

When the gun is fired the reaction force pushes
the barrel in opposite direction to projectile.
To bring it back, it made to translate with
critical damping so that it moves back quickly.
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PROBLEM 2
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PROBLEM 2 contd
Initial displacement is zero C1 X0
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PROBLEM 2 contd
If t2 denotes the time taken bt the gun to return
to a position 0.1 m from the initial position
The approximate value of t2 ? 0.75 s
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PROBLEM 3
A 500 kg vehicle is mounted on springs such that
the static deflection is 1.5 mm. What is the
damping coefficient of a viscous damper to be
added to the system in parallel with the springs,
such that the system is critically damped?
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UNDER DAMPED SYSTEM
In case of free vibrations equation of motion of
a spring mass damper is
or
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UNDER DAMPED contd
For the under damped system the term under the
radical sign is negative, leading to complex
conjugate roots as ? lt 1 or c lt cc.
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UNDER DAMPED contd
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UNDER DAMPED contd
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UNDER DAMPED contd
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UNDER DAMPED contd
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COMPARISON
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COMPARISON
Response for different values of damping 0 lt ? 1
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Problem 4
Between a solid mass 10 kg and the floor are kept
two slabs of isolators, natural rubber and felt,
in series, as shown. The natural rubber slab has
a stiffness of 3000 N/m and an equivalent viscous
damping coefficient of 100 N-s/m. The felt slab
has a stiffness of 12000 N/m and an equivalent
viscous damping coefficient of 330
N-s/m. Determine the damped natural frequency of
the system.
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Problem 4 contd
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PROBLEM 5

• Body of mass 5 kg is supported on a spring of
  stiffness1960 N/m and has a dashpot connected to
  it, which produces a resistance of 0.98 N at a
  velocity of 0.5 m/s.
• Find
• Damping ratio
• Displacement of mass, 0.5 sec after it was
  displaced through 20 mm and released.

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PROBLEM 5 contd
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PROBLEM 5 contd
Substituting the value of ? ? ? 90? and A 20 mm
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OVERSHOOT
It is the maximum displacement of the system at
the end of its first half cycle.
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PROBLEM 6
What is the minimum damping ratio for a system
such that it is subjected to no more than 5
overshoot?
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PROBLEM 6 contd
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RECAP

• Critically damped system
• Illustrations
• Under damped system
• Illustrations
• overshoot
• Illustrations

S. C. Pilli, KLE Belgaum