Viscously Damped Free Vibration

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Viscously Damped Free Vibration
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• Viscous damping force is expressed by the
  equation
• where c is a constant of proportionality.
• Symbolically. it is designated by a dashpot

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• From the free body diagram, the equation of
  motion is .seen to be

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• The solution of this equation has two parts.
• If F(t) 0, we have the homogeneous differential
  equation whose solution corresponds physically to
  that of free-damped vibration.
• With F(t) ? 0, we obtain the particular solution
  that is due to the excitation irrespective of the
  homogeneous solution.
• ? Today we will discuss the first condition

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• With the homogeneous equation
• the traditional approach is to assume a solution
  of the form
• where s is a constant.

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• Upon substitution into the differential equation,
  we obtain
• which is satisfied for all values of t when

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• Above equation, which is known as the
  characteristic equation, has two roots
• Hence, the general solution is given by the
  equation

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• where A and B are constants to be evaluated from
  the initial conditions
• and

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• Substitution characteristic equation into general
  solution gives

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• The first term, , is simply an
  exponentially decaying function of time.
• The behavior of the terms in the parentheses,
  however, depends on whether the numerical value
  within the radical is positive, zero, or
  negative.
• Positive ? Real number
• Negative ? Imaginary number

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• When the damping term (c/2m)2 is larger than k/m,
  the exponents in the previous equation are real
  numbers and no oscillations are possible.
• We refer to this case as overdamped.

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• When the damping term (c/2m)2 is less than k/m,
  the exponent becomes an
• imaginary number, .
• Because
• the terms within the parentheses are oscillatory.
  
• We refer to this case as underdamped.

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• In the limiting case between the oscillatory
• and non oscillatory motion
  ,
• and the radical is zero.
• The damping corresponding to this case is called
  critical damping, cc.

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• Any damping can then be expressed in terms of the
  critical damping by a non dimensional number ? ,
  called the damping ratio
• and

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• The three condition of damping depend on the
  value of ?
• i. ? lt 1 (underdamped)
• ii. ? gt 1 (overdamped)
• iii ? 1 (criticaldamped)

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• See Blackboard

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i. ? lt 1 (underdamped)

• The frequency of damped oscillation is equal to

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i. ? lt 1 (underdamped)

• the general nature of the oscillatory motion.

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ii. ? gt 1 (overdamped)

• The motion is an exponentially decreasing
  function of time

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iii ? 1 (criticaldamped)

• Three types of response with initial displacement
  x(0).

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STABILITY AND SPEED OF RESPONSE

• The free response of a dynamic system
  (particularly a vibrating system) can provide
  valuable information concerning the natural
  characteristics of the system.
• The free (unforced) excitation can be obtained,
  for example, by giving an initial-condition
  excitation to the system and then allowing it to
  respond freely.
• Two important characteristics that can be
  determined in this manner are
• 1. Stability
• 2. Speed of response

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STABILITY AND SPEED OF RESPONSE

• The stability of a system implies that the
  response will not grow without bounds when the
  excitation force itself is finite. This is known
  as bounded-input-bounded-output (BIBO) stability.
• In particular, if the free response eventually
  decays to zero, in the absence of a forcing
  input, the system is said to be asymptotically
  stable.
• It was shown that a damped simple oscillator is
  asymptotically stable.
• But an undamped oscillator, while being stable in
  a general (BIBO) sense, is not asymptotically
  stable. It is marginally stable.

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STABILITY AND SPEED OF RESPONSE

• Speed of response of a system indicates how fast
  the system responds to an excitation force.
• It is also a measure of how fast the free
  response
• (1) rises or falls if the system is
  oscillatory or
• (2) decays, if the system is non-oscillatory.
• Hence, the two characteristics stability and
  speed of response are not completely
  independent.
• In particular, for non-oscillatory (overdamped)
  systems, these two properties are very closely
  related.
• It is clear then, that stability and speed of
  response are important considerations in the
  analysis, design, and control of vibrating
  systems.

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STABILITY AND SPEED OF RESPONSE

• Level of stability
• Depends on decay rate of free response
• Speed of response
• Depends on natural frequency and damping for
  oscillatory systems and decay rate for
  non-oscillatory systems

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• Decrement Logarithmic