Viscously Damped Free
Vibration
• Viscous damping force is expressed by
the equation
where c is a constant of proportionality.
• Symbolically. it is designated by a dashpot
• From the free body diagram, the equation
of motion is .seen to be
• 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 freedamped 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
• With the homogeneous equation :
the traditional approach is to assume a
solution of the form :
where s is a constant.
• Upon substitution into the differential
equation, we obtain :
• which is satisfied for all values of t when
• Above equation, which is known as the
characteristic equation, has two roots :
Hence, the general solution is given by the
equation:
where A and B are constants to be
evaluated from the initial conditions
and
• Substitution characteristic equation into
general solution gives
• 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
• 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.
• 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.
• 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.
• Any damping can then be expressed in
terms of the critical damping by a non
dimensional number ζ , called the
damping ratio:
and


s1, 2       1  n
2
• The three condition of damping depend on
the value of ζ
i. ζ < 1 (underdamped)
ii. ζ > 1 (overdamped)
iii ζ = 1 (criticaldamped)
See Blackboard

i. ζ < 1 (underdamped)

s1, 2       1  n
2
• The frequency of damped oscillation is equal to :
i. ζ < 1 (underdamped)
• the general nature of the oscillatory
motion.
ii. ζ > 1 (overdamped)
• The motion is an exponentially decreasing function of time
iii ζ = 1 (criticaldamped)
• Three types of response with initial displacement x(0).
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
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.
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.
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
Decrement Logarithmic