Post on 23-Dec-2015
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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 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
• 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
• The three condition of damping depend on the value of ζ
i. ζ < 1 (underdamped)
ii. ζ > 1 (overdamped)
iii ζ = 1 (criticaldamped)
ns 122,1
See Blackboard
i. ζ < 1 (underdamped)
• The frequency of damped oscillation is equal to :
ns 122,1
• the general nature of the oscillatory motion.
i. ζ < 1 (underdamped)
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. Stability2. 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.
• 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
STABILITY AND SPEED OF RESPONSE
Decrement Logarithmic