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