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)

See Blackboard

i. ζ < 1 (underdamped) The frequency of damped oscillation is equal to :

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. 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.

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