# barcode generator vb.net source code VIBRATION AND CONTROL OF VIBRATION 31.5 in Software Print EAN13 in Software VIBRATION AND CONTROL OF VIBRATION 31.5

VIBRATION AND CONTROL OF VIBRATION 31.5
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FIGURE 31.3 Variation of the ratio of displacement maxima with damping.
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The equivalent viscous damping in a system is measured experimentally by using this principle. The system at rest is given an impact which provides initial velocity to the system and sets it into free vibration. The successive maxima of the ensuing vibration are measured, and by using Eq. (31.17) the damping ratio can be evaluated. The variation of the decaying amplitudes of free vibration with the damping ratio is plotted in Fig. 31.3 for different values of n. Critically Damped System (z = 1). When the system is critically damped, the roots of the characteristic equation given by Eq. (31.3) are equal and negative real quantities. Hence, the system does not execute oscillatory motion.The solution is of the form x = (A + Bt) exp ( nt) and after substitution of initial conditions, x = [x0 + (v0 + x0 n)t] exp ( nt) (31.19) (31.18)
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This motion is shown graphically in Fig. 31.4, which gives the shortest time to rest. Overdamped System (z > 1). When the damping ratio is greater than unity, there are two distinct negative real roots for the characteristic equation given by Eq. (31.3). The motion in this case is described by x = exp ( nt) [A exp nt 2 1 + B exp ( nt 2 1)] (31.20)
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FIGURE 31.4 damping.
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Free vibration of a single-degree-of-freedom system under different values of
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where A= and 0 = n 2 1 All four types of motion are shown in Fig. 31.4. If the mass is suspended by a spring and damper as shown in Fig. 31.5, the spring will be stretched by an amount st, the static deflection in the equilibrium position. In such a case, the equation of motion is m + cx + k(x + st) = mg x (31.21) 1 v0 + nx0 x0 + n 2 B= 1 2 x0 + nx0 0