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VIBRATION AND CONTROL OF VIBRATION 31.5
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VIBRATION AND CONTROL OF VIBRATION
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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
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VIBRATION AND CONTROL OF VIBRATION 31.7
VIBRATION AND CONTROL OF VIBRATION
FIGURE 31.5 Model of a single-degree-offreedom system showing the static deflection due to weight.
Since the force in the spring due to the static equilibrium is equal to the weight, or k st = mg = W, the equation of motion reduces to m + cx + kx = 0 x (31.22)
which is identical to Eq. (31.1). Hence the solution is also similar to that of Eq. (31.1). In view of Eq. (31.21) and since n = (k/m)1/2, the natural frequency can also be obtained by n = g st
(31.23)
An approximate value of the fundamental natural frequency of any complex mechanical system can be obtained by reducing it to a single-degree-of-freedom system. For example, a shaft supporting several disks (wheels) can be reduced to a single-degree-of-freedom system by lumping the masses of all the disks at the center and obtaining the equivalent stiffness of the shaft by using simple flexure theory.
31.2.2 Torsional Systems Rotating shafts transmitting torque will experience torsional vibrations if the torque is nonuniform, as in the case of an automobile crankshaft. In rotating shafts involving gears, the transmitted torque will fluctuate because of gear-mounting errors or tooth profile errors, which will result in torsional vibration of the geared shafts. A single-degree-of-freedom torsional system is shown in Fig. 31.6. It has a massless shaft of torsional stiffness k, a damper with damping coefficient c, and a disk with polar mass moment of inertia J. The torsional stiffness is defined as the resisting torque of the shaft per unit of angular twist, and the damping coefficient is the resisting torque of the damper per unit of angular velocity. Either the damping can be externally applied, or it can be inherent structural damping. The equation of motion of the system in torsion is given J + c + k = 0 (31.24)
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