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VIBRATION AND CONTROL OF VIBRATION 31.15
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FIGURE 31.13 Phase angle between transmitted and applied forces.
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FIGURE 31.14 Dynamic system subject to unbalanced excitation.
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Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
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FIGURE 31.15 A base excited system.
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where F0 = u0 (k2 + c2 2)1/2 and = tan 1 k c (31.49) (31.48)
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Equation (31.47) is identical to Eq. (31.25) except for the phase . Hence the solution is similar to that of Eq. (31.25). If the ratio of the system response to the base displacement is defined as the motion transmissibility, it will have the same form as the force transmissibility given in Eq. (31.38). Resonance, System Bandwidth, and Q Factor. A vibrating system is said to be in resonance when the response is maximum. The displacement and acceleration responses are maximum when = n(1 2 2)1/2 whereas velocity response is maximum when = n (31.51) (31.50)
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In the case of an undamped system, the response is maximum when = n, where n is the frequency of free vibration of the system. For a damped system, the frequency of free oscillations or the damped natural frequency is given by d = n (1 2 )1/2 (31.52)
In many mechanical systems, the damping is small and the resonant frequency and the damped natural frequency are approximately the same. When the system has negligible damping, the frequency response has a sharp peak at resonance; but when the damping is large, the frequency response near resonance will be broad, as shown in Fig. 31.8. A section of the plot for a specific damping value is given in Fig. 31.16. The Q factor is defined as Q= 1 = Rmax 2 (31.53)
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VIBRATION AND CONTROL OF VIBRATION 31.17
VIBRATION AND CONTROL OF VIBRATION
FIGURE 31.16 Resonance, bandwidth, and Q factor.
which is equal to the maximum response in physical systems with low damping. The bandwidth is defined as the width of the response curve measured at the halfpower points, where the response is Rmax/ 2. For physical systems with < 0.1, the bandwidth can be approximated by = 2 n = n Q (31.54)
Forced Vibration of Torsional Systems. In the torsional system of Fig. 31.3, if the disk is subjected to a sinusoidal external torque, the equation of motion can be written as (31.55) J + c + k = T0 sin t Equation (31.55) has the same form as Eq. (31.25). Hence the solution can be obtained by replacing m by J and F0 by T0 and by using torsional stiffness and torsional damping coefficients for k and c, respectively, in the solution of Eq. (31.25). 31.2.4 Numerical Integration of Differential Equations of Motion: Runge-Kutta Method When the differential equation cannot be integrated in closed form, numerical methods can be employed. If the system is nonlinear or if the system excitation can-
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VIBRATION AND CONTROL OF VIBRATION 31.18
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not be expressed as a simple analytical function, then the numerical method is the only recourse to obtain the system response. The differential equation of motion of a system can be expressed in the form = f(x,x ,t) x or x = y = F1(x,y,t) y = f(x, ,t) = F2(x,y,t) x x0 = x(0) x 0 = x (0) (31.56)
where x0 and x 0 are the initial displacement and velocity of the system, respectively. The form of the equation is the same whether the system is linear or nonlinear. Choose a small time interval h such that tj = jh for j = 0, 1, 2, . . .
Let wij denote an approximation to xi (tj) for each j = 0, 1, 2, . . . and i = 1, 2. For the initial conditions, set w1,0 = x0 and w2,0 = x 0. Obtain the approximation wij + 1, given all the values of the previous steps wij, as [31.1] 1 wi,j + 1 = wi,j + (k1,i + 2k2,i + 2k3,i + k4,i) i = 1, 2 (31.57) 6 where k1,i = hFi(tj + w1,j, w2,j) k2,i = hFi tj + k3,i = hFi tj + h 1 1 , w1,j + k1,1, w2i,j + k1,2 2 2 2 1 h 1 , w1,j + k2,1, w2,j + k2,2 2 2 2 i = 1, 2
(31.58)
k4,i = hFi(tj + h, w1,i + k3,1, w2, i + k3,2 )
Note that k1,1 and k1,2 must be computed before we can obtain k2,1. Example 1. Obtain the response of a generator rotor to a short-circuit disturbance given in Fig. 31.17. The generator shaft may be idealized as a single-degree-of-freedom system in torsion with the following values: 1 = 1737 cpm = 28.95(2 ) rad/s = 182 rad/s J = 8.5428 lb in s2 (25 kg m2) k = 7.329 106 lb in/rad (828 100 N m/rad) Solution Hence, J + k = f(t) = f(t) k = + J J k + f(t) = J J (31.59)
where f(t) is tabulated. Since 1 = 182 rad/s, the period = 2 /182 = 0.00345 s and the time interval h must be chosen to be around 0.005 s. Hence, tabulated values of f(t) must be available for t intervals of 0.005 s, or it has to be interpolated from Fig. 31.17.
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