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VIBRATION AND CONTROL OF VIBRATION 31.2
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FIGURE 31.1 Representation of a single-degreeof-freedom system.
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ficient of viscous damping c is the force provided by the damper opposing the motion per unit velocity. If the mass is given an initial displacement, it will start vibrating about its equilibrium position. The equation of motion is given by m + cx + kx = 0 x (31.1)
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where x is measured from the equilibrium position and dots above variables represent differentiation with respect to time. By substituting a solution of the form x = est into Eq. (31.1), the characteristic equation is obtained: ms 2 + cs + k = 0 The two roots of the characteristic equation are s = n i n(1 2)1/2 where n = (k/m)1/2 is undamped natural frequency = c/cc is damping ratio cc = 2m n is critical damping coefficient i= 1 (31.3) (31.2)
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Depending on the value of , four cases arise. Undamped System (z = 0). In this case, the two roots of the characteristic equation are s = i n = i(k/m)1/2 and the corresponding solution is x = A cos nt + B sin nt (31.5) (31.4)
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where A and B are arbitrary constants depending on the initial conditions of the motion. If the initial displacement is x0 and the initial velocity is v0, by substituting these values in Eq. (31.5) it is possible to solve for constants A and B. Accordingly, the solution is
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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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x = x0 cos nt +
v0 sin nt n
(31.6)
Here, n is the natural frequency of the system in radians per second (rad/s), which is the frequency at which the system executes free vibrations. The natural frequency is fn = n 2 (31.7)
where fn is in cycles per second, or hertz (Hz). The period for one oscillation is = 1 2 = fn n (31.8)
The solution given in Eq. (31.6) can also be expressed in the form x = X cos ( n ) where X = x2 + 0 v0 n
2 1/2
(31.9)
= tan 1
v0 n x0
(31.10)
The motion is harmonic with a phase angle as given in Eq. (31.9) and is shown graphically in Fig. 31.4. Underdamped System (0 < z < 1). When the system damping is less than the critical damping, the solution is x = [exp( nt)] (A cos dt + B sin dt) where d = n(1 2)1/2 (31.12) (31.11)
is the damped natural frequency and A and B are arbitrary constants to be determined from the initial conditions. For an initial amplitude of x0 and initial velocity v0, x = [exp ( nt)] x0 cos dt + which can be written in the form x = [exp ( nt)] X cos ( dt ) X = x2 + 0 and = tan 1 n x 0 + v0 d n x 0 + v0 d
2 1/2
nx0 + v0 sin dt d
(31.13)
(31.14)
An underdamped system will execute exponentially decaying oscillations, as shown graphically in Fig. 31.2.
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VIBRATION AND CONTROL OF VIBRATION 31.4
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FIGURE 31.2 Free vibration of an underdamped single-degree-of-freedom system.
The successive maxima in Fig. 31.2 occur in a periodic fashion and are marked X0, X1, X2,. . . . The ratio of the maxima separated by n cycles of oscillation may be obtained from Eq. (31.13) as Xn = exp ( n ) X0 where = 2 (1 2)1/2 (31.15)
is called the logarithmic decrement and corresponds to the ratio of two successive maxima in Fig. 31.2. For small values of damping, that is, << 1, the logarithmic decrement can be approximated by = 2 Using this in Eq. (31.14), we find Xn = exp ( 2 n ) X0 1 2 n (31.17) (31.16)
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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