00038___1707cc76b7866db6177d8e6d06853000 - Eq. (9) may be...

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Sec. 1.5 VIBRATIONS 17 body at a circular frequency of w rad/sec. It can exist without external load. Hence w is called the natural frequency of a free vibration. If the damping constant c were zero, and the forcing function i is periodic with the same frequency as the natural one, then the amplitude of the oscillation is unbounded, and we have the phenomenon of resonance. The solution of Eq. (4) must be an exponential function of time, because the derivative of an exponential function is another exponential function. Thus (8) On substituting (8) into Eq. (4), we have y(t) = Ae" implies y(t) = AXe" , y(t) = AX2ext. (9) MA2 + CX + k = 0. If we write (10) k/M = w2, & = c/(2&7), then the two roots of
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Unformatted text preview: Eq. (9) may be written as XI and X2: (11) X I = -&W + i W J S , Xz = -&W - i w J 1 - E 2 . When k and M are real positive numbers, w is real, and the solution of Eq. (9) under the initial conditions (12) y(0) = yo , y(0) = yo when t = 0 . is --EW t (13) y(t) = yoe cosw&2t Now, we can return to the solution of Eq. (3). A particular solution of Eq. (3) can be obtained by Laplace or Fourier transformation or other methods. By direct substitution, it can be verified that a particular solution satisfying the initial conditions y(t) = y(t) = 0 at t = 0 is: The general solution of Eq. (3) is the sum of the functions given in (13) and (14). From this solution we can examine the nature of the forced oscillation...
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This note was uploaded on 01/04/2012 for the course ENG 501 taught by Professor Thomson during the Fall '05 term at MIT.

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