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10 Nov 2019
Show that the general solution of d 2x/dt 2 + 2 lambda dx/dt + omega 2x = F 0 sin gamma t is x(t) = Ae -lambda t sin ( +phi) + F 0/ sin (gamma t + theta), where A = and the phase angles phi and theta are, respectively, defined by sin phi = c 1/A, cos phi = c 2/A and sin theta -2 lambda gamma/ cos theta = omega 2 - gamma 2/ The solution in part (a) has the form x(t) = x c(t) + x p(t). Inspection shows that x c(t) is transient, and hence for large values of time, the solution is approximated by x p(t) = g(gamma) sin (gamma t + theta), where g(gamma) = F 0/ Although the amplitude g(gamma) of x p(t) is bounded as t rightarrow infinity, show that the maximum oscillations will occur at the value gamma 1 = . What is the maximum value of g? The number 2 pi is said to be the resonance frequency of the system.
Show that the general solution of d 2x/dt 2 + 2 lambda dx/dt + omega 2x = F 0 sin gamma t is x(t) = Ae -lambda t sin ( +phi) + F 0/ sin (gamma t + theta), where A = and the phase angles phi and theta are, respectively, defined by sin phi = c 1/A, cos phi = c 2/A and sin theta -2 lambda gamma/ cos theta = omega 2 - gamma 2/ The solution in part (a) has the form x(t) = x c(t) + x p(t). Inspection shows that x c(t) is transient, and hence for large values of time, the solution is approximated by x p(t) = g(gamma) sin (gamma t + theta), where g(gamma) = F 0/ Although the amplitude g(gamma) of x p(t) is bounded as t rightarrow infinity, show that the maximum oscillations will occur at the value gamma 1 = . What is the maximum value of g? The number 2 pi is said to be the resonance frequency of the system.