Question 9.9: Solve the problem of Example 9.6 by the exponential window m...
Solve the problem of Example 9.6 by the exponential window method.
The response is calculated over a total time of T_{0}=2.4 \mathrm{~s}. To obtain a we use
a=\frac{2 \ln 10}{T_{0}}=1.92
We select a=2 and apply exponential scaling to both g(t) and h(t)=(1 / m \omega) \sin \omega t. The scaled functions are shown in Figure E9.9. As expected, they die rapidly as t approaches T_{0}. The two functions are sampled at 0.1 \mathrm{~s}. Discrete Fourier transforms of the sampled functions, calculated by using a standard computer program, provide \hat{G}(\Omega) and \hat{H}(\Omega). Response \hat{u}(t) is obtained by taking the inverse discrete Fourier transform of the product of \hat{G}(\Omega) and \hat{H}(\Omega). The true
Table E9.9 Response obtained by exponential window method.
response u(t) is now recovered from \hat{u}(t) by using Equation 9.136.
u(t) = e^{at} \hat{u}(t) \qquad (9.136)
The computed displacements are compared with the exact values in Table E9.9; the match between the two is quite good.
