For Problem 51, suppose the calculations need to be simplified. Can the autocorrelation of the force be replaced by a triangular spike around \(\tau=0\) ? Try this replacement, justify the simplification, and solve. Compare with the results of Problem 51.

Problem 51:

Suppose a simply supported beam is forced by a random load with mean value \(\mu_{F}\) and autocorrelation \(R_{F F}(\tau)=\exp (-\alpha \tau)\), where \(\alpha\) is a constant. Solve for the mean, autocorrelation, and spectral density of the response.