Question 9.P.1: A single degree of freedom system with frequency ω, stiffnes...
A single degree of freedom system with frequency \omega, stiffness k, and negligible damping is subjected to the periodic load shown in Figure P9.1. Obtain the response of the system to the applied load. Show that resonance takes place not only when \Omega=\omega but also when \Omega takes any of the values in the series \Omega=\omega / 3, \omega / 5, \omega / 7, \ldots..
\begin{aligned}&\begin{aligned}u(t)=\frac{p_{0}}{2 k}-\frac{4 p_{0}}{\pi^{2} k}\bigg[& \frac{1}{1-\beta_{1}^{2}} \cos \Omega t+\frac{1}{9} \frac{1}{\left(1-\beta_{3}^{2}\right)} \cos 3 \Omega t \\&\left.\quad+\frac{1}{25} \frac{1}{\left(1-\beta_{5}^{2}\right)} \cos 5 \Omega t+\cdots\right]\end{aligned} \\&\text { where } \beta_{n}=\frac{n \Omega}{\omega}\end{aligned}
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