2404 Exercises

Ex 1. For each spring-mass system, find whether pure resonance occurs, without actually calculating the solution.
a) $2x''+10x=F(t); F(t) = 1 \text{ on } (0,1), F(t)$ is odd, and of period 2;
a) $x''+4\pi^2x=F(t); F(t) = 2t \text{ on } (0,1), F(t)$ is odd, and of period 2;
a) $x''+9x=F(t); F(t) = 1 \text{ on } (0,\pi), F(t)$ is odd, and of period 2$\pi$;

Consider $$mx''+kx=F(t)$$ The natural frequency of this spring-mass system is $$\omega_0=\sqrt{\frac{k}{m}}$$ The typical term of the Fourier expansion of $F(t)$ is $\cos \frac{n\pi}{L}t$, $\sin \frac{n\pi}{L}t$; thus we get pure resonance if and only if the Fourier series has a term of the form $\cos \frac{n\pi}{L}t$ or $\sin \frac{n\pi}{L}t$, where $\frac{n\pi}{L} = \omega_0$.

a) $\omega_0=\sqrt{5}$ for spring-mass system, and $L = 1$. Fourier series is $\sum b_n\sin n\pi t$; $n\pi \neq \sqrt{5}$, so no resonance.
b) $\omega_0=2\pi$, $L = 1$. Fourier series is $\sum b_n\sin n\pi t$, and $n\pi=2\pi$ if $n=2$. Thus, do get resonance.
c) $\omega_0=3$, Fourier series is a sine series ($F(t)$ is odd): $F(t)=\sum b_n\sin n t$ - all odd $n$ occur ($b_n=0$ if $n$ is even), so $n = 3$ occurs and do get resonance.