Chapter 7.3, Problem 1.2SP

Resonance

Consider an undamped system exhibiting simple harmonic motion. In the real world, we never truly have an undamped system; -some damping always occurs. For theoretical purposes, however, we could imagine a spring-mass system contained in a vacuum chamber. With no air resistance, the mass would continue to move up and down indefinitely.

The frequency of the resulting motion, given by $f=\frac{1}{T}=\frac{\omega }{2\pi }$ , is called the natural frequency of the system. If an external force acting on the system has a frequency close to the natural frequency of the system, a phenomenon called resonance results. The external forte reinforces and amplifies the natural motion of the system.

2. Now suppose this system is subjected to an external force given by $f\left(t\right)=5\cos t$ . Solve the initial-value problem $x^{″}+x=5\cos t$ , $x\left(0\right)=0$ , $x^{\prime }\left(0\right)=1$ .