Chapter 7.3, Problem 1.1SP

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.

1. Consider the differential equation $x^{″}+x=0$ . Find the general solution. What is the natural frequency of the system?