- Solve equations for simple harmonic motion
- Solve equations for damped harmonic motion
- Solve equations for forced harmonic motion
- Apply differential equations to electrical circuits
Modeling Harmonic Motion and Resonance with Differential Equations
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 [latex]f=\frac{1}{T}=\frac{\omega}{2\pi}[/latex], 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 force reinforces and amplifies the natural motion of the system.
- Consider the differential equation [latex]x^{\prime\prime}+x=0[/latex]. Find the general solution. What is the natural frequency of the system?
- Now suppose this system is subjected to an external force given by [latex]f(t)=5\cos t[/latex]. Solve the initial-value problem [latex]x^{\prime\prime}+x=5\cos t[/latex], [latex]x(0)=0[/latex], [latex]x^\prime(0)=1[/latex].
- Graph the solution. What happens to the behavior of the system over time?
- In the real world, there is always some damping. However, if the damping force is weak, and the external force is strong enough, real-world systems can still exhibit resonance. One of the most famous examples of resonance is the collapse of the Tacoma Narrows Bridge on November 7, 1940. The bridge had exhibited strange behavior ever since it was built. The roadway had a strange “bounce” to it. On the day it collapsed, a strong windstorm caused the roadway to twist and ripple violently. The bridge was unable to withstand these forces and it ultimately collapsed. Experts believe the windstorm exerted forces on the bridge that were very close to its natural frequency, and the resulting resonance ultimately shook the bridge apart.
- Another real-world example of resonance is a singer shattering a crystal wineglass when she sings just the right note. When someone taps a crystal wineglass or wets a finger and runs it around the rim, a tone can be heard. That note is created by the wineglass vibrating at its natural frequency. If a singer then sings that same note at a high enough volume, the glass shatters as a result of resonance.