15.5 Damped oscillations  (Page 2/6)

The angular frequency for damped harmonic motion becomes

Recall that when we began this description of damped harmonic motion, we stated that the damping must be small. Two questions come to mind. Why must the damping be small? And how small is small? If you gradually increase the amount of damping in a system, the period and frequency begin to be affected, because damping opposes and hence slows the back and forth motion. (The net force is smaller in both directions.) If there is very large damping, the system does not even oscillate—it slowly moves toward equilibrium. The angular frequency is equal to

As b increases, $\frac{k}{m}-{\left(\frac{b}{2m}\right)}^2$ becomes smaller and eventually reaches zero when $b=\sqrt{4mk}$ . If b becomes any larger, $\frac{k}{m}-{\left(\frac{b}{2m}\right)}^2$ becomes a negative number and $\sqrt{\frac{k}{m}-{\left(\frac{b}{2m}\right)}^2}$ is a complex number.

[link] shows the displacement of a harmonic oscillator for different amounts of damping. When the damping constant is small, $b<\sqrt{4mk}$ , the system oscillates while the amplitude of the motion decays exponentially. This system is said to be underdamped    , as in curve (a). Many systems are underdamped, and oscillate while the amplitude decreases exponentially, such as the mass oscillating on a spring. The damping may be quite small, but eventually the mass comes to rest. If the damping constant is $b=\sqrt{4mk}$ , the system is said to be critically damped    , as in curve (b). An example of a critically damped system is the shock absorbers in a car. It is advantageous to have the oscillations decay as fast as possible. Here, the system does not oscillate, but asymptotically approaches the equilibrium condition as quickly as possible. Curve (c) in [link] represents an overdamped    system where $b>\sqrt{4mk}.$ An overdamped system will approach equilibrium over a longer period of time.

Critical damping is often desired, because such a system returns to equilibrium rapidly and remains at equilibrium as well. In addition, a constant force applied to a critically damped system moves the system to a new equilibrium position in the shortest time possible without overshooting or oscillating about the new position.

Summary

• Damped harmonic oscillators have non-conservative forces that dissipate their energy.
• Critical damping returns the system to equilibrium as fast as possible without overshooting.
• An underdamped system will oscillate through the equilibrium position.
• An overdamped system moves more slowly toward equilibrium than one that is critically damped.

Conceptual questions

Problems