Damped & Forced Oscillations

Damped & Forced Oscillations

Damping

Decrease in amplitude due to dissipative forces such as friction

Consider a frictional force proportional to vx:

So, for a body connected to a spring:

Recall that for b = 0, that is,

we showed that

with

With the presence of b in general, we have

where

Note that

a situation known as damped oscillation.

If a system with mass m connected to a spring with spring constant k is undergoing damped oscillation and you are given , then

Due to the factor e(-b/2m)t, the amplitude decreases with time (when b is large, the amplitude decreases quickly!):

6

Now, when the value of b is large enough such that

In other words, when

then

In other words, the system doesnt oscillate (which should be obvious from being zero). This situation is know as critically damped.

Again, to emphasize, the system is critically damped when

On the other hand, the system is underdamped when

while it is overdamped when

Critically Damped:

system doesnt oscillate and takes the least amount of time to return to equilibrium

Underdamped:

system oscillates with decaying amplitude

Overdamped:

system doesnt oscillate but takes a longer amount of time to return to equilibrium when compared to the case of critically damped

What type of damping do the curves a, b and c undergo?

Forced Oscillations

Consider a mass connected to a spring with spring constant k, then we drive it with a driving force given by Fmaxcos Dt.

In general, with friction, A doesnt go to infinity!

A has a maximum value when

a situation known as resonance.

Breaking Glass with Resonance

Broughton Bridge 1831

Example

Richard Feynman attached a of mass 50.0 g to a spring with k = 25.0 N/m. If its initial amplitude of A1 = 0.300 m decreases to A2 = 0.100 m in 5.00 s, find the damping coefficient b.

b = 0.0220 kg/s

Problem

A mass m = 2.20 kg attached to a spring with k = 250.0 N/m is observed to oscillate with a period of T = 0.615 s.

Is the system damped? If so, find b.

Determine whether the system is underdamped, critically damped or overdamped.