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Damped Harmonic Motion

Let's take stock. In the previous chapter we found that

satisfies the basic differential equation

defining damped motion (e.g. motion under the influence of a frictional force proportional to the velocity). We now have a solution to the equation of motion defining SHM,

where

Can we put these together to "solve" the more general (and realistic) problem of damped harmonic motion? The differential equation would then read

 (13.20)

which is beginning to look a little hard. Still, we can sort it out: the first term on the RHS says that there is a linear restoring force and an inertial factor. The second term says that there is a damping force proportional to the velocity. So the differential equation itself is not all that fearsome. How can we "solve" it?

As always, by trial and error. Since we have found the exponential function to be so useful, let's try one here: Suppose that

 (13.21)

where  x0  and  K  are unspecified constants. Now plug this into the differential equation and see what we get:

and

which can be true "for all  x" only if

which is in the standard form of a general quadratic equation for  K,  to which there are two solutions:

 (13.22)

Either of the two solutions given by substituting Eq. (22) into Eq. (21) will satisfy Eq. (20) describing damped SHM. In fact, generally any linear combination of the two solutions will also be a solution. This can get complicated, but we have found the answer to a rather broad question.

Next: Limiting Cases Up: Simple Harmonic Motion Previous: Imaginary Exponents
Jess H. Brewer - Last modified: Sun Nov 15 13:37:26 PST 2015