Damped Oscillators driven by an external source

An external force must be supplied to do work on a damped oscillator in an amount that is equal to or greater than the work done by the dissipative force.

An external force (source) is added to the homogeneous differential equation making it inhomogenous

making it

where f(t) represents the external force (source) that depends on time divided by the objects mass.

Differential equations in Operator form

In the previous sections we used the definition

to solve the second order linear differential equation.

Let's take this a step further with the following operator definition

then

becomes

Linear differential equations have coefficient that can constant or variable coefficients that can be transformed into constant coefficients.

Note
is a linear operator

meaning

the above is a property of differential calculus where

and

Solving the Inhomogeneous Diff. Eq.

To solve the Inhomogeneous problem we take advantage of the linear operator such that

Since the solution of the 2nd order differential equation requires exactly two arbitrary constant,

We can formulate the solution by tacking a particular solution onto the homogeneous solution that already has two arbitrary constants.

All we need to do is find the particular solution that satisfies

and we will have a complete set of solutions.

Break the equation up into a Homogeneous solution and a Particular Solution

Homogeneous Solution

Th Homogenous solution solved the equation

we know from the previous section that the homogenous solution has the form

where

Case 1 : f(t) is sinusoidal

Consider the case where the driving force is a sinusoidal function

We seek a solution to the particular equation

Remember
is not necessarily at the natural (resonant) frequency

Trick
If a differential equation has the cosine function as a solution then the sine function may also be a solution since the difference between the two is only a phase shift.

It must also be try that

You can construct a complex solution now such that

the constant C is determine by substituting the solution into the equation

The amplitude of the solution is a complex number that may be cast in terms of a real amplitude times the complex exponential such that

where the real amplitude A is given by

What is delta?

Since and are not imaginary variable

The phase and must be due to the remaining complex term

To detmine the angle we construct a right trianble in the complex plane where the leg along the real axis is given by and the other leg of the triangle is along the imaginary axis with a length of

The final complex solution si

with

where