We consider the damped, mass-spring system with a sinusoidal external force:

where .

By using the method of undetermined coefficients, one can shown that

where complementary solution, , is the solution of the corresponding homogeneous equation.

In real-world damped systems, we must have as , and therefore, is called the *transient part* of the solution. The transient part depends only on the parameters of the system (, , and ) and the initial conditions. The other term of the solution is called the *steady-state part* or the *steady-periodic part* of the solution.

The expression

is called the *gain factor*. Notice that the gain factor depends on the system parameters and the frequency of the external force.

Let's find the maximum value of .

Upon differentiating, we find that

and if and only if

We use to denote the frequency on the right.

If the system is over-damped () or critically damped (), then is imaginary, and we must have the maximum when . In this case, the maximum gain is .

If the system is under-damped (), we have the maximum when . In this case, the maximum gain is

and is called the *resonance frequency*. **For undamped systems, maximum gain occurs when the frequency of the external force is equal to the resonance frequency.** Notice that as , the gain increases without bound and ( is the natural frequency of the system).

Solve:

Here is the Sage code.

`t=var("t")`

`x=function("x")(t)`

`de=10*diff(x,t,2)+3*diff(x,t)+49*x==20*cos(4*t)`

`desolve(de,x,[0,5,0])`

The solution is

which can be written in the approximate form

The first term is the transient part. Notice that it decays with time. The second term is the steady-state part. It dominates the solution for large .

The graph of is shown here.

In this example, the resonance frequency is , which is not particularly close to the frequency of the external force (). The maximum gain is rather small.