Let us return to the forced oscillations. Consider a mass-spring system as before, where we have a mass \(m\) on a spring with spring constant \(k\text{,}\) with damping \(c\text{,}\) and a force \(F(t)\) applied to the mass. Suppose the forcing function \(F(t)\) is \(2L\)-periodic for some \(L > 0\text{.}\) We saw this problem in Chapter 2 with \(F(t) = F_0 \cos (\omega t)\text{.}\) The equation that governs this particular setup is

The general solution consists of the complementary solution \(x_c\text{,}\) which solves the associated homogeneous equation \(mx'' + cx' + kx = 0\text{,}\) and a particular solution of (9) we call \(x_p\text{.}\) For \(c > 0\text{,}\) the complementary solution \(x_c\) will decay as time goes by. Therefore, we are mostly interested in a particular solution \(x_p\) that does not decay and is periodic with the same period as \(F(t)\text{.}\) We call this particular solution the steady periodic solution and we write it as \(x_{sp}\) as before. What is new in this section is that we consider an arbitrary forcing function \(F(t)\) instead of a simple cosine.

For simplicity, let us suppose that \(c=0\text{.}\) The problem with \(c > 0\) is very similar. The equation

\begin{equation*}
mx'' + kx = 0
\end{equation*}

has the general solution

\begin{equation*}
x(t) = A \cos (\omega_0 t) +
B \sin (\omega_0 t) ,
\end{equation*}

where \(\omega_0 = \sqrt{\frac{k}{m}}\text{.}\) Any solution to \(mx''(t) + kx(t) = F(t)\) is of the form \(A \cos (\omega_0 t) + B \sin (\omega_0 t) + x_{sp}\text{.}\) The steady periodic solution \(x_{sp}\) has the same period as \(F(t)\text{.}\)

In the spirit of the last section and the idea of undetermined coefficients we first write

where \(a_n\) and \(b_n\) are unknowns. We plug \(x\) into the differential equation and solve for \(a_n\) and \(b_n\) in terms of \(c_n\) and \(d_n\text{.}\) This process is perhaps best understood by example.

Example4.5.1

Suppose that \(k=2\text{,}\) and \(m=1\text{.}\) The units are again the mks units (meters-kilograms-seconds). There is a jetpack strapped to the mass, which fires with a force of 1 newton for 1 second and then is off for 1 second, and so on. We want to find the steady periodic solution.

The equation is, therefore,

\begin{equation*}
x'' + 2 x = F(t) ,
\end{equation*}

Once we plug \(x\) into the differential equation \(x''+2x = F(t)\text{,}\) it is clear that \(a_n = 0\) for \(n \geq 1\) as there are no corresponding terms in the series for \(F(t)\text{.}\) Similarly \(b_n = 0\) for \(n\) even. Hence we try

We know this is the steady periodic solution as it contains no terms of the complementary solution and it is periodic with the same period as \(F(t)\) itself. See Figure 5.5.2 for the plot of this solution.

Subsection4.5.2Resonance

Just like when the forcing function was a simple cosine, resonance can still happen. Let us assume \(c=0\) and let us discuss only pure resonance. Again, take the equation

\begin{equation*}
m x''(t) + k x (t) = F(t) .
\end{equation*}

When we expand \(F(t)\) and find that some of its terms coincide with the complementary solution to \(mx''+kx=0\text{,}\) we cannot use those terms in the guess. Just like before, they disappear when we plug them into the left hand side and we get a contradictory equation (such as \(0=1\)). That is, suppose

\begin{equation*}
x_c = A \cos (\omega_0 t) + B \sin (\omega_0 t),
\end{equation*}

where \(\omega_0 = \frac{N \pi}{L}\) for some positive integer \(N\text{.}\) In this case we have to modify our guess and try

\begin{equation*}
x(t) = \frac{a_0}{2} +
t \left(
a_N \cos \left( \frac{N \pi}{L}\, t \right) +
b_N \sin \left( \frac{N \pi}{L}\, t \right) \right) +
\sum_{\substack{n=1\\n\not= N}}^\infty
a_n \cos \left( \frac{n \pi}{L}\, t \right) +
b_n \sin \left( \frac{n \pi}{L} \, t \right) .
\end{equation*}

In other words, we multiply the offending term by \(t\text{.}\) From then on, we proceed as before.

Of course, the solution is not a Fourier series (it is not even periodic) since it contains these terms multiplied by \(t\text{.}\) Further, the terms \(t \left( a_N \cos \left( \frac{N \pi}{L}\, t \right) +
b_N \sin \left( \frac{N \pi}{L}\, t \right) \right)\) eventually dominate and lead to wild oscillations. As before, this behavior is called pure resonance or just resonance.

Note that there now may be infinitely many resonance frequencies to hit. That is, as we change the frequency of \(F\) (we change \(L\)), different terms from the Fourier series of \(F\) may interfere with the complementary solution and cause resonance. However, we should note that since everything is an approximation and in particular \(c\) is never actually zero but something very close to zero, only the first few resonance frequencies matter in real life.

If we just try a Fourier series with \(\sin (n\pi t)\) as usual, the complementary equation eats our 3rd harmonic, that is, the term with \(\sin(3 \pi t)\) is in in our complementary solution. Therefore, we pull that term out and multiply it by \(t\text{.}\) We also add a cosine term to get everything right. That is, we try

When \(c > 0\text{,}\) you do not have to worry about pure resonance. That is, there are never any conflicts and you do not need to multiply any terms by \(t\text{.}\) There is a corresponding concept of practical resonance and it is very similar to the ideas we already explored in Chapter 2. We will not go into details here.

Subsection4.5.3Exercises

Exercise4.5.2

Let \(F(t) = \frac{1}{2} + \sum_{n=1}^\infty \frac{1}{n^2} \cos (n \pi t)\text{.}\) Find the steady periodic solution to \(x'' + 2 x = F(t)\text{.}\) Express your solution as a Fourier series.

Exercise4.5.3

Let \(F(t) = \sum_{n=1}^\infty \frac{1}{n^3} \sin (n \pi t)\text{.}\) Find the steady periodic solution to \(x'' + x' + x = F(t)\text{.}\) Express your solution as a Fourier series.

Exercise4.5.4

Let \(F(t) = \sum_{n=1}^\infty \frac{1}{n^2} \cos (n \pi t)\text{.}\) Find the steady periodic solution to \(x'' + 4 x = F(t)\text{.}\) Express your solution as a Fourier series.

Exercise4.5.5

Let \(F(t) = t\) for \(-1 < t < 1\) and extended periodically. Find the steady periodic solution to \(x'' + x = F(t)\text{.}\) Express your solution as a series.

Exercise4.5.6

Let \(F(t) = t\) for \(-1 < t < 1\) and extended periodically. Find the steady periodic solution to \(x'' + \pi^2 x = F(t)\text{.}\) Express your solution as a series.

Exercise4.5.101

Let \(F(t) = \sin(2\pi t) + 0.1 \cos(10 \pi t)\text{.}\) Find the steady periodic solution to \(x'' + \sqrt{2}\, x = F(t)\text{.}\) Express your solution as a Fourier series.

Let \(F(t) = \sum_{n=1}^\infty e^{-n} \cos(2 n t)\text{.}\) Find the steady periodic solution to \(x'' + 3 x = F(t)\text{.}\) Express your solution as a Fourier series.

Let \(F(t) = \lvert t \rvert\) for \(-1 \leq t \leq 1\) extended periodically. Find the steady periodic solution to \(x'' + \sqrt{3}\, x = F(t)\text{.}\) Express your solution as a series.

Let \(F(t) = \lvert t \rvert\) for \(-1 \leq t \leq 1\) extended periodically. Find the steady periodic solution to \(x'' + \pi^2 x = F(t)\text{.}\) Express your solution as a series.