Oscillator with external forcing.

In this chapter we consider an oscillator under the influence of an external sinusoidal force $F = \cos (\omega t + \psi)$. Why this particular form of the force? This is because nearly any arbitrary time varying force $F(t)$ can be decomposed into the sum of sinusoidal forces of different frequencies

$$F(t)=\sum_{n=1,...}^{\infty} F_{n} \cos(\omega_n t + \psi_{n})$$ (3.1)

Here $F_{n}$ and are respectively the amplitude and phase of the different frequency components. Such an expansion is called a Fourier series. The behaviour of the oscillator under the influence of the force $F(t)$ can be determined by separately solving

$$m \ddot{x_{n}}+ k x_{n} = F_{n} \cos(\omega_n t + \psi_{n})$$ (3.2)

for a force with a single frequency and then superposing the solutions

$$x(t)=\sum_{n} x_{n}(t) \,.$$ (3.3)

We shall henceforth restrict our attention to equation (3.2) which has a sinusoidal force of a single frequency and drop the subscript $n$ from $x_n$ and $F_{n}$. It is convenient to switch over to the complex notation

$$\ddot{\tilde{x}}+ \omega_0^2 \tilde{x}= \tilde{f}e^{i \omega t }$$ (3.4)

where $\tilde{f}=F e^{i \psi}/m$.