1 Modelling vibration by differential equation

Vibration problems are often modelled by ordinary differential equations with constant coefficients. For example the motion of a spring with stiffness $k$ and damping constant $c$ is modelled by

$m\frac{d^{2}y}{d{t}^2}+c\frac{dy}{dt}+ky=0$ (1)

where $y\left(t\right)$ is the displacement of a mass $m$ connected to the spring. It is well-known that if $c^2<4mk$ , usually referred to as the lightly damped case, then

$y\left(t\right)={\text{e}}^{-\alpha t}\left(Acos\omega t+Bsin\omega t\right)$ (2)

i.e. the motion is sinusoidal but damped by the negative exponential term. In (2) we have used the notation

$\alpha =\frac{c}{2m}\omega =\frac{1}{2m}\sqrt{4km-c^2}$ to simplify the equation.

The values of $A$ and $B$ depend upon initial conditions.

The system represented by (1), whose solution is (2), is referred to as an unforced damped harmonic oscillator .

A lightly damped oscillator driven by a time-dependent forcing function $F\left(t\right)$ is modelled by the differential equation

$m\frac{d^{2}y}{d{t}^2}+c\frac{dy}{dt}+ky=F\left(t\right)$ (3)

The solution or system response in (3) has two parts:

1. A transient solution of the form (2),
2. A forced or steady state solution whose form, of course, depends on $F\left(t\right)$ .

If $F\left(t\right)$ is sinusoidal such that

$F\left(t\right)=Asin\left(\Omega t+\varphi \right)$   where $\Omega$ and $\varphi$ are constants,

then the steady state solution is fairly readily obtained by standard techniques for solving differential equations. If $F\left(t\right)$ is periodic but non-sinusoidal then Fourier series may be used to obtain the steady state solution. The method is based on the principle of superposition which is actually applicable to any linear (homogeneous) differential equation. (Another engineering application is the series $LCR$ circuit with an applied periodic voltage.)

The principle of superposition is easily demonstrated:-

Let $y_1\left(t\right)$ and $y_2\left(t\right)$ be the steady state solutions of (3) when $F\left(t\right)=F_1\left(t\right)$ and $F\left(t\right)=F_2\left(t\right)$ respectively. Then

$m\frac{d^2{y}_1}{d{t}^2}+c\frac{d{y}_1}{dt}+k{y}_1=F_1\left(t\right)$

$m\frac{d^2{y}_2}{d{t}^2}+c\frac{d{y}_2}{dt}+k{y}_2=F_2\left(t\right)$

Simply adding these equations we obtain

$m\frac{d^2}{d{t}^2}\left(y_1+y_2\right)+c\frac{d}{dt}\left(y_1+y_2\right)+k\left(y_1+y_2\right)=F_1\left(t\right)+F_2\left(t\right)$

from which it follows that if $F\left(t\right)=F_1\left(t\right)+F_2\left(t\right)$ then the system response is the sum $y_1\left(t\right)+y_2\left(t\right)$ . This, in its simplest form, is the principle of superposition. More generally if the forcing function is

$F\left(t\right)={\sum }_{n=1}^N{F}_n\left(t\right)$

then the response is $y\left(t\right)={\sum }_{n=1}^N{y}_n\left(t\right)$ where $y_n\left(t\right)$ is the response to the forcing function $F_n\left(t\right)$ .

Returning to the specific case where $F\left(t\right)$ is periodic, the solution procedure for the steady state response is as follows:

Step 1 :

Obtain the Fourier series of $F\left(t\right)$ .

Step 2 :

Solve the differential equation (3) for the response $y_n\left(t\right)$ corresponding to the $n^{\text{ th}}$ harmonic in the Fourier series. (The response $y_o$ to the constant term, if any, in the Fourier series may have to be obtained separately.)

Step 3 :

Superpose the solutions obtained to give the overall steady state motion:

$y\left(t\right)=y_0\left(t\right)+{\sum }_{n=1}^N{y}_n\left(t\right)$

The procedure can be lengthy but the solution is of great engineering interest because if the frequency of one harmonic in the Fourier series is close to the natural frequency $\sqrt{\frac{k}{m}}$ of the undamped system then the response to that harmonic will dominate the solution.