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Section 14.2 Adding Damping

Imagine now that some sort of damping mechanism is added to the object-spring system as shown in Figure 14.2.1. (Such a system might be used as a simple model of a car shock absorber for example.)

Figure 14.2.1.
So long as the speed of the object is not too fast then a reasonable assumption is that the damping force acting on the object, \(\mathbf{F_d}\text{,}\) is proportional to the speed and acts in the direction opposing the motion of the object. Thus

\begin{equation*} F_d=-D\frac{dy}{dt} \end{equation*}

where \(D\gt 0\text{.}\) Thus the motion of the object can be modelled by the DE

\begin{equation*} m\frac{d^2y}{dt^2}=-ky-D\frac{dy}{dt} \end{equation*}

which can be rearranged as

\begin{equation*} y''+\left(\frac{D}{m}\right)y'+\left(\frac{k}{m}\right)y=0 \end{equation*}

i.e. a homogeneous 2nd order linear DE with constant coefficients. It is convenient in the subsequent algebra to let

\begin{equation*} \frac{D}{m}=2p \textrm{ and } \frac{k}{m}=q^2 \end{equation*}

and so the model becomes

\begin{equation} y''+2py'+q^2y=0,\hspace{5mm} y(0)=-L,\hspace{5mm} y'(0)=0, \hspace{5mm}p\gt 0,\hspace{5mm} q\gt 0\label{Eqn_14_2_1}\tag{14.2.1} \end{equation}

The characteristic equation for (14.2.1) is

\begin{equation*} r^2+2pr+q^2=0 \end{equation*}

which gives

\begin{equation*} r=\frac{-2p\pm \sqrt{4p^2-4q^2}}{2}=-p\pm \sqrt{p^2-q^2} \end{equation*}

Thus the nature of the general solution to (14.2.1) depends upon the sign of \(p^2-q^2\text{,}\) or equivalently, on the magnitude of the parameter \(\delta=\frac{p}{q}\text{.}\) \(\delta\) is called the damping parameter of the system.

If \(p^2-q^2\gt 0\) (or equivalently \(\delta=\frac{p}{q}\lt 1\)) then the general solution is

\begin{equation*} y(t)=e^{-pt}(C_1\cos(\omega t)+C_2\sin(\omega t)) \end{equation*}

where \(\omega=\sqrt{q^2-p^2}\) and hence the function will be like that shown in Figure 14.2.2. In this case the system is called under-damped.

Figure 14.2.2.

If \(p^2-q^2\gt 0\) (or equivalently \(\delta=\frac{p}{q}\gt 1\)) then the general solution is

\begin{equation*} y(t)=C_1e^{r_1t}+C_2e^{r_2t} \end{equation*}

where both \(r_1\) and \(r_2\) are negative and hence the function will be like that shown in Figure 14.2.3. In this case the system is called over-damped.

Figure 14.2.3.
Finally if \(p^2-q^2=0\) (or equivalently \(\delta=\frac{p}{q}=1\)) then the general solution is

\begin{equation*} y(t)=(C_1+C_2t)e^{-pt} \end{equation*}

and hence the function will be like that shown in Figure 14.2.4. In this case the system is called critically-damped.

Figure 14.2.4.
The critically-damped case is of practical interest. Often we want to add damping to a system so that it returns close to its equilibrium position "as quickly as possible", (for example in the shock absorber of a car). It turns out that this is achieved by critical damping.