# Over-Damped System (​ξ>1)

Fig-1 (Over damped system)

We know that the characteristic equation of the damped free vibration system is,

$mS^2 + cS + K = 0$

This is a quadratic equation having two roots $S_1$ and $S_2$;

$S_{1,2} = {-c \over 2m} \pm \sqrt{\bigg({-c \over 2m}\bigg)^2 - {K \over m}}$

In order to convert the whole equation in the form of $\xi$ , we will use two parameters, critical damping coefficient '$c_c$' and damping factor '$\xi$'. So the roots $S_1$ and $S_2$ can be written as follows;

\begin{aligned} \xi= {c \over c_c} \quad \text {OR} \quad \xi &= {c \over 2m\omega_0} \quad (\text {as} \quad c_c=2m\omega_n)\\ \therefore {c \over 2m} &= \xi \omega_n \end{aligned}

where, $\omega_n$ = natural frequency of undamped free vibration = $\sqrt{K \over m}$ rad/s

$\therefore \omega_n^2 = {K \over m}$

So we can write roots $S_1$ and $S_2$ and as;

\begin{aligned} S_{1,2} &= -\xi \omega_n \pm \sqrt{(\xi \omega_n)^2 - \omega_n^2}\\ \therefore S_{1,2} &= [-\xi \pm \sqrt{\xi^2 - 1}]\omega_n\\ \therefore \ \ S_{1} &= [-\xi + \sqrt{\xi^2 - 1}]\omega_n\\ \text{And, }\\ S_{2} &= [-\xi - \sqrt{\xi^2 - 1}]\omega_n \end{aligned}

## Overdamped system (ξ>1)

If the damping factor ‘$\xi$’ is greater than one or the damping coefficient ‘$c$’ is greater than critical damping coefficient ‘$c_c$’, then the system is said to be over-damped.

$\xi > 1 \quad \text{Or} \quad {c \over c_c}\quad \text{Or} \quad c > c_c$

In overdamped system, the roots are given by;

\begin{aligned} S_{1} &= [-\xi + \sqrt{\xi^2 - 1}]\omega_n\\ \text{And, }\\ S_{2} &= [-\xi - \sqrt{\xi^2 - 1}]\omega_n \end{aligned}

For $\xi > 1$ , we get $S_1$ and $S_2$ as real and negative so we get,

\begin{aligned} x &= Ae^{S_1t} + Be^{S_2t}\\ \therefore x &= Ae^{[-\xi + \sqrt{\xi^2-1}]\omega_nt} + Be^{[-\xi - \sqrt{\xi^2-1}]\omega_nt} \quad \dots \dots \text{(1)} \end{aligned}

Now differentiating equation (1) with respect to ‘t’, we get;

\begin{aligned} \mathring x = &Ae^{[-\xi + \sqrt{\xi^2-1}]\omega_nt}[-\xi + \sqrt{\xi^2-1}]\omega_n \ + \\ &Be^{[-\xi - \sqrt{\xi^2-1}]\omega_nt} [-\xi - \sqrt{\xi^2-1}]\omega_n\quad \dots \dots \dots \text{(2)} \end{aligned}

Now, let at
$t = 0$ : $x = X_0$
$t = 0$ : $\mathring x =0$

Substituting this value in equation (1) we get;

$X_0 = A+B\quad \dots \dots \dots \text{(3)}$

Substituting this value in equation (2) we get;

$0= A[-\xi + \sqrt{\xi^2-1}]\omega_n + B[-\xi - \sqrt{\xi^2-1}]\omega_n \quad \dots \dots \text{(4)}$

From equation (3), $B= X_0 - A$ and putting value of B in (4);

\begin{alignedat}{2} &\therefore \qquad 0 &=& \ A[-\xi + \sqrt{\xi^2-1}]\omega_n + (X_0 - A)[-\xi - \sqrt{\xi^2-1}]\omega_n\\ &\therefore \qquad 0 &=& \ -A\xi + A\sqrt{\xi^2-1} + X_0 [-\xi - \sqrt{\xi^2-1}]+A\xi+A\sqrt{\xi^2-1}\\ &\therefore \qquad 0 &=& \ 2A\sqrt{\xi^2-1} + X_0 [-\xi - \sqrt{\xi^2-1}]\\ &\therefore 2A\sqrt{\xi^2-1} \ &=& \ X_0 [\ \xi + \sqrt{\xi^2-1}]\\ &\therefore \qquad A &=& \ {X_0 [\xi + \sqrt{\xi^2-1}] \over 2\sqrt{\xi^2-1} }\quad \dots \dots \text{(5)} \end{alignedat}

Now,
Putting $A=X_0-B$, in equation (4);

\begin{alignedat}{2} &\therefore \qquad 0 &=& \ (X_0-B)[-\xi + \sqrt{\xi^2-1}]\omega_n + B[-\xi - \sqrt{\xi^2-1}]\omega_n\\ &\therefore \qquad 0 &=& \ X_0[-\xi + \sqrt{\xi^2-1}] + B\xi - B\sqrt{\xi^2-1} - B\xi - B\sqrt{\xi^2-1}\\ &\therefore \qquad 0 &=& \ X_0[-\xi + \sqrt{\xi^2-1}] - 2B\sqrt{\xi^2-1}\\ &\therefore 2B\sqrt{\xi^2-1} &=& \ X_0[-\xi + \sqrt{\xi^2-1}]\\ &\therefore \qquad B &=& \ {X_0[-\xi + \sqrt{\xi^2-1}] \over 2\sqrt{\xi^2-1}} \quad \dots \dots \text{(6)} \end{alignedat}

Now putting equation (5) and (6) in equation (1), we get;

\begin{aligned} x = &{X_0 [\xi + \sqrt{\xi^2-1}] \over 2\sqrt{\xi^2-1} } e^{[-\xi + \sqrt{\xi^2-1}]\omega_nt} +\\ &{X_0[-\xi + \sqrt{\xi^2-1}] \over 2\sqrt{\xi^2-1}} e^{[-\xi - \sqrt{\xi^2-1}]\omega_nt}\\ x = &{X_0 \over 2\sqrt{\xi^2-1}} \bigg [\bigg (\xi + \sqrt{\xi^2-1} \bigg) e^{[-\xi + \sqrt{\xi^2-1}]\omega_nt} + \\ &\bigg (-\xi + \sqrt{\xi^2-1} \bigg ) e^{[-\xi - \sqrt{\xi^2-1}]\omega_nt} \bigg ] \end{aligned}

Above equation represents the equation of motion for overdamped system.

The motion obtained by above equation is aperiodic (aperiodic motion motions are those motions in which the motion does not repeats after a regular interval of time i.e non periodic motion) and so the system does not shows vibrations.

This type of system does not show much damping, so this systems are used very rarely.