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DAMPING RATIO OR DAMPING FACTOR.
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It is defined as the ratio of actual damping co-efficient to the critical damping coefficient.

i.e. $\xi = \frac{c}{c_c}$

Hence $\xi = \frac{c}{2mw_n}$

$\therefore$ $\xi w_n = \frac{c}{2m}$

Then the roots of characteristics equation can be written as

$s_1 = (\xi w_n) + \sqrt{ (\xi w_n)^2 - w^2_n}$

$= [ - (\xi) + \sqrt{ - \xi^2 - 1]} w_n$

Similarly $s_2 = [ -\xi - \sqrt{ \xi^2 - 1]} w_n$

If $c_c \lt c$ then $\xi \gt 1$, the system is said to be over damped.

If $c_c = c$ then $\xi = 1$, then system is said to be critically damped.

If $c_c \gt c$ then $\xi \lt 1$, then system is said to be under damped.

From above analysis, depending upon damping factor, values of $s_1$ and $s_2$ are real and unequal, real and equal, complex conjugate respectively.

Case 1: General solution for over damped system, i.e. $(\xi \gt 1)$ :

$x = c_1e^{s1t} + c_2e^{s2t}$ - - -(1)

$c_1$ and $c_2$ are constants to be determined by known boundary condition.

Differentiating equation (1), we can write

$x = s_1 \ c_1e^{s1t} + s_2 \ c_2 \ e^{s2t}$ - - -(2)

then using boundary conditions.

At t = 0, x = $x_0$ and t = 0 x = $x_0$

Substituting 1st boundary condition in (1), we get

$x_0 = c_1 - c_2$

$\therefore$ $x_0 = x_0 - c_2$ - - - (3)

Substituting 2nd boundary condition in (2), we get

$x_0 = s_1 c_1 + s_2 c_2$ - - -(4)

From equations (3) and (4)

$x_0 = s_1 (x_0 - c_2) + s_2 c_2$

$= s_1 x_0 - s_1 c_2 + s_2 c_2$

$\therefore$ $s_1 c_2 - s_2 c_2 = s_1 x_0 - x_0$

$\therefore$ $c_2 = \frac{s_1 x_0 - x_0}{(s_1 - s_2)}$ - - -(5)

Then from equation (3),

$c_1 = x_0 - \frac{s_1 x_0 - x_0}{s_1 - s_2}$

$= \frac{x_0 s_1 - x_0 s_2 - s_1 x_0 + x_0}{s_1 - s_2}$

$c_1 = \frac{x_0 - x_0 s_2}{s_1 - s_2}$ - - -(6)

From equations (5), (6) and (1) , we get the general solution.

$x = (\frac{x_0 - x_0 s_2}{s_1- s_2}) e^{s1t} + (\frac{s_1 x_0 - x_0}{s_1 - s_2})e^{s2t} = \frac{1}{s_1 - s_2} [[( x_0 - x_0 s_2]e^{s1t} + [s_1 x_0 - x_0]e^{s2t}]$

Substituting $s_1 ( -\xi + \sqrt{\xi ^2 - 1})w_n$ and $s_2 = (-\xi - \sqrt{\xi ^2 - 1}) w_n$

$\therefore$ $x = \frac{1}{2 \sqrt{\xi ^2 - 1} w_n} [ [x_0 - x_0 ( - \xi - \sqrt{ \xi^2 - 1 }) w_n ] e^ {(-\xi + \sqrt{\xi ^2 - 1} w_nt]}$

$[ ( - \xi + \sqrt{ \xi ^2 - 1} w_n x_0 - x_0 ] e^{( - \xi - \sqrt{\xi ^2 - 1}) w_nt ]}$

$= \frac{e^{-\xi w_nt}}{2 \sqrt{ \xi ^2 - 1} w_n}$

$[ x_0 - w_n x_0 ( - \xi - \sqrt{ \xi ^2 - 1} ] e^ {(\sqrt{ \xi^2 - 1}) w_{nt} ]}$

$[ w_n x_0 (-\xi + \sqrt{ \xi ^2 - 1}) - x_0 ] e^{(-\xi - \sqrt{\xi ^2 - 1} ) w_{nt}]}$

$\therefore$ $x = \frac{x_0 e^{-\xi e_{nt}}}{2 \ sqrt{\xi^2 - 1}w_n}$

$[\frac{x_0}{x_0} - w_n ( \xi - \sqrt{\xi ^2 - 1} ] e^{ (\sqrt{\xi ^2 - 1)} w_{nt} ]}$

$[ w_n ( - \xi + \sqrt{\xi ^2 - 1}) - \frac{x_0}{x_0}] e ^{ (- \sqrt{\xi ^2 - 1}) w_{nt}]}$

Above equation represents the relation between displacement x and time t.

Then the response curve with respect to time for above case is as shown. From above graph following conclusions can be drawn.

1] The displacement x exponentially decreases as time increases.

2] Displacement will be zero when $t \rightarrow \infty$

Thus it can be said that over damped system motion is 'a periodic'.

3] In above graph it is seen that as $' \xi '$ increases displacement for same time is more as system becomes more sluggish.