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Over-Damped System (​ξ>1)

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over-damped-system Fig-1 (Over damped system)

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

mS2+cS+K=0mS^2 + cS + K = 0

This is a quadratic equation having two roots S1S_1 and S2S_2;

S1,2=c2m±(c2m)2KmS_{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 ξ\xi , we will use two parameters, critical damping coefficient 'ccc_c' and damping factor 'ξ\xi'. So the roots S1S_1 and S2S_2 can be written as follows;

ξ=cccORξ=c2mω0(ascc=2mωn)c2m=ξωn\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, ωn\omega_n = natural frequency of undamped free vibration = Km\sqrt{K \over m} rad/s

ωn2=Km\therefore \omega_n^2 = {K \over m}

So we can write roots S1S_1 and S2S_2 and as;

S1,2=ξωn±(ξωn)2ωn2S1,2=[ξ±ξ21]ωn  S1=[ξ+ξ21]ωnAnd, S2=[ξξ21]ωn\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 ‘ξ\xi’ is greater than one or the damping coefficient ‘cc’ is greater than critical damping coefficient ‘ccc_c’, then the system is said to be over-damped.

ξ>1OrcccOrc>cc\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;

S1=[ξ+ξ21]ωnAnd, S2=[ξξ21]ωn\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 ξ>1\xi > 1 , we get S1S_1 and S2S_2 as real and negative so we get,

x=AeS1t+BeS2tx=Ae[ξ+ξ21]ωnt+Be[ξξ21]ωnt(1)\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;

x˚=Ae[ξ+ξ21]ωnt[ξ+ξ21]ωn +Be[ξξ21]ωnt[ξξ21]ωn(2)\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=0t = 0 : x=X0x = X_0
t=0t = 0 : x˚=0\mathring x =0

Substituting this value in equation (1) we get;

X0=A+B(3)X_0 = A+B\quad \dots \dots \dots \text{(3)}

Substituting this value in equation (2) we get;

0=A[ξ+ξ21]ωn+B[ξξ21]ωn(4)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=X0AB= X_0 - A and putting value of B in (4);

0= A[ξ+ξ21]ωn+(X0A)[ξξ21]ωn0= Aξ+Aξ21+X0[ξξ21]+Aξ+Aξ210= 2Aξ21+X0[ξξ21]2Aξ21 = X0[ ξ+ξ21]A= X0[ξ+ξ21]2ξ21(5)\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}

Putting A=X0BA=X_0-B, in equation (4);

0= (X0B)[ξ+ξ21]ωn+B[ξξ21]ωn0= X0[ξ+ξ21]+BξBξ21BξBξ210= X0[ξ+ξ21]2Bξ212Bξ21= X0[ξ+ξ21]B= X0[ξ+ξ21]2ξ21(6)\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;

x=X0[ξ+ξ21]2ξ21e[ξ+ξ21]ωnt+X0[ξ+ξ21]2ξ21e[ξξ21]ωntx=X02ξ21[(ξ+ξ21)e[ξ+ξ21]ωnt+(ξ+ξ21)e[ξξ21]ωnt]\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.

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