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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)2−KmS_{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)2−ωn2∴S1,2=[−ξ±ξ2−1]ωn∴  S1=[−ξ+ξ2−1]ωnAnd, S2=[−ξ−ξ2−1]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+ξ2−1]ωnAnd, S2=[−ξ−ξ2−1]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+BeS2t∴x=Ae[−ξ+ξ2−1]ωnt+Be[−ξ−ξ2−1]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+ξ2−1]ωnt[−ξ+ξ2−1]ωn +Be[−ξ−ξ2−1]ωnt[−ξ−ξ2−1]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+ξ2−1]ωn+B[−ξ−ξ2−1]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=X0−AB= X_0 - A and putting value of B in (4);

∴0= A[−ξ+ξ2−1]ωn+(X0−A)[−ξ−ξ2−1]ωn∴0= −Aξ+Aξ2−1+X0[−ξ−ξ2−1]+Aξ+Aξ2−1∴0= 2Aξ2−1+X0[−ξ−ξ2−1]∴2Aξ2−1 = X0[ ξ+ξ2−1]∴A= X0[ξ+ξ2−1]2ξ2−1……(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}

Now,
Putting A=X0−BA=X_0-B, in equation (4);

∴0= (X0−B)[−ξ+ξ2−1]ωn+B[−ξ−ξ2−1]ωn∴0= X0[−ξ+ξ2−1]+Bξ−Bξ2−1−Bξ−Bξ2−1∴0= X0[−ξ+ξ2−1]−2Bξ2−1∴2Bξ2−1= X0[−ξ+ξ2−1]∴B= X0[−ξ+ξ2−1]2ξ2−1……(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+ξ2−1]2ξ2−1e[−ξ+ξ2−1]ωnt+X0[−ξ+ξ2−1]2ξ2−1e[−ξ−ξ2−1]ωntx=X02ξ2−1[(ξ+ξ2−1)e[−ξ+ξ2−1]ωnt+(−ξ+ξ2−1)e[−ξ−ξ2−1]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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Help us build Education Lessons

If our notes and videos are helpful to you, kindly support us by making a donation from our support page so we can continue making more content for students like you.

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Suggested Notes:

  1. Balancing of V-Engines
  2. Damped free Vibration - Numerical 1
  3. Critically Damped System (ξ = 1)
  4. Damped free Vibration - Numerical 4
  5. Logarithmic Decrement (​δ)
  6. Under-Damped System (​ξ < 1)
  7. Static and Dynamic Balancing
  8. Concept of Direct and Reverse Crank for V-engines & Radial engines
  9. Numerical - Multi Cylinder Inline Engine (with firing order)

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