Fig-1 (Over damped system)
We know that the characteristic equation of the damped free vibration system is,
m S 2 + c S + K = 0 mS^2 + cS + K = 0 m S 2 + c S + K = 0 This is a quadratic equation having two roots S 1 S_1 S 1 S 2 S_2 S 2
S 1 , 2 = − c 2 m ± ( − c 2 m ) 2 − K m S_{1,2} = {-c \over 2m} \pm \sqrt{\bigg({-c \over 2m}\bigg)^2 - {K \over m}} S 1 , 2 = 2 m − c ± ( 2 m − c ) 2 − m K
In order to convert the whole equation in the form of ξ \xi ξ c c c_c c c ξ \xi ξ S 1 S_1 S 1 S 2 S_2 S 2
ξ = c c c OR ξ = c 2 m ω 0 ( as c c = 2 m ω n ) ∴ c 2 m = ξ ω n \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} ξ = c c c OR ξ ∴ 2 m c = 2 m ω 0 c ( as c c = 2 m ω n ) = ξ ω n where, ω n \omega_n ω n K m \sqrt{K \over m} m K
∴ ω n 2 = K m \therefore \omega_n^2 = {K \over m} ∴ ω n 2 = m K So we can write roots S 1 S_1 S 1 S 2 S_2 S 2
S 1 , 2 = − ξ ω n ± ( ξ ω n ) 2 − ω n 2 ∴ S 1 , 2 = [ − ξ ± ξ 2 − 1 ] ω n ∴ S 1 = [ − ξ + ξ 2 − 1 ] ω n And, S 2 = [ − ξ − ξ 2 − 1 ] ω n \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} S 1 , 2 ∴ S 1 , 2 ∴ S 1 And, S 2 = − ξ ω n ± ( ξ ω n ) 2 − ω n 2 = [ − ξ ± ξ 2 − 1 ] ω n = [ − ξ + ξ 2 − 1 ] ω n = [ − ξ − ξ 2 − 1 ] ω n
If the damping factor ‘ξ \xi ξ c c c c c c_c c c
ξ > 1 Or c c c Or c > c c \xi > 1 \quad \text{Or} \quad {c \over c_c}\quad \text{Or} \quad c > c_c ξ > 1 Or c c c Or c > c c In overdamped system, the roots are given by;
S 1 = [ − ξ + ξ 2 − 1 ] ω n And, S 2 = [ − ξ − ξ 2 − 1 ] ω n \begin{aligned}
S_{1} &= [-\xi + \sqrt{\xi^2 - 1}]\omega_n\\
\text{And, }\\
S_{2} &= [-\xi - \sqrt{\xi^2 - 1}]\omega_n
\end{aligned} S 1 And, S 2 = [ − ξ + ξ 2 − 1 ] ω n = [ − ξ − ξ 2 − 1 ] ω n For ξ > 1 \xi > 1 ξ > 1 S 1 S_1 S 1 S 2 S_2 S 2
x = A e S 1 t + B e S 2 t ∴ x = A e [ − ξ + ξ 2 − 1 ] ω n t + B e [ − ξ − ξ 2 − 1 ] ω n t … … (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} x ∴ x = A e S 1 t + B e S 2 t = A e [ − ξ + ξ 2 − 1 ] ω n t + B e [ − ξ − ξ 2 − 1 ] ω n t … … (1) Now differentiating equation (1) with respect to ‘t’, we get;
x ˚ = A e [ − ξ + ξ 2 − 1 ] ω n t [ − ξ + ξ 2 − 1 ] ω n + B e [ − ξ − ξ 2 − 1 ] ω n t [ − ξ − ξ 2 − 1 ] ω n … … … (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} x ˚ = A e [ − ξ + ξ 2 − 1 ] ω n t [ − ξ + ξ 2 − 1 ] ω n + B e [ − ξ − ξ 2 − 1 ] ω n t [ − ξ − ξ 2 − 1 ] ω n … … … (2) Now, let att = 0 t = 0 t = 0 x = X 0 x = X_0 x = X 0 t = 0 t = 0 t = 0 x ˚ = 0 \mathring x =0 x ˚ = 0
Substituting this value in equation (1) we get;
X 0 = A + B … … … (3) X_0 = A+B\quad \dots \dots \dots \text{(3)} X 0 = A + B … … … (3) Substituting this value in equation (2) we get;
0 = A [ − ξ + ξ 2 − 1 ] ω n + B [ − ξ − ξ 2 − 1 ] ω n … … (4) 0= A[-\xi + \sqrt{\xi^2-1}]\omega_n + B[-\xi - \sqrt{\xi^2-1}]\omega_n \quad \dots \dots \text{(4)} 0 = A [ − ξ + ξ 2 − 1 ] ω n + B [ − ξ − ξ 2 − 1 ] ω n … … (4) From equation (3), B = X 0 − A B= X_0 - A B = X 0 − A
∴ 0 = A [ − ξ + ξ 2 − 1 ] ω n + ( X 0 − A ) [ − ξ − ξ 2 − 1 ] ω n ∴ 0 = − A ξ + A ξ 2 − 1 + X 0 [ − ξ − ξ 2 − 1 ] + A ξ + A ξ 2 − 1 ∴ 0 = 2 A ξ 2 − 1 + X 0 [ − ξ − ξ 2 − 1 ] ∴ 2 A ξ 2 − 1 = X 0 [ ξ + ξ 2 − 1 ] ∴ A = X 0 [ ξ + ξ 2 − 1 ] 2 ξ 2 − 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} ∴ 0 ∴ 0 ∴ 0 ∴ 2 A ξ 2 − 1 ∴ A = = = = = A [ − ξ + ξ 2 − 1 ] ω n + ( X 0 − A ) [ − ξ − ξ 2 − 1 ] ω n − A ξ + A ξ 2 − 1 + X 0 [ − ξ − ξ 2 − 1 ] + A ξ + A ξ 2 − 1 2 A ξ 2 − 1 + X 0 [ − ξ − ξ 2 − 1 ] X 0 [ ξ + ξ 2 − 1 ] 2 ξ 2 − 1 X 0 [ ξ + ξ 2 − 1 ] … … (5) Now,A = X 0 − B A=X_0-B A = X 0 − B
∴ 0 = ( X 0 − B ) [ − ξ + ξ 2 − 1 ] ω n + B [ − ξ − ξ 2 − 1 ] ω n ∴ 0 = X 0 [ − ξ + ξ 2 − 1 ] + B ξ − B ξ 2 − 1 − B ξ − B ξ 2 − 1 ∴ 0 = X 0 [ − ξ + ξ 2 − 1 ] − 2 B ξ 2 − 1 ∴ 2 B ξ 2 − 1 = X 0 [ − ξ + ξ 2 − 1 ] ∴ B = X 0 [ − ξ + ξ 2 − 1 ] 2 ξ 2 − 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} ∴ 0 ∴ 0 ∴ 0 ∴ 2 B ξ 2 − 1 ∴ B = = = = = ( X 0 − B ) [ − ξ + ξ 2 − 1 ] ω n + B [ − ξ − ξ 2 − 1 ] ω n X 0 [ − ξ + ξ 2 − 1 ] + B ξ − B ξ 2 − 1 − B ξ − B ξ 2 − 1 X 0 [ − ξ + ξ 2 − 1 ] − 2 B ξ 2 − 1 X 0 [ − ξ + ξ 2 − 1 ] 2 ξ 2 − 1 X 0 [ − ξ + ξ 2 − 1 ] … … (6) Now putting equation (5) and (6) in equation (1), we get;
x = X 0 [ ξ + ξ 2 − 1 ] 2 ξ 2 − 1 e [ − ξ + ξ 2 − 1 ] ω n t + X 0 [ − ξ + ξ 2 − 1 ] 2 ξ 2 − 1 e [ − ξ − ξ 2 − 1 ] ω n t x = X 0 2 ξ 2 − 1 [ ( ξ + ξ 2 − 1 ) e [ − ξ + ξ 2 − 1 ] ω n t + ( − ξ + ξ 2 − 1 ) e [ − ξ − ξ 2 − 1 ] ω n t ] \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} x = x = 2 ξ 2 − 1 X 0 [ ξ + ξ 2 − 1 ] e [ − ξ + ξ 2 − 1 ] ω n t + 2 ξ 2 − 1 X 0 [ − ξ + ξ 2 − 1 ] e [ − ξ − ξ 2 − 1 ] ω n t 2 ξ 2 − 1 X 0 [ ( ξ + ξ 2 − 1 ) e [ − ξ + ξ 2 − 1 ] ω n t + ( − ξ + ξ 2 − 1 ) e [ − ξ − ξ 2 − 1 ] ω n t ] 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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