Under-Damped System (​ξ < 1)

Under-Damped System (​ξ < 1)

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Fig_1_under-damped-system Fig-1

Underdamped system  (ξ<1)\ (\xi < 1)

If the damping factor  ξ\ \xi is less than one or the damping coefficient cc is less than critical damping coefficient ccc_c, then the system is said to be an under-damped system.

ξ<1ORccc<1    c<cc\xi < 1 \quad \text OR \quad {c \over c_c} < 1\implies c < c_c
  • We know that roots of differential equations are:
S1=[ξ+ξ21]ωnS2=[ξξ21]ωnS_1 = \big [-\xi + \sqrt{\xi^2 -1} \big] \omega_n \\ S_2 = \big [-\xi - \sqrt{\xi^2 -1} \big] \omega_n
  • But for  ξ<1\ \xi < 1; the roots for under-damped system are given by S1S_1 and S2S_2 as below:
S1=[ξ+i(1ξ2)]ωnS2=[ξi(1ξ2)]ωn\begin{aligned} S_1 = & \big [-\xi + i \sqrt{(1 - \xi^2)}] \omega_n \\ S_2 = & \big [-\xi - i \sqrt{(1 - \xi^2)}] \omega_n \\ \end{aligned}

Where i=1i = \sqrt{-1} is the imaginary unit of complex root

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  • The roots are complex and negative, so the solution of differential equation is given by
x=AeS1t+BeS2tx=Ae[ξ+i(1ξ2)]ωnt+Be[ξi(1ξ2)]ωntx=A.eξωnt.ei(1ξ2)ωnt+B.eξωnt.ei(1ξ2)ωntx=eξωnt[A.ei(1ξ2)ωnt+B.ei(1ξ2)ωnt]x=eξωnt[A.eiωdt+B.eiωdt][(1ξ2)ωn=ωd]\begin{aligned} x &= Ae^{S_1t} + Be^{S_2t} \\ \therefore x &= Ae^{[-\xi + i \sqrt{(1 - \xi^2)}] \omega_nt} + Be^{[-\xi - i \sqrt{(1 - \xi^2)}] \omega_nt} \\ \therefore x &= A.e^{-\xi\omega_nt}.e^{i \sqrt{(1 - \xi^2)} \omega_nt} + B.e^{-\xi\omega_nt}.e^{- i \sqrt{(1 - \xi^2)} \omega_nt} \\ \therefore x &= e^{-\xi\omega_nt} \big [A.e^{i \sqrt{(1 - \xi^2)} \omega_nt} + B.e^{- i \sqrt{(1 - \xi^2)} \omega_nt}] \\ \therefore x &= e^{-\xi\omega_nt} \big [A.e^{i\omega_dt} + B.e^{- i\omega_dt}] \quad \big [ \because \sqrt {(1 - \xi^2) \omega_n} = \omega_d] \end{aligned}
  • According to Euler’s theorem, above equation can be written as:
x=Xeξωnt[sin(ωdt+)]Where  Xand   are constantsx = Xe^{-\xi \omega_nt} \big [\sin (\omega_dt + \varnothing)] \\ \text Where \ \ X \text and \ \ \varnothing \text { are constants}
  • Above equation shows the equation of motion for an underdamped system, and the amplitude reduces gradually and finally becomes zero after some time.

  • Amplitude decreases by  X.eξωnt\ X.e^{-\xi \omega_nt}

  • The natural angular frequency of damped free vibrations is given by:

ωd=(1ξ2)ωn\omega_d = \sqrt {(1 - \xi^2)}\omega_n
  • Time period for under-damped vibration is given by:
tp=2πωd=2π(1ξ2)ωnsec.t_p = {2\pi \over \omega_d} = {2\pi \over \sqrt{(1 - \xi^2)}\omega_n} sec.

Fig_1_under-damped-system Fig-1

Underdamped system  (ξ<1)\ (\xi < 1)

If the damping factor  ξ\ \xi is less than one or the damping coefficient cc is less than critical damping coefficient ccc_c, then the system is said to be an under-damped system.

ξ<1ORccc<1    c<cc\xi < 1 \quad \text OR \quad {c \over c_c} < 1\implies c < c_c
  • We know that roots of differential equations are:
S1=[ξ+ξ21]ωnS2=[ξξ21]ωnS_1 = \big [-\xi + \sqrt{\xi^2 -1} \big] \omega_n \\ S_2 = \big [-\xi - \sqrt{\xi^2 -1} \big] \omega_n
  • But for  ξ<1\ \xi < 1; the roots for under-damped system are given by S1S_1 and S2S_2 as below:
S1=[ξ+i(1ξ2)]ωnS2=[ξi(1ξ2)]ωn\begin{aligned} S_1 = & \big [-\xi + i \sqrt{(1 - \xi^2)}] \omega_n \\ S_2 = & \big [-\xi - i \sqrt{(1 - \xi^2)}] \omega_n \\ \end{aligned}

Where i=1i = \sqrt{-1} is the imaginary unit of complex root

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  • The roots are complex and negative, so the solution of differential equation is given by
x=AeS1t+BeS2tx=Ae[ξ+i(1ξ2)]ωnt+Be[ξi(1ξ2)]ωntx=A.eξωnt.ei(1ξ2)ωnt+B.eξωnt.ei(1ξ2)ωntx=eξωnt[A.ei(1ξ2)ωnt+B.ei(1ξ2)ωnt]x=eξωnt[A.eiωdt+B.eiωdt][(1ξ2)ωn=ωd]\begin{aligned} x &= Ae^{S_1t} + Be^{S_2t} \\ \therefore x &= Ae^{[-\xi + i \sqrt{(1 - \xi^2)}] \omega_nt} + Be^{[-\xi - i \sqrt{(1 - \xi^2)}] \omega_nt} \\ \therefore x &= A.e^{-\xi\omega_nt}.e^{i \sqrt{(1 - \xi^2)} \omega_nt} + B.e^{-\xi\omega_nt}.e^{- i \sqrt{(1 - \xi^2)} \omega_nt} \\ \therefore x &= e^{-\xi\omega_nt} \big [A.e^{i \sqrt{(1 - \xi^2)} \omega_nt} + B.e^{- i \sqrt{(1 - \xi^2)} \omega_nt}] \\ \therefore x &= e^{-\xi\omega_nt} \big [A.e^{i\omega_dt} + B.e^{- i\omega_dt}] \quad \big [ \because \sqrt {(1 - \xi^2) \omega_n} = \omega_d] \end{aligned}
  • According to Euler’s theorem, above equation can be written as:
x=Xeξωnt[sin(ωdt+)]Where  Xand   are constantsx = Xe^{-\xi \omega_nt} \big [\sin (\omega_dt + \varnothing)] \\ \text Where \ \ X \text and \ \ \varnothing \text { are constants}
  • Above equation shows the equation of motion for an underdamped system, and the amplitude reduces gradually and finally becomes zero after some time.

  • Amplitude decreases by  X.eξωnt\ X.e^{-\xi \omega_nt}

  • The natural angular frequency of damped free vibrations is given by:

ωd=(1ξ2)ωn\omega_d = \sqrt {(1 - \xi^2)}\omega_n
  • Time period for under-damped vibration is given by:
tp=2πωd=2π(1ξ2)ωnsec.t_p = {2\pi \over \omega_d} = {2\pi \over \sqrt{(1 - \xi^2)}\omega_n} sec.

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