Damped oscillation of a spring-mass system
A damped simple harmonic motion is excited on a spring-mass system with an initial displacement \((x_0)\). This can be varied (drag the point on the sliders) along with spring stiffness \((k)\) and mass \((m)\).
Following quantities are indicated as outputs:
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Play buttons can be used for the following purposes:
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Equation of motion : \(m\ddot{x}+c\dot{x}+kx=0\)
Taking \(x(t)=Ce^{st}\Rightarrow\) Characteristic equation : \(ms^2+cs+k=0\) with roots \(s_{1,2}=\dfrac{-c\pm\sqrt{c^2-4km}}{2m}\) \((1)\)
Taking \(x(t)=Ce^{st}\Rightarrow\) Characteristic equation : \(ms^2+cs+k=0\) with roots \(s_{1,2}=\dfrac{-c\pm\sqrt{c^2-4km}}{2m}\) \((1)\)
Observations
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Displacement of mass with respect to time (for three damped cases)
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The roots in Eq. \((1)\) can be rewritten as \(s_{1,2}=(-\zeta\pm\sqrt{\zeta^2-1})\omega_n\). Therefore, the displacement of the mass at any time \('t'\) can be expressed as,
\(x(t)=C_1e^{s_1t}+C_2e^{s_2t}\) \((2)\)
Following table gives the displacement \((x(t))\) based on the different values of damping ratio \((\zeta)\).
Ref: S.S. Rao, "Mechanical vibrations", 6th Edition, Pearson (2017)
\(x(t)=C_1e^{s_1t}+C_2e^{s_2t}\) \((2)\)
Following table gives the displacement \((x(t))\) based on the different values of damping ratio \((\zeta)\).
Ref: S.S. Rao, "Mechanical vibrations", 6th Edition, Pearson (2017)
Damping ratio \((\zeta)\) |
Roots \((s_{1,2})\) | Nature of motion | Displacement \((x(t))\) |
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\(<1\) (Underdamped) |
\(s_{1,2}=(-\zeta\pm i\sqrt{1-\zeta^2})\omega_n\) | Damped \((Re<0)\) Oscillatory \((Im\ne0)\) |
\(e^{-\zeta\omega_nt}\bigg(x_0\cos\omega_dt+\dfrac{\dot{x_0}+\zeta\omega_nx_0}{\omega_d}\sin\omega_dt\bigg)\) |
\(=1\) (Critically damped) |
\(s_1=s_2=-\omega_n\) | Damped \((Re<0)\) Non oscillatory \((Im=0)\) |
\([{x_0}+(\dot{x_0}+\omega_nx_0)t]e^{-\omega_nt}\) |
\(>1\) (Overdamped) |
\(s_{1,2}=(-\zeta\pm\sqrt{\zeta^2-1})\omega_n\) and \(s_2\ll s_1\) | Damped \((Re<0)\) Non oscillatory \((Im=0)\) |
\(\dfrac{\dot{x_0}+x_0\omega_n(\zeta+\sqrt{\zeta^2-1})}{2\omega_n\sqrt{\zeta^2-1}}e^{s_1t}-\dfrac{\dot{x_0}+x_0\omega_n(\zeta-\sqrt{\zeta^2-1})}{2\omega_n\sqrt{\zeta^2-1}}e^{s_2t}\) |
\(0\) (Undamped) |
\(s_{1,2}=\pm i\omega_n\) | Undamped \((Re=0)\) Oscillatory \((Im\ne0)\) |
\(x_0\cos\omega_nt\) (for initial displacement case) \(\dfrac{\dot{x_0}}{\omega_n}\sin\omega_nt\) (for initial velocity case) |
The Logarithmic decrement \((\delta)\) can be expressed as \(\delta=\ln\dfrac{x_0}{x_1}=\dfrac{1}{n}\ln\dfrac{x_0}{x_n}=\dfrac{2\pi\zeta}{\sqrt{1-\zeta^2}}=\zeta\omega_nT_d=\dfrac{2\pi}{\omega_d}\dfrac{c}{2m}\) The value of damping ratio can be found from the logarithmic decrement (as given below) which can be obtained by a Non Destructive Test (NDT). \(\zeta=\dfrac{\delta}{\sqrt{(2\pi)^2+\delta^2}}\) |
Energies can be expressed as, Potential energy \((\mathrm{PE})=kx(t)^2/2\) Kinetic energy \((\mathrm{KE})=mv^2/2=m\dot{x}(t)^2/2\) Total energy \((\mathrm{TE})=\mathrm{PE|_{t=0}}=kx_0^2/2\) Energy loss \(=\mathrm{TE-(PE+KE)}\) |