Forced vibration
A forced undamped/damped simple harmonic motion is excited on a spring-mass system with an initial displacement \((x_0)\).
In the first interactive plot, only four specific cases are presented. For each of these cases, the input parameters (as given in Inputs table) are fixed with particular values. However, these input parameters can be varied (drag the point on the sliders) in the second interactive plot.
In the first interactive plot, only four specific cases are presented. For each of these cases, the input parameters (as given in Inputs table) are fixed with particular values. However, these input parameters can be varied (drag the point on the sliders) in the second interactive plot.
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: \({m\ddot x +c\dot x +kx=F_0\sin(\omega t)}\)
The above equation has general solution and a particular solution. General solution is of the form (transient response): \(x_{\mathrm{c}}(t)={X_0e}^{-\zeta\omega_nt}\sin{\left(\omega_dt+\phi_0\right)}\) Particular solution is assumed to be of the form (steady state response): \(x_{\mathrm{p}}(t)=X\sin\left(\omega t-\phi\right)\) where \(X=\dfrac{F_0}{\sqrt{{{(k-m\omega^2)}^2+\left(c\omega\right)}^2}}\) and \(\phi=\tan^{-1}{\left(\dfrac{c\omega}{k-m\omega^2}\right)}\). Note that \(\phi\) can also be written as \(\phi=\arctan (y,x)=\arctan{(c\omega,k-m\omega^2)}\). Hence, the total response is \(x(t)=x_{\mathrm{c}}(t)+x_{\mathrm{p}}(t)\). Given the following initial conditions: \(x(0)=x_0\,\&\,\dot{x}(0)=v_0\), the constants \(X_0\,\&\,\phi_0\) can be written as, \(X_0=\sqrt{\left(\dfrac{\zeta\omega_nx_{0}+v_0+\zeta\omega_nX\sin\phi-X\omega \cos\phi}{\omega_d}\right)^2+\left(x_{0}+X\sin\phi\right)^2}\) \(\phi_0=\tan^{-1}\left[\dfrac{\omega_d\left(x_{0}+X\sin{\phi}\right)}{\zeta\omega_nx_{0}+v_0+\zeta\omega_nX\sin{\phi}-X\omega\cos{\phi}}\right]\) |
Equation of motion: \({m\ddot x +c\dot x +kx=F_0\cos(\omega t)}\)
The above equation has general solution and a particular solution. General solution is of the form (transient response): \(x_{\mathrm{c}}(t)={X_0e}^{-\zeta\omega_nt}\cos{\left(\omega_dt-\phi_0\right)}\) Particular solution is assumed to be of the form (steady state response): \(x_{\mathrm{p}}(t)=X\cos\left(\omega t-\phi\right)\) where \(X=\dfrac{F_0}{\sqrt{{{(k-m\omega^2)}^2+\left(c\omega\right)}^2}}\) and \(\phi=\tan^{-1}{\left(\dfrac{c\omega}{k-m\omega^2}\right)}\). Note that \(\phi\) can also be written as \(\phi=\arctan (y,x)=\arctan{(c\omega,k-m\omega^2)}\). Hence, the total response is \(x(t)=x_{\mathrm{c}}(t)+x_{\mathrm{p}}(t)\). Given the following initial conditions: \(x(0)=x_0\,\&\,\dot{x}(0)=v_0\), the constants \(X_0\,\&\,\phi_0\) can be written as, \(X_0=\sqrt{\left(\dfrac{\zeta\omega_nx_{0}+v_0-\zeta\omega_nX\cos\phi-X\omega \sin\phi}{\omega_d}\right)^2+\left(x_{0}-X\cos\phi\right)^2}\) \(\phi_0=\tan^{-1}\left[\dfrac{\omega_d\left(x_{0}-X\cos{\phi}\right)}{\zeta\omega_nx_{0}+v_0-\zeta\omega_nX\cos{\phi}-X\omega\sin{\phi}}\right]\) |