A single-degree-of-freedom system with mass \(m\), stiffness \(k\), and damping ratio \(\zeta\) is subjected to a sinusoidal force \(F(t)=F_0\sin(\omega t)\). Determine and visualize the displacement response \(x(t)\), and study the steady-state frequency response.
Governing ODE: \(\ddot x + 2\zeta\omega_n \dot x + \omega_n^2 x = \dfrac{F_0}{m}\sin(\omega t)\), where \(\omega_n=\sqrt{k/m}\).
The steady-state amplitude under harmonic excitation is \[ |X(\omega)| = \frac{F_0/k}{\sqrt{(1-r^2)^2+(2\zeta r)^2}},\quad r=\frac{\omega}{\omega_n}. \] The phase lag is \[ \phi(\omega)=\tan^{-1}\!\left(\frac{2\zeta r}{1-r^2}\right). \]
Assume steady state \(x_p=A\sin(\omega t-\phi)\), substitute in ODE, match sine/cosine terms to get amplitude and phase. The complete response is \(x(t)=x_h(t)+x_p(t)\); the homogeneous part decays for \(\zeta>0\).
Compute the natural frequency and critical damping for the current parameters.