Generally, equation of motion in forced vibration.

$M\ddot{x} + c\dot{x} + kx$ = f sin wt.

$I \ddot{θ} + C \dot{θ} + Kθ $ = M sin wt.

Forced vibration with SHM.

1] $X \ = \ \frac{Xst}{\sqrt{(1-r^2)^2 \ + \ (2 \zeta r)^2}}$

Where, X = steady state amplitude or amplitude of forced vibration. Xst = static deflection = $\frac{Fo}{k} \rightarrow$ magnitude of external force (N)

2] Phase angle = $\phi \ = \ tan^{-1} [\frac{2 \zeta r}{1 – r^2}]$

NOTE: $\frac{x}{Xst}$ is called magnification factor.

3] Equation of motion is given by,

$M\ddot{x} + c\dot{x} + kx$ = Fo sin wt [in X – co-ordinate]

$I \ddot{θ} + C \dot{θ} + Kθ $ = \ Mo \ sin \ wt [ In $\theta$ - co-ordinate]

$r = \frac{w}{w_n}$

When $X_p$ is given or to be find out.

$\therefore$ $\frac{X_p}{X_{st}} = \frac{1}{2 \xi \sqrt{ 1 \ = \ \xi^2}}$

4] $\theta st = \frac{M_o}{K}$

For disc:

$\theta_{st} = \frac{M_o}{kt}$