Resonance: Damped driven mass-spring system

A mass $m$ is attached to a linear spring with a spring constant $k$. A drag force acts on the mass that is in the opposite direction as the velocity $F_{\text{drag}}=-bv_x$ where $b$ is the drag force constant. The system is driven by a periodic drive force $F_0\cos(\Omega t)$. The differential equation that describes the motion is,

\[ \begin{equation} m\frac{d^2x}{dt^2}+b\frac{dx}{dt}+kx=F_0\cos(\Omega t). \end{equation} \]

$m=$ 1 [kg]

$b=$ 0.2 [kg/s]

$k=$ 0.9 [N/m]

$F_0=$ 0.1 [N]

$\Omega=$ 1 [rad/s]

The resonance frequency is $\omega=\sqrt{k/m-b^2/4m^2}=$ 0.943 rad/s. The maximum response will be when the drive frequency equals the resonance frequency. The amplitude of the steady-state oscillations are $F_0/\sqrt{(k-m\Omega^2)^2+\Omega^2b^2}=$ 0.447 m.

   

 Numerical 2nd order differential equation solver 

$ \large \frac{dx}{dt}=$

$v_x$

$ \large a_x=\frac{F_x}{m}=\frac{dv_x}{dt}=$

Intitial conditions: 

$x(t_0)=$

 

$\Delta t=$

$v_x(t_0)=$

 

$N_{steps}$

$t_0=$

 

Plot:

vs.

 

 $t$       $x$      $v_x$