Numerical Methods

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Parametric oscillator

A parametric oscillator is a system that induces oscillations by periodically modulating some parameter of the system. A typical example is a child on a swing. For small amplitude oscillations of a swing, the restoring force is $-\frac{mg}{l}x$ where $m$ is the mass, $l$ is the length of the swing, and $g=9.81$ m/s² is the acceleration of gravity at the earth's surface. If the length of the swing is modulated periodically, this system is described by the differential equation,

\[ \begin{equation} m\frac{d^2x}{dt^2}+b \frac{dx}{dt}+\frac{mg}{l(1-A\cos(\omega t))}x=0. \end{equation} \]

Large amplitude oscillations are induced when the modulation is about twice the resonance frequency. For most parameters, no parametric amplification is observed.

$m=$ 1 [kg]

$b=$ 0.2 [kg/s]

$l=$ 0.5 [N/m]

$A=$ 0.4 [N]

$\omega=$ 8.3 [rad/s]

The resonance frequency is $\omega_0=\sqrt{g/l-b^2/4m^2}=$ 4.43 rad/s.

 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.

 

 $t$       $x$      $v_x$