Rotational Inertia: Calculate & Plot Function w/ Sinus Decay

[JPGraphX]

Hi,

I would like to calculate the function of a rotating disc that has 2 spring slowing it down. (look at my pictures to understand)

Let say I turn my disc 90 degrees clockwise and release it, it is going to oscillate a certain moment than stop.

I want to plot that function and to have a sinus function decreasing to 0 after a "x" number of period.

Thanks for your help,
Jean-Philippe

 

[K^2]

Doesn't work quite like that. A torsion spring pendulum is typically described by damped harmonic oscillator equation. For the equation of the following form.

$$\frac{d^2x}{dt^2} + 2 \zeta \omega_0 \frac{dx}{dt} + \omega_0^2 = 0$$

And for initial conditions $x(t) = x_0$, $x'(0) = 0$, the general solution has the following form.

$$x(t) = x_0 e^{-\zeta \omega_0 t} \left( cos(\omega t) + \frac{\zeta \omega_0}{\omega} sin(\omega t) \right)$$

Where the angular frequency is $\omega = \omega_0 \sqrt{1-\zeta^2}$. To figure out the parameters $\omega_0$ and $\zeta$, you can follow prescription in the torsional harmonic oscillators article. But basically, you want $0 < \zeta < 1$ if you want decaying sinusoidal motion. Higher value will result in faster decay, but there is no way to say that it will make N oscillations. Each oscillation will be smaller than the last by a fixed ratio, but it never goes perfectly to zero. And, of course, once you figured out $\zeta$, you can find $\omega_0$ that gives you desired frequency of oscillations.

 

[JPGraphX]

This is my project, it is not a pendulum.