Hello, I'm trying to figure out how solve the below for varying time and/or corresponding theta. I'm stumped.. any insight is greatly appreciated. Thank you
$\ddot{R} = −\alpha|v|\dot{R}−\beta_{s}\frac{|v|}{R}\dot{R}-\beta_{c}\frac{g}{|v|}\dot{R} + [R\dot{\theta}^2-\frac{5}{7}g(tan\delta cos\varepsilon -sin\varepsilon cos\theta)]cos^2\delta$
$\ddot{\theta} = −\alpha|v|\dot{\theta}−\beta_{s}\frac{|v|}{R}\dot{\theta}-\beta_{c}\frac{g}{|v|}\dot{\theta}- \frac{2\dot{R}\dot{\theta}}{R} - \frac{5}{7}\frac{g}{R}sin\varepsilon sin\theta$
with $|v|=\sqrt{\dot{R}^2/cos^2\delta + R^2\dot{\theta}^2} $
where
$\alpha = 0.00900350489819847$
$\beta_{s} = 0.0022919958783044724$
$\beta{c}=0.0062192039382357378$
$\delta=0.2574$
$R=0.242$
$g=9.807$
with the 'initial conditions' at t = 0:
$R(0) = R$
$\dot{R}(0) = 0$
$\theta(0) = 57.899552605659885$
$\dot{\theta}(0) = 2.7371516764119637$
Per PaPiro's suggestion, I've converted the above to first order ODE's and got the following:
Let $x1=\dot{R}$ and $\dot{x1}=\ddot{R}$ .
$x1=\dot{R}$
$\dot{x1} = −\alpha(\sqrt{x1^2/cos^2\delta + R^2\dot{\theta}^2} ) x1−\beta_{s}\frac{\sqrt{x1^2/cos^2\delta + R^2\dot{\theta}^2}}{R}x1-\beta_{c}\frac{g}{\sqrt{x1^2/cos^2\delta + R^2\dot{\theta}^2}}x1 + $
$[R\dot{\theta}^2-\frac{5}{7}g(tan\delta cos\varepsilon -sin\varepsilon cos\theta)]cos^2\delta = \ddot{R}$
Let $x2=\dot{\theta}$ and $\dot{x2}=\ddot{\theta}$ .
$x2=\dot{\theta}$
$\dot{x2} = −\alpha(\sqrt{\dot{R}^2/cos^2\delta + R^2 x2^2} )x2−\beta_{s}\frac{\sqrt{\dot{R}^2/cos^2\delta + R^2 x2^2}}{R}x2-\beta_{c}\frac{g}{\sqrt{\dot{R}^2/cos^2\delta + R^2 x2^2}}x2-$
$ \frac{2\dot{R}x2}{R} - \frac{5}{7}\frac{g}{R}sin\varepsilon sin\theta$
Per PaPiro's suggestion, I've rewritten the above $\dot{x1}$ as a function of x2 and the equation for $\dot{x2}$ as a function of x1:
$x1=\dot{R}$
$\dot{x1} = −\alpha(\sqrt{x1^2/cos^2\delta + R^2x2^2} ) x1−\beta_{s}\frac{\sqrt{x1^2/cos^2\delta + R^2x2^2}}{R}x1-\beta_{c}\frac{g}{\sqrt{x1^2/cos^2\delta + R^2x2^2}}x1 +$
$[Rx2^2-\frac{5}{7}g(tan\delta cos\varepsilon -sin\varepsilon cos\theta)]cos^2\delta = \ddot{R}$
$x2=\dot{\theta}$
$\dot{x2} = −\alpha(\sqrt{x1^2/cos^2\delta + R^2 x2^2} )x2−\beta_{s}\frac{\sqrt{x1^2/cos^2\delta + R^2 x2^2}}{R}x2-\beta_{c}\frac{g}{\sqrt{x1^2/cos^2\delta + R^2 x2^2}}x2-$
$ \frac{2 \cdot x1 \cdot x2}{R} - \frac{5}{7}\frac{g}{R}sin\varepsilon sin\theta = \ddot{\theta}$
Please let me know if this is correct and if I'm on the right track. I'll try plugging this into 4th Order Runge-Kutta next.
