Given a Non-Linear Equation that is:
$$I\ddot\theta = mgl \cdot \sin \theta + F_D \cdot l + k\theta $$
Where, $$F_D$$ is representative of Drag Force and is equal to:
$$-1/2C_D\rho Av^2\cdot \operatorname{sign}(v)$$
Meaning, $$I\ddot\theta = mgl \cdot \sin \theta -1/2C_D\rho A(\dot\theta l)^2 \cdot l + k\theta $$
Derive the finite difference equation for this. How would one go about a method of doing this. It is being used as a model for a Inverted Pendulum Setup.
Take a step back from $\dot{\theta}=\frac{d\theta}{dt}$ and $\ddot{\theta}=\frac{d^2\theta}{{dt}^2}$ and approximate it as
$$\dot{\theta}\approx\frac{\Delta\theta}{\Delta t}=\frac{\theta_{i+1}-\theta_i}{\Delta t}$$
$$\ddot{\theta}\approx\frac{\Delta}{\Delta t}\frac{\Delta \theta}{\Delta t}=\frac{\theta_{i+1}-2\theta_i+\theta_{i-1}}{\Delta t^2}$$
Now solve for $\theta_{i+1}$ to turn the differential equation into an iterative algebraic equation