I have been trying to build a simulator of motor dynamics, but am running into the wall of a seemingly recursive integral definition. Currently I am trying to derive a formula for the rpm of a motor at time t.
Here is my reasoning:
The torque output of a dc motor at time t is this: $$torque(t) = T + D*rpm(t)$$ Where T is the stall torque of the motor and D is the derivative of torque with respect to rpm. (because the torque of a DC motor decreases as rpm increases) And rpm(t) is the rpm of the motor at time t.
Now rpm(t) can be expressed as an integral of $torque(t)$ $$rpm(t) = \frac 1m \int^t_0 torque(i) di$$ where m is the mass of the load attached to the motor and $rpm(0) = 0$
Substituting in my formula for torque gains this:
$$rpm(t) = \frac 1m \int^t_0 T + D*rpm(i) di$$ Which reduces to this: $$rpm(t) = \frac{Tt}m+D\int^t_0 rpm(i)di$$ Here is where I am stuck. How can I simplify $\int_0^trpm(i)di$? Is this formula even possible to solve? Am I missing something?
Forgive me if this question has been answered already. I have searched numerous times with keywords related to "recursive integrals" and haven't found anything that seems to be relevant. Is there a better term for what I am encountering here?
Thanks!
Couldn't you just differentiate both sides to get ($rpm = r$)
$$r'(t) = \frac{T}{m} + D r(t)$$ $$r(0)=0$$
Then we may easily solve this using any of several methods. The result is
$$r(t) = \frac{T}{m D} \left (e^{D t}-1 \right )$$
Then
$$\int_0^t dt' \, r(t') = \frac{T}{m D} \left (\frac{e^{D t}-1}{D} - t \right ) $$