Computing the moment of inertia for a rotary inverted pendulum

[Remusco]

Homework Statement

Compute the moment of inertia seen by the motor shaft for a rotary inverted pendulum

Relevant Equations

I_{\mathrm{total}}=I_m+\frac{1}{3}m_al_a^2+\frac{1}{3}m_pl_p^2+m_p\left(l_a+\frac{l_p}{2}\cos{\left(\theta_{i-1}\right)}\right)^2

I looked all over the internet and I can't find a derivation of this. It is over my head to derive this.
This is my system:

I want to assume that the pendulum and motor arm are uniform rods. I want to ignore the motor shaft inertia and the rotary encoder inertia since they are negligible. Does anyone have the formula for J about the motor shaft? I imagine this is dependent on the angle of the pendulum.

 

[WWGD]

Homework Statement: Compute the moment of inertia seen by the motor shaft for a rotary inverted pendulum
Relevant Equations: I_{\mathrm{total}}=I_m+\frac{1}{3}m_al_a^2+\frac{1}{3}m_pl_p^2+m_p\left(l_a+\frac{l_p}{2}\cos{\left(\theta_{i-1}\right)}\right)^2

I looked all over the internet and I can't find a derivation of this. It is over my head to derive this.
This is my system:
View attachment 359080

I want to assume that the pendulum and motor arm are uniform rods. I want to ignore the motor shaft inertia and the rotary encoder inertia since they are negligible. Does anyone have the formula for J about the motor shaft? I imagine this is dependent on the angle of the pendulum.

Hi, please wrap your text with Latex tags ##'s.

 

[renormalize]

Relevant Equations: $I_{\mathrm{total}}=I_m+\frac{1}{3}m_al_a^2+\frac{1}{3}m_pl_p^2+m_p\left(l_a+\frac{l_p}{2}\cos{\left(\theta_{i-1}\right)}\right)^2$

You need to post a labeled diagram of your system. How can we possibly assist if we don't know the meanings of the symbols $I_m,m_a,l_a,m_p,l_p,\theta_{i-1}$?

 

[haruspex]

Relevant Equations: $I_{\mathrm{total}}=I_m+\frac{1}{3}m_al_a^2+\frac{1}{3}m_pl_p^2+m_p\left(l_a+\frac{l_p}{2}\cos{\left(\theta_{i-1}\right)}\right)^2$

You need to post a labeled diagram of your system. How can we possibly assist if we don't know the meanings of the symbols $I_m,m_a,l_a,m_p,l_p,\theta_{i-1}$?

I assume this is effectively a continuation of https://www.physicsforums.com/threads/simulating-a-rotary-inverted-pendulum-in-python.1079126/

 

[renormalize]

Relevant Equations: $I_{\mathrm{total}}=I_m+\frac{1}{3}m_al_a^2+\frac{1}{3}m_pl_p^2+m_p\left(l_a+\frac{l_p}{2}\cos{\left(\theta_{i-1}\right)}\right)^2$

OK, thanks to @haruspex I've found your previous post that defines most of the variables in your Relevant Equation:

# System Parameters
m_p = 0.1 # Mass of pendulum (kg)
m_a = 0.5 # Mass of arm (kg)
l_p = 0.2 # Length of pendulum (m)
l_a = 0.15 # Length of arm (m)
g = 9.81 # Gravity (m/s^2)
b = 0.00 # Damping coefficient
I_a = (1/3) * m_a * l_a**2 # Arm moment of inertia
I_p = (1/3) * m_p * l_p**2 # Pendulum moment of inertia

And from the photo of your apparatus in post #1 I surmise that the following is a reasonable representation of your inverted pendulum (but without the Mass at the end of the pendulum rod):

(From: https://www.st.com/content/dam/AME/...to_Integrated_Rotary_Inverted_Pendulum_v2.pdf)
If my surmise is correct, can you attempt to write down the formula for the moment-of-inertia $I_{\text{total}}$ seen by the Motor-shaft, in terms of the variables $r,l,\theta,\phi$ shown in Fig. 1 and the masses of the Rotor and Pendulum rods?

 

[Remusco]

OK, thanks to @haruspex I've found your previous post that defines most of the variables in your Relevant Equation:

And from the photo of your apparatus in post #1 I surmise that the following is a reasonable representation of your inverted pendulum (but without the Mass at the end of the pendulum rod):
View attachment 359098
(From: https://www.st.com/content/dam/AME/2019/Educational Curriculums/motor-control/Introduction_to_Integrated_Rotary_Inverted_Pendulum_v2.pdf)
If my surmise is correct, can you attempt to write down the formula for the moment-of-inertia $I_{\text{total}}$ seen by the Motor-shaft, in terms of the variables $r,l,\theta,\phi$ shown in Fig. 1 and the masses of the Rotor and Pendulum rods?

Yes this is what I am looking for. Thanks for clarifying.