# Inverted pendulum balance dynamics equations

1. Aug 30, 2015

### aerograce

1. The problem statement, all variables and given/known data
Hi guys! I am reading a paper on inverted pendulum. However, I cant understand the dynamics equations of this inverted pendulum. You may look at the images below. Basically the thruster is placed at top plate, while the middle plate is a base plate where pivot is placed. And then bottom plate is where you place your counterweight. Upon thruster firing, the pendulum arm will rotate.

The equations are attached in the pictures below. I(T) and I(CW) represent the moments of inertia of the thruster and counterweight respectively.

My question is, can any one help me explain these equations? I understand that the left side is the net force on horizontal direction based on newton's second law, and the left side has the thrust caused by pivots stiffness, weight and thrust. However, I dont understand where the momentum of inertia * angular acceleration / arm length comes in.

2. Relevant equations
Relevant equations for this question will be newton's second law, and also rotational dynamics.

3. The attempt at a solution
My attempt at understanding the equations are stated in the question part.

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Last edited: Aug 30, 2015
2. Aug 30, 2015

### TSny

As you say, the right hand side of equation (3.6) must be the sum of the horizontal forces acting on $m_{T}$. So, the complicated fraction must represent the horizontal component of a force acting on $m_{T}$. Conceptually, what are the forces that act on $m_{T}$?

3. Sep 8, 2015

### aerograce

Hello! Sorry for my late reply. I cant seem to get the forces acting on mT. The force acting on mT will be gravity, and then force caused by pivot stiffness, and then thrust. However, I cant seem to catch the meaning of that moment inertia * angular acceleration over arm length, isnt it denoting the total external force instead of one component?

4. Sep 8, 2015

### TSny

I think of the system as consisting of two point masses ( $m_T$ and $m_{CW}$), an upper rod of length $l_T$ and a lower rod of length $l_{CW}$. I think of the couples $k_{tp}$, $k_b$, and $k_{bp}$ as acting on the rods (not the point masses).

The upper mass $m_T$ is connected to the upper rod. So, the upper rod will exert a force on $m_T$. This force is unknown, but we can break it up into an unknown horizontal force $F_x$ and an unknown vertical force $F_y$. So, you can draw a free body diagram for the upper mass and set up $ΣF = ma$ for horizontal and vertical components.

You can also draw a free body diagram for the upper rod with the intention of setting up $\sum{\tau} = I \ddot{\theta}$. Apparently, you are to assume that half the couple $k_{b}$ acts on the upper rod and half acts on the lower rod. (I don't see why that would necessarily have to be the case unless the upper section of the system is identical to the lower section.)

5. Sep 8, 2015

### aerograce

Thank you so much!:)