I haven’t looked at this website in years, and I realized recently that it leans very heavily on design (mostly from grad school) with very little analysis. Design is fun and gratifying, but most of the time that I’ve spent doing engineering as a job has focused more on analysis and testing. So, to add a bit of balance I’m documenting a project with more “traditional” engineering - a reaction wheel pendulum.
Reaction wheels are not particularly common in everyday life, but they’re frequently used in satellites for attitude control. I’ve never designed one professionally, and I thought it sounded like a fun personal project that combines a small amount of mechanical design with a bigger focus on electronics and control. The basic premise is simple - attach a motor and a reaction wheel to the end of a pendulum, and use the reaction wheel to swing the pendulum from the bottom (perigee) to the top (apogee), and then use the reaction wheel to sustain balance at apogee.
Here is the completed system in action:
Code and CAD files are available on my Github repo here:
Mechanical parts were made in Onshape and are publicly available here:
Here is a look at the free body diagram and the relevant equations that govern its behavior:
Let’s define some variables:
\(m_p\) : mass of pendulum link
\(m_m\) : mass of motor
\(m_w\) : mass of reaction wheel
\(r_p\) : pendulum length from main pivot to reaction wheel center
\(r_w\) : radius of reaction wheel
\(\theta\) : angular displacement between pendulum and vertical axis
\(\varphi\) : angular displacement between reaction wheel and pendulum
\(J_p\) : mass moment of inertia of everything moving around main pivot
\(J_w\) : mass moment of inertia of reaction wheel relative to wheel center
\(\tau_m\) : motor torque
Newton’s 2nd law in the context of rotation is: \(\sum M = \dot L\), where \(L\) is angular momentum. The more common form is \(\sum M = J\alpha\), but that’s not technically accurate here since it assumes that everything rotating is a single rigid frame, and we have two bodies rotating.
If we just look at the reaction wheel itself, Newton’s 2nd law looks like this:
\(\tau_m = J_w \alpha_{wheel,absolute}\)
In order to correctly capture the angular acceleration of the wheel, we need to incorporate both the acceleration of the wheel relative to the pendulum \((\ddot{\varphi})\) and the center of the wheel relative to the fixed support \((\ddot{\theta})\). This gives us the equation:
\((1)\qquad \tau_m = J_w(\ddot{\varphi} + \ddot{\theta})\)
If we step back to look at the system overall (pendulum, motor, reaction wheel), the only external force acting on it is gravity - everything associated with the motor is internal. This gives us:
\(\sum M = \sum m_i g \ell_i \sin(\theta)\)
Where \(m_i\) and \(\ell_i\) are respective masses and distances to center of mass for individual components. We also know definitionally that:
\(L_{\text{sys}} = \underbrace{J_p\dot\theta}_{\text{everything swinging about the pivot}} + \underbrace{J_w\dot\varphi}_{\text{extra spin the wheel has relative to the body}}\)
Putting these together gives us:
\((2)\qquad \sum M = \dot L_{\text{sys}} = J_p\ddot\theta + J_w\ddot\varphi = \sum m_i g \ell_i \sin(\theta)\)
We can rearrange equation \((1)\) to show that:
\(J_w\ddot{\varphi} = \tau_m -J_w\ddot{\theta}\)
Combining this with equation \((2)\) yields:
\(J_p\ddot\theta + \tau_m -J_w\ddot{\theta} = \sum m_i g \ell_i \sin(\theta)\)