# Jitter Experienced by Spacecraft from Reaction Wheel Imbalances

I am working on a project trying to model a satellite's pointing error caused by jitter from the reaction wheels. I have found a couple of papers (Paper 1 and Paper 2) which implement the simple model I'm trying to build, but I am unable to reconcile the results (simplified model used in the first paper is the same as that used in the second model which is far less technical). I am able to model the static imbalance forces and torques from both static and dynamic imbalances in the wheels, but my issue is computing the actual angular displacements about the axes. My understanding is that the below equation can be rearranged solving for $$\alpha$$ and integrated twice to find angular displacement $$\theta$$, but I think my issue is setting up boundary conditions. Both static and dynamic torque functions are sinusoidal (written fully in the papers) which are simple enough to integrate analytically. I believe the governing equation is:

$$\vec{\tau}_{c} = [I]\vec{\alpha}$$ or

$$\vec{\alpha} = \ddot{\vec{\theta}}= [I]^{-1}\vec{\tau}_{c}$$

Enforcing steady state initial conditions results in an added constant term in the angular velocity expression $$\dot{\theta}$$ after the first integration which adds a polynomial term in $$\theta$$ which causes the angular displacement terms to continue to climb, when in each of the papers, the jitter (angular displacement) remain sinusoidal and continue to simply oscillate about $$0^{o}$$. I've reached out to the authors of each of them but haven't gotten responses, so I'm kind of at a loss as to how they managed to arrive at their results. It feels like a relatively simple problem, but I am clearly missing something.

If you're only interested in a sinusoidal disturbance (which, by definition, you are in this case), then just ignore the constant terms in the integrations. I.e., for $$\vec \tau_c = \sum \vec \tau_n \cos \omega_n$$ find that $$\Delta \dot {\vec \theta} = [I]^{-1} \int \sum \vec \tau_n \cos \omega_n dt = -[I]^{-1} \sum \frac 1 {\omega_n} \vec \tau_n \sin \omega_n$$ and that $$\Delta \vec \theta = \int \Delta \dot {\vec \theta} dt = -[I]^{-1} \sum \frac 1 {\omega_n^2} \vec \tau_n \cos \omega_n$$