I have a variable speed gear system with 4 primary parts. I need to find the relationship between input rotational speed (wi) to output rotational speed (wo).
Fig. 1 shows three of the primary parts - input wheel (radius ri) , satellite wheel (radius rs) and connecting arm with a slot. The connecting arm can rotate "relative" to the input wheel about its axis A. The satellite wheel is on other end of connecting arm mounted on a ratchet bearing that allows spinning only in counter-clockwise (CCW) at O. When "wi" is in clockwise (CW) direction satellite wheel spins about axis $O$ in CCW direction at free spinning speed $\omega_s$ without rotating about $A$. When $\omega_i$ is in CCW direction satellite wheel is locked by its ratchet bearing at $O$ and hence two wheels stay at the same teeth contact location and whole system rotates at $\omega_i$ in CCW as a rigid body about $A$, $\omega_c=\omega_i$ (see Fig. 2).
Fig. 3 shows the 4th part of the system – output arm (radius ro). Output arm axis (B) is first placed "in front of" (or behind) the axis “A” and its pin is slidably connected to the slot in the connecting arm as shown in Fig 4. When axes A and B are aligned the system either moves as explained in Fig. 1 or Fig. 2 based on the input wheel "wi" direction of rotation. Now, we shift the axis B by a distance “x” (Fig. 5) and the input wheel ROTATES at a constant speed (wi) in CCW direction. Then depending on its location (how far from B) the satellite wheel either locked and rotates about A or it rotates at aspeed "wc" about A while spinning (ws) about O at varying speeds. i.e. wc >= wi and ws >= 0. With this the output arm also rotates at variable speed (wo). Can someone please show me the relationship of wi to wo in this system? I appreciate if the key equations and steps are shown.