Calculating Gear Ratios and Torque Transmission in a 3-Gear Train System

[aasamt]

I have a gear train system where i have 3 gears meshing with each other (different sizes) gear A, B and C. Gear B is my input/driver and gears A and C are my output/driven. If i know the input RPM and input torque for gear B, what are the gear ratios/the amount of torque transmitted to gear A and C. I doubt that i can say that ratio between A and B = ratio between B and C.

Also, would it be correct to say T_A + T_C = T_B since gears A and C are in a sense parallel to each other. Use N for rpm and T for torque please. I would highly appreciate it if you just refrained from posting comments such "Google is a savior" and the sort.

 

[OldEngr63]

If I have understood you correctly, you actually have two simple trains,
B→A, and
B→C
that share the common input gear B.

The basic kinematic relation for rolling at the pitch point is
r_b θ_b = r_a θ _a= r_c θ_c
where the r's are pitch radii and the θ's are the angles turned by the respective gears.

The respective train ratios are then
θ _a/θ_b = r_b/r_a = n_b/n_a
θ _c/θ_b = r_b/r_c = n_b/n_c

The last equality follows because the tooth numbers are proportional to the pitch radii.

With this much given to you, let me suggest that you think a bit about the static equilibrium problem for this system in order to get the torque relations.

 

[aasamt]

Thank you for your reply. I have a follow up question.
I figured your first relationship r_bθ_b = r_aθ_a= r_cθ_c is based on the fact that the amount of displacement (arc length) should be the same for all gears, and when dividing θ by time t, i can get the ratios in terms of ω and radius, r (or D). However, I am kind of confused about the torque ratios. Based on your statement that the system is simply two gear trains, then would it be correct to say that D_B/D_A = T_B/T_A and the same relationship for gears B and C?

 

[OldEngr63]

I'm going to throw that one right back to you and say, "why would that be true?" Show me how you got it. Start out by writing the sum of the torques on each gear, and work through the math.