Relations between torque for system of pulleys

[BeeKay]

Homework Statement

In an attempt to find a transfer function of the system, I need to come up with equations that I can use to solve for unknowns. See the attached image to see the diagram of the pulley system. J is the moment of inertia, r is the radius. The smaller radius on pulley 2 is r₁. Let me know if you can't access the picture.

I believe the sum of the moments on the first pulley should equal that on the second and third, but I do not know

Homework Equations

Iθ'' = Σ M
M = r x F

The Attempt at a Solution

1) J₁ θ₁'' = τ + r₁(T₁-T₂)
2) J₂θ₂'' = r₂(T₃-T₄)
3) J₃θ₃'' = r₂(T₃-T₅)
4) x = r₂θ₂
5) x = r₂θ₃
6) mx'' = T₄-T₅ -kx

I feel that I can set equations 1 and 2 equal to each other because there is not a mass or anything between them. I do not think that I can set 2 and 3 equal to each other because it is connected to a mass. But I honestly do not know the actual reasons behind why I can and/or cannot equate them. Equations 4 and 5 tell me that θ₁ is equal to θ₂, but because they have different moments of inertia the sum of the moments cannot be the same.

Any help is appreciated regarding which equations I can set equal to each other and more importantly what allows me to do that. Thanks

 

[collinsmark]

Homework Statement

In an attempt to find a transfer function of the system, I need to come up with equations that I can use to solve for unknowns. See the attached image to see the diagram of the pulley system. J is the moment of inertia, r is the radius. The smaller radius on pulley 2 is r₁. Let me know if you can't access the picture.

I believe the sum of the moments on the first pulley should equal that on the second and third, but I do not know

Homework Equations

Iθ'' = Σ M
M = r x F

The Attempt at a Solution

1) J₁ θ₁'' = τ + r₁(T₁-T₂)
2) J₂θ₂'' = r₂(T₃-T₄)
3) J₃θ₃'' = r₂(T₃-T₅)
4) x = r₂θ₂
5) x = r₂θ₃
6) mx'' = T₄-T₅ -kx

I feel that I can set equations 1 and 2 equal to each other because there is not a mass or anything between them. I do not think that I can set 2 and 3 equal to each other because it is connected to a mass. But I honestly do not know the actual reasons behind why I can and/or cannot equate them. Equations 4 and 5 tell me that θ₁ is equal to θ₂, but because they have different moments of inertia the sum of the moments cannot be the same.

Any help is appreciated regarding which equations I can set equal to each other and more importantly what allows me to do that. Thanks

In equation 2 I think you are missing the torques attributed to the tensions $T_1$ and $T_2$. Keep in mind that this pulley has a total of four torques acting on it, one from each of the four belt/line segments: two from the bottom and two from the left.

Other than that, be mindful of your own convention (you get to decide) of what direction is positive and what is negative. In other words, is clockwise positive or is counterclockwise positive? This will make a difference in whether you express a term as, say, $r_2 \left( T_3 - T_5 \right)$ or instead as $r_2 \left( T_5 - T_3 \right)$ for example.