Kinematics Belt and Pulley Problem

[whitejac]

Homework Statement

I think I made a mistake somewhere..

Homework Equations

T = Jα
T = F*R

The Attempt at a Solution

A)
I started with T = Jα
Since there is no slip, α_m = α_L

Thus:
T_m / J_m = T_L / J_L

Plugging in, we find T_L = T_m * J_L / J_m = 2560

Now use T = F*R.

T_m = F_m * R_m
Plugging in shows F_m = 2666.67.

Since F has to be equal on both pulleys, we use T = F*R again to solve for R_L = 0.96m.
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B)
Now that we have those solved we can solve for t.

Θ = Θ_i + 0.5α*t²

The initial conditions are zero.

Since α must be the same for either, we can solve for just one scenario.

rα_m = R_Lα_L
α_m = T_m/J_m
α_L = r*T_m / R_L*J_m

plugging back into this we get
Θ = 0.5(r*T_m / R_L*J_m)*t²

Which when applying the condition Θ=1rad, we get
t as the square root of the reciprocal which shows
t = 0.08sec.

This is where I'm in trouble though. My velocity vs time for this equation will not have a maximum or a finish. I can get a minimum but my equation of motion Θ is a parabola without a finish line. What did I do wrong here because I can't calculate a maximum from that velocity graph.
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C)

This is where I'm in trouble

 

[haruspex]

Since there is no slip, αm = αL

The same angular acceleration? Why?

 

[whitejac]

Because each pulley has to rotate the same theta or else there would be slip. They also have to accelerate at the same rate or one old be going faster than the other.

 

[haruspex]

each pulley has to rotate the same theta or else there would be slip.

This is two pulleys connected by a belt, not sharing an axle, right?
If each pulley rotates through angle theta, what length of belt has gone around each pulley?

 

[whitejac]

This is two pulleys connected by a belt, not sharing an axle, right?
If each pulley rotates through angle theta, what length of belt has gone around each pulley?

Okay, I understand now why each pulley cannot have the same alpha. I got stuck in a way of thinking and then I asked someone who was supposedly good at this to look over it and they said that it looked good. Larger wheel has to have a lower angular velocity, rpm's, than the smaller wheel though the length of belt (RΘ) is the same. How then can I adapt my equations to be good for any theta, especially when ω is not constant, but rather increases and decreases with time? My whole foundation was on equating each of them through alpha.

 

[haruspex]

How then can I adapt my equations to be good for any theta,

By equating the belt movements. That relates displacement, velocity and acceleration in exactly the way.

 

[whitejac]

By equating the belt movements. That relates displacement, velocity and acceleration in exactly the way.

I guess I don't see how to relate the acceleration, which based on the maximum torque is T/J and the velocity in a generalized term.
Relating the diameter of each pulley with the velocity,
ND = ND, where N is the number of rotations. I guess I'm having trouble now seeing how T/J can have the derivative taken and give me a velocity because it's not very generalized.

 

[haruspex]

I guess I don't see how to relate the acceleration, which based on the maximum torque is T/J and the velocity in a generalized term.
Relating the diameter of each pulley with the velocity,
ND = ND, where N is the number of rotations. I guess I'm having trouble now seeing how T/J can have the derivative taken and give me a velocity because it's not very generalized.

If the two pulleys have radii r₁ and r₂, and rotate through angles θ₁, θ₂, the belt movement is r₁θ₁=r₂θ₂. To relate the angular velocities differentiate once, to relate the angular accelerations differentiate a second time.

 

[whitejac]

If the two pulleys have radii r₁ and r₂, and rotate through angles θ₁, θ₂, the belt movement is r₁θ₁=r₂θ₂. To relate the angular velocities differentiate once, to relate the angular accelerations differentiate a second time.

I guess I understand that but here let me show you how i see it,
I understand that
r₁Θ₁ = x
r₁Θ'₁ = x'
r₁Θ''₁ = x''

but we have T=Jα, where T and J are given and T is the max torque. That leaves α as T/J = 26,666.67 and if I differentiate it, as it's a constant, will equal zero.
Conversely, if i integrate r₁Θ₁ = x then I'm left with Θ = 1/2αt² + 0. This can't be the answer either as the velocity vs time relationship should ascend and the descend because it starts and stops at 0, unless it immediately stops the instant it reaches the desired position (which would theoretically 'minimize' the amount of time required). So I understand the general relationship between (angular) placement, velocity, acceleration, but I'm having trouble applying it.

 

[haruspex]

Θ = 1/2αt²

That equation is only valid for constant acceleration. You have to decide how the acceleration needs to vary with time to satisfy the question.

 

[whitejac]

That equation is only valid for constant acceleration. You have to decide how the acceleration needs to vary with time to satisfy the question.

You're right, I'm left with an even more generalized solution. r₁Θ₁ = x
The question ask me to minimize the time required to move through an angle theta, but it doesn't give me any relationship with time. It just gives me the size, weight and torque of one pulley and then the weight of another.
If I say r₁α₁=r₁α₁ then I could say that α₁ = T/J at max torque (and it is assumed that this is the maximum amount of force the pulley can exert, thus minimum time. However, this does not give me a t to manipulate. It just says that r₁α₁ = (0.1m)(0.0015kg-m^2) / (40N-m) and I have 2 unknowns and 1 expression. I could potentially use the knowledge that r₁θ₁ = r₂θ₂ but I don't really see how since the theta is arbitrary.

I'm sorry to sound like a broken record ( and make you sound like one too) but this deriving things is a serious challenge for me.

 

[haruspex]

I could say that α1 = T/J at max torque (and it is assumed that this is the maximum amount of force the pulley can exert, thus minimum time.

Right, but you cannot keep it at max torque in the same direction since it has to end stationary. So what can you do instead?

 

[whitejac]

Right, but you cannot keep it at max torque in the same direction since it has to end stationary. So what can you do instead?

accelerate it as fast as possible and then decelerate it just as fast at the 50% marker.

 

[haruspex]

accelerate it as fast as possible and then decelerate it just as fast at the 50% marker.

Exactly

 

[whitejac]

Exactly

To me, this would result in a piecewise function, where
For 0< θ < θ/2
α = T/J

For θ/2 < θ < θ
α = -T/J

 

[haruspex]

To me, this would result in a piecewise function, where
For 0< θ < θ/2
α = T/J

For θ/2 < θ < θ
α = -T/J

Right.

 

[whitejac]

Right.

Okay, then this does puts be back at the same problem though.
At in the first case, 0 < θ < θ/2
I'm in the original instance of α = t/j but this only shows that r₂ = (r * J) / (T *α₂)

 

[haruspex]

Okay, then this does puts be back at the same problem though.
At in the first case, 0 < θ < θ/2
I'm in the original instance of α = t/j but this only shows that r₂ = (r * J) / (T *α₂)

I find it hard to followyour algebra because you don't clearly distinguish the different radii, moments of inertia and torques.

If the angular acceleration of the load is α_L, what torque is being applied? What corresponding torque is the load exerting on the motor pulley? What is the angular acceleration of themotor pulley? What total torque isthe motor therefore generating?

 

[whitejac]

If the angular acceleration of the load is αL, what torque is being applied? What corresponding torque is the load exerting on the motor pulley? What is the angular acceleration of the motor pulley? What total torque is the motor therefore generating?

In order:

Given:
r = the radius of the motor
J_m = the moment of inertia for the motor
T_m = the torque of the motor
α_m = the angular acceleration of the motor, which = T_m / J_m

R = the radius of the pulley
J_L = the moment of inertia for the motor
T_L = the torque of the motor
α_L = the angular acceleration of the motor, which = T_L / J_L

rα_m = Rα_L as there is no slip.
Thus
α_L = rα_m / R = rT_m / RJ_mSolving for the torque... I have this:

T_m = F_m * r
T_L = F_L *R
Where F is the tension in the belt, and theoretically has to be the same or else the pulley/motor are moving.

So F_m = F_L
Thus
T_L = T_m *R /r = J_Lα_L

If
α_L = rα_m / R = rT_m / RJ_m
then
T_m *R /r = rT_m / RJ_m
and
R² = r² / J_m, which is finally solvable.

The angular acceleration of the motor pulley is T_m / r as stated above
The total torque would be... the maximum torque given in the problem? I am not quite sure what you mean by this.

 

[haruspex]

the angular acceleration of the motor, which = Tm / Jm

No, that is what it would be if there were no belt attached.

 

[whitejac]

No, that is what it would be if there were no belt attached.

okay, but you said here:

So I'm a bit confused as to where the distinction lies and how to see it in the future.

 

[haruspex]

okay, but you said here:
View attachment 195840

So I'm a bit confused as to where the distinction lies and how to see it in the future.

This is an example of why you need to be clear about what your variables are. I read that post as purely a generic statement that the acceleration should be max in one direction for half the distance and max in the other for the second half. I did not read the T and J as referring to a specific torque and MoI in the context of the problem, since there are more than one of each.

In the equation ΣF=ma, ΣF is the net force. In exactly the same way, we have Στ=Jα, where Στ is the net torque. The driving pulley is subject to two torques, that from the motor and that from the load.

 

[whitejac]

This is an example of why you need to be clear about what your variables are. I read that post as purely a generic statement that the acceleration should be max in one direction for half the distance and max in the other for the second half. I did not read the T and J as referring to a specific torque and MoI in the context of the problem, since there are more than one of each.

In the equation ΣF=ma, ΣF is the net force. In exactly the same way, we have Στ=Jα, where Στ is the net torque. The driving pulley is subject to two torques, that from the motor and that from the load.

I've been working on this most of the day between work.
Here's what I've come after studying some notes my professor put out and trying do it. This relates the torque with the two pulleys in terms of J and alpha.
His notes have a dynamic expression for belt driven pulleys with a rotating base:

T_m = J_mα_m +1/N * J_Lα_L, where N = R/r
And
rα_m = Rα_L <- this is the expression we've been tossing around.
=> α_L = r * α_m / R

Then
T_m = J_mα_m +r/R * J_Lr * α_m / R
(R² / r²)(T_m - J_mα_m) = J_L* α_m
=>
R = SQRT[((r²)J_L * α_m) / (r²)(T_m - J_mα_m)]

However, if T_m = J_mα_m only applies to a motor without a belt attached, then I am still left with 1 equation and 2 unknowns

 

[haruspex]

I am still left with 1 equation and 2 unknowns

You have not used the fact that the time is to be minimised.

 

[whitejac]

You have not used the fact that the time is to be minimised.

How would this look? From what my professor has told me, you set the torque to its max so that your arm/pulley/etc uses the minimum amount of time (though you'd factor in some safety in a real application because it's highly idealized)

 

[haruspex]

How would this look? From what my professor has told me, you set the torque to its max so that your arm/pulley/etc uses the minimum amount of time (though you'd factor in some safety in a real application because it's highly idealized)

Yes, but in this question you are asked to find the R which minimises the time.

 

[whitejac]

Yes, but in this question you are asked to find the R which minimises the time.

Well, in that case I would set R to the minimum it could go. Using the expression

R = SQRT[((r2)JL * αm) / (r2)(Tm - Jmαm)]

I would... set R to zero? That doesn't make sense.

 

[haruspex]

Well, in that case I would set R to the minimum it could go. Using the expression

I would... set R to zero? That doesn't make sense.

Why would that minimise the time?

 

[whitejac]

Why would that minimise the time?

Reduction in R reduces the length, thus the amount of time.

 

[haruspex]

Reduction in R reduces the length, thus the amount of time.

But it also changes the acceleration, so maybe you don't want the minimum length.

 

[whitejac]

But it also changes the acceleration, so maybe you don't want the minimum length.

I'm running out of variables here. The only other one not given is α_m, which is something I currently do not know how to find specifically.

 

[haruspex]

I'm running out of variables here. The only other one not given is α_m, which is something I currently do not know how to find specifically.

Concentrate on how the time depends on R. Your equation in post #23 does not include time. You need an equation that relates R to time.

 

[whitejac]

Concentrate on how the time depends on R. Your equation in post #23 does not include time. You need an equation that relates R to time.

I know it doesn't, that's why taking derivatives isn't something I can currently do. I don't know where to put the t because it seems like I'd just be placing it arbitrarily. R isn't related to time as the pulley isn't growing or shrinking with relation to time. The Radius reduces the acceleration linearly, as shown in the relationship but I have not seen from the beginning how I can conjure a t and justify it as a logical relationship.

 

[haruspex]

R isn't related to time as the pulley isn't growing or shrinking with relation to time.

The time taken would be different with a different R, so the two are related.
If the load accelerates from rest at α_L, how long will it take to turn through angle θ/2?

 

[whitejac]

The time taken would be different with a different R, so the two are related.
If the load accelerates from rest at α_L, how long will it take to turn through angle θ/2?

I simply don't know.