Start-up acceleration, max speed and torque of a transfer car

[r_prieto5]

Homework Statement

A transfer car moves concrete slabs.
Given values are wheel diameter D, shaft diameter d, rolling resistance F_L , speed v, car weight m_v , load weight m_k , output power P at rotation speed n, average torque M_m , motor moment of inertia J_m , fan moment of inertia J_t and efficiency of the mechanical transmission system η.

a) Calculate start-up acceleration and distance (up to maximum speed).
b) Calculate the torque of the driving shaft of the gear at start.

Homework Equations

None given... (this is the main problem). I am studying a course abroad which has support material in a foreign language.
I have found these so far:
P=M_m⋅ω
a=R⋅α
α=dω/dt

The Attempt at a Solution

I have tried to find the angular acceleration using power, torque, radius and speed, however, these don't work in the start-up case since v=0. As for torque, I think I first need to find the acceleration and the just input into torque=moment of inertia⋅angular acceleration
I also don't know how and what forces to sum to calculate the torque.

 

[billy_joule]

The Attempt at a Solution

I have tried to find the angular acceleration using power, torque, radius and speed, however, these don't work in the start-up case since v=0.

Show your attempt so we can see where you're stuck.
You can find acceleration at v=0 by assuming average torque is constant. Power is irrelevant when v=0 as no work is being done.

 

[r_prieto5]

@billy_joule The problem is I don't really know where to start, therefore, I don't really have an attempt as such. I have only come up with the conclusion that torque=angular acceleration x moment of inertia, but even then, which of the two moments of inertia am I supposed to use? I don't even know why I have two different moments of inertia
Once I solve that I will have the acceleration and from there I just use uniform acceleration formulas to find the distance reached at top speed.
As for start-up torque, I believe I need to calculate the sum of all weight plus rolling resistance forces and calculate the torque needed to move that? But then, with what formula do I relate angular with linear forces?

 

[billy_joule]

Is that the entire problem statement word for word?

@billy_joule The problem is I don't really know where to start, therefore, I don't really have an attempt as such. I have only come up with the conclusion that torque=angular acceleration x moment of inertia, but even then, which of the two moments of inertia am I supposed to use?

Well, do the fan and motor both need to be accelerated? Or to put it another way, would a fan or motor that didn't spin be any use at all?

Once I solve that I will have the acceleration and from there I just use uniform acceleration formulas to find the distance reached at top speed.

Yes, this approach will ignore P @ n and assume average torque is constant from v = 0 to v_max. There is really no way the given power variable can be used as it's only for a single RPM.

As for start-up torque, I believe I need to calculate the sum of all weight plus rolling resistance forces and calculate the torque needed to move that? But then, with what formula do I relate angular with linear forces?

T=Fr
The motor and fan are accelerated rotationally, the car and load are accelerated linearly. These two accelerations are constrained together as the linear velocity of the car depends on the rotational velocity of the motor ( ω_engine ∝ v _car ). To solve for linear acceleration, you need to form an expression that includes this fact.
It'll require Newtons second law:
F = ma
and it's rotational analogue:
T = Iα
You'll also need to account for torque loss in the transmission.

 

[r_prieto5]

I have attached a screenshot of the original problem.
My thoughts are both moments of inertia will only be used for the calculation of the torque on the shaft. They add each other?
I agree power won't be used for a) but isn't it used to find one more torque to add to b) with the formula P=Mm⋅ω?
How do I find torque loss? Does that involve rolling resistance and efficiency of the mechanical transmission system?

 

[billy_joule]

I have attached a screenshot of the original problem.
My thoughts are both moments of inertia will only be used for the calculation of the torque on the shaft. They add each other?

Yes they add, in just the same way you'd add masses constrained together to find linear acceleration.
eg
F = ma = (m_vehicle + m_load)a
T = Iα = (I_motor + I_fan)α

I agree power won't be used for a) but isn't it used to find one more torque to add to b) with the formula P=Mm⋅ω?

Well you're only given the power at a single RPM so it's not much use.
The question is rather vague, I think there are multiple way to interpret the situation:

1) The transmission is a 1:1 right angle drive (Fair assumption as transmission ratio isn't mentioned) - in which case the motor RPM is the same as the wheel RPM and is much lower than the given power value RPM making the provided power value completely irrelevant (as the motor never reaches that speed).

2) We are expected to assume the motor is running at the given power value RPM when traveling at top speed, this will give use a gear ratio that is not 1:1.
And it introduces a significant problem: The given power output does not match that found if we use the average torque value for that RPM eg

We are told P @ 2780RPM = 0.6 kW

and we know P = Tω (and 2780RPM = 291 rad/s) and are told average motor torque is 3.9 Nm
so P = (3.9 Nm) * (291 rad/s) = 1.13 kW

Which begs the question: When is the given motor torque valid? And when is the given motor power valid?
In conclusion, the question is very poor, I'd suggest skipping it or asking your tutor to clarify.

How do I find torque loss? Does that involve rolling resistance and efficiency of the mechanical transmission system?

You multiply the torque input into the transmission by efficiency (and the gear ratio) to find the torque output.

The rolling resistance is a resistive, external force on the car as a whole. So, we have two forces, the forward force due to the driving torque on the wheel:
T = Fr
T_{wheel shaft}=F_{wheel-road contact}r_{wheel radius}
so
F_{wheel-road contact} = T_{wheel shaft} / r_{wheel radius}

and the rearward rolling resistance force F_{rolling resistance}
If we apply Newtons second law we can solve for the net force

F = ma = F_{wheel-road contact} - F_{rolling resistance} = ma
Solve for a

a = (F_{wheel-road contact} - F_{rolling resistance} ) / (m_vehicle + m_load)

 

[r_prieto5]

We are told P @ 2780RPM = 0.6 kW

and we know P = Tω (and 2780RPM = 291 rad/s) and are told average motor torque is 3.9 Nm
so P = (3.9 Nm) * (291 rad/s) = 1.13 kW

I had also delved around this thought, it does not make any sense...

You multiply the torque input into the transmission by efficiency (and the gear ratio) to find the torque output.

According to this you say and some calculations I managed to make last night, I came up with an equation, I hope it works for calculating the torque on the shaft once I find the angular acceleration:
M_s=(J_m+J_t-((D-d)/2)⋅F_L)ηα (considering transmission efficiency not only affects power)
The problem now would be finding either the angular or linear acceleration since the equation you gave needs for me to find the shaft torque. Is it possible to find the angular acceleration from the fact that I know the moment of inertia and torque of the motor? I would simply solve M_m=(J_m+J_t)α, or α=M_m/(J_m+J_t) but then again, that is not affecting the shaft or wheels...

Edit: what if I use I=mr²/2 using the total mass of the cart plus slabs, and then find α=Torque of motor/I ?

 

[billy_joule]

According to this you say and some calculations I managed to make last night, I came up with an equation, I hope it works for calculating the torque on the shaft once I find the angular acceleration:

Well, which shaft? If we assume the power flow goes motor > Fan > transmission > Wheels and there is a shaft between each step then torque will decrease at each step (assuming no gearing).

M_s=(J_m+J_t-((D-d)/2)⋅F_L)ηα (considering transmission efficiency not only affects power)

That equation is not dimensionally consistent. You can't add a moment of inertia to a torque. And 'd' should not appear in your working, the wheel diameter is the only relevant distance. Also, F_L does not produce a torque about the wheels.

Is it possible to find the angular acceleration from the fact that I know the moment of inertia and torque of the motor? I would simply solve M_m=(J_m+J_t)α, or α=M_m/(J_m+J_t) but then again, that is not affecting the shaft or wheels...

That would be the correct solution of the motor were only accelerating itself and it's fan, but we know that's not the case.

Edit: what if I use I=mr²/2 using the total mass of the cart plus slabs, and then find α=Torque of motor/I ?

The cart and slab aren't rotating so their α is zero and moment of inertia (about any axis) isn't relevant.

I'll post a more complete explanation of the correct approach tomorrow to get you on the right track.

 

[r_prieto5]

Well, which shaft? If we assume the power flow goes motor > Fan > transmission > Wheels and there is a shaft between each step then torque will decrease at each step (assuming no gearing).

The question says on the driving shaft of the gear. I assume this means the point of the shaft between the wheels and motor-fan set-up.

That equation is not dimensionally consistent. You can't add a moment of inertia to a torque. And 'd' should not appear in your working, the wheel diameter is the only relevant distance. Also, FL does not produce a torque about the wheels.

You are right, I made a mistake there, this should be the equation I meant Ms=(Jm+Jt)ηα-((D-d)/2)⋅FL and I think the shaft diameter does affect since the moment created by the rolling resistance times the radial distance (torque=force x distance) starts from the end of the shaft's radius until the end of the wheels right?

That would be the correct solution of the motor were only accelerating itself and it's fan, but we know that's not the case.

Ok then, if F=ma, Torque=F x distance and the mass is given, a=Torque / (Distance x m). Is this any help?

 

[r_prieto5]

Ok after analyzing your past messages I came up with a new formula.
a=(T/r-F_L)⋅mη
This considers Fwheel-road contact = Twheel shaft / rwheel radius, however T=moment of inertia x angular acceleration; how do i find the latter?