Drive Line Out Of Gear Correction

[ricky.helgess]

I am working on a vehicle based computer game. The focus right now is to get the internal physics of each drive line component as physically correct as possible. Since a few weeks, I am stuck with a problem that I cannot get past by myself. I have narrowed it down to a specific issue. It should not be too hard to solve but I am stuck so please help. :)

Prerequisites

* I use a tree of physical components to represent the drive line. To simplify my question, let's see each component in the drive line tree as several two-disc-gears, welded together (I am not sure if I use the English language correctly here - I hope you can follow my reasoning anyway).
* Each two-disc-gear has an inertia of 0.1.
* The time step in calculating is 1/50 = 0.02.
* The first (left) gear's small disc is connected to the second gear's large disc and so on and so on.
* The radius of the large discs is 1 meter.
* The radius of the small discs is 1/3 meter.
* There are three gears in my example.
* The first (left (index 1 below)) and last (right (index 3 below)) gears have a weight attached to them at the same height as the fulcrum, both at 1 meter from the fulcrum.
* The weights generates 10 N each.
* The gears are at rest at t0.

Problem

What should my formula look like to find the angular velocity of each gear at t1 (0.02 seconds later)?

Let's take a look at the last (right (index 3 below)) gear and find how much it's angular velocity should change:

* The resulting torque multiplied with the time step should equal the system's change in angular momentum.
* \tau\Deltat = \Delta\omega₁I+\Delta\omega₂I+\Delta\omega₃I
* Since the gears are interconnected, their velocities are linked to each other:
\Delta\omega₁ = 3\Delta\omega₂
\Delta\omega₂=3\Delta\omega₃
* Meaning: \tau\Deltat=9\Delta\omega₃I+3\Delta\omega₃I+\Delta\omega₃I
* Meaning: \tau\Deltat=13\Delta\omega₃I
* Meaning: \Delta\omega₃=(\tau\Deltat)/(13I)

Now, WHAT is the resulting torque? If I calculate from left to right I get 80 Nm. If I calculate from right to left, I get 8.888...

It seems I should use 8.888... but why? If the rightmost gear was connected to the ground, I would move the vehicle with 80 Nm/tire radius (thinking about the rightmost gear as a tire on the ground). But when not all the torque is added to the ground and I must use the excess torque to speed up the drive line tree, all of a sudden, I cannot use 80 Nm. Why? Where am I going wrong?

If I have torques added on different places in the drive line tree, how do I use these when speeding up the drive line tree?

The calculations I use work very well when the rightmost gear (tire) has full grip but when I have no grip, I just get too large values if I use the gear-ratio-multiplied torques... so, how should I think?

/Ricky

 

[ricky.helgess]

Since I did not get any answers, I will change this into a simpler example:

You have two gears (not welded double gears as in the first example, just two simple gears). The gears are connected to each other using a weightless chain belt.

Each of the two gears have the following quantities:

Inertia (I).
Radius (r).
An external torque acting on the gear (\tau).

We have a time step (\Deltat).

The goal is to find the change of angular momentum of the gears (\Delta\omega).

What is the formula to find the angular momentum change of each of the gears?

I tried the following but somewhere I go wrong:

The angular velocities of the two gears have a relationship:
\omega_{2} = \omega_{1}\frac{r_{1}}{r_{2}}

The total torque multiplied with delta time should equal the total system's change of angular momentum:

\tau_{total}\Delta t = \sum\Delta\omega_{i}I_{i}

Now, looking from the first gears perspective, I try the following:

(\tau_{1}+\tau_{2}\frac{r_{1}}{r_{2}})\Delta t = \Delta\omega_{1}I_{1}+\Delta\omega_{1}\frac{r_{1}}{r_{2}}I_{2}

...which means that \Delta\omega_{1} should be:

\Delta\omega_{1} = \frac{(\tau_{1}+\tau_{2}\frac{r_{1}}{r_{2}})\Delta t}{I_{1}+\frac{r_{1}}{r_{2}}I_{2}}

This does not seem to work though, because when I enter numbers in the equation, I always get the same change in angular velocities on both gears and that is obviously not right.

Please give me some direction! :-)

/Ricky

 

[ricky.helgess]

I found the error! :)

The velocity of the second gear is \Delta\omega_{1}\frac{r_{1}}{r_{2}} and the torque needed to rotate it is \tau \frac{r_{1}}{r_{2}} which means that the formula should be:

<br /> (\tau_{1}+\tau_{2}\frac{r_{1}}{r_{2}})\Delta t = \Delta\omega_{1}I_{1}+\Delta\omega_{1}\left(\frac{r_{1}} {r_{2}}\right)^{2}I_{2}<br />

Finally I can continue with other stuff in the game.

/Ricky