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Rear wheel spacers for B2710
Does anyone know if it is possible to widen the rear wheels an inch or so on each side by putting in a spacer? What are these usually made of  a thick disk of steel? Or aluminum? I was given this suggestion today and would like to try it. But I am also a bit paranoid about those cantilevered wheel axles sticking out so far already. Has anyone seen a bent rear axle before?
Here is a discussion that I found:
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Rear wheel spacers for B2710
No one seems to be weighing in on the matter, so I got out a pencil and made a few guesses.
Let's say that a tractor has its center of gravity (CG) up off the ground at 3', and centered between the wheels. And the rear wheels are each 2' out from the midline. This would give a tipping angle (where the CG is over the wheel's contact point) of 33.4 degrees (inverse tangent of 2/3). There are many variables here, but I just want to make a comparison.
So let's put a 4" spacer in, to increase the wheels to 2.33'. This now gives a tipping angle of 37.9 degrees, or 4 degrees better. Just by coincidence, it turns out in this case that an inch of spacer results in a degree of improvement, as a rough rule of thumb.
How does that affect the stress on the axle and bearings? On the B2710, it looks like the two axle bearings are 6" apart, and that the axle cantilevers out 1' from the end bearing. This would mean that, if the tire had 1000lbs of load on it, the inner bearing would have twice that, 2000lbs, and the end bearing would then have the sum of those, or 3000lbs. With that 4" spacer in there, the inner bearing would now have 2666lbs, and the outer 3666lbs, or a 22.2% increase.
Is the small improvement in tipping angle worth the increase in stress for this type of design? Death or broken bearings, which is worse?


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Rear wheel spacers for B2710
I would agree with mango. You would be better to add rear wheel weights than add spacers. Rear wheel weights will not add to additonal stress on the bearings or bending stresses in the axle.
If fact, mango if you have the time, run the numbers and see how much weight would have to be added to a wheel to accomplish the same tipping angle. It isn't much.


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Rear wheel spacers for B2710
OK, good idea. Carrying on with the hypothetical example of a CG 3' off the ground (which on the B2710 looks like it would be the case with the bucket up a ways, for ex.) on wheels 2' out from center. In order for the tipping angle to be improved by a little over 4 degrees, the same as provided by the 4" wheel spacer, the tractor's CG would need to be lowered by about 5".
The axles are about 18" off the ground. To get the 5" lowering of CG, given the tractor weighs 2400 lbs, you would need 923 lbs of weight if they were added concentric to the wheels. That is a fair chunk of mass.
The rear tire measures about 20 gallons of volume. If partially filled with water, so as to obtain the greatest lowering of CG, I estimate that the tires would hold about 100 lbs each (~2/3 full). This would place the water's CG down lower than the axle, say at 12" off the ground. This only gives a lowering of the tractor's overall CG of just less than 2", not so much. It is a little complicated calculation to determine the CG of a partially filled tire, so I am guessing a bit here, but you may not get the maximum lowering of overall CG by filling the tires all the way full, since some of the weight ends up so high.
So it looks like perhaps a smaller spacer, and some wheel weight in combination might be a good comprimise. The 3PH is rated for 1700 lb. If I never carry anything that heavy, I should be within safe limits for bearing load.
Hope someone can check my backofenvelope, early morning math...


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Rear wheel spacers for B2710
Mango, your calculations seem reasonable. I don't think a full integral analysis would give much higher accuracy. But here's something to think about. Widening the track would help in a rollover situation such as mowing along a sidehill. There the forces are sideways. But my worst sphincter puckering has happened when I have a loaded bucket and one front wheel falls into a depression. The rear end comes up and if the rear wheels are off the ground, it doesn't matter how wide they are, tipping could occur. Counter weight is necessary. I think your thoughts of a combination of widening and weight are right on.
Dave


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Rear wheel spacers for B2710
Mango, I don't think you took into account the couterweight effect of the wheel weights. It seems that the lowering of the center of gravity calculation is reasonable, but I don't think the couterweight effect is in your calculations. The wheel weights would be about 2'0 from the xaxis center of gravity, offering significant couterweight effect. In doing a tipping calculation, the entire spread of the rear wheels with the wieghts would be used. Assuming a rear wheel weight of 250lbs, and a rear wheel spread of 4', taking a moment about the downhill wheel would lead to a couterbalancing moment if 1000ftlbs. The yaxis contributing moment would be 375ftlbs.
Plug this moment into the overall stability calculation, and you will see that it effects the overtuning moment significantly. When taking moments for stability, assume the rear wheel weight as a secondary load, not part of the tractors mass. Use your same assumptions as before for the center of gravity for the tractor mass, and couteract it with the secondary load induced moment. Compare that to the 4" increase in the moment arm (since it is from the CG to the tipping point) from the wheel spacers. I know these are all approximates, but it will give orders of magnitude.


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Rear wheel spacers for B2710
I'd agree that the *initial* resistance to rolling would be increased by the moment of wheel weights, and that the dynamics would also be improved by the increase in rotational inertia, leading perhaps to a lessened sphincter response when dropping a front wheel into a hole, as mentioned. If it feels better, it probably is better.
But the all critical tipping angle is just that point when the moment is gone, and the combined CG is above the pivot. If my aging synapses have got this point correctly, in rigid body analysis, you can subsume all the distributed masses into one overall point mass for all intents and purposes. So those two 250lb wheel weights, one on each side, would then be equivalent in CG to 500lb, centered at the middle of the axle. That then combines with the CG of the tractor.
Is that counterintuitive? It's different from a sailboat, for instance, where as you heal over, your pivot point (center of bouyancy) moves out to the side away from the the CG. So ballast effects increase with heal angle. with the tractor, ballast effect decrease with tipping angle, which is why you want to give a good safety margin to allow for those holes. Again, resistance to shock rolling by increase of rotational inertia (weights far from center) is a Good Thing (tm).


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Rear wheel spacers for B2710
All the same, agreed. The reason why I was using a rotational moment approach was because the yaxis CG point is unknown. Therefore, treat the difference betweeen spreading the wheels and adding wheel weights as a seperate issue, eliminating the unknown ydimension of the CG, and look at the differences in rotational moment from these two influences only.
The Center of Gravity is by definition what you suggest, the center of all rotational moments of the mass. For the entire system, this would determine the tipping point of the tractor. It is just a little difficult to calculate where it is for the yaxis. But I would disagree that counterrotational moments are effective only in dynamic loading conditions. The static equilibrium point is justified as is the dynamic.


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Rear wheel spacers for B2710
All the physics and dynamics aside. The big problem with a tractor's stability arises from the pivoting front axle. When you are carrying a load on the FEL it becomes even worse.
If you happen to be close to the critical point and a front wheel drops into a hole or something then the sudden weight transfer may be more than enough momentum to throw all those calc.'s out the window.
Best of luck.


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Rear wheel spacers for B2710
This is an interesting thread. I need to upgrade my membership so I can read all the postings. When I was a senior in engineering school, my design project (and also paying job working for my advisor) was to develop a "Stability Index" for tractors. The models I came up with took into effect CG height, track width, front axle pivot height, turning rate, slope, and vehicle speed. Surprisingly, the weight of the tractor reduces out to not being a factor. The final number to represent stability was a number from 0 (lifting a rear tire) to 100 (motionless on level ground). I rigged a Ford 800 with sensors and a datalogger, drove around turning on hills, then analyzed the data. If I remember, the lowest "Stability Index" I recorded was around 45 when operating on a 10 degree slope. Of course, I didn't really want to roll Colorado State's tractor over so I didn't get too carried away!


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