How to Cut Narrow Body and Install Weight

[FatSebastian]

Before the curve and after the curve, there is zero rotation of the car nose and hence zero energy of rotational motion tied up in such rotations.

Before the curve the nose is pointing downward (say, 30 degrees, or whatever the initial slope is). After the curve the nose is pointing level, so a permanent net rotation of the body has occurred.

[Darin McGrew]

Question is, is that a permanent loss of energy, or just a temporary borrowing?

In theory, it's conceivable that you could get the energy back. In practice, there really isn't any mechanism to get the energy back. One exception might be the rotational energy in the spinning wheels, except that the flywheel effect won't help without a much longer track than anyone uses.

[FatSebastian]

It is possible that all of the rotational energy gets converted back to linear motion energy.

Zero angular velocity (pitch rate) before and after the transition could imply that energy was spent and subsequently returned, but it could also imply that energy was consumed twice: once in order to start pitching, and once again to stop pitching. Doc Jobe effectively states this on p. 110 "...half the rotational energy is used to rotate the car to the 1/2-alpha position and the other half to decelerate the rotation from 1/2-alpha to alpha" [alpha being the angular difference in slope before and after transition].

Consider a heavy ballistic pendulum. If a shot is fired into it, then in a frictionless and airless environment it would swing forever. However, in theory the swinging could be stopped by firing a perfectly-timed, identical shot from the opposite direction. In both cases translational kinetic energy was consumed to start the pendulum, and return the pendulum to its former (stopped) state.

Thought experiment time. If this was an ideal roller coaster with one car and the track had a long series of humps rather than just a single concave section --- would a barbell-weighted car suffer permanent losses at every hill and dale and come to an absolute stop? Or would it continue on to the end just like a central-weight-only car? I think it would continue!

I would like to think this too. But in this Gedankenexperiment, the orientation of the roller-coaster cars are cyclically returned to their former state of pitch by the conservation of energy under uniform gravity (not unlike a pendulum). I agree with Darin that in the usual PWD track configuration, there is no mechanism by which to get back energy spent pitching the nose upward. But if a PWD track were shaped like an ogee, or S-curve, then seemingly there would be a mechanism by which to regain that energy, because the nose could be allowed to return to its initial orientation.

[Duane]

Doc Jobe effectively states this on p. 110 "...half the rotational energy is used to rotate the car to the 1/2-alpha position and the other half to decelerate the rotation from 1/2-alpha to alpha" [alpha being the angular difference in slope before and after transition].

So he's saying that energy is permanently lost at both transitions. Did he confirm those particular speculations with actual experiments?

His wording makes it sounds like the mid-alpha angle is special. That's misleading, for Besttrack-style curves. The energy transfers are during the acceleration/decelerations of the rotation rate, not in the changed rotation angle or the duration of the coasting phase of rotation. If the curve is circular and extends say 45 degrees, there is little change to rotation rates or energy of rotation as the car navigates most of those 45 degrees. (The rotation rate does speed up a bit more due to the car continuing to roll faster.) Most of the rotational acceleration and all of the deceleration occurs in the sudden transitions from straight to curved, and then back. If the track instead curves just 15 degrees total but has the same abrupt transitions, then the cars experience exactly the same brutal rotational thumps, at only the start and end of the curve; no difference during the circular curve itself.

Do Besttrack curves go from straight to 48" radius curves all at once at a point? That would be brutal for cars. Or is there some transitional bending (eg wider radius) that slows down the induced rotational accelerations?

The rate at which the car gets accelerated upwards, then downwards, is partially determined by the wheelbase distance, because this helps average out the slopes seen by front and rear wheels. If Besttracks have no transitional curves, then the car's wheelbase is the only thing determining the duration and rate of rotational accelerations.

The variations in axle pressures, and resulting steering problems, would be worst in cars with short wheelbase and barbell weighting. The pressure variations would be least in cars with long wheelbase and com-compacted weighting.

[Duane]

Before the curve and after the curve, there is zero rotation of the car nose and hence zero energy of rotational motion tied up in such rotations.

Before the curve the nose is pointing downward (say, 30 degrees, or whatever the initial slope is). After the curve the nose is pointing level, so a permanent net rotation of the body has occurred.

Yes. I should have said "there is zero rotational speed of the car nose".

[Duane]

Consider a heavy ballistic pendulum. If a shot is fired into it, then in a frictionless and airless environment it would swing forever. However, in theory the swinging could be stopped by firing a perfectly-timed, identical shot from the opposite direction. In both cases translational kinetic energy was consumed to start the pendulum, and return the pendulum to its former (stopped) state.

That example assumes an "inelastic collision", where the bodies merge and lots of kinetic energy gets turned permanently into heat. In billiards, the collisions are "elastic", meaning the balls rebound perfectly with almost no heat loss. In that case, the shot bullet bounces back towards the gun. The target's changed momentum and changed energy of motion gets transferred to the other object, causing a higher speed if it was lighter. Nasa often uses Jupiter and the Earth as heavy magic pendulums to re-fling spacecraft at much higher speeds than we can get by hauling rocket fuel that far into space.

Lots of discussions about PWD physics assert that various shakes and bumps are causing energy loss to heat. I'm not yet convinced that the nose-up rotations of track curves are one of those. The collisional interactions of car and track curve could be mostly elastic; all of the materials involved are fairly or very elastic and nothing suffers a permanent change of shape.

[FatSebastian]

So he's saying that energy is permanently lost at both transitions.

He is? Rather than speak for him, I can only repeat what I said earlier:

"I agree with Darin that in the usual PWD track configuration, there is no mechanism by which to get back energy spent pitching the nose upward. But if a PWD track were shaped like an ogee, or S-curve, then seemingly there would be a mechanism by which to regain that energy, because the nose could be allowed to return to its initial orientation."

I therefore suppose there is a difference between having energy "permanently lost" (to say, heat) versus having energy conserved in a form which cannot be converted back into translational kinetic energy (forward speed).

Did he confirm those particular speculations with actual experiments?

Generally speaking, the motivation for Doc's work seemingly intends to align theory (to the degree that it can be modeled) with observation. As stated before, the conclusion seemed to be that the loss of speed caused by transitional pitching is barely detectable on a millisecond timer; if so, it would not likely be possible to detect experimentally.

His wording makes it sounds like the mid-alpha angle is special.

The quote actually starts "We can imagine that half the rotational energy..."; that is, this is the way he chose to frame his thinking about the problem in order to approximate a solution. The expression for rotational kinetic energy is:



Jobe approximates omega-squared as the angular distance (alpha) squared divided by the time it takes to travel through alpha (the transition) squared. [This is Eq. (5.135) on p. 110.]

I should note that this a topic to which he devotes at least six pages of analysis, only to seemingly conclude that the energy loss due to transitional pitching is barely perceptible. The limited context of my presentation here may be the actual culprit that is misleading; due to your above-average interest in this topic I strongly encourage you to acquire and consult his book (which will be more productive for you than my imperfect recollections and partial quotations in this forum).

Do Besttrack curves go from straight to 48" radius curves all at once at a point?

I do not own one, but my understanding is that a Best Track goes from straight to a 48" radius curve then back to straight.

I did not understand what was meant by "all at once at a point." As the section of an arc of fixed radius gets shorter and shorter, the angle of transition between the ends becomes less and less. When the length of the transition arc becomes a point (zero), there is no transition at all and the angle change is zero.

The rate at which the car gets accelerated upwards, then downwards, is partially determined by the wheelbase distance, because this helps average out the slopes seen by front and rear wheels. If Besttracks have no transitional curves, then the car's wheelbase is the only thing determining the duration and rate of rotational accelerations.

I am getting lost by the terminology. By "rate of rotational acceleration," do we mean the time derivative of angular acceleration (positive or negative) of the body? I don't quite know how these higher order derivatives fit into the energy calculation. And "If Besttracks have no transitional curves", then it would seem to me that no rotation due to transition is occurring, such that there can be no rotational accelerations.

That example assumes an "inelastic collision"...

I'm glad it was obvious from the description that this was assumed.

Lots of discussions about PWD physics assert that various shakes and bumps are causing energy loss to heat. I'm not yet convinced that the nose-up rotations of track curves are one of those.

I noticed no claims about whether nose-up rotations due to track curvature cause energy loss to heat. However, I might suggest simply that if the orientation of an object changed against gravity, then that seemingly was a consequence of Work (or Torque), therefore Energy had to be redirected for that purpose. If so, the most relevant question is the amount of energy that is redirected due to transitional pitch, and whether that amount can be appreciably altered by varying the mass moment of inertia of the body.

As far as I can tell, based on evidence available so far, the degree to which energy can be redirected during transitional pitch by varying the mass moment of inertia of the body does not seem to be appreciable. Since there is experimental evidence to suggest that the mass moment of inertia matters, then either the effect of transitional rotation is much greater than theory predicts, or one needs to start looking beyond transitional rotation to explain the experimental evidence. Personally I would tend to look elsewhere (such as rail impacts) as a possible source. I also cannot rule out that there might be unknown issues with the limited experimental evidence presently available.

[AlabamaDan]

OK. I'm lost.

[FatSebastian]

OK. I'm lost.

To simply summarize (I think)...

Some experimental evidence suggests that concentrated mass improves elapsed time. Why it makes a difference is not obvious.

[Duane]

To simply summarize (I think)... Some experimental evidence suggests that the mass distribution makes a difference in the elapsed time. Why it makes a difference is not obvious.

In particular, the most experienced builders here and elsewhere in the PWD universe have found, by many builds and races, that they get even better cars if they concentrate the added weights. And minimizing wood allows even more concentration and even faster cars. How come, and how much?

My personal guess today is that barbell weighting's total effect on energies etc is small, but that there is a nontrivial effect on axle pressures at the very beginning & very end of Besttrack curves. And that change in axle pressure messes up steering enough to trigger fishtailing, for cars with very aggressive com offset. Concentrating the weight allows the builder to safely make the com offset more aggressive than in barbell-weighted cars. (end of guess.)

I promise to buy and study Doc Jobe's book firsthand.
Meanwhile ...

When cars are going through the concave-curved part of a PWD track, they are temporarily spinning nose-upward, from their initial 30-degree nose-down start ramp position to their final nose-level position on the flats. This pitching/spinning action involves some rotational kinetic energy. As with the rotational energy in the 3 or 4 wheels, this takes away from the kinetic energy otherwise available for linear downramp motion of the com. (Perhaps permanently, perhaps temporarily.) The amount of diverted energy might be a trivially small part of the total. There is also some pushing actions (from front wheels at start of curve, and from rear wheels at end of curve) to initiate that nose-up rotation and then to end that rotation. The total rotation also affects the com's altitude above the front wheel footprint, which affects potential energy.

The only direct physical effect of barbell weighting versus concentrated, is that it increases the car's 'angular moment', its resistance to being accelerated rotationally by the nose axle pressures. All other effects from the barbell weighting must be some kind of side effect from that changed resistance. But how?

On Besttrack curves, the track curve stays fixed at a 48-inch-radius throughout the curve. If the car's speed remained constant (rather than actually accelerating a bit), then the car's rate of rotation would remain constant throughout the curve. That rotation is not accelerating nor decelerating. So there are no extra nose-up pushes being experienced by the car axles while in the curve. But at the very start of the curved section, the car is rapidly switched from no rotation to max-rotation rate. And at the very end of the curved section, the car is rapidly switched back from max-rotation rate to no rotation. This switching involves accelerations, angular moments, forces, and extra axle pressures. The switching is fortunately not instantaneous, because of the car's wheelbase separation. (If the switching were instantaneous, the car nose and tail would be experiencing very high G forces, like with a car dropped nose-down onto the floor.) The transient axle pressures could add frictions that affect steering, or could partially unload a wheel and affect steering.

Similar things happen on non-Besttrack tracks, but at a much gentler scale. The rotations are slower, over a longer distance, and the rate of rotation accelerates from zero to max over a longer time and longer distance, with lesser forces and pressures.

Acceleration is a change in speed, over time. It is always proportional to current forces, divided by the mass (or angular moment, for rotational speed). The forces and accelerations on an object can change suddenly in an instant, in a discontinuous way, as in impacts. But an object's speed and position can only change slowly and smoothly in a continuous way, as a cumulative result of its accelerations over time.

Most forces and accelerations in PWD are generally gentle and changing smoothly. But the nose-up forces at the start and end of Besttrack curves are apparently not so gentle.

The indirect problems from barbell weighting, are either steering loss, or friction loss, or energy getting diverted to car rotation, or some combination. All three could be affected by the car's angular moment, attempting to resist the curve motions that can't be resisted.

[FatSebastian]

In particular, the most experienced builders here and elsewhere in the PWD universe have found, by many builds and races, that they get even better cars if they concentrate the added weights. And minimizing wood allows even more concentration and even faster cars. How come, and how much?

Remembering MaxV's interesting tests where he first concluded that increasing MOI was beneficial, but later concluded that MOI has no appreciable effect, here are some added thoughts...

1. The most-experienced builders place the CoM toward the rear of the car. This configuration pretty much requires that the mass moment of inertia of a 5-ounce body be near the lower end of possible MOI values. So reduced mass MOI is in many cases simply an artifact of having a competitively placed CoM.

2. The most-experienced builders also rail ride (thus reducing yaw impacts) and often raise one of the front wheels, minimizing variations in the car's performance from trial to trial.

3. Experiments such as MaxV's and Cory's seem to place the CoM toward the center of the car. This is presumably done to extend the range of available mass MOI in order to exaggerate and detect its effect. But this results in an unrealistically high MOI and also a highly non-optimal CoM compared to a real racing environment.

4. Cory's tests ran with all four wheels down and to the best of my knowledge did not employ rail riding. MaxV's test car rode the rail with all wheels down. The extra wheel down will change test speeds and possibly affect steering. A lack of rail riding would certainly increase the chances of multiple rail collisions (including rear-wheel contact) that would not ordinarily happen with today's rail-riders.

These comments are not intended to disparage previous test work; rather, it is just an observation that other influential variables may alter the relevance of test data, especially with respect to today's most-experienced builders and their practices. Duane has already noted air drag as a possible variable within Cory's tests. It seems to me that MaxV's latest test is the one which controls the variables of drag profile and rail riding most tightly, and his conclusion was that the benefit of reduced MOI was almost undetectable. My understanding of Doc Jobe's theory is that the redirection of energy caused by transitional pitch is also almost undetectable. This evidence - both theoretical and experimental - is in keeping with Duane's initial instincts and continues to suggest that optimizing mass MOI is a secondary or tertiary design consideration that no one should lose sleep over.

Seemingly, further testing of variable MOI on high performance bodies (aft-weighted, 3-wheeled rail-riders) would be more useful than armchair physics at this point. A nagging question is how well the CoM has been maintained between the high MOI test body and the low MOI test body in such testing already. Precise CoM location is best characterized by the amount of weight on the front and rear wheels as measured by a scale, which I'm not sure was done in any of the tests cited.

[stang68]

We use a wood planer to plan down the bodies to about 3/16.We cutout everything except 1/4" upfront and leave about 7/16" at the axle slots,sides rails 3/16"wide,then we strengthen the body with epoxy.Lead is poured into matching body molds,then epoxied into car.These cars are the fastest ones we have come up with so far. I DO NOT RECOMMEND POURING LEAD TO ANYONE THAT DOES NOT KNOW HOW TO DO IT SAFTLY.

[Duane]

It requires energy to rotate a massive object, and the greater the mass moment of inertia, the more energy is required, for a given angular velocity. Therefore, via conservation of energy some translational speed must be lost due to this rotation. The question is whether the loss is significant enough to account for the experimental values.

Thanks, I wasn't yet seeing the diversion of energy into car rotation. Yes, the harder-to-flip car must be putting more of the current energy into rotational momentum rather than just down-the-track linear momentum, than does the easy-to-flip car. Question is, is that a permanent loss of energy, or just a temporary borrowing?

Okay, I've done some calculations that are likely roughly right.
The diversion of some kinetic energy into nose-up rotations is a real but tiny effect.

For a car dropped down a 48"-high ramp with no curve (oops!), at the bottom it is travelling at about 15 feet/sec. About 1/15th of the car's total Kinetic Energy is diverted into spinning its BSA wheels. (The ratio between wheel spin KE and total-body linear motion KE is basically the ratio between wheel tread weight versus total car weight.) With weightless wheels, the car's linear KE would be higher by 15/14, and its current linear speed would be higher by sqrt(15/14).

With circular curves like Besttracks, the 30-degree curve has a track length (partial circumference) of 25 inches. The car crosses the curve in 0.14 seconds. While going through the curve, the car experiences a centripetal force of 56 ft/sec*sec, or 1.75 G's. This is in addition to gravitational forces, which are 0.87G at the 30-degree start of the curve, and 1.0G at the level exit of the curve. So when starting the curve, the front axle sees a sudden change from 0.87G's of pressure, to 2.62G's of pressure. During the curve, this pressure grows slightly and smoothly. When exiting the curve, the front axle sees a sudden change from 2.75G's to 1.0G's of pressure. Similar things happen at the rear axle. The centripetal axle loadings are greatly affected by com position but only slightly by MOI effects.

At no point does either axle become underloaded. In prior posts, I imagined that the change in accelerations at the start and end of the curve were accompanied by big shocks to the axle. I don't think that anymore. There is a sudden jump or drop in pressures and accelerations, but nothing extreme like in impacts. In prior posts, I imagined that the wheelbase made a different to the duration and intensity of the imagined shocks, but I now think wheelbase is irrelevant here.

During the curve, some of the car's total Kinetic Energy is also diverted (temporarily) into spinning the entire car nose-up. The amount of diversion depends on the car's Moment of Inertia for its waistline axis. During the curve, the car is spinning at a rate of 1.7 seconds for a full 360-degree flip. When measured at the axles (for 4.5inch BSA wheelbase), the axles are moving circularly at 0.6 ft/sec. If the car's MOI is quite bad (2.5oz concentrated at each axle, with com exactly in between), then 0.2% of the car's total Kinetic Energy is now in that motion rather than in wheel spinning or in progress down the track.

If that diversion were permanent (or lost to friction) and if the MOI could be somehow entirely eliminated in other cars, the bad car's lowered KE is equivalent to lowering the start height by 0.1 inch, or increasing the com's offset from rear axle by 0.2 inch. But no-one builds race cars with such bad MOI, and wheel weights prevent building of cars with zero MOI. And the energy diversion lasts only as long as the curve itself. I estimate that the practical effect of typical MOI versus best-achieved MOI is more like changing the com offset by only 0.01 inch or so.

I have two armchair arguments for why the Kinetic Energy in whole-body rotation is only temporarily diverted.

1. People have guessed that this KE gets permanently lost. The only destination for permanent loss is heat, via axle frictions. But friction losses would be greatly affected by whether axle bearings were unlubed untrimmed nails, or well-lubed ideal machined bearings. But the KE supposedly lost to friction would be the same in either case, which can't be.

2. The way for whole-body rotational energies to get converted back to linear motion was not obvious. Imagine two Besttracks with flats removed, joined together toe-to-toe at the bottoms of their curved sections. Overall shape is a V, with a rounded bottom. Use a car with ideal bearings and with front/back symmetry. Remove all air and air drag. Videotape a run. The car goes down, and then climbs up all the way to the 4-ft-high point on the far end. The positions, speeds, angles, accelerations, and forces on the second upward half are identical to those on the first downward half, but reversed in time. Play the videotape backwards. The car's descent on the first half now looks like an ascent, with exactly the same positions, motions, angles, etc as seen on the second half of the forward-time video. What this shows is that the mechanisms for un-spinning the car are identical to those that start the nose-up spinning of the car, just reversed in time or direction. Furthermore, exactly the same motions happen whether the car has zero MOI or heavy MOI. This thought experiment does not prove that there are no frictional effects in real cars, but does show that everything observed can be explained in the ideal case, without frictions.

[Stan Pope]

Duane,

I think that #1 is missing consideration of exit speed. With that adjustment, the KE losses to heat are different.

[Stan Pope]

I'm still thinking about #2, Duane. It makes the assumption that the heights on each side will be the same, and while that sounds plausible, I'd like to understand more.

I've got a feeling that the formula for this idealized car-track is like the formula for the pendulum ... the pendulum approximates to "simple harmonic motion" until the angle of displacement gets large. Then it gets complicated. My univ. physics prof didn't want to go there. But he was happy to explain the equations of relativity.

Still, I think that the two motions are analogous, i.e. described by the same equations of motion if the bob on the pendulum is larger than a point mass and the displacement is large. With 30 degrees of displacement (on each side) I think that this hypothetical BestTrack pendulum is into the complicated range, too.

With that analogy in mind, I asked Google about the "complicated case" pendulum formula, but so far it has mimiced my profs. Maybe I can find 'em in the "big green book!"