March 19th, 2005, 10:10 AM  #11  
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Straights
The Physics of Racing, Part 9: Straights Brian Beckman physicist and member of No Bucks Racing Club P.O. Box 662 Burbank, CA 91503 ©Copyright 1991 We found in part 5 of this series, "Introduction to the Racing Line," that a driver can lose a shocking amount of time by taking a bad line in a corner. With a sixfootwide car on a tenfootwide course, one can lose sixteen hundredths by "blowing" a single rightangle turn. This month, we extend the analysis of the racing line by following our example car down a straight. It is often said that the most critical corner in a course is the one before the longest straight. Let's find out how critical it is. We calculate how much time it takes to go down a straight as a function of the speed entering the straight. The results, which are given at the end, are not terribly dramatic, but we make several, key improvements in the mathematical model that is under continuing development in this series of articles. These improvements will be used as we proceed designing the computer program begun in Part 8. The mathematical model for travelling down a straight follows from Newton's second law:
where F is the force on the car, m is the mass of the car, and a is the acceleration of the car. We want to solve this equation to get time as a function of distance down the straight. Basically, we want a table of numbers so that we can look up the time it takes to go any distance. We can build this table using accountants' columnar paper, or we can use the modern version of the columnar pad: the electronic spreadsheet program. To solve equation (1), we first invert it:
Now a, the acceleration, is the rate of change of velocity with time. Rate of change is simply the ratio of a small change in velocity to a small change in time. Let us assume that we have filled in a column of times on our table. The times start with 0 and go up by the same, small amount, say 0.05 sec. Physicists call this small time the integration step. It is standard practice to begin solving an equation with a fixed integration step. There are sometimes good reasons to vary the integration step, but those reasons do not arise in this problem. Let us call the integration step . If we call the time in the ith row t_{i}, then for every row except the first,
We label another column velocity, and we'll call the velocity in the ith row v_{i}. For every row except the first, equation (2) becomes:
We want to fill in velocities as we go down the columns, so we need to solve equation (4) for v_{i}. This will give us a formula for computing v_{i} given v_{i1} for every row except the first. In the first row, we put the speed with which we enter the straight, which is an input to the problem. We get:
We label another column distance, and we call the distance value in the ith row x_{i}. Just as acceleration is the rate of change of velocity, so velocity is the rate of change of distance over time. Just as before, then, we may write:
Solved for x_{i}, this is:
Equation (7) gives us a formula for calculating the distance for any time given the previous distance and the velocity calculated by equation (5). Physicists would say that we have a scheme for integrating the equations of motion. A small detail is missing: what is the force, F? Everything to this point is kinematic. The real modelling starts now with formulas for calculating the force. For this, we will draw on all the previous articles in this series. Let's label another column force, and a few more with drag, rolling resistance, engine torque, engine rpm, wheel rpm, trans gear ratio, drive ratio, wheel torque, and drive force. As you can see, we are going to derive a fairly complete, if not accurate, model of accelerating down the straight. We need a few constants:
and a few variables:
All the example values are for a late model Corvette. Slugs are the English unit of mass, and 1 slug weighs about 32.1 lbs at sea level (another manifestation of F = ma, with F in lbs, m in slugs, and a being the acceleration of gravity, 32.1 ft/sec^{2}). The most basic modelling equation is that the force we can use for forward acceleration is the propelling force transmitted through the wheels minus drag and rolling resistance:
The force of drag we get from Part 6:
Note that to calculate the force at step i, we can use the velocity at step i. This force goes into calculating the acceleration at step i, which is used to calculate the velocity and distance at step i + 1 by equations (5) and (7). Those two equations represent the only "backward references" we need. Thus, the only inputs to the integration are the initial distance, 0, and the entrance velocity, v_{0}. The rolling resistance is approximately proportional to the velocity:
This approximation is probably the weakest one in the model. I derived it by noting from a Corvette book that 8.2 hp were needed to overcome rolling resistance at 55 mph. I have nothing else but intuition to go on for this equation, so take it with a grain of salt. Finally, we must calculate the forward force delivered by the ground to the car by reaction to the rearward force delivered to the ground via the engine and drive train:
This equation simply states that we take the engine torque multiplied by the rear axle ratio and the transmission drive ratio in the kth gear, which is the torque at the drive wheels, T_{W}, and divide it by the radius of the wheel, which is half the diameter of the wheel, d. To calculate the forward force, we must decide what gear to be in. The logic we use to do this is the following: from the velocity, we can calculate the wheel rpm:
From this, we know the engine rpm:
At each step of integration, we look at the current engine rpm and ask "is it past the torque peak of the engine?" If so, we shift to the next highest gear, if possible. Somewhat arbitrarily, we assume that the torque peak is at 4200 rpm. To keep things simple, we also make the optimistic assumption that the engine puts out a constant torque of 330 ftlbs. To make the model more realistic, we need merely look up a torque curve for our engine, usually expressed as a function of rpm, and read the torque off the curve at each step of the integration. The current approximation is not terrible however; it merely gives us artificially good times and speeds. Another important improvement on the logic would be to check whether the wheels are spinning, i.e., that acceleration is less than about ½g, and to "lift off the gas" in that case. We have all the ingredients necessary to calculate how much time it takes to cover a straight given an initial speed. You can imagine doing the calculations outlined above by hand on columnar paper, or you can check my results (below) by programming them up in a spreadsheet program like Lotus 123 or Microsoft Excel. Eventually, of course, if you follow this series, you will see these equations again as we write our Scheme program for simulating car dynamics. Integrating the equations of motion by hand will take you many hours. Using a spreadsheet will take several hours, too, but many less than integrating by hand. To illustrate the process, we show below the times and exit speeds for a 200 foot straight, which is a fairly long one in autocrossing, and a 500 foot straight, which you should only see on race tracks. We show times and speeds for a variety of speeds entering the straight from 25 to 50 mph in Table 1. The results are also summarized in the two plots, Figures (1) and (2). Table 1: Exit speeds and times for several entrance speeds
The notable facts arising in this analysis are the following. The time difference resulting from entering the 200' straight at 27 mph rather than 25 mph is about 6 hundredths. Frankly, not as much as I expected. The time difference between entering at 31 mph over 25 mph is about 2 tenths, again less than I would have guessed. The speed difference at the end of the straight between entering at 25 mph and 50 mph is only 8 mph, a result of the fact that the car labours against friction and higher gear ratios at high speeds. It is also a consequence of the fact that there is so much torque available at 25 mph in low gear that the car can almost make up the difference over the relatively short 200' straight. In fact, on the longer 500' straight, the exit speed difference between entering at 25 mph and 50 mph is not even 5 mph, though the time difference is nearly a full second. This analysis would most likely be much more dramatic for a car with less torque than a Corvette. In a Corvette, with 330 ftlbs of torque on tap, the penalty for entering a straight slower than necessary is not so great as it would be in a more typical car, where recovering speed lost through timidity or bad cornering is much more difficult. Again, the analysis can be improved by using a real torque curve and by checking whether the wheels are spinning in lower gears. 

March 19th, 2005, 10:10 AM  #12 
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Grip Angle
The Physics of Racing, Part 10: Grip Angle Brian Beckman physicist and member of No Bucks Racing Club P.O. Box 662 Burbank, CA 91503 ©Copyright 1991 In many ways, tyre mechanics is an unpleasant topic. It is shrouded in uncertainty, controversy, and trade secrecy. Both theoretical and experimental studies are extremely difficult and expensive. It is probably the most uncontrollable variable in racing today. As such, it is the source of many highs and lows. An improvement in modelling or design, even if it is found by lucky accident, can lead to several years of domination by one tyre company, as with BFGoodrich in autocrossing now. An unfortunate choice of tyre by a competitor can lead to frustration and a disastrous hole in the budget. This month, we investigate the physics of tyre adhesion a little more deeply than in the past. In Parts 2, 4, and 7, we used the simple friction model given by F µW, where F is the maximum traction force available from a tyre; µ, assumed constant, is the coefficient of friction; and W is the instantaneous vertical load, or weight, on a tyre. While this model is adequate for a rough, intuitive feel for tyre behaviour, it is grossly inadequate for quantitative use, say, for the computer program we began in Part 8 or for race car engineering and set up. I am not a tyre engineer. As always, I try to give a fresh look at any topic from a physicist's point of view. I may write things that are heretical or even wrong, especially on such a difficult topic as tyre mechanics. I invite debate and corrections from those more knowledgeable than I. Such interaction is part of the fun of these articles for me. I call this month's topic "grip angle." The grip angle is a quantity that captures, for many purposes, the complex and subtle mechanics of a tyre. Most writers call this quantity "slip angle." I think this name is misleading because it suggests that a tyre works by slipping and sliding. The truth is more complicated. Near maximum loads, the contact patch is partly gripping and partly slipping. The maximum net force a tyre can yield occurs at the threshold where the tyre is still gripping but is just about to give way to total slipping. Also, I have some difficulties with the analyses of slip angle in the literature. I will present these difficulties in these articles, unfortunately, probably without resolution. For these reasons, I give the quantity a new name. A tyre is an elastic or deformable body. It delivers forces to the car by stretching, compressing, and twisting. It is thus a very complex sort of spring with several different ways, or modes, of deformation. The hypothetical tyre implied by F µW with constant µ would be a nonelastic tyre. Anyone who has driven hard tyres on ice knows that nonelastic tyres are basically uncontrollable, not just because µ is small but because regular tyres on ice do not twist appreciably. The first and most obvious mode of deformation is radial. This deformation is along the radius of the tyre, the line from the centre to the tread. It is easily visible as a bulge in the sidewall near the contact patch, where the tyre touches the ground. Thus, radial compression varies around the circumference. Second is circumferential deformation. This is most easily visible as wrinkling of the sidewalls of drag tyres. These tyres are intentionally set up to deform dramatically in the circumferential direction. Third is axial deformation. This is a deflection that tends to pull the tyre off the (nonelastic) wheel or rim. Last, and most important for cornering, is torsional deformation. This is a difference in axial deflection from the front to the back of the contact patch. Fundamentally, radial, circumferential, and axial deformation furnish a complete description of a tyre. But it is very useful to consider the differences in these deflections around the circumference. Let us examine exactly how a tyre delivers cornering force to the car. We can get a good intuition into the physics with a pencil eraser. Get a block eraser, of the rectangular kind like "Pink Pearl" or "Magic Rub." Stand it up on a table or desk and think of it as a little segment of the circumference of a tyre. Think of the part touching the desk as the contact patch. Grab the top of the eraser and think of your hand as the wheel or rim, which is going to push, pull, and twist on the segment of tyre circumference as we go along the following analysis. Consider a car travelling at speed v in a straight line. Let us turn the steering wheel slightly to the right (twist the top of the eraser to the right). At the instant we begin turning, the rim (your hand on the eraser), at a circumferential position just behind the contact patch, pushes slightly leftward on the bead of the tyre. Just ahead of the contact patch, likewise, the rim pulls the bead a little to the right. The push and pull together are called a force couple. This couple delivers a torsional, clockwise stress to the inner part of the tyre carcass, near the bead. This stress is communicated to the contact patch by the elastic material in the sidewalls (or the main body of the eraser). As a result of turning the steering wheel, therefore, the rim twists the contact patch clockwise. The car is still going straight, just for an instant. How are we going to explain a net rightward force from the road on the contact patch? This net force must be there, otherwise the tyre and the car would continue in a straight line by Newton's First Law. Consider the piece of road just under the contact patch at the instant the turn begins. The rubber particles on the left side of the patch are going a little bit faster with respect to the road than the rest of the car and the rubber particles on the right side of the patch are going a little bit slower than the rest of the car. As a result, the left side of the patch grips a little bit less than the right. The rubber particles on the left are more likely to slide and the ones on the right are more likely to grip. Thus, the left edge of the patch "walks" a little bit upward, resulting in a net clockwise twisting motion of the patch. The torsional stress becomes a torsional motion. As this motion is repeated from one instant to the next, the tyre (and the eraserI hope you are still following along with the eraser) walks continuously to the right. The better grip on the right hand side of the contact patch adds up to a net rightward force on the tyre, which is transmitted back through the sidewall to the car. The chassis of the car begins to yaw to the right, changing the direction of the rear wheels. A torsional stress on the rear contact patches results, and the rear tyres commence a similar "walking" motion. The wheel (your hand) is twisted more away from the direction of the car than is the contact patch. The angular difference between the direction the wheel is pointed and the direction the tyre walks is the grip angle. All quantities of interest in tyre mechanicsforces, friction coefficients, etc., are conventionally expressed as functions of grip angle. In steady state cornering, as in sweepers, an understeering car has larger grip angles in front, and an oversteering car has larger grip angles in the rear. How to control grip angles statically with wheel alignment and dynamically with fourwheel steering are subjects for later treatment. The greater the grip angle, the larger the cornering force becomes, up to a point. After this point, greater grip angle delivers less force. This point is analogous to the idealized adhesive limit mentioned earlier in this series. Thus, a real tyre behaves qualitatively like an ideal tyre, which grips until the adhesive limit is exceeded and then slides. A real tyre, however, grips gradually better as cornering force increases, and then grips gradually worse as the limit is exceeded. The walking motion of the contact patch is not entirely smooth, or in other words, somewhat discrete. Individual blocks of rubber alternately grip and slide at high frequency, thousands of times per second. Under hard cornering, the rubber blocks vibrating on the road make an audible squalling sound. Beyond the adhesive limit, squealing becomes a lower frequency sound, "squalling," as the point of optimum efficiency of the walking process is bypassed. There is a lot more to say on this subject, and I admit that my first attempts at a mathematical analysis of grip angle and contact patch mechanics got bogged down. However, I think we now have an intuitive, conceptual basis for better modelling in the future. Speaking of the future, summarizing briefly the past of and plans for the Physics of Racing series. The following overlapping threads run through it: Tyre Physics concerns adhesion, grip angle, and elastic modelling. This has been covered in Parts 2, 4, 7, and 10, and will be covered in several later parts.Car Dynamics concerns handling, suspension movement, and motion of a car around a course; has been covered in Parts 1, 4, 5, and 8 and will continue.Drive Line Physics concerns modelling of engine performance and acceleration. Has been covered in Parts 3, 6, and 9 and will also continue.Computer Simulation concerns the design of a working program that captures all the physics. This is the ultimate goal of the series. It was begun in Part 8 and will eventually dominate discussion.The following is a list of articles that have appeared so far:
and the following is a tentative list of articles I have planned for the near future (naturally, this list is "subject to change without notice"): Springs and Dampers, presenting a detailed model of suspension movement (suggested by Bob Mosso)Transients, presenting the dynamics of entering and leaving corners, chicanes, and slaloms (this one suggested by Karen Babb)Stability, explaining why spins and other losses of control occurSmoothness, exploring what, exactly, is meant by smoothnessModelling Car Data in a computer program; in several articlesModelling Course Data in a computer program; also in several articlesIn practice, I try to keep the lengths of articles about the same, so if a topic is getting too long (and grip angle definitely did), I break it up in to several articles. 
March 19th, 2005, 10:11 AM  #13  
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Braking
The Physics of Racing, Part 11: Braking Brian Beckman physicist and member of No Bucks Racing Club P.O. Box 662 Burbank, CA 91503 ©Copyright 1991 I was recently helping to crew Mark Thornton's effort at the Silver State Grand Prix in Nevada. Mark had built a beautiful car with a theoretical top speed of over 200 miles per hour for the 92 mile time trial from Lund to Hiko. Mark had no experience driving at these speeds and asked me as a physicist if I could predict what braking at 200 mph would be like. This month I report on the backoftheenvelope calculations on braking I did there in the field. There are a couple of ways of looking at this problem. Brakes work by converting the energy of motion, kinetic energy, into the energy of heat in the brakes. Converting energy from useful forms (motion, electrical, chemical, etc.) to heat is generally called dissipating the energy, because there is no easy way to get it back from heat. If we assume that brakes dissipate energy at a constant rate, then we can immediately conclude that it takes four times as much time to stop from 200 mph as from 100 mph. The reason is that kinetic energy goes up as the square of the speed. Going at twice the speed means you have four times the kinetic energy because 4 = 2^{2}. The exact formula for kinetic energy is ½mv^{2}, where m is the mass of an object and v is its speed. This was useful to Mark because braking from 100 mph was within the range of familiar driving experience. That's pretty simple, but is it right? Do brakes dissipate energy at a constant rate? My guess as a physicist is "probably not." The efficiency of the braking process, dissipation, will depend on details of the friction interaction between the brake pads and disks. That interaction is likely to vary with temperature. Most brake pads are formulated to grip harder when hot, but only up to a point. Brake fade occurs when the pads and rotors are overheated. If you continue braking, heating the system even more, the brake fluid will eventually boil and there will be no braking at all. Brake fluid has the function of transmitting the pressure of your foot on the pedal to the brake pads by hydrostatics. If the fluid boils, then the pressure of your foot on the pedal goes into crushing little bubbles of gaseous brake fluid in the brake lines rather than into crushing the pads against the disks. Hence, no brakes. We now arrive at the second way of looking at this problem. Let us assume that we have good brakes, so that the braking process is limited not by the interaction between the pads and disks but by the interaction between the tyres and the ground. In other words, let us assume that our brakes are better than our tyres. To keep things simple and backoftheenvelope, assume that our tyres will give us a constant deceleration of The time t required for braking from speed v can be calculated from: t = v / a which simply follows from the definition of constant acceleration. Given the time for braking, we can calculate the distance x, again from the definitions of acceleration and velocity: Remembering to be careful about converting miles per hour to feet per second, we arrive at the numbers in Table 1.
Table 1: Times and Distances for barking to zero from various speeds We can immediately see from this table (and, indeed, from the formulas) that it is the distance, not the time, that varies as the square of the starting speed v. The braking time only goes up linearly with speed, that is, in simple proportion. The numbers in the table are in the ballpark of the braking figures one reads in published tests of high performance cars, so I am inclined to believe that the second way of looking at the problem is the right way. In other words, the assumption that the brakes are better than the tyres, so long as they are not overheated, is probably right, and the assumption that brakes dissipate energy at a constant rate is probably wrong because it leads to the conclusion that braking takes more time than it actually does. My final advice to Mark was to leave lots of room. You can see from the table that stopping from 210 mph takes well over a quarter mile of very hard, precise, threshold braking at 1g! 

March 19th, 2005, 10:11 AM  #14  
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CyberCar, Every Racer's DWIM Car?
The Physics of Racing, Part 12: CyberCar, Every Racer's DWIM Car? Brian Beckman physicist and member of No Bucks Racing Club P.O. Box 662 Burbank, CA 91503 ©Copyright 1991 The cybernetic DWIM car is coming. DWIM stands for "Do What I Mean."^{1} It is a commonplace term in the field of Humanmachine Interfaces, and refers to systems that automatically interpret the user's intent from his or her inputs. Cybernetics (or at least one aspect of it) is the science of unifying humans and machines. The objective of cybernetics is usually to amplify human capability with "intelligent" machines, but sometimes the objective is the reverse. Most of the work in cybernetics has been under the aegis of defence, for building advanced tanks and aircraft. There is a modest amount of cybernetics in the automotive industry, as well. Antilock Braking (ABS), Acceleration Slip Reduction (ASR), Electronic Engine Management, and Automatic Traction Control (ATC) are cybernetic DWIM systemsof a kindalready in production. They all make "corrections" on the driver's input based on an assumed intention. Steerbywire, Continuously Variable Transmissions (CVT), and active suspensions are on the immediate horizon. All these features are part of a distinct trend to automate the driving experience. This month, we take a break from hard physics to look at the better and the worse of increased automation, and we look at one concept of the ultimate result, CyberCar. Among the research directions in cybernetics are advanced sensors for human inputs. One of the more incredible is a system that reads brain waves and figures out what a fighter pilot wants to do directly from patterns in the waves. A major challenge in the fighter cockpit is information overload. Pilots have far too many instruments, displays, horns, buzzers, radio channels, and idiot lights competing for their attention. In stressful situations, such as high speed dogfights, the pilot's brain simply ignores inputs beyond its capacity, so the pilot may not hear a critical buzzer or see a critical warning light. In the "intelligent cockpit," however, the pilot consciously suppresses certain displays and auditory channels, thus reducing sensory clutter. By the same token, the intelligent cockpit must be able to override the pilot's choices and to put up critical displays and to sound alarms in emergencies. In the reduced clutter of the cockpit, then, it is much less likely that a pilot will miss critical information. How does the pilot select the displays that he^{2} wants to see? The pilot cannot afford the time to scroll through menus like those on a personal computer screen or huntandpeck on a button panel like that on an automatic bank teller machine. There are already sensors that can read a pilot's brain waves and anticipate what he wants to look at next. Before the pilot even consciously knows that he wants to look at a weapon status display, for example, the cybernetic system can infer the intention from his brain waves and pop up the display. If he thinks it is time to look at the radar, before he could speak the command, the system reads his brain waves, pops up the radar display, and puts away the weapon status display. How does it work? During a training phase, the system reads brain waves and gets explicit commands through a button panel. The system analyses the brain waves, looking for certain unique features that it can associate with the intention (inferred from the command from the button panel) to see the radar display, and other unique features to associate with the intention to look at weapon status, and so on. The system must be trained individually for each pilot. Later, during operation, whenever the system sees the unique brain wave patterns, it "knows" what the pilot wants to do. The implications of technology like this for automobiles is amazing. Already, things like ABS are a kind of rudimentary cybernetics. When a driver stands all over the brake pedal, it is assumed that his intention is to stop, not to skid. The ABS system "knows," in a manner of speaking, the driver's intention and manages the physical system of the car to accomplish that goal. So, instead of being a mere mechanical linkage between your foot and the brakes, the brake pedal becomes a kind of intentional, DWIM control. Same goes for traction control and ASR. When the driver is on the gas, the system "knows" that he wants to go forward, not to spin out or do doughnuts. In the case of TC, the system regulates the torque split to the drive wheels, whether there be two or four. In the case of ASR, the system backs off the throttle when there is wheel spin. Cybernetics again. ABS, TC, and ASR exist now. What about the future? Consider steerbywire. CyberCar, the total cybernetic car, infers the driver's intended direction from the steering wheel position. It makes corrections to the actual direction of the steered wheels and to the throttle and brakes much more quickly and smoothly than any driver can do. Coupled with slip angle^{3} sensors [1] and inertial guidance systems, perhaps based on miniaturized laser/fibre optic gyros (no moving parts), cybernetic steering, throttle, and brake controls will make up a formidable racing car that could drive a course in practically optimal fashion given only the driver's desired racing line. In an understeering situation, when a car is not turning as much as desired, a common driver mistake is to turn the steering wheel more. That is a mistake, however, only because the driver is treating the steering wheel as an intentional control rather than the physical control it actually is. In CyberCar, however, the steering wheel is an intentional control. When the driver adds more lock in a corner, CyberCar "knows" that the driver just wants more steering. Near the limits of adhesion, CyberCar knows that the appropriate physical reaction is, in fact, some weight transfer to the front, either by trailing throttle or a little braking, and a little less steering wheel lock. When the fronts hook up again, CyberCar can immediately get back into the throttle and add a little more steering lock, all the while tracking the driver's desires through the intentional steering wheel in the cockpit. Similarly, in an oversteer situation, when the driver gives opposite steering lock, CyberCar knows what to do. First, CyberCar determines whether the condition is trailing throttle oversteer (TTO) or power oversteer (PO). It can do this by monitoring tyre loads through suspension deflection and engine torque output over time. In TTO, CyberCar adds a little throttle and counter steers. When the drive wheels hook up again, it modulates the throttle and dials in a little forward lock. In PO, CyberCar gently trails off the throttle and counter steers. All the while, CyberCar monitors driver's intentional inputs and the physical status of the car at the rate of several kilohertz (thousands of times per second). The very terms "understeer" and "oversteer" carry cybernetic implication, for these are terms of intent. Understeer means the car is not steering as much as wanted, and oversteer means it is steering too much. The above description is within current technology. What if we get really fantastic? How about doing away with the steering wheel altogether? CyberCar, version II, knows where the driver wants to go by watching his eyes, and it knows whether to accelerate or brake by watching brain waves. With Virtual Reality and teleoperation, the driver does not even have to be inside the car. The driver, wearing binocular video displays that control incar cameras (or even synthetic computer graphics) via head position, sits in a virtual cockpit in the pits. Now we must ask how much cybernetics is desirable? Autocrossing is, largely, a pure driver skill contest. Wheeltowheel racing adds race craftdrafting, passing, deception, etc. to car control skills. Does it not seem that cybernetics eliminates driver skill as a factor by automating it? Is it not just another way for the "haves" to beat the "havenots" by outspending them? Drivers who do not have ABS have already complained that it gives their competition an unfair advantage. On the other hand, drivers who do have it have complained that it reduces their feel of control and their options while braking. I think they doth protest too much. In the highest forms of racing, where money is literally no object, cybernetics is already playing a critical role. The clutchless seven speed transmissions of the Williams/Renault team dominated the latter half of the 1991 Formula 1 season. But for some unattributable bad luck, they would have won the driver's championship and the constructor's cup. Carrol Smith, noted racing engineer, has been predicting for years that ABS will show up in Formula 1 as soon as systems can be made small and light enough [2]. It seems inevitable to me that cybernetic systems will give the unfair advantage to those teams most awash in money. However, autocrossers, club racers, and other grass roots competitors will be spared the expense, and the experience of being relieved of the enjoyment of car control, for at least another decade or two. Acknowledgements Thanks to Phil Ethier for giving me a few tips on car control that I might be able to teach to CyberCar and to Ginger Clark for bringing slip angle sensors to my attention. Notes ^{1} And the word play on 'dream' was too much to resist. ^{2} Everywhere, 'he' means 'he or she,' 'his' means 'his or her,' etc. ^{3} Also known as grip angle; see Part 10 of this series. References


March 19th, 2005, 10:11 AM  #15 
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Transients (The missing episode)
Transients (The missing episode)

March 19th, 2005, 10:12 AM  #16 
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Why Smoothness?
Why Smoothness?

March 19th, 2005, 10:12 AM  #17 
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Bumps In The Road
Bumps In The Road

March 19th, 2005, 10:12 AM  #18 
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RARS, A Simple Racing Simulator
RARS, A Simple Racing Simulator

March 19th, 2005, 10:13 AM  #19 
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"Slowin, Fastout!" or, Advanced Analysis of the Racing Line
"Slowin, Fastout!" or, Advanced Analysis of the Racing Line

March 19th, 2005, 10:13 AM  #20 
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"Slowin, Fastout!" or, Advanced Analysis of the Racing Line, Continued
"Slowin, Fastout!" or, Advanced Analysis of the Racing Line, Continued


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