These are the formulas that transpose the centrifugal force on the center of gravity to some amount of Body-roll and Tire Load. First we will explain the general physical mechanics between Roll Centers, Roll Axis, the Center of Gravity, Tread Widths (car width), etc. Also, each of these will be explained in detail in later chapters. Then we'll show the formulas and relate them to the Body-roll, Tire Loads and Camber.
The center of gravity ( cg) of a car is an imaginary point where all the masses of the car can be summed to for calculation purposes. That's center of gravity in a nutshell. Here are a few more details. The cg can be expressed as a mass at a distance from a reference in all 3 dimensions (x, y, and z), such as, 60 pounds, 175 inches forward of the rear axle, 15 inches from the ground, and 36 inches to the left of the car's centerline. There are many different masses (motor, transmission, battery, driver, etc.) in a racing car. These masses can be summed together, once in each dimension to get the three cg dimensions. This following table and figure are an example of summing four masses together for one dimension of the cg. Expressing this math in words: The sum of all the "weight times distances" in that dimension (plane), divided by the center of gravity weight, is the center of gravity distance. The center of gravity distance is the length from the reference, such as a rear axle, to the center of the mass or object. The center of gravity weight is the sum of all the masses in that dimension (plane). In this example, there is 170 pounds 36.76 in the plus direction from the reference. This is done in each of the dimensions to locate the cg mathematically.
| mass 1 | 60 | 75 | 4500 |
| mass 2 | 50 | 25 | 1250 |
| mass 3 | 20 | -25 | -500 |
| mass 4 | 40 | 25 | 1000 |
| Totals | 170 | 6250 | |
| Total of (wt * x) / Total wt | 170 lbs 36.76 inches | ||
Figure 7: CG Example Masses and Distances
The roll centers are said to be imaginary because they are not something tangible you can touch, but mathematically they are real and very predictable. There are two roll centers per car, one at the front of the car and one at the back. The chassis/body of the car pivots on the roll centers. Exactly how the roll center pivots on the roll center could be called the roll center characteristics. The roll center characteristics would include things like the roll center height and the camber changes caused by the roll center. In general terms, there are many types of suspensions, each as unique roll center characteristics.
The following discussion shows two of the types that are commonly used. Figure 8 is a picture of a Roll Center with two non-parallel suspension arms. This is one type of suspension. It has unique roll characteristics. In this picture the rounded rectangles are the tires, the dots are the "A" frame pivot points on the chassis/body, the vertical dashed line is the center line of the car, the colored rectangles are chassis/bodies. The chassis/bodies are shown pivoting around the car's roll center. The roll center is the circle on the car's centerline. The actual suspension arms are not shown in this picture, just the suspension arm locations on the chassis/bodies. These items are not shown because the objective of the picture is to show that the chassis/body rotates relative to the roll center, even though the roll center is only an imaginary point in space.
Figure 8: Roll Center with Non Parallel Arms
The following picture is a another type of suspension with its own unique roll center characteristics. Many race cars use it along with a live axle for the rear suspension. I've totally redrawn the picture to match the new parameters. The dots are now the pivot points of a "pan hard bar". The live axle is not shown. The roll center is where the pan hard bar passes through the center line of the car. Again, the objective of the picture is to show that the chassis/body rotates relative to the roll center, even thought the roll center is only an imaginary point in space. Note that the "pan hard bar" connections to the chassis/body move with the body, and therefore the roll center would in fact be moving up and down slightly with body roll. For the scope of the book, these changes in roll center height are ignored.
Figure 9: Roll Center of a Car with a Pan Hard Bar
The roll axis is an imaginary line that passes through the two roll centers. When the chassis/body leans, it rolls around the roll axis. Stated another way, the chassis/body rolls on the two roll centers which make the roll axis.
It is time to learn how the center of gravity relates to the roll centers, roll axis and the Cts (grip). First, the center of gravity is an imaginary point where all the masses of the car are summed to, for calculation purposes. And the roll centers are imaginary points that the car's chassis/body roll on or pivot on during cornering. The roll centers are said to be imaginary because they are not something tangible you can touch, but mathematically they are real and very predictable. We need to stress the word real in that last sentence. The roll axis is an imaginary line that passes through the two real roll centers. Since the roll axis passes through the two real roll centers, it should be thought of as real , too. When the chassis/body leans, it rolls (leans) around the roll axis. Stated another way, the chassis/body rolls (leans) on these two roll centers which is the roll axis.
The position of the roll axis relative to the center of gravity is very key to the performance a car. Let's see why. Force arrows are shown in this following picture of a cg inside of a mass. The long force arrow on the cg is our reference force, we can assume it to be stationary because it the largest. It is not going to move. Therefore, because of the short force arrow, the mass will rotate as shown by the curved force arrow. Now if we redraw this picture to show the short force directly apposing the long force arrow, then the mass will not rotate. The curved force arrow is not shown because there is no rotation. And a third instance of this picture with the short force on the lower portion of the mass. Again the mass will rotate as shown by the curved force arrow, except now it is in the opposite direction from the first picture. These three pictures are the beginning of the explanation of how the roll axis and roll center interact. As you read the next few sections, keep these pictures in mind. It will make it easier to visualize the concepts.
Figure 10: Roll Centers and Roll Axis, Side View
These pictures show the roll axis crossing through the car's two roll centers. These pictures show you where these items are in the car. The large dot is the rear roll center. The circle is the front roll center. The diamond is the center of gravity and the dotted line is the roll axis. This is the same picture (same idea, different view), but from the Rear View. These are the major components of the car, the two roll centers, the roll axis and the center of gravity.
Figure 11: Roll Centers and Roll Axis, Rear View
Ok, now that we can picture these major components of the car in out minds, imagine what is happens when a cornering force acts on the center of gravity. The car rotates around the roll axis with the outside of the car gaining load. Unless the cg happens to be the same height as the roll axis, then the car will not rotate at all. And of course, if the cg is below the roll axis, the cornering force rotates the car such that the inside of the car gains load. Interesting.
The next picture will explain that last paragraph. In short, a cornering force on the center of gravity is transmitted through the chassis/body via roll axis as torque and loads the tires. Remember the roll axis may not be tangible, but is real and transmits the torque that loads the tires. To show this, we'll redraw the rear view with a triangle in it that represents the car chassis/body. The arrow in Figure 12 from the center of gravity (the diamond) is the cornering load. The other arrows are the up and down loads on the tires because the chassis/body rolls around the roll axis and loads the tires. Notice that the horizontal line of the triangle does not pass through either dot. It is on the roll axis between the roll centers about half way up the car.
Equation 7: Transferred Weight, Total
Now that you can picture the dynamics of the car, here is the math that goes with that last discussion. This is a simplified, but useful formula relating the roll axis, the center of gravity, transferred weight (tire loads) and the TreadWidth. The exact nomenclature of the names in Equation 7 are: CG ht is short for center of gravity height. RollAxis ht is the height of the roll axis line just below the center of gravity. So "CG ht - RollAxis ht " is the length of that wrench on the roll axis. The TreadWidth is the chassis/body width in the last discussions. And the RollAxis ht (the height of the rollaxis where it passes beneath the center of gravity) can be found with Equation 8 . The quantity ("CG ht -RollAxis ht " divided by ".5 times the TreadWidth") is the mechanical advantage (leverage) that the side load has on the tires. G is a coefficient of Equation 7 and is the side load (force) expressed in Gs. This is all to determine how much of the sprung weight of the car is transferred "to and from" the tires.
Equation 8: Finding the Roll Axis Height
Put some values into Equations 7 and 8 . The units are in inches and pounds. G values could be as high as four or five. See Equation See Centrifugal Force in Gs to find the exact value of G for your circumstances. Play around a little with Equation 7 to get the feel of it. Especially note that as the value "CG ht - RollAxis ht " approaches zero, less weight is transferred to the tires. And when "CG ht - RollAxis ht " does reach zero, no weight is transferred to the tires.
OK, so that is how the transferred weight is calculated. Now how much does the body roll at the tread width because of the transferred weight? In other words, what is the wheel travel? In Equation 9 the front and rear wheel rates are summed to get the total wheel rate for that side. Then just divide the Weight Transferred by the total Wheel Rate to get the Wheel Travel.
Equation 9 is the wheel travel of the chassis/body. The chassis is rigid, therefore the travel for front and back wheels because of a cornering force is the same. For example, if the chassis/body rolls 1.5 inch as measured at the tread width, the front wheel rate is 300 pounds per inch and the rear wheel rate is 250 pounds per inch. Then the right front tire loads change by 450 pounds each. And the right rear tire loads change by 375 pounds each.
Also remember that the camber changes with body roll. I will not show you the math to calculate camber. It would be about the size of this booklet all by itself. It is not that hard, just a lot of it. All I want you to remember is that the camber changes with of body roll. You knew that, but now you can measure the car's camber changes, now that you know how much the body roll is.
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