Determing The Go Kart Center Of Mass - Purdue University

Go-Kart Activity Forces in Go Kart Racing Student Name: Determining the Kart Center of Gravity/Mass (CG/CM) Guiding Question: How can the center of gravity (CG) of an evGrand Prix kart be set so that there is a 47%/53% weight distribution from front to back and a 50%/50% from side to side? What is meant by the center of gravity? Center of gravity, also known as center of mass, is that point at which a system or body behaves as if all its mass were centered at that point. Where the weight, and also all accelerative forces of acceleration, braking and cornering act through it. Center of gravity location can be defined as: - The balance point of an object - The point through which a force will cause pure translation - The point about which gravity moments are balanced (see Fig.1) - The point which if the body is hanged from it will stay balanced (leveled as it is on the ground). Fig. 1 Summation of moments of parts weights around any point is equal to the moment of the summation of weights around this point. W. XCG (W1.X1 W2. X2 W3. X3 W4. X4 W5. X5 .) XCG Σ (W.X) / Σ (W), where Xi is the distance in x direction between the point i and that point. What is the importance of the CG? When making an analysis of the forces applied on the kart, the CG is the point to place the kart weight, and the centrifugal forces when the kart is turning or when accelerating or decelerating. Any force that acts through the CG has no tendency to make the kart rotate. The center of mass height, relative to the track, determines load transfer (related to weight transfer) from side to side and causes body lean. When the tires of a kart provide a centripetal force to pull it around a turn, the momentum of the kart actuates load transfer in a direction going from the kart's Version: 11.25.19 2019 Purdue University All Rights Reserved Page 1

current position to a point on a path tangent to the kart's path. This load transfer presents itself in the form of body lean. Body lean can be controlled by lowering the center of weight or by widening the kart track, it can also be controlled by springs, anti-roll bars or the roll center heights. Kart topples Kart does not topple over Fig.2 Increasing the height of the CG or decreasing the width of the kart track will cause the kart to topple. The center of mass height, relative to the wheelbase, determines load transfer between front and rear. The kart's momentum acts at its center of mass to tilt the kart forward or backward, respectively during braking and acceleration. Since it is only the downward force that changes and not the location of the center of mass, the effect on over/under steer is opposite to that of an actual change in the center of mass. A lower center of mass is a principal performance advantage of sports cars, compared to sedans and (especially) SUVs. Some cars have body panels made of lightweight materials partly for this reason. Obtaining the position of the kart CG: A kart is not symmetrical in shape or mass from front to rear. Many karts are symmetrical left to right in shape but not in mass. Fig. 2 Weight (W), Reaction (R) Fig. 3 The CG location in the side and front view The difference between weight W and Reaction R: From Fig.2, W is the weight of the tire (always vertical and downward), while R is the reaction from the ground (always perpendicular to the ground surface and away from it). The tire is balanced in the vertical direction under its weight (W) and the ground reaction (R), that means the summation of forces in that direction is zero (W – R 0), which gives W R. In our analysis we measure the weights, but when studying the kart balance we use the reactions supporting the kart from the ground (where: Wf Rf, Wr Rr, Ww Rw and WL RL). The symbols in Fig. 3 are donated to: W - is the kart weight, Version: 11.25.19 2019 Purdue University All Rights Reserved Page 2

Rf - is the ground reaction of the frontal axle weight Rr - is ground reaction of the rear axle weight RR - is ground reaction of the kart right wheels weight RL - is ground reaction of the kart left wheels weight L - is the kart wheel base (distance between the front and rear kart wheels/axles) T - is the kart track (distance between the center of the wheels on the same axle) a - is the location of the CG behind the front axle b - is the location of the CG in front of the rear axle x - is the location of the CG away from the right wheels y - is the location of the CG away from the left wheels Materials Required: 4 Digital Bathroom Scales Assembled go kart Z-axis Blocks (Instructions Included) Setup and Procedures: Step 1 – Put scales underneath each of the kart tires and make sure that all the tires create a flat plane that balances the weight of the unloaded shelf evenly. Do this by ensuring that the surface on which the scales are setting is level. Have the driver sit in the seat with hands on the steering wheel and feet on the pedals. Fig. 4 Measure the front and rear axle weights (Wf ,Wr), and the wheel base length (L) Fig. 5 The car on a level surface, L is the wheel base, the front and rear axles weights (Wf, Wr) and (a and b are unknown longitudinal distances of the CG) Step 2 – Measure the wheel base, L (the distance between the center of the front and rear wheels), Fig. 3. Record this value on the Kart Center of Gravity Calculations Form. L a b (1) a L b (2) Version: 11.25.19 2019 Purdue University All Rights Reserved Page 3

W ( R f Rr ) 0 (3) Then W ( R f Rr ) (4) Step 3 – Measure the kart front axle weight Wf Rf (Fig. 3) by adding the readings of the scales under the front wheels of the kart and record these values in the Kart Center of Gravity Calculations Form. Step 4 – Measure the kart rear axle weight Wr Rr (Fig. 3) by adding the readings of the scales under the rear wheels of the kart and record these values in the Kart Center of Gravity Calculations Form. Step 5 – Determine the total kart weight W (Equation 4) by adding the front axle weight and the rear axle weight together and record this value in the Kart Center of Gravity Calculations Form. In order to locate the center of gravity in relation to the rear axle you will need to determine the “moment” around the rear axle (E) using the following formula R f L Wb 0 R f L Wb , then b L( Rf W ) (5) Step 6 – Use eq. (5) to find the distance b. Record this value at the bottom of the Kart Center of Gravity Calculations Form. Step 7 – Determine the percentage of weight on the front axle of the kart and the rear axle of the kart and record these values in the Kart Center of Gravity Calculations Form. Calculate the percentage using the following equation. Axle Weight 100 % Total Kart Weight Step 8 – Substitute the value of b in eq. (2) to get the distance a. Record this value at the bottom of the Kart Center of Gravity Calculations Form. NOTE: In Eq. 5, the units of Rf and W can be expressed in unit force (N, lb) or unit mass (kg), the units of L and b can be expressed in (m, cm, mm, or ft, in). Fig. 8 Finding the CG position in the frontal view, distance x, y. Version: 11.25.19 2019 Purdue University All Rights Reserved Page 4

Step 9 – Measure the wheel track, T (the distance between the center of the front wheels), Fig. 8. Record this value on the Kart Center of Gravity Calculations Form. T x y (6) x T y (7) Then W ( RR RL ) (8) Step 10 – Measure the kart right side weight WR RR (Fig. 8) by adding the readings of the scales under the wheels on the right side of the kart and record this value in the Kart Center of Gravity Calculations Form. Step 11 – Measure the kart left side weight WL RL (Fig. 8) by adding the readings of the scales under the wheels on the left side of the kart and record these values in the Kart Center of Gravity Calculations Form. In order to locate the center of gravity in relation to the rear axle you will need to determine the “moment” around the right side wheels using the following RRT Wy 0 RRT Wy , then y T( RR ) W (9) Step 12 – Determine the percentage of weight on the right side of the kart and the left side of the kart and record these values in the Kart Center of Gravity Calculations Form. Calculate the percentage using the following equation. Side Weight 100 % Total Kart Weight Step 13 – Use eq. (9) to find the distance y. Record this value at the bottom of the Kart Center of Gravity Calculations Form Step 14 – Substitute the value of y in eq. (7) to get the distance x. Record this value at the bottom of the Kart Center of Gravity Calculations Form. NOTE: In Eq. 9, the units of RR and W can be expressed in unit force (N, lb) or unit mass (kg), the units of T and y can be expressed in (m, cm, mm, or ft, in). Most karts are asymmetric in weight distribution in the front view (RR is not equal to RL). So that the CG position will not be in the middle. Determine the height above ground of CG: The weight of the rear axle (Wr1) will be weighed while the front pair of wheels are raised up quite a small distance H (or h1) (as shown in figure 6). Step 15 – Construct two 9 inch by 7 inch Z-axis wheel blocks by cutting four 7 inch pieces of 2x6, two 6 inch pieces of 2x6, and two 7 inch pieces of 2x2 lumber. Version: 11.25.19 2019 Purdue University All Rights Reserved Page 5

Step 16 – Make a rectangular box by placing two of the 7inch pieces, parallel to one another and 6 inches apart. Place two 6 inch pieces between them. Screw together at corners. Predrilling these holes will make it easier to attach these together. Step 17 – Place the remaining two 7 inch pieces of 2x6 material on the top of the newly constructed box and screw into place. Step 18 – Finish of the Z-axis wheel blocks by attaching the two 7 inch 2x2 pieces at each end of the box top. NOTE: your Z-axis wheel block should look something like the image to the right. Fig.6 Measure the rear axle weight (Wr1) and the distance of front raise (H) Fig. 7 The front wheels are raised up a small distance H (h1), r is the wheel radius, (h is the unknown distance of the CG height) Step 19 – Place Z-axis wheel blocks on front tire scales, then place front tires of kart on Z-axis blocks. Have the driver sit in the seat with hands on the steering wheel and feet on the pedals. Step 20 – Measure the height of the Z-axis wheel blocks and record this value in the bottom of the Kart Center of Gravity Calculations Form. Step 21 – Measure the elevated front axle weight Wf1 Rf1 (Fig. 7) by subtracting the weight of the Z-axis wheel blocks from the readings of the scales under the Z-axis wheel blocks and record this value in the Kart Center of Gravity Calculations Form. In order to locate the center of gravity in relation to the chassis of the kart you will need to determine the summation of vertical forces in the y direction using the following. (R f1 Rr1 ) W 0 Version: 11.25.19 2019 Purdue University All Rights Reserved Page 6