Problem set 2 -week 3 challenge : Anti-Characteristics

1. To calculate the parameters

• Front end deflection under load.
• Rear end deflection under load.
• Pitch angle.
• Pitch gradient.

need to calculate the total load transfer, static weights and dynamic weights.

Calculations:

1. Static Weights:

• Step 1: Calculate the static weight supported by the front axle in kilograms.
• Formula: Static Weight Front (kg) = Total Weight (kg) * Weight Distribution (Front)
• Step 2: Convert the static front weight to Newtons.
• Formula: Static Weight Front (N) = Static Weight Front (kg) * Acceleration due to Gravity (m/s²)
• Step 3: Calculate the static weight supported by the rear axle in kilograms.
• Formula: Static Weight Rear (kg) = Total Weight (kg) * (1 - Weight Distribution (Front))
• Step 4: Convert the static rear weight to Newtons.
• Formula: Static Weight Rear (N) = Static Weight Rear (kg) * Acceleration due to Gravity (m/s²)

2. Load Transfer:

• Step 1: Calculate the load transfer during acceleration.

Formula: ΔW = (a * h * W) / (wb)

3. Dynamic Axle Weights:

• Step 1: Calculate the dynamic front axle weight.
• Formula: W_front_dynamic = W_front - ΔW
• Step 2: Calculate the dynamic rear axle weight.
• Formula: W_rear_dynamic = W_rear + ΔW

4. Wheel Rates:

• Step 1: Calculate the front wheel rate.
• Formula: Front Wheel Rate (N/m) = Front Spring Rate (N/m) * (Front Motion Ratio)²
• Step 2: Calculate the rear wheel rate.
• Formula: Rear Wheel Rate (N/m) = Rear Spring Rate (N/m) * (Rear Motion Ratio)²

5. Deflections:

• Step 1: Calculate the front end deflection.
• Formula: Front End Deflection (m) = (Dynamic Front Axle Weight (N) - Static Weight Front (N)) / Front Wheel Rate (N/m)
• Step 2: Calculate the rear end deflection.
• Formula: Rear End Deflection (m) = (Dynamic Rear Axle Weight (N) - Static Weight Rear (N)) / Rear Wheel Rate (N/m)

6. Pitch Angle:

• Step 1: Calculate the pitch angle in radians.
• Formula: Pitch Angle (radians) = atan((Front End Deflection (m) - Rear End Deflection (m)) / Wheelbase (m))
• Step 2: Convert the pitch angle to degrees.
• Formula: Pitch Angle (degrees) = Pitch Angle (radians) * (180 / π)

7. Pitch gradient:

• Calculate the pitch gradient with Formula: pitch gradient = pitch angle (degree) / (throttle acceleration/acceleration due to gravity)

And the results are as follows

the detailed calculations are in excel sheet.

2, Now with compensation and using IC location information, the parameters needed to calculate are

• front anti lift angle
• rear anti squat angle
• front swing arm angle for 100%antilift
• rear swing arm angle for 100%antisquat
• front anti lift %
• rear antisquat%

8. Anti-Lift and Anti-Squat Angles:

• Front Anti-Lift Angle:

Formula: Front Anti-Lift Angle (degrees)= atan((Front IC vertical dist. from CP - tire radius/Front IC horizontal dist. from CP)) * 180/pi

• Rear Anti-Squat Angle:

Formula: Rear Anti-Squat Angle (degrees) = atan((Rear IC vertical dist. from CP -tire radius)/ Rear IC horizontal dist. from CP) * 180/pi

9. Swing Arm Angles for 100% Anti-Lift/Squat:

For 100% anti-lift or anti-squat, the swing arm angle needs to be such that the line connecting the wheel centre to the IC is parallel to the ground.

• Front Swing Arm Angle (100% Anti-Lift): atan(CG height/wheel base) * 180/pi
• Rear Swing Arm Angle (100% Anti-Squat): atan(CG height/wheel base) * 180/pi

10. Anti-Lift and Anti-Squat Percentages:

• Front Anti-Lift %: (Front anti-lift angle * Drive ratio) / (Front swing arm angle))
• Rear Anti-Squat %: (Rear anti-squat angle * ((100-drive ratio)/100) / Rear swing arm angle)*100
• 
  

11.Parameters with compensation:

Using the anti-characteristics now need to calculate the frontend deflection with compensation, rear end deflection with compensation, pitch angle with compensation, pitch angle with compensation degrees.

· Frontend Deflection With Compensation= Front end deflection * (100-front anti lift) / 100

· Rear End Deflection With Compensation= Rear end deflection * (100-rear anti squat lift) / 100

· Pitch Angle With Compensation = θ = atan((δ_rear +δ_front) / wb)

· Pitch Angle With Compensation Degrees = θ = atan((δ_rear +δ_front) / wb) * (100/pi)

· Pitch Gradient = pitch angle (degree) / (throttle acceleration/acceleration due to gravity)

11.Percentage Compensation:

· Now percentage compensation is calculated it explains how effectively the anti-lift and anti-squat systems are counteracting the natural pitching motion of the vehicle during acceleration or braking.

· It is calculated using the formula Pitch Compensation % = (Pitch Angle (before compensation) - Pitch Angle (after compensation)) / Pitch Angle (before compensation) * 100

· Ideally, the pitch compensation percentage needs to be as close to 100% as possible. This would mean the anti-lift/squat system has eliminated the pitching motion.

detailed calculation in excel sheet.

12.Percentage Compensation increase:

• while fixing the Rear IC vertical distance from CP with 16in as stated in question. And varying the Rear IC horizontal distance from CP as 9.5in.
• then the percentage compensation will be 99.65%.
• Values above 100% are theoretically possible but would imply the suspension is somehow exaggerating the pitching motion in the opposite direction, which is generally undesirable.
• Practical Limitations: Achieving exactly 100% compensation is often difficult or impossible due to design constraints and other factors. A value above 80% is typically considered very good.
• So in this problem Rear IC horizontal distance from CP between 9.5in to 12in is desirable because it maintains the percentage between 80 to 100%

The parameters change after changing the Rear IC horizontal distance from CP as 9.5in.

Please find the excel sheet :

https://docs.google.com/spreadsheets/d/1OaIR2nQcb1eV9dyWGiQRO8hrSj-dR1UV/edit?usp=sharing&ouid=106354191505670406233&rtpof=true&sd=true