Vehicle Dynamics - Suspension System

Question:

A detailed description of the project can be found in this link.

https://drive.google.com/file/d/1SCMtRsRwEXTJvzREyqS4qcZ5CmBUrmJl/view?usp=sharing

*Additional Values:

Front Roll Center Height: 0.497 inches

Rear Roll Center Height: 0.896 inches

Objective:

Select springs and anti-roll bars, and set up swing arm geometry to meet targets for ride frequency, roll gradient, and pitch gradient, while staying within certain constraints.

Answer:

RIDE FREQUENCY:

Rear Ride Frequency = 1/((1/F_Front)-Time Lag)

Time Lag = Wheelbase / Speed

= 0.086 s

Rear Ride Frequency = 1.53 Hz

Octave script is used to get values from multiple iterations with the Rear Ride Frequency. Rear Ride Frequency is assumed by plotting amplitude to optimize the point where both lines merge, through the octave.

Therefore, Rear Ride Frequency = 1.544 Hz

We know,

Front Ride Frequency = 1/2π*√(Ride Rate F/Sprung Mass F)

From this equation, we get

Ride Rate = 4π2 *Ride frequency2*sprung mass

Ride Rate - Front = 28.12 N/mm

Ride Rate - Rear = 28.47 N/mm

We Know,

Ride Rate = (Wheel Rate F * Tyre Ratio F)/(Wheel Rate F + Tyre Ratio F)

From this equation, we get

Wheel Rate = (Ride Rate * Tyre Rate)/(Tyre Rate -Ride Rate)

Wheel rate - Front = 31.33 N/mm

Wheel rate - Rear = 31.36 N/mm

We Know,

Wheel Rate = Spring Rate * Motion Ratio2

From this equation, we get

Spring Rate = (Wheel Rate / Motion Ratio2)

Spring Rate - Front = 34.72 N/mm

= 198.39 lb/in

Spring Rate – Rear = 49.00 N/mm

= 279.98 lb/in

ROLL ANALYSIS:

1. Sprung Mass Information:

Distance from Front Axle to CG = Total Weight Rear * (Wheelbase/ Total Weight)

= 44.10 in

Distance from Rear Axle to CG = Total Weight Front * (Wheelbase/ Total Weight)

= 53.90 in

Roll Moment Arm (h_arm) = CG Height - Front Roll Centre Height -

( Front Axle to CG * (Rear Roll Centre Height - Front Roll Centre Height))

Roll Moment Arm (h_arm) = 16.33 in

= 414.71 mm

Total Sprung Mass = Front Sprung Mass + Rear Sprung Mass

Total Sprung Mass = 3060.00 lb

= 1387.99 kg

2. Roll Moment:

Roll Moment - Lateral Component = Sprung Mass * Sprung mass Roll moment Arm

Roll Moment - Lateral Component = 49960.75 in-lb/g

Roll Moment - Gravity Component = Sprung Mass * Sprung mass Roll moment Arm

* Roll Gradient*(π/180)

Roll Moment - Gravity Component = 3313.52 in-lb/g

Total Roll Moment = Lateral Component + Gravity Component

Total Roll Moment = 53274.27 in-lb/g

3. Lateral Load Transfer and Components:

Total Lateral Load Transfer = Total Weight * (CG Height/Trackwidth)

Total Lateral Load Transfer = 465.32 kg

Lateral Load Transfer - Front= Total Lateral Load Transfer * (Front Weight Distribution %+5%)

(5% added in order to gain understeer, as per given in statement)

Lateral Load Transfer - Front = 279.19 kg

Front Lateral Load Transfer due to Sprung Mass = Front Sprung Mass

* (Front Roll Centre Height/ Trackwidth)

Front Lateral Load Transfer due to Sprung Mass = 6.70 kg

Front Lateral Load Transfer due to Unsprung Mass = Front Unsprung Mass

* (Front Loaded Radius / Trackwidth)

Front Lateral Load Transfer due to Unsprung Mass = 42.31 kg

Front Lateral Load Transfer Due to Roll (Elastic component) = ((Front Sprung Mass * h_arm)

+(Front Sprung Mass * h_arm * Roll Gradient * (π/180))/Trackwidth

Front Lateral Load Transfer Due to Roll (Elastic component) = 235 kg

Rear Lateral Load Transfer = Total Lateral Load Transfer * (Rear Weight Distribution %)

Rear Lateral Load Transfer = 209.40 kg

Rear Lateral Load Transfer due to Sprung Mass = Rear Sprung Mass

* (Rear Roll Centre Height/ Trackwidth)

Rear Lateral Load Transfer due to Sprung Mass = 20.62 kg

Rear Lateral Load Transfer due to Unsprung Mass = Rear Unsprung Mass

* (Rear Loaded Radius / Trackwidth)

Rear Lateral Load Transfer due to Unsprung Mass = 50.77 kg

Rear Lateral Load Transfer Due to Roll (Elastic component) = ((Rear Sprung Mass * h_arm)

+(Rear Sprung Mass * h_arm * Roll Gradient * (π/180))/Trackwidth

Rear Lateral Load Transfer Due to Roll (Elastic component) = 181.77 kg

4. Roll Stiffness:

Roll Stiffness = Roll Moment / Roll Gradient

Roll Stiffness = 14019.54 in-lb/deg

Front Roll Stiffness = (Front Lateral Load Transfer Due to Roll * Trackwidth)/ Roll Gradient

Front Roll Stiffness = 7903.17 in-lb/deg

Rear Roll Stiffness = Total Roll Stiffness - Front Roll Stiffness

Rear Roll Stiffness = 6116.37 in-lb/deg

Roll Stiffness - Tyre Contribution = Tire Stiffness *( (Trackwidth2/2)*(π/180))

Front Roll Stiffness - Tyre Contribution = 46001.54 in-lb/deg

Rear Roll Stiffness - Tyre Contribution = 51784.76 in-lb/deg

Roll Stiffness at WC = (Tyre Contribution * Total Front Tyre Stiffness)

/ (Tyre Contribution - Total Front Tyre Stiffness)

Front Roll Stiffness at WC = 9542.62 in-lb/deg

Rear Roll Stiffness at WC = 6935.54 in-lb/deg

Roll Stiffness at WC due to spring = Wheel rate *( (Trackwidth2/2)*(π/180))

Front Roll Stiffness at WC due to spring = 5110.09 in-lb/deg

Rear Roll Stiffness at WC due to spring = 5119.92 in-lb/deg

Roll Stiffness at WC due to Roll bar = Roll Stiffness at WC - Roll Stiffness at WC due to Spring

Front Roll Stiffness at WC due to Roll bar = 4432.53 in-lb/deg

Rear Roll Stiffness at WC due to Roll bar = 1815.62 in-lb/deg

5. Auxiliary Roll Stiffness:

Auxilliary Roll Stiffness = (Roll Stiffness at WC due to Roll Bar / Roll Stiffness)*100

Front Auxilliary Roll Stiffness = 56.09%

Rear Auxilliary Roll Stiffness = 29.68%

* The derived Auxilliary Roll stiffness is well below the given constraints.

6. Roll Bar Diameters:

We Know,

Front anti-roll bar stiffness = (Front Anti-roll spring rate for roll * Trackwidth2)/(180/π)

Where,

Anti Roll Spring Rate = 0.5 *((500000*(OD4-ID4))/((0.4422*A2*B)+(0.2264*C3)))*MR2

So,

Front anti-roll bar stiffness = ((0.5*(500000*(OD4-ID4))

/((0.4422*A2*B)+(0.2264*C3)))*MR2* Trackwidth2)/(180/π)

From this equation, we get

OD-ID =(((Anti - Roll bar Stifness*2)/(500000*MR2*Trackwidth2))

*(((0.4422*A2*B)+(0.2264*C3))*(180/PI())))0.25

Since ID = 0 (Given),

OD = (((Anti - Roll bar Stifness*2)/(500000*MR2*Trackwidth2))

*(((0.4422*A2*B)+(0.2264*C3))*(180/PI())))0.25

Front Roll Bar Outer Diameter = 1.01 in

= 25.62 mm

Rear Roll Bar Outer Diameter = 0.71 in

= 18.14 mm

PITCH COMPONENTS:

1. Load Transfer:

Longitudinal Load Transfer = (Total weight * CG Height*Throttle Acceleration)

/(Wheelbase*Acceleration due to Gravity )

Longitudinal Load Transfer = 138.98 kg

= 306.40 lb

Front end deflection under load = (Load Transfer *0.5) / Front Wheel Rate

Front end deflection under load = 0.8801 in

Rear end deflection under load = (Load Transfer *0.5) / Rear Wheel Rate

Rear end deflection under load = 0.8794 in

Pitch Angle = tan -1 ((Front End Deflection + Rear End Deflection )/ Wheelbase )*(180/π)

Pitch Angle (No Compensation) = 1.029 deg

Pitch Gradient = Pitch Angle / Throttle Acceleration per g

Pitch Gradient (No Compensation) = 2.04 deg/g

2. Anti-Characteristics - Acceleration:

Considering the constraints provided, the below values are assumed.

Front IC Vertical Distance from Contact Patch (FICV) =14.1 in

Front IC Horizontal Distance from Contact Patch (FICH) = 70 in

Rear IC Vertical Distance from Contact Patch (RICV) = 15 in

Rear IC Horizontal Distance from Contact Patch (RICH) = 70 in

Front Anti-Lift Angle = tan -1 ((FICV - Tire Radius )/ FICH )*(180/π)

Front Anti-Lift Angle = 1.50 deg

Rear Anti-Squat Angle = tan -1 ((RICV - Tire Radius )/ RICH )*(180/π)

Rear Anti-Squat Angle = 2.23 deg

Swing arm angle = tan -1 (CG Height/ Wheelbase )*(180/π)

Front swing arm angle for 100% anti-lift = 9.84 deg

Rear swing arm angle for 100% anti-squat = 9.84 deg

Front anti-lift % = (Front anti-lift angle * Drive ratio - Front) / FV Swing Arm Angle

Front anti-lift % = 11%

Rear Anti-Squat % = (Rear anti-Squat angle * ((100 - Drive ratio - Rear)/100))

/ RV Swing Arm Angle

Rear Anti-Squat % = 9.08%

Front End Deflection (with Compensation) = (Front End Deflection *(100-Anti-Lift))/100

Front End Deflection (with Compensation) = 0.87 in

Rear End Deflection (with Compensation) = (Rear End Deflection *(100-Anti-Squat))/100

Rear End Deflection (with Compensation) = 0.86 in

Pitch Angle = tan -1 ((Front End Deflection + Rear End Deflection )/ Wheelbase )*(180/π)

Pitch Angle (With Compensation) = 1.009 deg

Pitch Gradient = Pitch Angle / Throttle Acceleration per g

Pitch Gradient (With Compensation) = 2.00 deg/g

*The value of the Pitch gradient for acceleration is equal to the target value.

3. Anti-Characteristics - Braking:

Front Anti-Lift Angle = tan -1 ((FICV - Tire Radius )/ FICH )*(180/π)

Front Anti-Lift Angle = 1.50 deg

Rear Anti-Squat Angle = tan -1 ((RICV - Tire Radius )/ RICH )*(180/π)

Rear Anti-Squat Angle = 2.23 deg

Swing arm angle = tan -1 (CG Height/ Wheelbase )*(180/π)

Front swing arm angle for 100% anti-lift = 9.84 deg

Rear swing arm angle for 100% anti-squat = 9.84 deg

Front anti-lift % = (Front anti-lift angle * Drive ratio - Front) / FV Swing Arm Angle

Front anti-lift % = 10.65%

Rear Anti-Squat % = (Rear anti-Squat angle * ((100 - Drive ratio - Rear)/100))

/ RV Swing Arm Angle

Rear Anti-Squat % = 9.08%

Front End Deflection (with Compensation) = (Front End Deflection *(100-Anti-Lift))/100

Front End Deflection (with Compensation) = 0.88 in

Rear End Deflection (with Compensation) = (Rear End Deflection *(100-Anti-Squat))/100

Rear End Deflection (with Compensation) = 0.86 in

Pitch Angle = tan -1 ((Front End Deflection + Rear End Deflection)/ Wheelbase)*(180/π)

Pitch Angle (With Compensation) = 1.017deg

Pitch Gradient = Pitch Angle / Braking Deacceleration

Pitch Gradient (With Compensation) = 1.0166 deg/g

*The value of the Pitch gradient for acceleration is equal to the target value.

WRITEUP:

In this project, we calculated the Ride Frequency & its components, Roll Analysis and Pitch Components having the constraints and targets in consideration.

The major targets are Auxiliary Roll stiffness and Pitch Gradient of Acceleration & Braking.

To achieve the targets and stay within constraints, we made iterations to optimize the values of the following,

1. Rear Ride Frequency
2. Front IC Vertical Distance from Contact Patch (FICV)
3. Front IC Horizontal Distance from Contact Patch (FICH)
4. Rear IC Vertical Distance from Contact Patch (RICV)
5. Rear IC Horizontal Distance from Contact Patch (RICH)

i. Considered Input Values:

1. Rear Ride Frequency:

We considered a frequency of 1.544 Hz at the rear based on the iteration we did use octave script to plot the amplitude. This frequency had a better overlapping of front and rear amplitudes which happened at 0.576s.

2. Distance from Instance Centre:

All the values of distance from IC are optimized to the below-mentioned values by iterating multiple values to achieve the given constraints.

ii. Deliverables:

The deliverables are mentioned below.

1. Front Spring Rate - 198.39 lb/in
2. Rear Spring rate - 279.98 lb/in
3. Front Roll Bar Outer Diameter - 1.01 in
4. Rear Roll Bar Outer Diameter - 0.71 in
5. Rear Ride Frequency (@ 65mph) – 1.544 Hz
6. Front Auxiliary Roll Stiffness - 56.09%
7. Rear Auxiliary Roll Stiffness - 29.68%
8. Flat Ride Plot (attached as an image)
9. Front IC Vertical Distance from Contact Patch (FICV) - 14.1 in
10. Front IC Horizontal Distance from Contact Patch (FICH) – 70 in
11. Rear IC Vertical Distance from Contact Patch (RICV) - 15 in
12. Rear IC Horizontal Distance from Contact Patch (RICH) – 70 in
13. Front anti-lift % - Acceleration - 11%
14. Rear Anti-Squat % - Acceleration – 9.08%
15. Pitch Angle – Acceleration – 1.009 deg
16. Pitch Gradient – Acceleration – 2.0002 deg/g
17. Front anti-lift % - Braking - 10.65%
18. Rear Anti-Squat % - Braking – 9.08%
19. Pitch Angle – Braking – 1.017 deg
20. Pitch Gradient – Braking – 1.0166 deg/g

Google Drive Link for Excel Sheet:

https://docs.google.com/spreadsheets/d/1g7b8U03NKieDk37wyjBjqS-6M8Y22M8m/edit?usp=sharing&ouid=102891323117469303946&rtpof=true&sd=true

Google Drive Link for Octave Script - Flat Ride Plot:

https://drive.google.com/file/d/1B0KSp26ou91ZzLJgXUk02SIu3NKzSzgM/view?usp=sharing