design analysis of a 2011 subaru wrx stipaws.kettering.edu/~amazzei/final_presentation_sti.pdf ·...
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Design Analysis of a 2011Subaru WRX STI
MECH 542 Final ProjectKettering UniversityProfessor A. Mazzei
Team Members:
Jason Berger Jacob Kornas Jordan Nizza
Overview
Vehicle History
Vehicle Specifications
Vehicle Architecture
Competency 1—Weight Distribution and Tire Patch Forces
Competency 2—Suspension
Competency 3—Steering Systems
Competency 4—Rollover
Competency 5—Tires
Vehicle HistoryFirst Generation: 1992-2000
Second Generation: 2001-2007
Third Generation: 2008-2011
Fourth Generation: 2012-2015
Vehicle History The first generation Subaru Impreza WRX was introduced in Japan in
1992. In 1994, the Impreza WRX STI was released in Japan with an increased power rating, stronger suspension, and a stronger transmission compared to the WRX.
In 1995 the Subaru Impreza wagon, badged as the Outback Sport, was introduced to the United States.
For the 2000 model year, the Subaru Impreza was redesigned with a notably larger footprint.
The Subaru Impreza WRX STI was introduced in the United States in 2004 with a power rating of 300 horsepower.
In 2008, the Subaru Impreza was redesigned and the STI version was only available as a hatchback.
In 2011, the STI was once again available as a sedan, boasting 305 horsepower produced by a 2.5L turbocharged 4-cylinder boxer engine.
In 2012, the Subaru Impreza was redesigned with small body styling and interior changes.
https://en.wikipedia.org/wiki/Subaru_Impreza http://jalopnik.com/a-brief-history-of-the-subaru-wrx-461608007
Manufacturer Specifications
Overall Length 180.3"
Overall Width 70.7"
Overall Height 57.9"
Wheelbase 103.3"
Track f/r 60.2"/60.6"
Ground Clearance 5.9"
Curb Weight 3,384 lbs
Dimensions and Weights
Bore 3.92" x 3.11"
Compression Ratio 8.2:1
Redline 6700 RPM
Maximum Boost 14.7 psi
Power 305 hp @ 6000 RPM
Engine
Turbocharged 2.5L boxer DOHC, 4-cylinder, 16 valves
1st 3.636
2nd 2.235
3rd 1.521
4th 1.137
5th 0.971
6th 0.756
Front Final Drive 3.900
Rear Final Drive 3.545
Transfer Gear 1.1
6-speed manual transmission
Drivetrain
All wheel drive
Electronic stability control
Driver controlled center differential
Front and rear limited-slip differentials
City 17 mpg
Highway 23 mpg
Economy
Disc Diameter f/r 13.0"/12.6"
Calipers f/r 4 piston/2 piston
Brakes
Front and rear anti-lock brakes
Quick Ratio 15.0:1
Lock-to-Lock 2.8 turns
Curb-to-Curb 36.1'
Wall-to-Wall 38.7'
Steering
Rack and pinion
Turning Radius
Wheels (f and r) 18" x 8.5"
Tires (f and r) 245/40R18
Wheels and Tires
Front
Forged aluminum alloy lower L-arms,
inverted struts, cross member stiffener,
stabilizer bar
Rear Double-wishbone, stabilizer bar
Suspension
http://www.cars101.com/subaru/impreza/wrxsti2011.html
Vehicle Architecture: Overview
http://pressroom.subaru.pl/photo/2010m_wrxsti/
245/40R18 TiresRdyn,F = Rdyn,R = 322.6 mm
Front MacPherson Strut Suspension
Rear Multi-Link Suspension
Turbocharged 2.5L, 4-Cylinder, DOHC
Boxer Engine
AWD Drivetrain
Vehicle Architecture: Front
http://jp.autoblog.com/photos/2011-subaru-impreza-wrx-sti-first-drive-2/ http://eibach.com/m-america/en/eibach-news/subaru-wrx-sti-2015-plus-pro-kit
http://perrinperformance.com/i-13324020-front-sway-bar-for-2008-14-sti-09-14-wrx.html http://pressroom.subaru.pl/photo/2010m_wrxsti/
Lower L-Arm
Strut AssemblySpring Rate : 58.0 N/mm
Front CV Shaft13” Brake Rotor with
4-Piston Caliper
Stabilizer Bar
Steering Tie Rod
Steering Rack Assembly
Vehicle Architecture: Rear
http://jp.autoblog.com/photos/2011-subaru-impreza-wrx-sti-first-drive-2/ http://eibach.com/m-america/en/eibach-news/subaru-wrx-sti-2015-plus-pro-kit
http://pressroom.subaru.pl/photo/2010m_wrxsti/
Stabilizer Bar
Lower Wishbone
12.6” Brake Rotor with 2-Piston Caliper
Rear CV Shaft
Upper Wishbone
Strut AssemblySpring Rate : 60.0 N/mm
Toe Control Arm
Competency 1—Weight Distribution
Objectives
Define vehicle coordinate system
Calculate vehicle system center of gravity
Calculate sprung system weight and center of gravity
Calculate unsprung system weight and center of gravity
Competency 1—Weight Distribution
Vehicle dynamics depend directly on weight distribution
Weight distribution affects the yaw, roll, and pitch motions of a vehicle during corning, braking, and acceleration
The location of the center of gravity of the vehicle affects weight transfer during braking and acceleration and the cornering performance of a vehicle
Coordinate System
Vehicle ISO Coordinate System
Z
Y
X
EquationsWeight Distribution
𝑖𝑊𝐷,𝐹 =𝑙𝑣,𝑅𝑙
=𝐹𝑣,𝐹𝐹𝑣,𝑡
𝑖𝑊𝐷,𝑅 =𝑙𝑣,𝐹𝑙
=𝐹𝑣,𝑅𝐹𝑣,𝑡
Vehicle Center of Gravity Height
ℎ𝑣,𝑡 = ℎ𝑣,0 + ∆ℎ𝑙𝑜𝑎𝑑
ℎ𝑣,0 = 𝑖𝑢𝑙ℎ𝑢𝑙
Unsprung and Sprung Weight
𝐹𝑢,𝐹 𝑜𝑟 𝑅 =𝑖𝑚,𝐹 𝑜𝑟 𝑅𝐹𝑣,𝑡,𝑜𝑖𝑊𝐷,𝐹 𝑜𝑟 𝑅
1 + 𝑖𝑚,𝐹 𝑜𝑟 𝑅
𝐹𝐵𝑜,𝐹 𝑜𝑟 𝑅 = 𝐹𝑣,𝑡𝑖𝑊𝐷,𝐹 𝑜𝑟 𝑅 − 𝐹𝑈,𝐹 𝑜𝑟 𝑅
𝑖𝑚,𝐹 𝑜𝑟 𝑅 =𝐹𝑢,𝐹 𝑜𝑟 𝑅𝐹𝐵𝑜,𝐹 𝑜𝑟 𝑅
Sprung Weight
𝐹𝑢,𝐹 𝑜𝑟 𝑅 =𝑖𝑚,𝐹 𝑜𝑟 𝑅𝐹𝑣,𝑡,𝑜𝑖𝑊𝐷,𝐹 𝑜𝑟 𝑅
1 + 𝑖𝑚,𝐹 𝑜𝑟 𝑅
Dynamic Tire Radius
𝑟𝑑𝑦𝑛,𝐹 𝑜𝑟 𝑅 =1
225.4𝑑𝐹 𝑜𝑟 𝑅 +
2
100
𝐻
𝑊 𝐹 𝑜𝑟 𝑅𝑊𝐹 𝑜𝑟 𝑅 − ∆𝑟𝐹 𝑜𝑟 𝑅
Body Center of Gravity Height
ℎ𝐵𝑜 =𝐹𝑣,𝑡ℎ𝑣,𝑡−𝐹𝑈,𝐹𝑟𝑑𝑦𝑛,𝐹−𝐹𝑈,𝑅𝑟𝑑𝑦𝑛,𝑅
𝐹𝐵𝑜
Vehicle Center of Gravity Lateral Position
𝑏𝑣 =1
𝐹𝑣,𝑡
𝑏𝐹2
𝐹𝑣,𝐹𝐿−𝐹𝑣,𝐹𝑅 +𝑏𝑅2
𝐹𝑣,𝑅𝐿−𝐹𝑣,𝑅𝑅
Vehicle Weight Distribution
http://www.cars101.com/subaru/impreza/wrxsti2011.html
Parameter Definition Symbol Value Unit Value Unit
Vehicle curb weight Fv,t,0 3384 lbf 15053 N
Analysis Weight, Two people Fv,t 3781 lbf 16819 N
Test weight - 3451 lbf 15351 N
Front left corner test weight Fv,FL 952 lbf 4235 N
Front right corner test weight Fv,FR 1018 lbf 4528 N
Rear left corner test weight Fv,RL 762 lbf 3390 N
Rear right corner test weight Fv,RR 719 lbf 3198 N
Vehicle wheelbase l 103.3 in. 2623.8 mm
Vehicle front track width bF 60.2 in. 1529.1 mm
Vehicle rear track width bR 60.6 in. 1539.2 mm
Height of unloaded vehicle hul 57.9 in. 1470.7 mm
Loaded vehicle change in height Δhload 0.39 in. 10 mm
Front tire wheel Diameter dF 18 in. 457.2 mm
Front tire width WF 9.65 in. 245 mm
Front tire aspect ratio (H/W)F 40 - - -
Front tire deflection ΔrF 0.16 in. 4 mm
Rear tire wheel Diameter dR 18 in. 457.2 mm
Rear tire width WR 9.65 in. 245 mm
Rear tire aspect ratio (H/W)R 40 - - -
Rear tire deflection ΔrR 0.16 in. 4 mm
Front axle weight distribution iWD,F 0.57 - - -
Rear axle weight distribution iWD,R 0.43 - - -
Front unsprung to sprung weight ratio im,F 0.12 - - -
Rear unsprung to sprung weight ratio im,R 0.14 - - -
Vehicle CG height coefficient iul 0.36 - - -
Given Parameters: 2011 Subaru WRX STIParameter Definition Symbol Value Unit Value Unit
Vehicle weight at front axle Fv,F 2158.3802 lbf 9600.9534 N
Vehicle weight at rear axle Fv,R 1622.6198 lbf 7217.7726 N
Loction of vehicle CG from the front axle plane lv,F 44.3313 in. 1126.0149 mm
Location of the vehicle CG from the rear axle plane lv,R 58.9687 in. 1497.8051 mm
Front axle unsprung weight Fu,F 206.9735 lbf 920.6642 N
Rear axle unsprung weight Fu,R 178.3461 lbf 793.3230 N
Front body (sprung) weight FBo,F 1951.4066 lbf 8680.2892 N
Rear body (sprung) weight FBo,R 1444.2737 lbf 6424.4495 N
Total vehicle body (sprung) weight FBo 3395.6803 lbf 15104.7387 N
Unloaded vehicle CG height hv,0 20.8440 in. 529.4376 mm
Loaded vehicle CG height hv,t 21.2377 in. 539.4376 mm
Front dynamic tire radius rdyn,F 12.7008 in. 322.6000 mm
Rear dynamic tire radius rdyn,R 12.7008 in. 322.6000 mm
Loaded body CG height hBo 22.2064 in. 564.0429 mm
Lateral position of vehicle CG bv -0.1981 in. -5.0322 mm
Loction of the body CG from the front axle plane lBo,F 43.9363 in. 1115.9809 mm
Location of the body CG from the rear axle plane lBo,R 59.3637 in. 1507.8391 mm
Calculated Values
Laboratory Measurements:Test weight = 3451 lbf
FL Corner Weight = 952 lbf
FR Corner Weight = 1018 lbf
RL Corner Weight = 762 lbf
RR Corner Weight = 719 lbf
Competency 1—Tire Patch Forces
Objectives
Define vehicle system tire patch forces
Calculate braking tire patch forces
Calculate acceleration tire patch forces
Calculate cornering tire patch forces
Competency 1—Tire Patch Forces
All forces acting on a vehicle, minus aerodynamic forces, act at the tire contact patch
The weight transfer during braking and acceleration depends on the location of the center of gravity of the vehicle and the vehicle wheelbase
For example, in a dragster, weight transfer is small because the relatively low center of gravity height and the long wheel base
Tire patch forces depend on the coefficient of friction between the tire and road
Braking Equations
Maximum Braking Coefficient of Static Friction
𝑔𝑥,𝐵 = 0.0334𝑣2
𝑠
Front Tire Patch Normal Forces
𝐹𝑧,𝑊,𝐵,𝐹 =1
2𝐹𝑣,𝑡𝑖𝑊𝐷,𝐹 + 𝐹𝑣,𝑡𝑔𝑥,𝐵
ℎ𝑣,𝑡
𝑙
Rear Tire Patch Normal Forces
𝐹𝑧,𝑊,𝐵,𝑅 =1
2𝐹𝑣,𝑡𝑖𝑊𝐷,𝑅 − 𝐹𝑣,𝑡𝑔𝑥,𝐵
ℎ𝑣,𝑡
𝑙
Rear Tire Patch Ideal Braking Forces
𝐹𝑥,𝑊,𝐵,𝑅 =1
2𝐹𝑣,𝑡𝑖𝑊𝐷,𝑅 − 𝐹𝑣,𝑡𝑔𝑥,𝐵
ℎ𝑣,𝑡
𝑙𝑔𝑥,𝐵
Front Tire Patch Ideal Braking Forces
𝐹𝑥,𝑊,𝐵,𝐹 =1
2𝐹𝑣,𝑡𝑖𝑊𝐷,𝐹 + 𝐹𝑣,𝑡𝑔𝑥,𝐵
ℎ𝑣,𝑡
𝑙𝑔𝑥,𝐵
Acceleration Equations
Maximum Drive Off Coefficient of Static Friction
𝑔𝑥,𝐴 = 0.0334𝑣2
𝑠
Front Tire Patch Normal Forces
𝐹𝑧,𝑊,𝐴,𝐹 =1
2𝐹𝑣,𝑡𝑖𝑊𝐷,𝐹 − 𝐹𝑣,𝑡𝑔𝑥,𝐴
ℎ𝑣,𝑡
𝑙
Rear Tire Patch Normal Forces
𝐹𝑧,𝑊,𝐴,𝑅 =1
2𝐹𝑣,𝑡𝑖𝑊𝐷,𝑅 + 𝐹𝑣,𝑡𝑔𝑥,𝐴
ℎ𝑣,𝑡
𝑙
Rear Tire Patch Drive Off Forces
𝐹𝑥,𝑊,𝐴,𝑅 =1
2𝐹𝑣,𝑡𝑖𝑊𝐷,𝑅 + 𝐹𝑣,𝑡𝑔𝑥,𝐴
ℎ𝑣,𝑡
𝑙𝑔𝑥,𝐴
Front Tire Patch Drive Off Forces
𝐹𝑥,𝑊,𝐴,𝐹 =1
2𝐹𝑣,𝑡𝑖𝑊𝐷,𝐹 − 𝐹𝑣,𝑡𝑔𝑥,𝐴
ℎ𝑣,𝑡
𝑙𝑔𝑥,𝐴
Cornering Equations
Front Outside Tire Patch Normal Force
𝐹𝑧,𝑊,𝑜,𝐹 =1
2𝐹𝑣,𝑡𝑖𝑊𝐷,𝐹 + 𝐹𝑣,𝑡𝑖𝑊𝐷,𝐹𝑔𝑦,𝐶
ℎ𝑣,𝐹
𝑏𝐹
Front Inside Tire Patch Normal Force
𝐹𝑧,𝑊,𝑖,𝐹 =1
2𝐹𝑣,𝑡𝑖𝑊𝐷,𝐹 − 𝐹𝑣,𝑡𝑖𝑊𝐷,𝐹𝑔𝑦,𝐶
ℎ𝑣,𝐹
𝑏𝐹
Front Outside Tire Patch Cornering Force
𝐹𝑦,𝑊,𝑜,𝐹 =1
2𝐹𝑣,𝑡𝑖𝑊𝐷,𝐹 + 𝐹𝑣,𝑡𝑖𝑊𝐷,𝐹𝑔𝑦,𝐶
ℎ𝑣,𝐹
𝑏𝐹𝑔𝑦,𝐶
Front Inside Tire Patch Cornering Force
𝐹𝑦,𝑊,𝑖,𝐹 =1
2𝐹𝑣,𝑡𝑖𝑊𝐷,𝐹 − 𝐹𝑣,𝑡𝑖𝑊𝐷,𝐹𝑔𝑦,𝐶
ℎ𝑣,𝐹
𝑏𝐹𝑔𝑦,𝐶
Rear Outside Tire Patch Normal Force
𝐹𝑧,𝑊,𝑜,𝑅 =1
2𝐹𝑣,𝑡𝑖𝑊𝐷,𝑅 + 𝐹𝑣,𝑡𝑖𝑊𝐷,𝑅𝑔𝑦,𝐶
ℎ𝑣,𝑅
𝑏𝑅
Rear Inside Tire Patch Normal Force
𝐹𝑧,𝑊,𝑖,𝑅 =1
2𝐹𝑣,𝑡𝑖𝑊𝐷,𝑅 − 𝐹𝑣,𝑡𝑖𝑊𝐷,𝑅𝑔𝑦,𝐶
ℎ𝑣,𝑅
𝑏𝑅
Rear Inside Tire Patch Cornering Force
𝐹𝑦,𝑊,𝑖,𝑅 =1
2𝐹𝑣,𝑡𝑖𝑊𝐷,𝑅 − 𝐹𝑣,𝑡𝑖𝑊𝐷,𝑅𝑔𝑦,𝐶
ℎ𝑣,𝑅
𝑏𝑅𝑔𝑦,𝐶
Rear Outside Tire Patch Cornering Force
𝐹𝑦,𝑊,𝑜,𝑅 =1
2𝐹𝑣,𝑡𝑖𝑊𝐷,𝑅 + 𝐹𝑣,𝑡𝑖𝑊𝐷,𝑅𝑔𝑦,𝐶
ℎ𝑣,𝑅
𝑏𝑅𝑔𝑦,𝐶
Vehicle Tire Patch Forces
http://paws.kettering.edu/~amazzei/DataPanel/CT_2008-Mitsubishi-Lancer-Evolution-GSR-vs-2008-Subaru-Impreza-WRX-STI_data.pdf
Parameter Definition Symbol Value Unit Value Unit
Vehicle speed v 60 mph 97 kph
Vehicle stopping distance s 119 ft 36 m
Drive off acceleration gx,A 1.010 g - -
Longitudinal coefficient of static friction μx,W 1.010 - - -
Cornering acceleration gy,C 0.9 g - -
Given Parameters: 2011 Subaru WRX STI
Parameter Definition Symbol Value Unit Value Unit
Braking acceleration gx,B 1.0104 g - -
Front tire patch normal force Fz,B,F 2891.6602 lbf 12862.7455 N
Rear tire patch normal force Fz,B,R 822.3398 lbf 3657.9496 N
Braking weight transfer ΔFz,v 771.5269 lbf 3431.9227 N
Front tire patch normal force per wheel Fz,W,B,F 1445.8301 lbf 6431.3727 N
Rear tire patch normal force per wheel Fz,W,B,R 411.1699 lbf 1828.9748 N
Ideal total longitudinal braking force Fx,v,B 3752.7005 lbf 16692.8435 N
Front tire patch ideal braking force per wheel Fx,W,B,F 1460.8959 lbf 6498.3887 N
Rear tire patch ideal braking force per wheel Fx,W,B,R 415.4544 lbf 1848.0330 N
Parameter Definition Symbol Value Unit Value Unit
Front tire patch normal force Fz,A,F 1348.6064 lbf 5998.9000 N
Rear tire patch normal force Fz,A,R 2365.3936 lbf 10521.7950 N
Acceleration weight transfer ΔFz,v 771.5269 lbf 3431.9227 N
Front tire patch normal force per wheel Fz,W,A,F 674.3032 lbf 2999.4500 N
Rear tire patch normal force per wheel Fz,W,A,R 1182.6968 lbf 5260.8975 N
Ideal total longitudinal acceleration force Fx,v,A 3752.7005 lbf 16692.8435 N
Front tire patch ideal drive off force per wheel Fx,W,A,F 681.3295 lbf 3030.7048 N
Rear tire patch ideal drive off force per wheel Fx,W,A,R 1195.0207 lbf 5315.7170 N
Parameter Definition Symbol Value Unit Value Unit
Front axle tire patch cornering force Fy,v,F 1908.1200 lbf 8487.7405 N
Front axle outside tire patch normal force Fz,W,o,F 1733.2241 lbf 7709.7651 N
Front axle inside tire patch normal force Fz,W,i,F 386.9092 lbf 1721.0577 N
Front axle outside tire patch cornering force Fy,W,o,F 1559.9017 lbf 6938.7886 N
Front axle inside tire patch cornering force Fy,W,i,F 348.2182 lbf 1548.9519 N
Rear axle tire patch cornering force Fy,v,R 1434.4800 lbf 6380.8851 N
Rear axle outside tire patch normal force Fz,W,o,R 1299.6571 lbf 5781.1627 N
Rear axle inside tire patch normal force Fz,W,i,R 294.2096 lbf 1308.7096 N
Rear axle outside tire patch cornering force Fy,W,o,R 1169.6914 lbf 5203.0464 N
Rear axle inside tire patch cornering force Fy,W,i,R 264.7887 lbf 1177.8387 N
Cornering
Calculated ValuesBraking
AccelerationRoad and Track Data:
Stopping Speed= 60 mphStopping Distance= 952 lbf
Maximum Cornering Acceleration = 0.90 g
Braking Tire Patch Forces
0
500
1000
1500
2000
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Ve
rtic
al F
orc
e (
lbf)
Braking g's
Vertical Force vs. Braking Acceleration
FZ,W,B,F FZ,W,B,R
0
500
1000
1500
2000
2500
3000
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Lon
gitu
din
al F
orc
e (
lbf)
Braking g's
Longitudinal Force vs. Braking Acceleration
FX,W,B,F FX,W,B,R
Weight transfer increases with acceleration
Cornering Tire Patch Forces
-500
0
500
1000
1500
2000
2500
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Ve
rtic
al F
orc
e (
lbf)
Braking g's
Vertical Force vs. Cornering Acceleration
FZ,W,O,F FZ,W,I,F FZ,W,O,R FZ,W,I,R
-1000
0
1000
2000
3000
4000
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Late
ral F
orc
e (
lbf)
Braking g's
Lateral Force vs. Cornering Acceleration
FY,W,O,F FY,W,I,F FY,W,O,R FY,W,I,R
Lift off occurs when vertical forces equal zero, onset of rollover
Competency 2—Suspensions
Objectives
Define suspension types
Determine full anti-squat geometry
Determine full anti-pitch geometry
Determine pitch rate of the vehicle
Competency 2—Suspensions
Vehicle suspension isolates the chassis from rough road surfaces and maintains tire contact patch with the road surface
Vehicle suspension maintains the proper steering alignment angles
Vehicle suspension reacts to tire patch forces
Suspension forces act at a single virtual point
2011 Subaru WRX STI Suspension
Front MacPherson Strut
Advantageous for packaging
Requires taller hood
Rear Multi-link
Links provide lateral and longitudinal control
Utilize ball-joints and bushing to eliminate bending moments
Suspension Equations
100% Anti-Squat Condition, Rear
𝑒 − 𝑟𝑑𝑦𝑛
𝑑=ℎ𝑣,𝑡𝑙
*Note: A simplified independent RWD case is considered instead of considering the AWD power distribution of the 2011 Subaru WRX STI
100% Anti-Pitch Condition, Rear
𝑒 − 𝑟𝑑𝑦𝑛
𝑑=ℎ𝑣,𝑡𝑙+ℎ𝑣,𝑡𝑙
𝐾𝐹𝐾𝑅
Pitch Rate, Rear
𝜃𝑝𝑎𝑥
=1
𝑙
𝐹𝑣,𝑡𝑔
1
𝐾𝑅
ℎ𝑣,𝑡𝑙−
1
𝐾𝑅
𝑒 − 𝑟
𝑑+
1
𝐾𝑓
ℎ𝑣,𝑡𝑙
Gillespie, Thomas: Fundamentals of Vehicle Dynamics
Spring Rate Calculations
F (lbf) Δx (in) F (N) Δx (mm)
0 0.00 0 0.0
150 0.25 667 6.4
200 0.38 890 9.5
250 0.50 1112 12.7
300 0.56 1334 14.3
Average KF (lbf/in) 541.7 Average KF (N/mm) 94.9
Table: Average front spring rate calculated from laboratory measurements
Sample Calculation: 𝐾𝐹 =𝐹
∆𝑥=
1334 𝑁
14.3 𝑚𝑚= 93.3 𝑁/𝑚𝑚
http://eibach.com/m-america/en/eibach-news/subaru-wrx-sti-2015-plus-pro-kit
For the suspension calculations, the following spring rates provided by Eibach for the 2015 Subaru WRX STI will be used:
𝐾𝐹 = 58.0 𝑁/𝑚𝑚 and 𝐾𝑅 = 60.0 𝑁/𝑚𝑚
Suspension Calculations
http://eibach.com/m-america/en/eibach-news/subaru-wrx-sti-2015-plus-pro-kit
Parameter Definition Symbol Value Unit Value Unit
Suspension Type -
Front suspension spring rate (per corner) KF 331.19 lbf/in 58.00 N/mm
Rear suspension spring rate (per corner) KR 342.61 lbf/in 60.00 N/mm
Given Parameters: 2011 Subaru WRX STI
Independent, RWD
Parameter Definition Symbol Value Unit Value Unit
Full anti-squat (e-rdyn)/d 0.2056 - - -
Full anti-pitch (e-rdyn)/d 0.4183 - - -
Pitch rate at full anti-squat θp/ax 0.0112 rad/g 0.6394 deg/g
Calculated Values
Competency 3—Steering
Objectives
Define steering system types
Define steering system characteristics including Ackerman geometry, caster, camber, and toe
Explore the relationship between steering geometry and suspension geometry
Calculate the moments acting on the steering system
Competency 3—Steering The steering system controls the direction of the vehicle
based on driver input
Steering angles are affected by steering system geometry, suspension geometry, and drivetrain geometry and reactions
The most common steering system on passenger vehicles today is rack and pinion
In handling, understeer is desired and oversteer can be dangerous
The steering system consists of multiple elements, each with their own compliance, making steering systems hard to model
Steering Equations
Moment Due to Vertical Force
𝑀𝑉 = − 𝐹𝑧𝐿 + 𝐹𝑧𝑅 𝑟𝜎 sin λ sin 𝛿 + 𝐹𝑧𝐿 − 𝐹𝑧𝑅 𝛿 sin ν cos 𝛿
Moment Due to Lateral Force
𝑀𝐿 = − 𝐹𝑦𝐿 + 𝐹𝑦𝑅 𝑟𝑑𝑦𝑛 tan ν
𝑟𝜎 = Scrub Radiusλ = Lateral Inclination Angle𝛿 = Steering Angleν = Caster Angle
Steering Calculation Parameters
Laboratory Measurements
λ = 10°ν = 6°
Assumed Based on Common Values
𝑟𝜎 = 10 mm
Steering Angle
Calculations will be performed for a steering angle of δ = 5°
Weight Distribution Values
Maximum Vertical Forces: 𝐹𝑧𝐿 + 𝐹𝑧𝑅 = 𝐹𝑧,𝐵,𝐹 = 2891.6602 𝑙𝑏𝑓Maximum Lateral Forces: 𝐹𝑦𝐿 + 𝐹𝑦𝑅 = 𝐹𝑦,𝑣,𝐹 = 1908.1200 𝑙𝑏𝑓
Steering Calculations
Parameter Definition Symbol Value Unit Value Unit
Scrub radius rσ 0.3937 in 10 mm
Lateral inclination angle λ 10.00 deg 0.17 rad
Caster angle ν 6.00 deg 0.10 rad
Steering angle δ 5.00 deg 0.09 rad
Given Parameters: 2011 Subaru WRX STI
Parameter Definition Symbol Value Unit Value Unit
Total moment due to vertical force MV -1.4358 ft∙lbf -1.9467 N∙m
Total moment due to lateral force ML 2.8510 ft∙lbf -287.7906 N∙m
Calculated Values
Steering
Competency 4—Rollover
Objectives
Understand how vehicle geometry and weight distribution affects rollover
Calculate rollover threshold for a rigid and suspended vehicle model
Discuss transient rollover and the effects of vehicle damping on rollover threshold
Competency 4—Rollover
The onset of rollover occurs when one tire lifts off the ground
Rollover is defined as the situation where the vehicle rotates 90 degrees or more about its longitudinal axis, resulting in the vehicle body contacting the ground
Road cross-slope angle attempts to balance tire patch forces
Many roll over events result because the vehicle is ‘tripped’
During rollover, the center of gravity of the vehicle is transient, thus, the rollover threshold decreases with increasing roll angle
Rollover Equations
Quasi-Static Rollover Threshold: Rigid Vehicle
𝑎𝑦
𝑔=
𝑡
2ℎ𝑣,𝑡+ 𝜑
𝑡 = Track Width, 𝑏𝐹 or 𝑏𝑅𝜑 = Cross-Slope Angle𝑅Ф = Roll Rate (rad/g)ℎ𝑟 = Roll center height at longitudinal CG location
Quasi-Static Rollover Threshold: Suspended Vehicle
𝑎𝑦
𝑔=
𝑡
2ℎ𝑣,𝑡
1
1 + 𝑅Ф 1 −ℎ𝑟ℎ𝑣,𝑡
Rollover Calculation Parameters
Road Conditions
𝜑 = 0°
Assumed Based on Common Values
𝑅Ф = 3.0 deg/g ℎ𝑟 = 10.0 in.
Vehicle Specifications
𝑏𝐹 = 60.2 in.𝑏𝑅 = 60.6 in.
Rollover Calculations
Parameter Definition Symbol Value Unit Value Unit
Cross-slope angle φ 0.00 deg 0.00 rad
Roll center height hr 10.00 in 254.0000 mm
Roll Rate Rφ 3.00 deg/g 0.0524 rad/g
Given Parameters: 2011 Subaru WRX STI
Parameter Definition Symbol Value Unit Value Unit
Quasi-static rollover threshold, front (no φ) gR,F 1.4173 g - -
Quasi-static rollover threshold, rear (no φ) gR,R 1.4267 g - -
Suspended vehicle rollover threshold, front gR,F 1.3791 g - -
Suspended vehicle rollover threshold, rear gR,R 1.3882 g - -
Calculated Values
Rollover
Competency 5—Tires
Objectives
Understand how tires are constructed and the difference between bias, belted bias, and radial tires
Define tire terminology including tires planes and the forces and moments acting on the tire
Define how forces are generated at the tire contact patch and the pressure distribution over the contact patch
Understand the Magic Formula and its application for characterizing tire behavior
Competency 5—Tires
Tires are visco-elastic toroids
The flexible tire carcass is constructed of high-tensile-strength cords fastened to steel cable rim beads
Adhesion and hysteresis are the two primary mechanisms for the friction coupling at the tire-road interface
The Magic Formula can be used to determine cornering force and slip angle, self-aligning torque and slip angle, and braking effort and skid relationships
Empirical data is used to understand the complex, nonlinear nature of tires
Tire Equations
The Magic Formula
𝑦 𝑥 = 𝐷 sin 𝐶 tan−1 𝐵𝑥 − 𝐸 𝐵𝑥 − tan−1𝐵𝑥
Peak Factor
𝐷 = 𝑎1 𝐹𝑧2 + 𝑎2𝐹𝑧
Mechanics of Pneumatic Tires
Cornering Stiffness
𝐵𝐶𝐷 = 𝑎3 sin 𝑎4 tan−1 𝑎5𝐹𝑧
Longitudinal Stiffness
𝐵𝐶𝐷 =𝑎3 𝐹𝑧
2 + 𝑎4𝐹𝑧𝑒𝑎5𝐹𝑧
Shape Factor
𝐶 = 1.30 for cornering force – slip angle relationship𝐶 = 2.40 for self-aligning torque – slip angle relationship𝐶 = 1.65 for braking effort – skid relationship
Stiffness Factor
𝐵 =𝐵𝐶𝐷
𝐶𝐷
Curvature Factor
𝐸 = 𝑎6 𝐹𝑧2 + 𝑎7𝐹𝑧 + 𝑎8
*Fz is in kN
Tire Calculations
Parameter Definition Symbol Value Unit Value Unit
Maximum vertical tire patch force, braking Fz,W,B,F 1445.83 lbf 6431.37 N
Maximum vertical tire patch force, cornering Fz,W,o,F 1733.22 lbf 7709.7651 N
Skid percentage - -20.00 %
Given Parameters: 2011 Subaru WRX STI
Parameter Definition Symbol Value Unit Value Unit
Cornering stiffness BCD 1038.0034 N/deg - -
Maximum lateral force (peak factor) D 6480.9379 N - -
Parameter Definition Symbol Value Unit Value Unit
Stiffness factor B 0.2105 - - -
Shape factor C 1.6500 - - -
Peak factor D 6476.4680 - - -
Curvature factor E 0.5980 - - -
Longitudinal stiffness BCD 2248.9122 N/% skid - -
Longitudinal force Fx -5986.8423 N - -
Calculated ValuesCornering
Braking
a1 a2 a3 a4 a5 a6 a7 a8
Fy, N -22.1 1011 1078 1.82 0.208 0.000 -0.354 0.707
Mz, N -2.72 -2.28 -1.86 -2.73 0.110 -0.070 0.643 -4.04
Fx, N -21.3 1144 49.6 226 0.069 -0.006 0.056 0.486
Table 1.7: Values of Coefficients a1 to a8 for a car tire (Fz in kN)
Mechanics of Pneumatic Tires
*Assumed -20.0% Skid
Friction forces are typically maximum at -15 to -20% skid
Longitudinal Brake Force
0
1000
2000
3000
4000
5000
6000
7000
0 10 20 30 40 50 60 70 80 90 100
Ne
gati
ve B
rake
Fo
rce
(N
)
Negative Skid (%)
Longitudinal Brake Force vs. Skid
The Magic Formula can be used to determine the per tire longitudinal brake force versus skid during maximum braking tire patch forces. Below is the
longitudinal brake force per front tire for the 2011 Subaru WRX STI.
The initial slope is the longitudinal stiffness of the tire
CarSim Inputs
WOT Tire Patch Forces
WOT Graphs
WOT Graphs Continued
Cornering Tire Patch Forces
Cornering Graphs
Cornering Graphs Continued
SuspensionSim Inputs
Suspension Graphs
Suspension Graphs Continued