Design Analysis of a 2011 Subaru WRX STI

MECH 542 Final Project Kettering University Professor 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 History First Generation: 1992-2000 Third Generation: 2008-2011

Second Generation: 2001-2007 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

Dimensions and Weights Engine Drivetrain

Overall Length 180.3" Turbocharged 2.5L boxer DOHC, 4-cylinder, 16 valves All wheel drive Overall Width 70.7" Bore 3.92" x 3.11" Electronic stability control Overall Height 57.9" Compression Ratio 8.2:1 Driver controlled center differential Wheelbase 103.3" Redline 6700 RPM Front and rear limited-slip differentials Track f/r 60.2"/60.6" Maximum Boost 14.7 psi Ground Clearance 5.9" Power 305 hp @ 6000 RPM 6-speed manual transmission 1st 3.636 Suspension Curb Weight 3,384 lbs 2nd 2.235 3rd 1.521 Wheels and Tires Forged aluminum alloy lower L-arms, 4th 1.137 Front inverted struts, cross member stiffener, 5th 0.971 Wheels (f and r) 18" x 8.5" stabilizer bar 6th 0.756 Tires (f and r) 245/40R18 Rear Double-wishbone, stabilizer bar

Economy Steering Front Final Drive 3.900 Rear Final Drive 3.545 Transfer Gear 1.1 City 17 mpg Rack and pinion Highway 23 mpg Quick Ratio 15.0:1 Brakes Lock-to-Lock 2.8 turns Front and rear anti-lock brakes Turning Radius Disc Diameter f/r 13.0"/12.6" Curb-to-Curb 36.1' Calipers f/r 4 piston/2 piston Wall-to-Wall 38.7' http://www.cars101.com/subaru/impreza/wrxsti2011.html Vehicle Architecture: Overview

Front MacPherson Turbocharged 2.5L, Strut Suspension 4-Cylinder, DOHC Boxer Engine

245/40R18 Tires AWD Drivetrain Rdyn,F = Rdyn,R = 322.6 mm

Rear Multi-Link Suspension

http://pressroom.subaru.pl/photo/2010m_wrxsti/ Vehicle Architecture: Front

Strut Assembly Spring Rate : 58.0 N/mm

Lower L-Arm

Steering Tie Rod

Steering Rack Assembly

Stabilizer Bar 13” Brake Rotor with Front CV Shaft 4-Piston Caliper

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/ Vehicle Architecture: Rear

Strut Assembly Spring Rate : Stabilizer Bar 60.0 N/mm

Lower Wishbone Upper Wishbone

12.6” Brake Rotor Rear CV Shaft Toe Control Arm with 2-Piston Caliper 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/ 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 Equations

Vehicle Center of Gravity Height Weight Distribution

ℎ푣,푡 = ℎ푣,0 + ∆ℎ푙표푎푑 푙푣,푅 퐹푣,퐹 𝑖푊퐷,퐹 = = 푙 퐹푣,푡 ℎ푣,0 = 𝑖푢푙ℎ푢푙 푙푣,퐹 퐹푣,푅 𝑖푊퐷,푅 = = Body Center of Gravity Height 푙 퐹푣,푡

퐹푣,푡ℎ푣,푡−퐹푈,퐹푟푑푦푛,퐹−퐹푈,푅푟푑푦푛,푅 ℎ퐵표 = 퐹퐵표 Sprung Weight

𝑖푚,퐹 표푟 푅퐹푣,푡,표𝑖푊퐷,퐹 표푟 푅 Vehicle Center of Gravity Lateral Position 퐹푢,퐹 표푟 푅 = 1 + 𝑖푚,퐹 표푟 푅 1 푏퐹 푏푅 푏푣 = 퐹푣,퐹퐿−퐹푣,퐹푅 + 퐹푣,푅퐿−퐹푣,푅푅 퐹푣,푡 2 2 Unsprung and Sprung Weight

𝑖푚,퐹 표푟 푅퐹푣,푡,표𝑖푊퐷,퐹 표푟 푅 Dynamic Tire Radius 퐹푢,퐹 표푟 푅 = 1 + 𝑖푚,퐹 표푟 푅 1 2 퐻 퐹 = 퐹 𝑖 − 퐹 푟푑푦푛,퐹 표푟 푅 = 25.4푑퐹 표푟 푅 + 푊퐹 표푟 푅 − ∆푟퐹 표푟 푅 퐵표,퐹 표푟 푅 푣,푡 푊퐷,퐹 표푟 푅 푈,퐹 표푟 푅 2 100 푊 퐹 표푟 푅 퐹푢,퐹 표푟 푅 𝑖푚,퐹 표푟 푅 = 퐹퐵표,퐹 표푟 푅 Vehicle Weight Distribution

Given Parameters: 2011 Subaru WRX STI Calculated Values Parameter Definition Symbol Value Unit Value Unit Parameter Definition Symbol Value Unit Value Unit

Vehicle curb weight Fv,t,0 3384 lbf 15053 N Vehicle weight at front axle Fv,F 2158.3802 lbf 9600.9534 N

Analysis Weight, Two people Fv,t 3781 lbf 16819 N Vehicle weight at rear axle Fv,R 1622.6198 lbf 7217.7726 N

Test weight - 3451 lbf 15351 N Loction of vehicle CG from the front axle plane lv,F 44.3313 in. 1126.0149 mm

Front left corner test weight Fv,FL 952 lbf 4235 N Location of the vehicle CG from the rear axle plane lv,R 58.9687 in. 1497.8051 mm Front right corner test weight F 1018 lb 4528 N v,FR f Front axle unsprung weight Fu,F 206.9735 lbf 920.6642 N Rear left corner test weight F 762 lb 3390 N v,RL f Rear axle unsprung weight Fu,R 178.3461 lbf 793.3230 N Rear right corner test weight F 719 lb 3198 N v,RR f Front body (sprung) weight FBo,F 1951.4066 lbf 8680.2892 N Vehicle wheelbase l 103.3 in. 2623.8 mm Rear body (sprung) weight FBo,R 1444.2737 lbf 6424.4495 N Vehicle front track width bF 60.2 in. 1529.1 mm Total vehicle body (sprung) weight FBo 3395.6803 lbf 15104.7387 N Vehicle rear track width bR 60.6 in. 1539.2 mm Unloaded vehicle CG height hv,0 20.8440 in. 529.4376 mm Height of unloaded vehicle hul 57.9 in. 1470.7 mm Loaded vehicle CG height hv,t 21.2377 in. 539.4376 mm Loaded vehicle change in height Δhload 0.39 in. 10 mm Front dynamic tire radius rdyn,F 12.7008 in. 322.6000 mm Front tire wheel Diameter dF 18 in. 457.2 mm Rear dynamic tire radius rdyn,R 12.7008 in. 322.6000 mm Front tire width WF 9.65 in. 245 mm Loaded body CG height hBo 22.2064 in. 564.0429 mm Front tire aspect ratio (H/W)F 40 - - - Lateral position of vehicle CG bv -0.1981 in. -5.0322 mm Front tire deflection ΔrF 0.16 in. 4 mm Loction of the body CG from the front axle plane lBo,F 43.9363 in. 1115.9809 mm Rear tire wheel Diameter dR 18 in. 457.2 mm Location of the body CG from the rear axle plane lBo,R 59.3637 in. 1507.8391 mm Rear tire width WR 9.65 in. 245 mm Rear tire aspect ratio (H/W)R 40 - - - Laboratory Measurements: Rear tire deflection ΔrR 0.16 in. 4 mm

Front axle weight distribution iWD,F 0.57 - - - Test weight = 3451 lbf Rear axle weight distribution iWD,R 0.43 - - - FL Corner Weight = 952 lbf Front unsprung to sprung weight ratio i 0.12 - - - m,F FR Corner Weight = 1018 lbf Rear unsprung to sprung weight ratio im,R 0.14 - - - RL Corner Weight = 762 lbf Vehicle CG height coefficient iul 0.36 - - - RR Corner Weight = 719 lbf http://www.cars101.com/subaru/impreza/wrxsti2011.html 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

푣2 𝑔 = 0.0334 푥,퐵 푠

Front Tire Patch Normal Forces Front Tire Patch Ideal Braking Forces

1 ℎ 1 ℎ 퐹 = 퐹 𝑖 + 퐹 𝑔 푣,푡 퐹 = 퐹 𝑖 + 퐹 𝑔 푣,푡 𝑔 푧,푊,퐵,퐹 2 푣,푡 푊퐷,퐹 푣,푡 푥,퐵 푙 푥,푊,퐵,퐹 2 푣,푡 푊퐷,퐹 푣,푡 푥,퐵 푙 푥,퐵

Rear Tire Patch Normal Forces Rear Tire Patch Ideal Braking Forces

1 ℎ푣,푡 1 ℎ푣,푡 퐹 = 퐹 𝑖 − 퐹 𝑔 퐹푥,푊,퐵,푅 = 퐹푣,푡𝑖푊퐷,푅 − 퐹푣,푡𝑔푥,퐵 𝑔푥,퐵 푧,푊,퐵,푅 2 푣,푡 푊퐷,푅 푣,푡 푥,퐵 푙 2 푙 Acceleration Equations

Maximum Drive Off Coefficient of Static Friction

푣2 𝑔 = 0.0334 푥,퐴 푠

Front Tire Patch Normal Forces Front Tire Patch Drive Off Forces 1 ℎ푣,푡 1 ℎ 퐹 = 퐹 𝑖 − 퐹 𝑔 𝑔 퐹 = 퐹 𝑖 − 퐹 𝑔 푣,푡 푥,푊,퐴,퐹 2 푣,푡 푊퐷,퐹 푣,푡 푥,퐴 푙 푥,퐴 푧,푊,퐴,퐹 2 푣,푡 푊퐷,퐹 푣,푡 푥,퐴 푙

Rear Tire Patch Normal Forces Rear Tire Patch Drive Off Forces 1 ℎ푣,푡 1 ℎ 퐹푥,푊,퐴,푅 = 퐹푣,푡𝑖푊퐷,푅 + 퐹푣,푡𝑔푥,퐴 𝑔푥,퐴 퐹 = 퐹 𝑖 + 퐹 𝑔 푣,푡 2 푙 푧,푊,퐴,푅 2 푣,푡 푊퐷,푅 푣,푡 푥,퐴 푙 Cornering Equations

Front Outside Tire Patch Normal Force Rear Outside Tire Patch Normal Force

1 ℎ푣,퐹 1 ℎ푣,푅 퐹푧,푊,표,퐹 = 퐹푣,푡𝑖푊퐷,퐹 + 퐹푣,푡𝑖푊퐷,퐹𝑔푦,퐶 퐹푧,푊,표,푅 = 퐹푣,푡𝑖푊퐷,푅 + 퐹푣,푡𝑖푊퐷,푅𝑔푦,퐶 2 푏퐹 2 푏푅

Front Inside Tire Patch Normal Force Rear Inside Tire Patch Normal Force

1 ℎ푣,퐹 1 ℎ푣,푅 퐹푧,푊,𝑖,퐹 = 퐹푣,푡𝑖푊퐷,퐹 − 퐹푣,푡𝑖푊퐷,퐹𝑔푦,퐶 퐹푧,푊,𝑖,푅 = 퐹푣,푡𝑖푊퐷,푅 − 퐹푣,푡𝑖푊퐷,푅𝑔푦,퐶 2 푏퐹 2 푏푅

Front Outside Tire Patch Cornering Force Rear Outside Tire Patch Cornering Force

1 ℎ푣,퐹 1 ℎ푣,푅 퐹푦,푊,표,퐹 = 퐹푣,푡𝑖푊퐷,퐹 + 퐹푣,푡𝑖푊퐷,퐹𝑔푦,퐶 𝑔푦,퐶 퐹푦,푊,표,푅 = 퐹푣,푡𝑖푊퐷,푅 + 퐹푣,푡𝑖푊퐷,푅𝑔푦,퐶 𝑔푦,퐶 2 푏퐹 2 푏푅

Front Inside Tire Patch Cornering Force Rear Inside Tire Patch Cornering Force

1 ℎ푣,퐹 1 ℎ푣,푅 퐹푦,푊,𝑖,퐹 = 퐹푣,푡𝑖푊퐷,퐹 − 퐹푣,푡𝑖푊퐷,퐹𝑔푦,퐶 𝑔푦,퐶 퐹푦,푊,𝑖,푅 = 퐹푣,푡𝑖푊퐷,푅 − 퐹푣,푡𝑖푊퐷,푅𝑔푦,퐶 𝑔푦,퐶 2 푏퐹 2 푏푅 Vehicle Tire Patch Forces

Given Parameters: 2011 Subaru WRX STI Calculated Values Braking Parameter Definition Symbol Value Unit Value Unit Parameter Definition Symbol Value Unit Value Unit Vehicle speed v 60 mph 97 kph Braking acceleration gx,B 1.0104 g - - Vehicle stopping distance s 119 ft 36 m Front tire patch normal force Fz,B,F 2891.6602 lbf 12862.7455 N

Drive off acceleration gx,A 1.010 g - - Rear tire patch normal force Fz,B,R 822.3398 lbf 3657.9496 N Braking weight transfer ΔF 771.5269 lbf 3431.9227 N Longitudinal coefficient of static friction μx,W 1.010 - - - z,v Front tire patch normal force per wheel Fz,W,B,F 1445.8301 lbf 6431.3727 N Cornering acceleration gy,C 0.9 g - - 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 Road and Track Data: Rear tire patch ideal braking force per wheel Fx,W,B,R 415.4544 lbf 1848.0330 N Acceleration Stopping Speed= 60 mph Parameter Definition Symbol Value Unit Value Unit Front tire patch normal force F 1348.6064 lbf 5998.9000 N Stopping Distance= 952 lbf z,A,F Maximum Cornering Acceleration = 0.90 g 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 Cornering 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 http://paws.kettering.edu/~amazzei/DataPanel/CT_2008-Mitsubishi-Lancer-Evolution-GSR-vs-2008-Subaru-Impreza-WRX-STI_data.pdf Braking Tire Patch Forces

Vertical Force vs. Braking Acceleration . Weight transfer increases 2000 with acceleration

1500

1000

500 VerticalForce (lbf) 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Braking g's

FZ,W,B,F FZ,W,B,R

Longitudinal Force vs. Braking Acceleration 3000 2500 2000

Force(lbf) 1500 1000 500 0 Longitudinal 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Braking g's

FX,W,B,F FX,W,B,R Cornering Tire Patch Forces

Vertical Force vs. Cornering Acceleration . Lift off occurs when 2500 vertical forces equal 2000 zero, onset of rollover 1500 1000 500

VerticalForce (lbf) 0 -500 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Braking g's

FZ,W,O,F FZ,W,I,F FZ,W,O,R FZ,W,I,R

Lateral Force vs. Cornering Acceleration 4000

3000

2000

1000

LateralForce (lbf) 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 -1000 Braking g's

FY,W,O,F FY,W,I,F FY,W,O,R FY,W,I,R 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 *Note: A simplified independent RWD case is considered instead of considering the AWD power distribution of the 2011 Subaru WRX STI

100% Anti-Squat Condition, Rear

푒 − 푟푑푦푛 ℎ = 푣,푡 푑 푙

100% Anti-Pitch Condition, Rear

푒 − 푟푑푦푛 ℎ ℎ 퐾 = 푣,푡 + 푣,푡 퐹 푑 푙 푙 퐾푅

Pitch Rate, Rear

휃푝 1 퐹 1 ℎ 1 푒 − 푟 1 ℎ = 푣,푡 푣,푡 − + 푣,푡 푎푥 푙 𝑔 퐾푅 푙 퐾푅 푑 퐾푓 푙

Gillespie, Thomas: Fundamentals of Vehicle Dynamics Spring Rate Calculations

Table: Average front spring rate calculated from laboratory measurements

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

퐹 1334 푁 Sample Calculation: 퐾 = = = 93.3 푁/푚푚 퐹 ∆푥 14.3 푚푚

For the suspension calculations, the following spring rates provided by Eibach for the 2015 Subaru WRX STI will be used:

퐾퐹 = 58.0 푁/푚푚 and 퐾푅 = 60.0 푁/푚푚

http://eibach.com/m-america/en/eibach-news/subaru-wrx-sti-2015-plus-pro-kit Suspension Calculations

Given Parameters: 2011 Subaru WRX STI

Parameter Definition Symbol Value Unit Value Unit Suspension Type - Independent, RWD

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

Calculated Values

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

http://eibach.com/m-america/en/eibach-news/subaru-wrx-sti-2015-plus-pro-kit 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

푀푉 = − 퐹푧퐿 + 퐹푧푅 푟휎 sin λ sin 훿 + 퐹푧퐿 − 퐹푧푅 훿 sin ν cos 훿

Moment Due to Lateral Force

푀퐿 = − 퐹푦퐿 + 퐹푦푅 푟푑푦푛 tan ν

푟휎 = Scrub Radius λ = Lateral Inclination Angle 훿 = Steering Angle ν = Caster Angle Steering Calculation Parameters

Laboratory Measurements λ = 10° ν = 6°

Assumed Based on Common Values

푟휎 = 10 mm

Steering Angle Calculations will be performed for a steering angle of δ = 5°

Weight Distribution Values

Maximum Vertical Forces: 퐹푧퐿 + 퐹푧푅 = 퐹푧,퐵,퐹 = 2891.6602 푙푏푓 Maximum Lateral Forces: 퐹푦퐿 + 퐹푦푅 = 퐹푦,푣,퐹 = 1908.1200 푙푏푓 Steering Calculations

Given Parameters: 2011 Subaru WRX STI 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

Calculated Values

Steering 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 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

푎푦 푡 = + 휑 𝑔 2ℎ푣,푡

Quasi-Static Rollover Threshold: Suspended Vehicle

푎푦 푡 1 = 𝑔 2ℎ푣,푡 ℎ푟 1 + 푅Ф 1 − ℎ푣,푡

푡 = Track Width, 푏퐹 or 푏푅 휑 = Cross-Slope Angle 푅Ф = Roll Rate (rad/g) ℎ푟 = Roll center height at longitudinal CG location Rollover Calculation Parameters

Road Conditions 휑 = 0°

Assumed Based on Common Values

푅Ф = 3.0 deg/g ℎ푟 = 10.0 in.

Vehicle Specifications

푏퐹 = 60.2 in. 푏푅 = 60.6 in. Rollover Calculations

Given Parameters: 2011 Subaru WRX STI 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

Calculated Values

Rollover 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 - - 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 푦 푥 = 퐷 sin 퐶 tan−1 퐵푥 − 퐸 퐵푥 − tan−1 퐵푥

Cornering Stiffness Peak Factor Curvature Factor

−1 2 2 퐵퐶퐷 = 푎3 sin 푎4 tan 푎5퐹푧 퐷 = 푎1 퐹푧 + 푎2퐹푧 퐸 = 푎6 퐹푧 + 푎7퐹푧 + 푎8

Longitudinal Stiffness Stiffness Factor 푎 퐹 2 + 푎 퐹 퐵퐶퐷 퐵퐶퐷 = 3 푧 4 푧 퐵 = 푒푎5퐹푧 퐶퐷

Shape Factor 퐶 = 1.30 for cornering force – slip angle relationship 퐶 = 2.40 for self-aligning torque – slip angle relationship 퐶 = 1.65 for braking effort – skid relationship *Fz is in kN

Mechanics of Pneumatic Tires Tire Calculations

Given Parameters: 2011 Subaru WRX STI *Assumed -20.0% Skid Parameter Definition Symbol Value Unit Value Unit Maximum vertical tire patch force, braking Fz,W,B,F 1445.83 lbf 6431.37 N Friction forces are typically Maximum vertical tire patch force, cornering Fz,W,o,F 1733.22 lbf 7709.7651 N maximum at -15 to -20% skid Skid percentage - -20.00 %

Calculated Values Cornering Parameter Definition Symbol Value Unit Value Unit Cornering stiffness BCD 1038.0034 N/deg - - Maximum lateral force (peak factor) D 6480.9379 N - - Braking 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 - -

Table 1.7: Values of Coefficients a1 to a8 for a car tire (Fz in kN)

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

Mechanics of Pneumatic Tires Longitudinal Brake Force

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.

Longitudinal Brake Force vs. Skid 7000

6000

5000

4000

3000

2000 Negative Negative BrakeForce (N) 1000

0 0 10 20 30 40 50 60 70 80 90 100 Negative Skid (%)

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