Design Analysis of a 2011 Subaru WRX STIpaws.kettering.edu/~amazzei/Final_Presentation_STI.pdf · Design Analysis of a 2011 Subaru WRX STI MECH 542 Final Project Kettering University

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

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

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


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


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

http://jp.autoblog.com/photos/2011-subaru-impreza-wrx-sti-first-drive-2/



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


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


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


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


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


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

http://paws.kettering.edu/~amazzei/DataPanel/CT_2008-Mitsubishi-Lancer-Evolution-GSR-vs-2008-Subaru-Impreza-WRX-STI_data.pdf

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


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



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


Independent, RWD


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


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



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


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


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


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



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


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


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


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 %



Cornering stiffness BCD 1038.0034 N/deg - -

Maximum lateral force (peak factor) D 6480.9379 N - -


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

Design Analysis of a 2011 Subaru WRX STIpaws.kettering.edu/~amazzei/Final_Presentation_STI.pdf · Design Analysis of a 2011 Subaru WRX STI MECH 542 Final Project Kettering University

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