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Vehicle Dynamic Systems Final Portfolio

1987 BMW 325is

Summer 2015

Professor Arnaldo Mazzei

Kevin Sallee

Cody Clarke

Jiaqi Huang

http://ppttemplate.net/?utm_source=ppt&utm_medium=logo&utm_term=ppt&utm_content=NNNN&utm_campaign=ppt

Contents

• Vehicle History

• Vehicle Information

• Competency 1: Weight Distribution & Tire Patch Forces

• Competency 2: Acceleration

• Competency 3: Braking

• Competency 4: Ride

• Competency 5: Low-Speed Steering

• Competency 6: High-Speed Steering (Handling)

*All Calculations done for Light Loading condition


Vehicle History - The BMW 3 Series

• BMW's sporty compact executive sedan



• In continuous production since 1975, it has seen six generations: E21, E30, E36, E46, E90, and F30

E21 ('75-'82) E30 ('83-'91) E36 ('92-'99) E46 ('99-'05) E90 ('06-'11)

F30 ('12-present)


Vehicle Comparison1987 BMW 325is• Base Price

– $27,300

• 2015 Price

– $58,012

• Engine/Power

– 2.5 I6 / 168 HP

• Weight/Distribution

– 53% / 47%

1987 Volvo 760 Turbo• Base Price

– $28,340

• 2015 Price

– $60,222

• Engine/Power

– 2.3 I4 / 160 HP

• Weight/Distribution

– 56% / 44%



• It has always used a semi-trailing arm rear suspension setup, which aids in the car's highly acclaimed handling characteristics


Other Vehicle Information

Garage Measurements

• Weight: 2825 lbs

• Distribution: 53.5%/46.5%

• Rear Deflection/Weight:

– 1.0 Inch / 200 lbs

• Front Deflection/Weight:

– 0.75 Inch / 200 lbs


Weight Distribution & Tire Patch Forces

OBJECTIVES: 1. Define vehicle coordinate system

2. Define weight distribution

3. Describe weight center location

4. Define the vehicle system tire patch forces

5. Calculate tire patch forces during acceleration

Vehicle iso coordinate system

Calculations base on isocoordinate system

• X-axis-linear velocity

• Y-axis-lateral velocity

• Z-axis-yaw velocity

Weight distribution & Weight center location

• In a vehicle which relies on gravity in some way, weight distribution directly affects a variety of vehicle characteristics, including handling, acceleration, traction, and component life. For this reason weight distribution varies with the vehicle's intended usage.

• The height of the vehicle’s center of gravity affects the vehicle’s acceleration and braking based on the amount of weight transfer.

• Vehicles with lower centers of gravity are able to create larger cornering forces, since they hold lower rolling moments on the X-Axis.

Vehicle system tire patch forces

• Tire patch forces are the amount of force transmitted from the body to the tires, which ultimately is passed on to the road.

• The tire patch forces are a function of the weight distribution and the vehicle’s acceleration in the X and Y Axis.

• Longitudinal tire patch forces provide acceleration and braking, while lateral tire patch forces are responsible for cornering.

Weight Distribution & Tire Patch Forces

Weights• Total Weight: 2825 lbs• Curb Weight: 2823 lbs• Corner Weights:

738 - 1510 - 773LF - Front - RF

Front / Rear = 53% / 47%•

LR - Rear - RR657 - 1314 - 657

Vehicle SpecificationsWheelbase (l) Wheel Diameter, front

101.2 inches 14inches

2570.48mm Tire Width, frontTrack, front 195mm

55.4 inches Tire Sidewall Ratio, front

1407.16mm 65

Track, rear Δr, rear

55.7 inches 4mm

1414.78mm Dynamic Radius, front (rdyn,f)

Height (Hul) 300.55mm54.3 inches Wheel Diameter, rear

1379.22mm 14inches

Height (HV,0) Tire Width, rear20.634 inches 195mm

524.1036mm Tire Sidewall Ratio, rear

Curb Weight (Fv,t) 652823 lbs Δr, rear

12557.32962N 4mm

Dynamic Radius, rear (rdyn,r)300.55mm

Weight Distribution

Front Rear

53% 47%Individual Wheel Weights (lbs)

FL FR

738 773

RL RR657 657

Δhload

10mm0.393700787 inches

Weight Distribution CalculationsLight Load (Fv,t,2p)

3153 lbs

14025.24 N

Vehicle Center of Gravity - Curb Vehicle Center of Gravity - Light LoadFv,f 1496.19 lb Fv,r 1326.81 lb Fv,f,2p 1671.09 lb Fv,r,2p 1481.91 lb

6655.385 N 5901.945 N 7433.379 N 6591.864 N

lv,f 47.564 inches lv,r 53.636 inches lv,f,2p 47.564 inches lv,r,2p 53.636 inches

1208.126 mm 1362.354 mm 1208.126 mm 1362.354 mm

hv,t 21.0277 inches bv 0 inches hv,t, 2p 21.0277 inches

534.1036 mm 0 mm 534.1036 mm

Body (Sprung) & Axle (Unsprung) Weight Body (Sprung) & Axle (Unsprung) WeightFU,f 160.3061 lb FU,r 162.9416 lb FU,f 160.3061 lb FU,r 162.9416 lb

713.0769 N 724.8003 N 713.0769 N 724.800 N

FBo,f 1335.884 lb FBo,r 1163.868 lb FBo,f 1510.784 lb FBo,r 1318.968 lb

5942.308 N 5177.145 N 6720.302 N 5867.064 N

lBo,f 47.11806 inches lBo,r 54.08194 inches lBo,f 47.17007 inches lBo,r 54.02993 inches

1196.799 mm 1373.681 mm 1198.12 mm 1372.36 mm

hBo 564.3049 mm hBo, r 566.8011 mm hBo, 2p 560.7828 mm hBo, r 562.9561 mm

22.21673 inches hBo, f 562.13 mm 22.07806 inches hBo, f 558.8855 mm

Tire Patch Braking Normal Forces at 60 mph

0

500

1000

1500

2000

2500

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

No

rmal

Fo

rce(

N)

Braking g's

FRONT

REAR

Ideal Tire Patch Forces (X-Axis) at 60 MPH

0.00

500.00

1000.00

1500.00

2000.00

2500.00

3000.00

3500.00

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

No

rmal

Fo

rce

(N)

Brakeing g's

Front

Rear

Vehicle

Acceleration Dynamics

Customer Focus• Quickly increase speed

from stop

• Easily accelerate in traffic

• Easily move from stationary on a hill

• Accelerates Smoothly

• Easily accelerate when loaded

Engineering Focus• Time-to-Speed

• Time-to-Distance

• Passing distance

• Acceleration capacity

• Grade ability

• Drag coefficient

• Projected frontal area

• Power-to-Weight ratio

1987 BMW 325is Acceleration Dynamics

• Acceleration Limit, 𝑔𝑥,𝐴 =𝜇𝑥,𝑤 𝑖𝑤𝑑,𝑟 −𝑓𝑅𝑜

𝐾𝑚−𝑢𝑥,𝑤(ℎ𝑣,𝑡𝑙)= 0.45𝑔

Drag Coefficient 0.39

Frontal Area (m) 1.751434239

Acceleration Limit

0.451829867g

Rotational Inertia Coefficient, Km 1.09Incline, α 0degrees

Rolling Resistance, coefficient 0.015210.3786413N

Gravitational Resistance0N


F_max V_max@ Max Power Aero DragAcceleration

CapacityN kph m/s N g's

1st 9512.14825 46.00110233 12.77808 67.02925473 0.604070746

2nd 5463.897167 80.08373723 22.24548 203.1498832 0.330359043rd 3477.02547 125.8458728 34.95719 501.6558341 0.180865959

4th 2483.589621 176.1842219 48.94006 983.2454348 0.0843803315th 2011.707593 217.5113851 60.41983 1498.6213 0.019800972

V_max @ Max Torque Aero DragAcceleration

Capacitykph m/s N g's

34.104266 9.473407088 36.84217955 0.60604536959.372426 16.49234052 111.6599685 0.336343656

93.299526 25.91653511 275.731759 0.195644298

130.61934 36.28314915 540.4342476 0.113345876161.25844 44.79401129 823.7071294 0.063949036


0

20

40

60

80

100

120

140

160

180

200

220

240

0 10 20 30 40 50 60 70 80 90 100

Spee

d (

kph

)

Time (sec)

Time-to-Speed

Max Power

Max Torque


0200400600800

1000120014001600180020002200240026002800300032003400360038004000420044004600

0 10 20 30 40 50 60 70 80 90 100

Dis

tan

ce (

m)

Time (sec)

Time-to-Distance

Max Power

Max Torque

1987 BMW 325is Acceleration - CarSim

Full Throttle Acceleration

1987 BMW 325is Acceleration - CarSimFull Throttle Acceleration

Braking Dynamics

• Brake system reduces vehicle speed by converting kinetic energy (vehicle motion) to thermal energy that can be dissipated to the atmosphere

• Sub-systems:

– Brake transmission fluid

– Actuation

– Foundation brakes

– Electronic braking assists

– Parking brake

Braking Dynamics

Customer Focus• Confident/Comfortable Braking

– Short stopping distance

– Pedal feel

– Consistency

– Fast response

– No noise or vibrations

• Controlled Braking– Ability to steer and decelerate in all

conditions

• Low maintenance– Long rotor/lining life

Engineering Requirements• Government Regulations

• Stopping distance

• Pedal feel

• Brake system response

• Brake balance & bias

• Thermal management

• Lift & dive performance

• Combined cornering & braking performance

1987 BMW 325is Brake System

• Vacuum assisted with anti-lock brake control

• Front: 10.2 x 0.9 inch vented discs

• Rear: 10.2 x 0.4 inch vented discs

Front brake rotor and caliper


• System Braking ratio assumed to be 75/25 (front/rear) = 3

• Ideal Braking Ratio

– Coefficient of friction, assumed for dry pavement, 𝜇 = 0.9

• 𝑖𝐼𝐵𝑅 =𝑖𝑊𝐷,𝑓+𝜇 (

ℎ𝑉,𝑡𝑙)

𝑖𝑊𝐷,𝑟−𝜇(ℎ𝑉,𝑡𝑙)= 2.417

• 𝑖𝐼𝐵𝑅 < 𝑖𝑆𝐵𝑅 ∴ 𝐹𝑟𝑜𝑛𝑡 𝑆𝑘𝑖𝑑 𝐿𝑖𝑚𝑖𝑡𝑒𝑑


0

0.5

1

1.5

2

2.5

3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

IDEA

L B

RA

KIN

G R

ATI

O

COEFFICIENT OF FRICTION, Μ

IDEAL BRAKING RATIO FOR VARYING COEFFICIENTS OF FRICTION

1987 BMW 325is Braking Dynamics

0

2000

4000

6000

8000

10000

12000

14000

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

AX

LE V

ERTI

CA

L FO

RC

ES (

N)

BRAKING G'S

AXLE VERTICAL FORCES DURING STEADY BRAKING

FZ,B,f FZ,B,r


0

5000

10000

15000

20000

25000

30000

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

LON

GIT

UD

INA

L B

RA

KIN

G F

OR

CE

(N)

BRAKING G'S

LONGITUDINAL AXLE FORCES DURING STEADY BRAKING

FX,B,f FX,B,r FX,B,V


0 20 40 60 80 100 120 140 160 180 200

0

1

2

3

4

5

6

7

8

9

0

50

100

150

200

250

0 20 40 60 80 100 120 140 160 180 200

Tim

e (

sec)

Dis

tan

ce (

m)

Speed (kph)

STOPPING TIME & STOPPING DISTANCE AT VARIOUS SPEEDS

Distance (Max) Distance (Min) Time (Max) Time (Min)

1987 BMW 325is Braking- CarSim

100 kph - 0

1987 BMW 325is Braking- CarSim

Ride Dynamics

• Customer Focus:– Smooth ride on bumpy roads

– No shake or vibration over bumpy roads

– Absence of road & wind noise

– Absence of vehicle noise (squeaks & rattles)

• Engineering Ride Metrics– Front ride frequency

– Ride frequency ratio

– Vertical Damping

– Impact isolation

– Single bump disturbances

– Pitch (braking & drive-off)

– Pitch damping

1987 BMW 325is Suspension

Front Suspension

• Independent, lower control arm with strut and anti-roll bar

Rear Suspension

• Independent, semi-trailing arm with coil springs and anti-roll bar

1987 BMW 325is Suspension

• Ride Rates, as measured in the Garage (𝐾′𝑅/𝑓 =2 ∗ 𝐾𝑅/𝑓):– Front: 200 lbs to compress

0.75” = 266.67 lb/in = 46.7 N/mm

• 𝐾𝑅,𝑓 = 23.35 N/mm

– Rear: 200 lbs to compress 1.00” = 200 lb/in = 35.0 N/mm

• 𝐾𝑅,𝑟 = 17.5 N/mm

1987 BMW 325is Ride Metrics• Sprung Mass Frequencies: 𝒇𝑩𝒐,𝒇/𝒓 =

𝟓𝟎𝟎𝑲𝑹,𝒇/𝒓𝒈

𝝅𝟐𝑭𝑩𝒐,𝒇/𝒓

– 𝑓𝐵𝑜,𝑓 =500∗23.35

𝑁

𝑚𝑚∗9.81 𝑚/𝑠2

𝜋2∗6720.3 𝑁= 1.314 𝐻𝑧

– 𝑓𝐵𝑜,𝑟 =500∗17.50

𝑁

𝑚𝑚∗9.81 𝑚/𝑠2

𝜋2∗5867.06 𝑁= 1.217 𝐻𝑧

–𝑓𝐵𝑜,𝑟

𝑓𝐵𝑜,𝑓=

1.217 𝐻𝑧

1.314 𝐻𝑧= .926

– This ratio should be approximately 1.10. The error is due to poor Ride Rate measurement

technique in the garage. Further Calculations estimate 𝑓𝐵𝑜,𝑟 based on this ratio

• 𝑓′𝐵𝑜.𝑟 = 1.445 Hz

1987 BMW 325is Ride Metrics

• Suspension Rates: 𝑲𝒇/𝒓 =𝑲𝑻,𝒇/𝒓𝑲𝑹,𝒇/𝒓

𝑲𝑻,𝒇/𝒓−𝑲𝑹,𝒇/𝒓

• Unsprung Bounce Frequencies: 𝒇𝑼,𝒇/𝒓 =𝟏

𝟐𝝅

𝟐𝟎𝟎𝟎 𝑲𝑻,𝒇 𝒐𝒓 𝒓 +𝑲𝒇 𝒐𝒓 𝒓 𝒈

𝑭𝑼,𝒇𝒐𝒓 𝒓

K_R,f 23.3475N/mm K_R,r 24.6637N/mmK_f 26.4332N/mm K_r 28.1330N/mmK_Sp,f 27.2508N/mm K_Sp,r 29.0031N/mmf_U,f 12.5624Hz f_U,r 12.5070Hz

1987 BMW 325is Ride Metrics• Bounce & Pitch frequencies

– 𝜔𝑛2 =

𝛼+𝛾

2±

𝛼−𝛾

2

2+

𝛽

𝑟𝐽

2

Bounce & Pitch K'_f 46694.9841N/mm K'_r 49327.3340N/mm

a 1.1968m b 1.3737mα 74.8353rad/sec2 ω_n,1 8.3083rad/sec 1.3223Hzγ 78.2896rad/sec2 ω_n,2 9.1705rad/sec 1.4595Hzβ 9.2552rad/sec2 Z/ϴ_1 -1.5936m Bounce Center nat freq 1 = Bounce Frequency

Z/ϴ_2 0.9993m Pitch Center nat freq 2 = Pitch Frequency

Low Speed Steering

• “low” speed steering is concerned with parking maneuvers and fundamental steering axis geometry.

• Vehicle system turning at low speeds

• Lock to lock turns of steering wheel

• Ability to return to vehicle system path

Definition

• Define Ackermann steering geometry

• Calculate the steering deviation

• Calculate the percent Ackermann

• Calculate the curb-to-curb turning circle

• Compose an overview of “low” speed steering

Objective

Ackermann Steering Geometry

• Ackermann steering geometry is a geometric arrangement of linkages in the steering of a car or other vehicle designed to solve the problem of wheels on the inside and outside of a turn needing to trace out circles of different radius.

Calculations i=45° bf=1407.16mm rr=10mm l=2570.48mm is=15.9F=o-A,O

Where

cotan( A,O) = cotan(i+𝒃𝒇−𝟐𝒓𝒓

𝒍),

A,O=27.56°

is=𝑯𝒎

=15.9

whereH=180*DNltl=702°

Hence,m=44.15°

m=𝑖+𝑜2

, o=43.3°

Steering DeviationF=o-A,O=43.3°-27.56°=15.74°

Percent Ackermann

PA=100(𝒊+𝒐𝒊−𝑨,𝑶

)=9.74%

Curb-to-curb turning circle

DTC,CB=1

500

𝑙

𝑠𝑖𝑛𝐴,𝑂+ 𝑟𝑟 +

(−0.1𝐹)+(10−3𝐵𝑓𝑀𝐴𝑋)=9.76 m

Low Speed Steering PerformanceSteering Deviation ,F 15.74 Degree

Percent Ackermann ,PA 9.74 %

Curb-to-curb turning circle 9.76 m

Inside angle, i 45.00 DegreeOutside front road wheel steering angle,

A,O 27.56 Degree

steering ratio , is 15.90

Mean road wheel steer angle𝒎 44.15 Degree

outside road angleo 43.30 DegreeTurns, lock to lock 3.90

Steering wheel displacement H 702.00 Degree

wheel base 2570.48 mm

rr 10 mm

Tire width 195 mm

Δr 4 mm

High Speed Steering (Handling)

Customer Focus• Handling characteristics

allow for “fast” cornering• Good handling and stability

at highway speeds• Good handling in all

weather conditions• Maneuvers in and out of

traffic with ease• Responsive to steering

wheel inputs• Handling provides a good

feel of the road

Engineering Focus• Maximum lateral acceleration

• Lateral acceleration response time

• Yaw velocity damping

• Understeer gradient

• Roll gradient

• Steering sensitivity

• Steering sensitivity ratio

• Roll damping

• On-center steering performance

High Speed Steering (Handling)

• Lateral acceleration, 𝐹𝑌,𝑉 = 𝐹𝑉,𝑡𝑔𝑦

• Understeer Gradient, 𝑈𝑆𝐺 =𝑖𝑊𝐷,𝑓𝐹𝑉,𝑡

2𝐶𝛼,𝑓−

𝑖𝑊𝐷,𝑟𝐹𝑉,𝑡

2𝐶𝛼,𝑟=

𝛿𝑚−180

𝜋(𝑙

𝑅𝑣)

𝑔𝑦

– Where 𝐶𝛼 = 𝑡𝑖𝑟𝑒 𝑐𝑜𝑟𝑛𝑒𝑟𝑖𝑛𝑔 𝑠𝑡𝑖𝑓𝑓𝑛𝑒𝑠𝑠 𝑖𝑛 𝑁/𝑑𝑒𝑔

• Vehicle Roll Gradient, 𝑅𝐺𝑉 =𝐹𝐵𝑜(ℎ𝐵𝑜−ℎ𝑅𝑜,𝐵)

𝐾𝜑,𝑉

– Where 𝐾𝜑,𝑉 = 𝐶ℎ𝑎𝑠𝑠𝑖𝑠 𝑆𝑦𝑠𝑡𝑒𝑚 𝑅𝑜𝑙𝑙 𝑆𝑡𝑖𝑓𝑓𝑛𝑒𝑠𝑠

1987 BMW 325is Handling

g_c (g_y) 0.81g RG_V,r 3.1deg/g h_Ro,r 0.11m h_Bo,r 0.5630m

R_v 1800inches RG_V,f 3.1deg/g h_Ro,f 0.08m h_Bo,f 0.5589m

F_Bo,r 5867.0638N b_r 1.4148m δ_H 106.5degrees

F_Bo,f 6720.3017N b_f 1.4072m δ_M 5.1951degrees

φ_Bo,r 2.5110degrees φ_Bo,f 2.5110degreesM_roll,r 2152.5933Nm M_roll,f 2606.7865NmK_φ,r 491.4065Nm/deg K_φ,f 456.7565Nm/degK_φ,r,t 857.2654Nm/deg K_φ,f,t 1038.1467Nm/degK_s,φ,r 365.8588Nm/deg K_s,φ,f 581.3902Nm/degUndersteer Gradient (USG) 2.4368 deg/g_y

1987 BMW 325is Handling - CarSim

Double-Lane-Change Maneuver

1987 BMW 325is Handling - CarSimDouble-Lane-Change Maneuver

vehicle dynamic systems final portfolio - kettering universitypaws.kettering.edu › ~amazzei ›...

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