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Vehicle Dynamic Systems Final Portfolio
1987 BMW 325is
Summer 2015
Professor Arnaldo Mazzei
Kevin Sallee
Cody Clarke
Jiaqi Huang
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
Vehicle History - The BMW 3 Series
• 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%
Vehicle History - The BMW 3 Series
• 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
1987 BMW 325is Acceleration Dynamics
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
1987 BMW 325is Acceleration Dynamics
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
1987 BMW 325is Acceleration Dynamics
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
1987 BMW 325is Brake System
• 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
1987 BMW 325is Brake System
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
1987 BMW 325is Braking Dynamics
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
1987 BMW 325is Braking Dynamics
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
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