Vehicle Dynamics Steve Muench
Outline 1. Resistance a. Aerodynamic b. Rolling c. Grade. Tractive Effort 3. Acceleration 4. Braking Force 5. Stopping Sight Distance (SSD)
Main Concepts Resistance Tractive effort Vehicle acceleration Braking Stopping distance F ma + R + R + a rl R g
Resistance Resistance is defined as the force impeding vehicle motion 1. What is this force?. Aerodynamic resistance 3. Rolling resistance 4. Grade resistance F ma + R + R + a rl R g
Aerodynamic Resistance R a Composed of: 1. Turbulent air flow around vehicle body (85%). Friction of air over vehicle body (1%) 3. Vehicle component resistance, from radiators and air vents (3%) from National Research Council Canada R P a R a ρ ρ C C D D 1 hp 550 A V f A V f ft lb sec 3
Rolling Resistance R rl Composed primarily of 1. Resistance from tire deformation ( 90%). Tire penetration and surface compression ( 4%) 3. Tire slippage and air circulation around wheel ( 6%) 4. Wide range of factors affect total rolling resistance 5. Simplifying approximation: R rl f rl W P R f rl rl ft lb 1 hp 550 sec WV f rl V 0.01 1 + 147
Grade Resistance R g R g W sinθ For small angles, R g W tanθ g sinθ g tanθ g G R g WG tan g θ g R g θ g W θ g
Available Tractive Effort The minimum of: 1. Force generated by the engine, F e. Maximum value that is a function of the vehicle s weight distribution and road-tire interaction, F max Available tractive effort min (, F ) F e max
Tractive Effort Relationships
Engine-Generated Tractive Effort Force F e M e ε 0 η r d F e Engine generated tractive effort reaching wheels (lb) M e Engine torque (ft-lb) ε 0 Gear reduction ratio η d Driveline efficiency r Wheel radius (ft) Power ( ft lb) ft lb torque engine rpm hp 550 π sec 550 sec 60 min
Vehicle Speed vs. Engine Speed ( 1 i) πrne V ε 0 V velocity (ft/s) r wheel radius (ft) n e crankshaft rps i driveline slippage ε 0 gear reduction ratio
Typical Torque-Power Curves
Maximum Tractive Effort Front Wheel Drive Vehicle F max µ W ( l + f h) r 1+ L µ h L rl Rear Wheel Drive Vehicle What about 4WD? F max ( l f h) f µ W L µ h 1 L rl
Diagram R a ma h h θg R rlf F bf W f l f W R rlr l r W r F br L θ g
Vehicle Acceleration Governing Equation F R γ m ma Mass Factor (accounts for inertia of vehicle s rotating parts) γ 1.04 + 0.005ε m 0
Example A 1989 Ford 5.0L Mustang Convertible starts on a flat grade from a dead stop as fast as possible. What s the maximum acceleration it can achieve before spinning its wheels? µ 0.40 (wet, bad pavement) 1989 Ford 5.0L Mustang Convertible Torque 300 @ 300 rpm Curb Weight 3640 Weight Distribution Front 57% Rear 43% Wheelbase 100.5 in Tire Size P5/60R15 Gear Reduction Ratio 3.8 Driveline efficiency 90% Center of Gravity 0 inches high
Braking Force Front axle F bf max µ W l [ + h( µ + f )] r L rl Rear axle F br max µ W l [ h( µ + f )] f L rl
Braking Force Ratio BFR l l r f + h h ( µ + f ) rl ( µ + f ) rear rl front Efficiency η b gmax µ
Braking Distance Theoretical ignoring air resistance S g ( γ ) b V1 V ( η µ + f ± sinθ ) b rl g Practical Perception d V 1 g V a g ± G For grade 0 d V V a 1 Total d p V1t p d d + s d p
Stopping Sight Distance (SSD) Worst-case conditions Poor driver skills Low braking efficiency Wet pavement Perception-reaction time.5 seconds Equation V1 SSD + V1t r a g ± G g
Stopping Sight Distance (SSD) from ASSHTO A Policy on Geometric Design of Highways and Streets, 001 Note: this table assumes level grade (G 0)
SSD Quick and Dirty 1. Acceleration due to gravity, g 3. ft/sec. There are 1.47 ft/sec per mph 3. Assume G 0 (flat grade) d ( a g ± G) ( 1.47 V ) ( 0) V1 V 1 1.47 1 V V V 1.075 1. 075 g 3. 11. d 1.47 V1 t 1. 47Vt p p ( 3. + 0) 11. 11. a p V d 1.075 + 1. 47 a s Vt p V V 1 in mph a deceleration, 11. ft/s in US customary units t p Conservative perception / reaction time.5 seconds
Primary References Mannering, F.L.; Kilareski, W.P. and Washburn, S.S. (005). Principles of Highway Engineering and Traffic Analysis, Third Edition). Chapter American Association of State Highway and Transportation Officals (AASHTO). (001). A Policy on Geometric Design of Highways and Streets, Fourth Edition. Washington, D.C.