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Rolling Resistance During Cornering - Impact of Lateral Forces for Heavy- Duty Vehicles

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Rolling Resistance During Cornering - Impact of Lateral Forces for Heavy- Duty Vehicles DEGREE PROJECT IN MASTER;S PROGRAMME, APPLIED AND COMPUTATIONAL MATHEMATICS 120 CREDITS, SECOND CYCLE STOCKHOLM, SWEDEN 2015 Rolling resistance during cornering - impact of lateral forces for heavy- duty vehicles HELENA OLOFSON KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF ENGINEERING SCIENCES Rolling resistance during cornering - impact of lateral forces for heavy-duty vehicles HELENA OLOFSON Master’s Thesis in Optimization and Systems Theory (30 ECTS credits) Master's Programme, Applied and Computational Mathematics (120 credits) Royal Institute of Technology year 2015 Supervisor at Scania AB: Anders Jensen Supervisor at KTH was Xiaoming Hu Examiner was Xiaoming Hu TRITA-MAT-E 2015:82 ISRN-KTH/MAT/E--15/82--SE Royal Institute of Technology SCI School of Engineering Sciences KTH SCI SE-100 44 Stockholm, Sweden URL: www.kth.se/sci iii Abstract We consider first the single-track bicycle model and state relations between the tires’ lateral forces and the turning radius. From the tire model, a relation between the lateral forces and slip angles is obtained. The extra rolling resis- tance forces from cornering are by linear approximation obtained as a function of the slip angles. The bicycle model is validated against the Magic-formula tire model from Adams. The bicycle model is then applied on an optimization problem, where the optimal velocity for a track for some given test cases is determined such that the energy loss is as small as possible. Results are presented for how much fuel it is possible to save by driving with optimal velocity compared to fixed average velocity. The optimization problem is applied to a specific laden truck. Keywords: Bicycle model, lateral forces, rolling resistance, velocity optimiza- tion, fuel consumption iv Rullmotstånd på kurvig väg - inverkan av laterala krafter för tunga fordon Sammanfattning Vi betraktar först den enspåriga cykelmodellen och ställer upp samband mellan däckens sidokrafter och kurvradien. Genom däcksmodellen fås ett samband för hur sidokrafterna beror av slipvinklarna. De extra rullmotståndskrafterna för kurvor fås via linjär approximation som funktion av slipvinklarna. Cykelmo- dellen valideras mot en däcksmodell från Adams. Cykelmodellen tillämpas sedan på ett optimeringsproblem där den optimala hastigheten längs en bana för några givna testfall bestäms så att energiförlus- ten blir så liten som möjligt. Resultat presenteras för hur mycket bränsle det är möjligt att spara genom att köra med optimal hastighet jämfört med fix medelhastighet. Optimeringsproblemet tillämpas på en specifik lastad lastbil. Nyckelord: Cykelmodell, sidokrafter, rullmotstånd, hastighetsoptimering, bräns- leförbrukning v Acknowledgements I would like to thank my supervisor Anders Jensen at Scania for the assistance and cooperation and Henrik Wentzel for his continual guidance and support. I would also like to thank Xiaoming Hu at the Division of Optimization and Systems Theory at KTH for the help and encouragement. I am also grateful for all the other people who have, directly or indirectly, supported me. Helena Olofson Stockholm, 2015 Contents Contents vi List of Figures viii 1 Introduction 1 1.1 Motivation . 1 1.2 Organization . 2 2 Background 3 2.1 Notation . 3 2.2 Vehicle coordinate system . 5 2.3 Forces . 5 2.3.1 Vertical force . 5 2.3.2 Longitudinal force . 5 2.3.3 Lateral force . 7 2.4 Bicycle model . 9 2.4.1 Low-speed cornering . 10 2.4.2 High-speed cornering . 10 3 Bicycle Model 13 3.1 Model . 13 3.1.1 Model with aerodynamic drag force . 13 3.1.2 Rolling resistance . 15 4 Validation of bicycle model 17 4.1 Purpose and method . 17 4.2 Truck specification . 17 4.3 Adams tire model simulation . 18 4.4 Comparison bicycle model to Adams tire simulation . 19 4.4.1 Lateral force . 20 4.4.2 Rolling resistance . 21 4.4.3 Weight shift . 24 4.5 Summary . 25 vi CONTENTS vii 5 Optimization problem 27 5.1 Problem formulation . 27 5.2 Method . 27 5.3 Solution and simulation . 30 5.3.1 Radius . 32 5.3.2 Turnlength . 34 5.3.3 Average velocity . 35 6 Conclusions and discussion 37 6.1 Future work . 38 Bibliography 39 A Graphs from the simulations 41 A.1 Validation of bicycle model using Adams tire simulation . 41 A.1.1 Slip angle . 42 A.1.2 Cornering stiffness analysis . 46 A.1.3 Longitudinal forces . 48 A.1.4 Lateral forces . 51 A.1.5 Weight shift . 52 List of Figures 2.1 Vehicle coordinate system DIN 70000/ISO 8855 . 5 2.2 Tire during forward motion . 6 2.3 Slip angle and lateral force of one tire . 8 2.4 Camber angle of one tire . 8 2.5 Transverse deformation of a tire . 9 2.6 Induced aligning moment, Mzi , on a tire from the lateral force, Fyi , acting a distance Pt from the contact point. 9 2.7 Kamm circle showing the traction limit for the tire forces Fxi and Fyi . 10 2.8 Bicycle model for low-speed cornering . 11 2.9 Bicycle model for high-speed cornering, without longitudinal force . 11 3.1 Bicycle model, with aerodynamic drag force, FA, and longitudinal force, Fxr. ...................................... 14 3.2 Bicycle model, with the rolling resistance forces . 15 4.1 Truck with maximum technical weight at front- and rear axle . 18 4.2 Total lateral force from all tires, Fytot , versus turning radius from bicycle model and Adams tire model simulation for v = 50 km/h, v = 70 km/h and v = 90 km/h using Cαf1 = 4360 N/deg and Cαr1 = 4940 N/deg (laden truck). 20 4.3 Total front lateral force, Fyf , and total rear lateral force, Fyr, versus turning radius from bicycle model and Adams tire model simulation for v = 70 km/h using Cαf1 = 4360 N/deg and Cαr1 = 4940 N/deg (laden truck). 21 4.4 Power loss due to rolling resistance from straight driving, Prr, versus turning radius from bicycle model and Adams tire model simulation for the laden truck. For small radii, the power loss Prr adds to the extra rolling resistance power loss from cornering, Prr,tu+. 22 4.5 Power loss due to rolling resistance from straight driving, Prr, versus turning radius from bicycle model and Adams tire model simulation for the unladen truck. For small radii, the power loss Prr adds to the extra rolling resistance power loss from cornering, Prr,tu+. 22 viii List of Figures ix 4.6 Extra rolling resistance power loss from cornering, Prr,tu+, versus turning radius from bicycle model and Adams tire model simulation using Cαf1 = 4360 N/deg and Cαr1 = 4940 N/deg (laden truck). 23 4.7 Extra rolling resistance power loss from cornering, Prr,tu+, versus turning radius from bicycle model and Adams tire model simulation using Cαf1 = 3320 N/deg and Cαr1 = 1290 N/deg (unladen truck). 23 4.8 Weight shift for each tire versus turning radius from Adams tire model simulation for v = 70 km/h (laden truck). 24 5.1 The optimization problem track . 27 5.2 Track consisting of one straight section and one turn with step length h. 28 5.3 Optimal velocity and fixed average velocity, vm, in each 5 m section along the track. The optimal velocity is lower than vm in the turns. However, at the straight sections, the optimal velocity is higher than vm in order to fulfill the time constraint. 31 5.4 Power loss in each 5 m section along the track. The power loss for the optimal velocity, is lower in the turns compared to the power loss for the fixed average velocity, vm. However, at the straight sections the power loss is higher for the optimal velocity. 32 5.5 Optimal velocity and fixed average velocity, vm, in each 5 m section along the track. 33 5.6 Power loss for the optimal velocity (blue line) and fixed average velocity vm (red line) in each 5 m section along the track. 33 5.7 Optimal velocity and fixed average velocity, vm, in each 5 m section along the track. 34 5.8 Power loss for the optimal velocity (blue line) and fixed average velocity vm (red line) in each 5 m section along the track. 35 5.9 Optimal velocity and fixed average velocity, vm = 80 km/h, in each 5 m section along the track. 36 5.10 Power loss along the track, for the optimal velocity and vm = 80 km/h. 36 A.1 Total front slip angle, αf,tot, and total rear slip angle, αr,tot, from bicycle model and Adams tire model simulation versus turning radius for v = 70 km/h using Cαf1 = 4710 N/deg and Cαr1 = 4940 N/deg (laden truck). 43 A.2 Total front slip angle, αf,tot, and total rear slip angle, αr,tot, from bicycle model and Adams tire model simulation versus turning radius for v = 50 km/h using Cαf1 = 4710 N/deg and Cαr1 = 4940 N/deg (laden truck). 43 A.3 Total front slip angle, αf,tot, and total rear slip angle, αr,tot, from bicycle model and Adams tire model simulation versus turning radius for v = 90 km/h using Cαf1 =4710 N/deg and Cαr1 = 4940 N/deg (laden truck). 44 A.4 Best correspondence between bicycle model and Adams tire model sim- ulation for v = 70 km/h using Cαf1 = 4360 N/deg and Cαr1 = 4940 N/deg (laden truck). 44 x List of Figures A.5 Slip angle versus turning radius for each tire from Adams tire model simulation for v = 70 km/h (laden truck). 45 A.6 Front- and rear slip angle (αf and αr) from bicycle model versus turning radius for v = 70 km/h using Cαf1 = 4360 N/deg and Cαr1 = 4940 N/deg (laden truck).
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