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i DATA ALGORITHM FOR VEHICLE DYNAMICS AND TIRE MODELING by Imevbore A. Ojebuoboh B.S., University of Colorado, 2010 A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Master of Science Mechanical Engineering Fall 2012
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ii This thesis for the Master of Science degree by Imevbore A. Ojebuoboh has been approved for the Mechanical Engineering Program by John A. Trapp, Chair Samuel W.J. Welch, Advisor Peter E. Jenkins Date: 11/1 6/2012
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iii Ojebuoboh, Imevbore A. (M.S., Mechanical Engineering) Data Algorithm for Vehicle Dynamics and Tire Modeling Thesis directed by Professor John A. Trapp ABSTRACT Without tires a car would be incomplete, tires are the main support of any car. This thesis is designed to portray the dynamic analysis of a race car and the behavior of its tire via tire modeling and data produced by Optimum G Results relating the vehicle to its driver and the road a re determined a ccurately by using Hans Pacejka magic formula to determine the longitudinal and lateral forces produced by the tire, its yaw, roll and pitch moments acting on the each wheel. Matlab is the primary software used for all calculations in this thesis work; a 3 D dynamic simulation is conducted implementing Newt equations of motions are very useful in understanding the vehicles motion. Analyses are performed to show the behavior and charact eristics of the tire and essentially the vehicle using different cases to determine such behavior such as; effect of different types of camber angle applied to its tires, effect of increase and decrease in steer i ng ratio to the vehicles handling and a skid p ad test in the form of with constant angular velocity on both rear wheels T o develop understand ing on how tire loads characterize vehicle handling is the main purpose of this thesis. The form and content of this abstract are approved. I recommend its publication. Approved: Professor John A. Trapp
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iv DEDICATION I dedicate this thesis to God almighty for the strength and will to complete this thesis work and conclude my graduate studies at the great University of Colorado Denver. I also dedicate this thesis to my fam ily; Dad, Mom, Aira and Ite, my extended family Sam, Tobechi, and my unborn nephew, and lastly my Uncle Funsho for their constant push to never give up, love, support, beliefs and prayers. A special dedication to the memory of my older brother Imonitie O jebuoboh, who I miss everyday, whom without his guidance and advises growing up it would have been more difficult to grow as a student and an individual. He taught me to never give up, and I use that advice to keep me moving and to reach and complete my go als
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v ACKNOWLEDGEMENT This thesis would have never existed without two very important parties; Professor John A. Trapp who I would like to thank specially for giving me his undivided attention and some advice on how to go about in completing t his thesis I would like to also thank him for helping and guiding me on this thesis work and ensuring I understood the materal correctly and also making sure I did the correct and accurate calculation The second party I would like to thank specially is a race car consulting company named Optimum G who without data and expertise provided by its very smart members and owner Claude Rouelle, the thesis would have never existed to begin with, let alone be completed. I would also like to thank the members of this thesis committee; Dr. Samuel W.J. Welch and Dr. Peter E. Jenkins who agreed to be on my thesis committee because they believed I could and would complete such a huge task. I would like to t hank you to Joe Cullen for granting me the permission to audit his vehicle dynamics course, so I could learn more about vehicle dynamics. Last but not least, I would like to thank the department of Mechanical Engineering at the University of Colorado Denver, College of Engineering and applied Sciences and its Profess ors and staff for guiding me through this incredible and unbelievable journey.
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vi TABLE OF CONTENTS Chapter 1. Introduction ................................ ................................ ................................ ................................ .......................... 1 1.1 Background ................................ ................................ ................................ ................................ ................. 1 1.2 Research Approach ................................ ................................ ................................ ................................ ..... 2 1.3 Research Objective ................................ ................................ ................................ ................................ ..... 2 2. Li terature Review ................................ ................................ ................................ ................................ ............... 4 2.1 Vehicle Dynamics ................................ ................................ ................................ ................................ ....... 4 2.1.1 Cornering ................................ ................................ ................................ ................................ ................... 4 2.1.2 Ae rodynamic Effect ................................ ................................ ................................ ................................ 9 2.1.3 Wheelbase and Track ................................ ................................ ................................ .......................... 11 2.1.4 Handling ................................ ................................ ................................ ................................ .................. 13 2 .1.4.1 Neutral Steer ................................ ................................ ................................ ................................ ....... 16 2.1.4.2 Oversteer ................................ ................................ ................................ ................................ ............. 16 2.1.4.3 Understeer ................................ ................................ ................................ ................................ ........... 17 2.2 Pneumatic Tire ................................ ................................ ................................ ................................ .......... 17 3. Newton Euler ................................ ................................ ................................ ................................ ................ 17 3.1 Equations of Motion ................................ ................................ ................................ ................................ 17 3.1.1 Light Link ................................ ................................ ................................ ................................ ............... 20 3.1.2 Spring and Damper ................................ ................................ ................................ .............................. 21 3.1.3 Ball and Socket ................................ ................................ ................................ ................................ ..... 25 3.1.4 Wheel on ground ................................ ................................ ................................ ................................ .. 26 4. Tire Modeling ................................ ................................ ................................ ................................ ................... 28 4.1 Forces ................................ ................................ ................................ ................................ .......................... 31 4.1.1 Longitudinal Forces ................................ ................................ ................................ ............................. 31 4.1.2 Lateral Forces ................................ ................................ ................................ ................................ ........ 35 4.2 Moments ................................ ................................ ................................ ................................ ..................... 39 4.2.1 Aligning Forces ................................ ................................ ................................ ................................ ..... 40 4.3 Friction Circle ................................ ................................ ................................ ................................ ........... 43 4.4 Cases ................................ ................................ ................................ ................................ ............................ 45 4.4.1 Effect of positive, negative and zero camber angle ................................ ................................ .... 45 4.4.1.1 Zero camber angle ................................ ................................ ................................ ............................ 47 4.4.1.2 Positive camber angle ................................ ................................ ................................ ...................... 48
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vii 4.4.1.3 Negative camber angle ................................ ................................ ................................ .................... 49 4.4.2 Skid Pad Test ................................ ................................ ................................ ................................ ......... 50 4.4.2.2 Constant Velocit y ................................ ................................ ................................ ............................. 54 5. Conclusion and Recommendation ................................ ................................ ................................ .......... 59 Appendix ................................ ................................ ................................ ................................ ................................ 61 A. Nomenclatur e ................................ ................................ ................................ ................................ .............. 61 B. MatLab Code ................................ ................................ ................................ ................................ ............... 64 REFERENCES ................................ ................................ ................................ ................................ ..................... 68
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vii i LIST OF FIGURES Figure 2.1: A vehicle with lateral forces applied to its wheels moving on a track ................................ .... 5 2.2: A tire showing slip angle ................................ ................................ ................................ ......... 5 2.3: Lateral force versus slip angle plot ................................ ................................ .......................... 6 2.4: Cornering stiffness versus normal ................................ ................................ ............. 8 2.5: ................................ ................................ 11 2.6: ................................ ................................ ................. 11 2.7: Full Trac ................................ ................................ ................................ .......................... 12 2.8: Wheelbas ................................ ................................ ................................ ......................... 13 2.9: Curvature Response of different values of undeersteer coefficient ................................ 16 3 .1 : Light link between two bodies ................................ ................................ ............................... 20 3.2: Spring and Dashpot forces with time, cornering exist ................................ ........................... 24 3.3: Spring forces with velocity ................................ ................................ ................................ .... 25 3.4: Ball and Socket joint ................................ ................................ ................................ .............. 26 3.5: Wheel on a flat surface ................................ ................................ ................................ .......... 27 4 .1: SAE tire axis ................................ ................................ ................................ ..... 29 4.2: Magic Formula graphical representation ................................ ................................ ............... 30 4.3: Rolling and Slipping of a ................................ ................................ ......................... 32 4.4: plot of Longitudinal force against Slip ratio ................................ ................................ .......... 34 4.5: plot of Longitudinal force against time ................................ ................................ .................. 35 4.6: plot of lateral force against slip angle ................................ ................................ .................... 37 4.7: Plot of normal force against time ................................ ................................ ........................... 38
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ix 4.8: Plot o f lateral force against time ................................ ................................ ............................ 38 4.9: Plot of tire angles and velocity against time ................................ ................................ .......... 39 4.10: Pneumatic ................................ ................................ ................................ ............ 40 4.11: Plot of aligning torque against slip angle ................................ ................................ ............. 42 4.12: Plot of aligning torque and pneumatic trail against time ................................ ..................... 43 4.13: Friction circle at initial run ................................ ................................ ................................ .. 44 4.14: Friction circle at final run ................................ ................................ ................................ .... 44 4.15: Race Car a t initial run ................................ ................................ ................................ .......... 45 4.16: Camber ................................ ................................ ................................ .................. 45 4.17: Types of camber angle (a.) zero, (b.) positive, (c.) ................................ .... 46 4.18: Lateral force with zero camber angle ................................ ................................ .................. 47 4.19: zero camber angle ................................ ................................ ................................ ................ 47 4.20: Lateral force with positive camber angle ................................ ................................ ............. 48 4.21: Positive camber angle ................................ ................................ ................................ .......... 48 4.22: Lateral force with negative camber angle ................................ ................................ ............ 49 4.23: Negative camber angle ................................ ................................ ................................ ......... 49 4.24: steering rack and pinion of a vehicle ................................ ................................ ...... 51 4.25: Vehicle on a track at normal steer ratio ................................ ................................ ............... 51 4.26: Radius made by the vehicle at normal steer ratio ................................ ................................ 51 4.27: V ehicle on a track at 20% less steer ratio ................................ ................................ ............ 52 4.28: Radius made by the vehicle at 20% less steer ratio ................................ ............................. 52 4.29: Vehicle on a track at 20% more steer angle ................................ ................................ ......... 53 4.30: Radius made by the vehicle at 20% more steer ratio ................................ ........................... 53
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x 4.31: Constant rotational moment on rear wh eels ................................ ................................ ........ 54 4.32: Non constant rotational moment on rear wheels ................................ ............................... 55 4.33: Constant rotational moment on rear wheels lateral forces ................................ ................ 56 4.34: non constant rotational moment on rear wheels lateral forces ................................ ....... 56 4.35: Constant rotational moment on rear wheels rad ius ................................ ........................... 57 4.36: non constant rotational moment on rear wheels radius ................................ .................. 57 4.37: Constant rotational moment on rear wheels aerodyn amics ................................ .............. 58 4.38: non constant rotational moment on rear wheels aerodynamics ................................ ....... 58
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1 1. I ntroduction Race cars are fast, smooth on the ground and just fun to wat ch on the track That is why engineers make it their priority to design a safe and fast race car to win the race. A race car and steady, stable and of course fast. OptimumG is an international vehicle dynamics consultant group that works with automotive companies and motorsports teams to enhance their understanding of vehicle dynamics through seminars, consulting and software This thesis will model forces acting on the tires which are needed to make the car make a turn or a cornering manuver (lateral forces), accelerate and brake (longitudinal force) while absorbing the load, balancing and stabilizing the car (normal force). 1.1 Background A tire to a layman is a round inflated tube of rubber that moves the vehicle from point A to point B To an automotive or mechan ical engineer, it is a lot more! There are a lot of details to take into accou nt in a tire. There are several tire manufacturing companies in the world today such names like ; Michelin, G ood year, F irestone, Dunlop etc who work non stop in des i gning, testing and manufacturing tires for all forms of vehicles When researching on tire modeling there is an individual who everyone takes good note from or need to take from at least. Hans B. Pacejka is an individual who developed a mathematical equations to understand the mechanical behavior of pneumatic tires He designed the widely used Magic formula tire model which will also be used in this thesis work; this Magic formula will be analysed and discussed in full details below.
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2 1.2 Research Approach The approach of this rese arch is a step by step approach. Understanding of how matlab ope rates and works is very imperative. There are several analysis completed but before any of which is done, these had to happen: i) Working with thesis adviser and Professor, John Trapp Ph.D. to understand, debug and edit three dimensional Newton Euler code d eveloped by Dr. Trapp used to solve typical dynamics problem. ii) C ar d ata collection from the Optimu m G team for a typical Lemans car, this prop rie tary data is generic and its proprietary natur e There are graph representations that will show the behavior of these tires iii) Implementing the magic formula for characterizing pneumatic tire behavior from Optimum G test data iv) Editing and manipulation of 3 D code to run for various input and test for beh avior at different cases. v) Running the code and plotting accu r ate graph s to show and explain the vehicles behavior 1.3 Research Objective The main purpose of this paper is to enhance the understanding of tire and model the lateral, longitudinal, normal for ces and the moments about their axis such as yaw, pitch and roll by modification of an iterative computational model for calculating such values in matlab. Analysis to show better understanding of these force and moments would be performed to show how the tire behav es with time. Several other cases will be considered such cases which will have an effect on the vehicle handling and cornering will include; behavior and comparison of tire s with camber a ngles and without camber angles, skid pad test with cons tant velocity applied
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3 on the rear wheels and effect of increase or decrease in steering to determine the steer coefficient of the vehicle. Th e developed three dimensional Newton Euler code can b e used to study the behavior of any moving vehicle, so long the data is corr ect, t he little changes that would be made would be the constraints, number of bodies and its components to get the correct degree of freedom
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4 2. Literature Review The literature review briefly describes certain major terms n eeded to be known in order to understand this thesis. Vehicle dynamics and its various sub sections such as cornering, aerodynamics effect, wheelbase and track of the vehicle and handling will be discussed in the literature review. 2.1 Vehicle Dynamics Th is is a very broad topic and has b een discussed in several papers. The effect of corn ering, aerodynamics effect, wheelbase size, steady state handling & steering, and many more are constantly considered when designing a vehicle, these are used to find bett er ways to improve the quality of any vehicle. To do any research in this understanding of some basic vehicle dynamics is imperative. 2.1.1 C ornering This is the term used when a vehicle negotiates a turn at a corner or in this case when racing. When corn ering occurs, depending on the turn (right or left) the normal force acting on the outer side tires are larger than that of the inner side tires. In order to negotiate this turn a lateral force is needed a lateral force is an applied tire force that originates at the center of the tire contact with the and lie s in the horizontal road plane and is perpendicular to the direction in which the wheel is moving towards. The figure 2 1 below illustrates the direction the lateral force acting on each wheel, with velocity direction of the vehicle and turning radius.
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5 Figure 2 1 : A vehicle with lateral forces applied to its wheels moving on a track The lateral force applied produces a slip angle; a slip angle has been defined in several ways, defines a slip angle as the angle between the centerline and the direction of travel or as describes it as the movement in the direction at an angle to the wheel plane. The figure 2 .2 below shows a wheel and its slip angle. Figure 2 2 : A tire showing slip angle Y X Direction of Travel Centerline V R Velocity
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6 Maurice Olley who to the world of the automobile industry is an outstanding pioneer and innovator in the ride and handling area of a car develops a model that can describe the re lationship between lateral force and slip angle, the equation is derived below: ( 1 ) Where a = weight transfer length of wheelbase in the front section of the car in meters, m b = weight transfer length of wheelbase in th e rear section of the car in meter, m Slip angle in degrees Lateral Force in Ne w t on, N The plot below can show the behavior of a tire with lateral force relating to slip angle, the slope of this graph at origin is the cornering stiffness be shown relating the lateral force of a tire to its slip angle, the plot in figure 2. 3 is shown in the next page, Figure 2 3 : Lateral force versus slip angle plot Initial Slope = C
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7 (2) ( 3 ) ( 4 ) = 0 ( 5 ) ( 6 ) The initial slope as see n in the graph called cornering stiffness is defined as such : ( 7 ) ( 8 ) After integrating, the lateral force is derived (9 ) An experiment performed lead to the following; t he cornering stiffness is plotted with the nowmal force which gives a parabola shape A similar graph figure 2. 4 is drawn t o explain the equation below
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8 Figure 2 4 : Cornering stiffness versus normal and characterize the tire and are used to derive and at = (10) (11) (12) (13) Now at and we get the fully defined equation of cornering stiffness (14) (15) Now th at a has been solved for, the equation (13) above can be redefined as,
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9 In order to get the at recalling becomes the redefined equation and (16) Where Normal Force in Newton, N Cornering Stiffness in N/deg = Maximum slip angle reached in degrees = nominal load of vehicle in Newton, N An expanded model arbitrary lateral force model was developed by Maurice Olley in his appendi x A in the Chassis Design text by William F. Milliken and Douglas L. Milliken. 2.1.2 Aerodynamic Effect Aerodynamics is the study of air motion and its effect on solid matter; it can be studied more in fluid dynamics. Race car designers and manufacturers take into account aerodynamics, the force of drag which r efers to the forces which act on a solid object in the direction opposing t he posi tive longitudinal direction, a downforce is also acknowledge and it is the force acting opposite the normal force of the vehicle, it is designed to accomplish stability and cornering in a vehicle. There are other aerodynamics forces to understand, s uch as; side force (wind), the forces in the x, y and z plane have moments also. The pitching, rolling and yawing moment are also taken into account. The general formulas for drag for each axis are shown below; the coefficients of the various drags are go tten from result from the wind tunnel testing.
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10 Drag Force = 0.5 (17) Down Force = 0.5 (18) Side Force = 0.5 (19) Where: The equation for is as follows; (20) Where is the mass of the vehicle. To get the moment, the various drag force multiply the wheel base essentially, but n ote is taken on the weight distribution of the vehicle. Pitching Moment = 0.5 Wheelbase (2 1 ) Rolling Moment = 0.5 Wheelbase (2 2 ) Yaw Moment = 0.5 Wheelbase (2 3 ) The equations above are the typical equations to use when calculating drag forces (N) and moments (Nm) the coefficient of drag are gotten from the wind tunnel testing. These equations are applied in the simulation for the testing of the behavior of the vehicle and the tire in this thesis for accurate results Figure 2. 5 below shows a race car and it s force and moment acting on it and Figure 2.6 shows how the aerodynamic forces vary with time for the simulated model used in this thesis at constant velocity.
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11 Figure 2 5 : Figure 2 6 : 2.1.3 Wheelbase and Track The wheelb ase is the horizontal distance between the center of the front wheel and the center of the rear wheel. A track is the distance between the centreline of two wheels on the
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12 same axle, each on the other side of the vehicle. Things to consider when choosing a wheelbase and track for a ; l ateral and longitudinal weight transfer, yaw moment of inertia, aerodynamics, packaging, type of car, type of race track and rules of the race. describes some of these considerations in relation to track and wheelbase; weight transfer is a function of track width (lateral) and wheelbase length (longitudinal), a wide track will cause less lateral weight transfer during cornering and a long wheelbase will cause less longitudinal weight transfer in braking and accelerating. An increase in track or wheelbase will cause the masses of the non suspended components (e.g. rim, tire, hub, upright, brake etc.) to be placed further from the centre of gravity of the car, this will increase the cars yaw moment of inertia. In the case of aerodynamics an increase in track width will cause an increase in drag Track changes in open wheel race cars will affect the entire flow pattern of the car. Figure 2. 7 shows the full track of a race car. Figure 2 7 : Full Trac Figure 2. 8 below shows the wheelbase of a race car, where the sum of a and b gives the wheelbase
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13 Figure 2 8 : Wheelbas 2.1.4 H andling Without steering there will be no handling, steering affects handling effect of the vehicle. Vehicl e handling describes how the motion of the vehicle is controlled taking into consideration the vehicles primary six degrees of freedom and how the effect of external disturbance such as wind on stability of the vehicle can be reduced. Steering is made poss ible by yaw motion (rotation in the z axis) reference figure 2.5 above; a 3D body can be rotated about three orthogonal axes, the yaw rotation, pitch rotation and the roll rotation. T he equation for rotational matrices and steer angle are defined below : (24) (25) a b
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14 (26) The rates of the above matrices are found and derived giving the and as the roll pitch and yaw rates known as body angular velocities (2 7 ) The steering angle for a vehicle is shown as (2 8 ) From equation ( 7 ) at and after inte grating can be defined as (2 9 ) is defined for each of the vehicle, the equation variables are described below: = steer angle in degrees = wheelbase in m = turning radius in m Equation (24) can be redefined as And eventually we get,
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15 Where is the understeer coef ficient and is defined below, ( 30 ) The describes if a vehicle is at oversteer, neutral steer or understeer. While neutral steer and understeer is still considered stable, oversteer tends to be unstable. A skid pad test would be performed to see the handling behavior of the vehicle by adjusting the steer angle and the other test will be the effect of wind force on the lateral side of the vehicle and paying attention to lateral velocity of the veh icle Figure 2. 9 shows a physical interpretation of this understeer coefficient values In this figure, understeer coefficient is explained; an increase in speed when cornering a vehicle with no change in steering angle is shown fo r three cases to explain handling. When the vehicle maintains a constant radius while cornering with increase in speed with constant steering angle, there is a neutral steer, when the vehicle has an increase in the radius while c ornering there is an unders teer, when there is a decrease in that radius there is an oversteer. Understeer occurs when the front wheels of the car lose traction before the rear wheels. The car is difficult to turn and pushes toward the outside of a turn. Oversteer is the opposite co ndition. The rear tires lose traction before the front tires. Hence, the rear of the car is loose. It slides toward the outside of the turn, and the car feels like it is going to spin out. A skid pad test is performed and results are gotten based on the un dersteer coefficient analogy.
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16 Figure 2 9 : Curvature Response of different values of undeersteer coefficient More description of the characteristics of the different understeer coefficien t value is described in the sections next 2.1.4.1 N eutral Steer The side force acts at the center of gravity, the cause of neutral steer while cornering ; turning radius R is remains constant, no increase or decrease. Steer angle is independent of forward speed V. The underste er coef fi cient is equal to zero 2.1.4.2 Oversteer The side force acts at the center of gravity, the cause of over steer while cornering ; turning radius R decreases Steer angle decrease with increase in forward speed V. The understeer coef fi cient is less than zero, Understeer Neutral Steer Oversteer
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17 2.1.4.3 U ndersteer The side force acts at the center of gravity, the cause of oversteer while cornering; turning radius R increases Steer angle increases with increase in forward speed V. The unders teer coef fi cient is greater than zero, 2.2 P neumatic T ire In the case of this thesis, understanding of a pneumatic tire is needed and shown below. describes the purpose of a pneumatic tire as functioning to support the vehicles weight on the ground, cushioning the vehicle over surface with an irregularity (which is really saying absorbing shock from the ground contact), provide sufficient traction for braking and accelerating, and also proving steering control and direction stabi lity. These descriptions clearly explain the main purposes of a tire on a vehicle. Now, these are just sayings, as engineers we back our words with mathematical calculations and accurate results. This leads to the next chapter of equations of motion and ti re modeling. 3. Newton Euler 3.1 Equations of Motion ( 3 1 ) Where is mass, is acceleration, and is The forces acting on the body which could be applied (gravity force, spring ad dashpot forces etc) or constrained forces acting on the joints of the body (ball and socket joints, point pl ane s etc). motion for a system of particles is defined as, ( 3 2 )
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18 Where is moment of inertia in is angular velocity in rad/sec and is rotation about point B of the particle acting on the d escribes the steps taken to solve all dynamics problems. There are usually 7 steps : 1. A sketch of the rigid body with a center of mass position vector, wit h only one world frame and several body frames depending on number of rigid body. 2. Free body diagrams for each rigid body showing applied forces and reaction/constraint forces. Applied forces are given forces such as; gravity [m/s 2 ], springs forces [N/m] et c. Reaction/ constraint forces are unknown magnitude that has to be found that has a known direction. 3. 4. Geometry of the systems; use primary variables from step 1 to formulate t he geometric constraint between each pair of bodies, one joint at a time (body coordinate formulation) and then write the primary variable sin term of a reduced generalized coordinates (joint coordinate formulations). 5. Number of unknown variables must equal the number of equations to make this system of equations solvable. List all the unknowns present. 6. Reduce the system of ordinary differential equations and geometric constraint equations to a smaller set if possible and solve. 7. Examine the solution and ens ure it makes sense and it is accurate. Equation ( 31 ) and ( 32 ) above are the simple general forms of the N E equation, for a three dimensional body, the equations are more complex. Account of the forces acting in all directions, in the longitudinal (x), la teral (y) and normal (z) axis as well as the moments along
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19 the roll (x), pitch (y), and yaw (z) axis is taken into effect. The steps are very essential and useful in any dynamics analysis. Appying the Euler method, the N E can be re written as ; ( 3 3 ) (3 4 ) (3 5 ) (3 6 ) (3 7 ) (3 8 ) The 1, 2 and 3 represent x, y and z respectively, the variables in the equation are defines as; m = mass of the object in kg M = moment acting on each axis in Nm I = moment of inertia in = angular velocity in rad/sec In the 3 D equation above the n denotes old time explicit evaluation ( e xplicit methods calculate the state of a system at a later time from the state of the system at the current time) for applied forces and moments term, i.e. spring, dashpots, aerodynamics forces etc. The n+1 term denotes new implicit evaluation ( implicit methods find a solution by solving an equation involving both the current state of the system and the later one) for re action/constraint forces and moments.
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20 The N E equation is used in this next section to describe a light link, distance between two or more solid objects 3.1.1 Light Link In the 3 D simulation performed in this thesis, the light links represent the frame and the suspension of the vehicle, the matlab function builds linearized geometric constraint equations for a weightless link between the two bodies. The corresponding constraint force is also loaded into the coefficient array. The figure 3.1 below shows a representation of a light link between two bodies B and C Figure 3 1 : Light link between two bodies For a typical light link body calculation betwe en two bodies in space the N E steps above are followed then the equation is added to the simulation for each body connection in the vehicle, the length of the light link is, B C Body 1 Body 2 L
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21 (3 9 ) ( 40 ) ( 4 1 ) The re is a normal force s acting on the surface of both body on point and the magnitude of the normal force is then dotted to the unit normal which lies on the plane to get the normal force vector T he equation is added to the Newton section of the equation and a moment from the force is added to the Euler part ( 4 2 ) ( 4 3 ) (4 4 ) (4 5 ) 3.1.2 Spring and Damper The suspension in the vehicle makes use of torsional springs and dashpot s Suspension systems serve a dual purpose, t he main purpose of the suspension is stabil ity and handling of the vehicle and it connects a vehicle to its wheels and allows relative motion between the two [ 11 ] as well as isolating the road n oise, bumps, and vibrations,etc from the driver. The spring for ce and dashpot are calculated with the equation shown below; (4 6 ) Where,
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22 = the displacement vector the distance and direction the spring is deformed from its equilibrium length. = the resulting force vector the magnitude and direction of the restoring force the spring exerts k = the rate spring constant or force constant of the spring, a constant that depends on the spring's material and construction. A dashpot on the other hand is a mechanical device, a damper which resists motion via viscous friction [1 2 ] The resulting force is proportional to the velocity but acts in the opposite direction [ 13 ] slowing the motion and absorbing energy. It is commonly used in conjunction with a spring (which acts to resist displacement). (4 7 ) Where, c = damping coefficient in N s/m. To apply the equation of both the spring and dashpot to the N E 3 D code it is very similar to that of the light link equation, except for some force application changes, the equations are shown below The matlab function implements a l inear spring or damper elem ent by loading the applied force and moment terms to the right hand side of the Newton Euler equations to model a spring and a dashpot ; again a point p on body 1 is connected to point q on body 2. Take note that there are four wheels, so essentially there will be 5 bodies, the main car body and the spring and dashpot connect eachs each light link body which is then connected to the wheel. ( 4 8 )
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23 ( 4 9 ) ( 50 ) The force is again dotted with its unit normal, the equation is added to the Newton section of the equation and a moment of from the force is added to the Euler part Equation (4 6 ) is re written as shown for forces acting on a spring and dashpot. ( 5 1 ) The dashpot becomes, ( 52 ) Both equation (48) and (49) are added to the Force section of the N E 3 D code and for the moment equation added to the Euler section, there is (5 3 ) (5 4 ) The graph below shows the forces of both the spring and dashpot against time, this is gotten f rom the simulation result. Since a four wheel vehicle is in question, there are four plots
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24 and these plots are shown below. The simulation result is gotten while the vehicle is negotiating a turn makes a 360 degree turn, forming a circle ; the front left wh eel creates more normal force, because when making a turn, the outside wheels have more normal forces, more will be explained in the tire modeling section of this thesis. The dashpot force in the front wheel s behaves a lot different compared to the rear wh eels, a higher amount is calculated and seen on the graph. The Force of the spring has opposite values from the outer wheels compared to the inner wheels; this is due to cornering of the vehicle. The behavior is seen to be similar on each side of the car, just differnent values because of the wheel negotiating a turn. The black color shows the dashpot forces and the red color shows the spring forces as seen in figure 3.2 below, Figure 3 2 : Spring and Dashpot forces with time cornering exist The figure above describes the spring and dashpot forces a s the vehicle is moving in real time, the initial fluctuation shows transient behavior then steady state. The behavior of the springs of the vehicle is shown the s prin g forces increase with velocity in the figure below.
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25 Figure 3 3 : Spring forces with velocity 3.1.3 B all and Socket A ball and socket joint between two bodies or between a body and the world. This fun ction generates 3 constraint equations and 3 force reactions. This is used in the matlab code to calculate forces in the joints and its function is used to connect part of the suspension system to the car body e.g the rocker. The reaction forces are the fo rces on body 1 in the world frame coordinates e1, e2, and e3 directions at the point with body frame coordinates p1, p2, p3. There are equal but opposite forces on body 2 at the point with body 2 coordinates q1, q2 and q3. The bodies could be and in genera l will be subject to (i) other constraints and (ii) applied forces and The equation is basical ly the N E 3 D equation where it calculates the forces from the two bodies. Equation ( 31 ) and ( 32 ) above shows these equations An illustration of two bodies connected by ball and socket joint is shown below, a body P connected to a body P (point mass)
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26 Figure 3 4 : Ball and Socket joint The distance between body P and body Q is there is a n equal and opposite force F holding both bodies together this is in the world frame which is added with i ts moment to the N E 3 D equation. Note the conversion from body frame to world frame has to be calculated. 3.1.4 W heel on ground A wheel matlab function connects the body (a tire) and the ground surface in the world frame. This function builds the appro piate constraint equations (using five extra variables) between a wheel and the ground. The function generates 6, 7, or 8 constraint equations five (because of the extra variables and 1, 2, or 3 additional constraints for normal contact, drive direction no slip, or transverse direction no slip. This set of constraint equations generates the three constraint forces (depending upon the case) at the wheel ground contact point on the wheel. This is a very essential part of this thesis, since the idea is to mo del a tire and understand its behavior via many different test, constraints and disturbances. Figure 3.5 below shows one wheel trying to make contact on a surface which is the ground. There are four wheels in this race car and each of them will have to be in contact with the ground and this function in matlab makes
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27 this possible and a simulation where the vehicle can drive in a straight road or attempt cornering is made possible. Figure 3 5 : Wheel on a flat surface There are certain steps to follow to create the wheel function, this is shown below; 1. Get the surface ground equation as for a flat surface, where ground height value 2. A point on the road for body ground with dimensions is established and also the point on the wheel that would be in contact with this point is 3. Equation of a circle is use d, equation is R is the radius and and both x and h are points in he middle of the circle. A derivation of the function is R
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28 calculated this is used t o get the unit vector which is 4. The force is dotted to the unit vector and added to the N E 3 D equation and subseq uently the moment is defined also. This is used to calculate any form of wheel on any type of ground, bumpy or smoo th as the case may be. 4. T ire Modeling Tire modeling is a method of developing various forces and moments acting on a tire in order to drive and maneuver the vehicle. A great understanding of a tire and its relation between the road, driver, and vehicle is imperative in order to model a tire. Hans B. an expert in vehicle dynamics with more focus on tire modeling provides an opportunity to better understand the behavior of pneumatic tire and its impact on vehicle dynamics. An empirical method to calculate these forces and moments acting on a tire is called Magic has developed model s in both steady and non steady transient s t ates to improve tire behavior. A simulation designed with this Magic Formula is very useful to be able to see and understand that behavior of the tire a nd make it easier to locate pot ential problems that can be fixed. The Magic Formula has several components tha t make the formula, these components can be us ed to calculate several forces and moments acting on a tire, that is why it is very widely used in the automotive industry. The formula which holds for given values of vertical load and camber angle will be broken down to longitudinal force (pure and comb ined longitudinal slip), lateral force (pure and combined side slip) and aligning torque (pure and combined side slip), some of the other
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29 moments which are not taken into account in this thesis is the overturning moment and the roll resistance moment. Test data provided by Optimum G do not include the coefficients values needed to analyze these moments, although the simulation, if data is produced can run and describe the moments. The Magic Formula is capable of producing propitious characteristics that clo sely match measured curves for the side, acceleration and braking force. The behavior of this formula is typically seen on a graph; a curve is produced passing throught the origin, reaching a maximum point like amplitude and then tends to a horizontal Figure 4 1 : SAE tire axis The formula is shown below; (5 5 )
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30 (5 6 ) (5 7 ) Where, Y = Output variable F x, F y and M z B = stiffness factor D = peak value of the curve C = shape factor (determines the shape of the peak) E = Curvature Factor and = shifting values; they shift the curve horiz ontally and vertically. An idea of how the graph should look like is shown in the graph below. This is a typical lateral force slope angle graph a data will produce Figure 4 2 : Magic Formula graphical re presentation
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31 The factors of the magic formula have equations and the se are shown below, reference the appendix MatLab code to read how these values got ten from the equation influence the main force and moment result s gotten from the tires. Other influences of the main output force and moment include camber angle, normal force, a normalized normal force and parameter values gotten from tire testing. The simple form of the formula is shown below (5 8 ) (5 9 ) ( 60 ) ( 61 ) Where and are gotten from the parameter values of the tire, values gotten from testing of the tires. 4 .1 F orces The external forces acting on the tire of the car are defined below according to and the longitudinal force, lateral force and normal force 4.1.1 Longitudinal Forces Longitudinal Force ( ) is the component of the force vector in the x direction. This is the force responsible for the braking and driving force of a vehic le, the positive for the driving and the equal but negative force is needed for the braking. T he Mag ic Formula becomes, ( 62 )
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32 Where longitudinal slip with horizontal shift in axis and the longitudinal slip ratio is defined as ( 63 ) ( 64 ) Figure 4 3 : Rolling and Slipping of a Where is the velocity of the wheel in the x direction is the slip velocity in the x direction is the contact velocity in the x d irection is the longitudinal running speed is the center of the rotation of motion or slip point
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33 is the imaginary point of the center of the wheel is the contact center of the tire is the angular speed is the angular speed of rolling tire is the loaded tires radius is the e ffe ctive rolling radius when the tire is in motion It can be seen where longitudinal slip ratio is calculated from slip is the relative motion between a tire and the road surface it is moving on T he above ratio is very essential in calculating the longitudinal force and then plotted with this force to observe the tires behavior. The stiffness factors (B), peak value of the curve (D), shape factor (C) and Curvature Factor (E) are variables that are very essential to this formula and are calculated from variables gotten during tire testing, the values are different compare d to that of the lateral force equation. Pure longitudinal force and combine d longitudinal force differ from each other; in combined longitudinal force, both the pure longitudinal force and lateral sli p which is gotten from the pure lateral force calculation is needed and a graph showing both forces is shown below, the pure longitudinal force (green) can be seen to be greater than the combined longitudinal force (blue) be cause of the ad diton of slip ang le values they do not depend on camber angle They are both equal when slip angle added is zero, the fourth graph seen with the highest amplitude is the behavior when more normal forces are acting on the tire.
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34 Figure 4 4 : plot of Longitudinal force against Slip ratio The simulation is run for a vehicle driving on a straight road that turn, so no cornering force is applied; the behavior of the longitudinal force on the tire is seen in time. The above plot is gotten for an instant time and it shows the relationship between force and the slip ratio, the plot below shows what that force behaves like in time and also the slip ratio behavior in time
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35 Figure 4 5 : plot of Longitudinal force against time 4.1.2 Lateral Forces Lateral (cornering) Force ( ) is the component of the force vector in the y direction. This is the force responsible for the cornering force of a vehicle, when a car tries to negotiate a turn whether in a race or just turning a curve into a street, the cornering force is needed to make that turn. The Magic Formula becomes, ( 65 ) Where and the lateral slip angle are defined as ( 66 ) ( 6 7 ) Where is the contact velocity in the x direction and is the contact velocity in the y direction, the above slip angle is very essential in calculating the lateral force and then plotted with this force to observe the tires behavior. Again, t he stiffness factor (B), peak value of the
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36 curve (D), shape factor (C ) and Curvature Factor (E) is variables that are very essential to this formula and are calculated from variables gotten tire testing, the values are different compared to that of the lateral force equation. Pure lateral force a nd combine lateral force differ from each other; in combined lateral force, both the pure lateral force and longitudinal slip which is gotten from the pure longitudinal force calculation is needed and a graph showing both forces i s shown below, they both depend on camber angle The effect of positive and negative camber can be seen on the graph, positive camber angle increase the lateral force and negative camber angle decreases the lateral force, shown in cyan (positive ) and green (negative ). This is essential in design of race cars and other vehicles, depends on how sharp a turn the designer wants the vehicle to make. T he pure lateral force ( blue ) can be seen to be greater than the combined longitudinal force ( red ) because of the ad diton of slip ratio values. They are both equal when slip ratio added is zero, the highest amplitude graph seen, is the behavior when more normal forces are acting on the tire.
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37 Figure 4 6 : plot of lateral force against slip angle The simulation is run for a vehicle driving on a road and negotiating a turn, the cornering force is applied; the behavior of the lateral force on th e tire is seen in time. The above plot is gotten for an instant time and it shows the relationship between force and the angle, the plot below shows what that force behaves like in time and also the slip angle. When negotiating a curve, the outer wheel pro duce more normal forces which in instantly increases the lateral force, the outer force is seen below as the higher normal forces. Note tire 1 FL (red), tire 2 FR (blue), tire 3 RL (green), tire 4 RR (black)
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38 Figure 4 7 : Plot of normal force against time Figure 4 8 : Plot of lateral force against time Some of the other relevant plots such as the plot for slip angle, slip ratio, camber angle and velocity plo tted with time are shown below. These graphs shows the behaviors of the tire
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39 Figure 4 9 : Plot of tire angles and velocity against time 4 .2 M oments The external moments acting on the car are defined below, a) Overturning moment (M x ) is the component of the moment vector tending to rotate the vehicle about the x axis, positive clockwise when looking in the positive direction of the x axis. b) Rolling resistance Moment (M y ) is the component of the moment vector tending to rotate the vehicle about the y axis, positive clockwise when looking in the positive direction of the y axis. c) Aligning torque (M z ) is the component of the moment vector tending to rotate the vehicle about the z axis, positive clockwise wh en looking in the positive direction of the z axis. For the purpose of this thesis as explained earlier in page 3 4 aligning force would only be discussed
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40 4. 2 1 Aligning Forces This is the moment responsible for the cornering force of a vehicle, when a car tries to negotiate a turn whether in a race or just turning a curve into a street, the moment about the z axis helps the vehicle make that turn. For the aligning torque to work, a pneumatic trail is the distance between the s ide force and the cornering force Figure 4 10 : Pneumati c There are different eqauations for the pure pneumatic trail and combined pneumatic trail and they are listed below, the combined term is, ( 6 8 ) The pure term is, (6 9 ) Where and the slip angle for pneumatic trail are defined as ( 70 )
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41 ( 71 ) Where and are constant values gotten from the tire testing and the final aligning torque equation is then the product of the lateral force with the pneumatic trail and then a dded to a residual torque ( 72 ) ( 73 ) The final equation is, ( 74 ) Pure aligning torque and combine lateral force differ from each other; in combined aligning force, values from calculating the pure lateral force, pure longitudinal force and aligning torque are used. A graph showing both moments is shown below; they both depend on camber angle The effect of positive and negative camber like that of the lateral forces can be seen on the graph, positive camber angle increase the pure aligning torque and negative camber angle decreases the aligning torque shown in cyan (positive ) and green (negative ). This is essential in design of race cars and other vehicles, depends on how sharp a turn the designer wants the vehicle to make. The pure aligning torque (blue) can be seen to be gre ater than the combined longitudinal force ( yellow ) because of the additon of slip ratio values. They are both equal when slip ratio added is zero, the highest amplitude graph seen, is the behavior when more normal forces are acting on the tire.
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42 Figure 4 11 : Plot of aligning torque against slip angle The simulation is run for a vehicle driving on a road and negotiating a turn, the cornering force and with it an aligning moment is applied; the behavior of th e lateral force on the tire is seen in time. The above plot is gotten for an instant time and it shows the relationship between torque and the angle, the plot below shows what that torque behaves like in time. A similar behavior should be see n like that of the lateral force in time, but different values and units.
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43 Figure 4 12 : Plot of aligning torque and pneumatic trail against time 4 .3 F riction Circle The lateral force and longitudinal force as functions of slip angle and traction/braking slip ratio is described in a friction circle. According to Milliken et al at the origin the tire is free rolling straight ah ead with no longitudinal force, t he slip angle and lateral force are zero. The vertical axis represents longitudinal force and various slip ratios are marked along the scale. Points on the vertical axis could be used to develop the curves of slip ratio vs. longitudinal force. The simulation runs a function that creates a friction circle, the plot below shows friction circle at initial run and at the end of the run for a cornering
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44 Figure 4 13 : Friction circle at initial run The blue line shows the direction of the velocity, the green line divides t he circle into four quadrants to be able to show the different lateral force on the side and the longitudinal force along the velocity direction the diameter of the circle shows the normal forces behavior The graph below shows the graph at final run afte r cornering has been made. As the wheels move, an angle is created which is the slip angle, the angle between the velocity line (blue) and quadrant (green). Recall a slip angle as occurring when the steering wheel is turned from straight forming an angle b etween where the tire is pointed and where the car is actually going, the outer normal force is bigger compared to that of the inner tires. Figure 4 14 : Friction circle at final run
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45 Figure 4 15 : Race Car at initial run 4.4 C ases Different case runs are made to see how different values can affect the behavior of the tire, such include; comparing cam ber angle to non camber angle and handling and steering 4.4.1 Effect of positive, negative and zero c amber angle Simulation is run for cornering and comparing the forces and moment acting on the tire with time and understands the different behavior. Figure 4 16 : Camber
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46 Wheel camber is defined as the angle between the wheel center plane and the normal to the road, the graph above shows the front view of a vehicle and its applied positive camber angle on the tires. The camber angle as discussed in the lateral force section above influences the force and moment acting on the tire when cornering, test cases are shown to know how the effect of camber angle whether zero, positive or negative can affect cornering forces. The equati on for camber angle is shown below ( 75 ) Where is the camber angle known as the ground surface normal vector pointing up from the ground in the positive ground grad ient direction is the vector normal to the wheel center plane Figure 4 17 : Types of camber angle ( a .) zero, ( b .) positive, ( c .)
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47 4.4.1.1 Zero cambe r angle When zero camber angles are applied to the tires of the race car, the tires are straight and not bent inwards or outwards as shown in f igure 4.18 above The influences of camber angle is referenced in the appendix B. matlab code below, both latera l force and alignin g torque equations are influenced by camber angle in the form of a term named spin_camber. The graph below shows the lateral force without the application of camber angle below Figure 4 18 : Lateral force with zero camber angle The graph below shows the tire angle at zero camber Figure 4 19 : zero camber angle
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48 4.4.1.2 Positive camber angle The lateral forces are plotted also for positive cam ber angle, the two figures shown below describes the effect of positive camber angle, an increase in the lateral force can be seen compared to the zero camber, hence allowing for more cornering. The second graph shows the plot of tire angles with time and the camber angle plot can be seen Figure 4 20 : Lateral force with positive camber angle Figure 4 21 : Positive camber angle
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49 4.4.1.3 Negative camber angle Th e lateral forces are plotted also for negative camber angle, the tw o figures shown below describe the effect of negative camber angle, a decrease in the lateral force can be seen compared to the zero camber and positive camber angle, hence allowing for les s cornering. The second graph shows the plot of tire angles with time and the camber angle plot can be seen Figure 4 22 : Lateral force with negative camber angle Figure 4 23 : Negative camber angle
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50 4.4.2 S kid Pad Test In this case, the steering ratio is adjusted to determine the behavior of the vehicle. A rack and pinion gearset is enclosed in a metal tube, with each end of the rack protruding from the tube. A rod, called a tie rod connects to each end of the rack. The pinion gear is attached to the steering shaft When you turn the steering wheel, the gear spins, moving the rack. The tie rod at each end of the rack connects to the steering arm on the spindle The rack and pinion gearset converts the rotational motion of the steering wheel into the linear motion needed to turn the wheels. The steering ratio is the ratio of how far you turn the steering wheel to how far the wheels For instance, if o ne complete revolution (360 degrees) of the steering wheel results in the wheels of the car turning 10 degrees, then the steering ratio is 360 divided by 1 0, or 36 :1. The steering ratio is gotten by a steer steer test, this is completed by using a lignmen t tables with a steering scale. The load and ride height of the car should be known. A ride height is the amount of space between the base of an automobile tire and the underside of the The steering wheel is turned to the right at even intervals, noting the steer angles of both front wheels. The test continues by rotating the steering wheel back to the center, stopping at each angle and noting the front wheel for any hysteresis. Hysterisis is the influence of the previous history or treatment of a body on its subsequent response to a given force or changed The skid pad test was performed for both 20% less steer ratio and 20% more steer ratio This is done to determine the handling of the car, to be able to see if the vehicle is oversteer, understeer or neutral steer when the steering is adjusted while driving. Using section 2.1.4 which defines underste er coefficient, this case can determine the effect of steering on the vehicle. Six plot s are seen below to explain this
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51 Figure 4 24 : steering rack and pinion of a vehicle Figure 4 25 : Vehicle on a track at normal steer ratio Figure 4 26 : Radius made by the vehicle at normal steer ratio
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52 The case where the steer ratio is decreased by 20% has values plotted below and can be seen it is stable as the case with less steer angle by viewing both graphs below the vehicle uses lesser lateral force to negotiate a turn and the radius of the curve is larger. This is a meth od to fix understeer in vehicles. Figure 4 27 : Vehicle on a track at 20% less steer ratio Figure 4 28 : Radius made b y the vehicle at 20% less ste er ratio
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53 The case where the steer ratio is increased by 20% has values plotted below and can be seen it is stable as the case with more steer angle by viewing both graphs below the vehicle uses more lateral force to negotiate a turn and the radius of the curve is smaller. This is a method to fix under oversteer in vehicles. Figure 4 29 : Vehicle on a track at 20% more steer angle Figure 4 30 : Radius made b y the vehicle at 20% more steer ratio
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54 4.4.2 .2 Constant Velocity Another case is performed to determine the handling properties of the vehicle; a constant velocity skid pad test is performed in MatLab. Comparison is made between constant velocities of the two rear wheels to non constant velocity of the two rear wheels, behavior of the vehicle can be read in the graphs shown below. The two rear wheels are linked together to have a constant rotational moment this then allows for the two rear wheels to move at the same rate. The graph below shows the cornering forces, radius, and velocity plots for both constant velocity and non constant velocity cases. Figure 4 31 : Constant rotatio nal moment on rear wheels
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55 Figure 4 32 : Non constant rotational moment on rear wheels From the two graphs above, differences can be read from the slip angle being affected all the way to the slip ratio, the tire velocity reduces in the constant rotational graph.T here are other graphs shown below to help explain which case is better for the vehicle and why. The lateral force graphs below show a large difference that can affect cornering of the vehicle.
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56 Figure 4 33 : Constant rotational moment on rear wheels lateral forces Figure 4 34 : non constant rotational moment on rear wheels lateral forces In t he constant rotation moment plot for the lateral forces, smaller lateral force s are acting on the rear wheels with lesser velocity in order to negotiate a more accurate turn. With so much lateral force acting on the rear wheels while trying to negotiate a turn, it will reduce the smoothness of the cornering of the vehicle.
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57 A comparison of both the radius made while turning and aerodynamic drag and lift can be read in the graph below also Figure 4 35 : Constan t rotational moment on rear wheels radius Figure 4 36 : non constant rotational moment on rear wheels radius
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58 Figure 4 37 : Constant rotational moment o n rear wheels aerodynamics Figure 4 38 : non constant rotational moment on rear wheels aerodynamics With constant moment applied to the rear wheels, a constant in aerodynamic forces can be maintaine d and in the non constant moment case a drop in the aerodynamic forces can be seen. It can be seen from the graphs that for constant moment acting on the rear wheels, handling
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59 characteristics a r e better compared to the non constant moment case the co nstant moment radius graph shows an oversteer compared to that of the non constant moment radius. 5. Conclusion and Recommendation The goal for this thesis was to determine the behavior of pneumatic tires of a Lemans race car and be able to see that behav ior with different cases, this done in order to design better, faster and safer tires for the racer. The three different simulation cases were performed in order to have a better understanding of the race car tire, the theories have been met. The camber a ngle case shows how camber angle can affect cornering of a vehicle, most standard race car vehicles have negative camber angle applied to their tires, even this race car I am working on does. In order to understand why they have those angles applied a case was made, simulations were run to show the differences. It can be seen that the negative camber angles produce more lateral forces applications and other vehi cles might have the right use for zero camber and positive camber angle. The other case was the comparison of steering ratio, typically looking on how much a driver can steer and how it affects handling of a car. The more steering provided b y the driver, depending on the design of the steering rack and pinion, the chassis of the vehicle etc., more steering can affect the vehicles handling and make it oversteer which makes the car unstable. Again cases were made to have a physical understanding of how incre asing or decreasing steering angle and ratio can affect the vehicle. The last case is performed by designing the vehicle to have its rear wheels have constant angular moment and note is taken on how much difference when compared to that of a vehicle
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60 with no constant angular moment. The lateral forces, aerodyanamic forces and velocity are all affected by the difference. The matlab code can essentially be used for other vehicle type, used to calculate the off road ground vehicle, two wheel vehicle, a train, trailer, cart etc, several components might have to be changed and adjusted, that is because all vehicles are not the same and several things need to be considered when designing vehicles. Although this chapter of the thesis is completed, there is still a work to be done, the stability of the vehicle for a linear and non linear steady state and transient state, also making the code run a bit faster than its current time. Although, there are so many different functions attached to one main code (reason w hy it runs a bit slow), this can be fixed by differentiation and changing the method to second order which will make calculation faster and give the same result. A more in depth experience with automotive engineering will also be a good step to follow.
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61 Appendix A. Nomenclature
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62
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63
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64 B. MatLab Code function [Fx,Fx0,Fy,Fy0,Mx,My,Mz,Mz0] = Magic_Formula(kappa,Fz,ntire,spin_camber, sangle) % %Input new values for Pacejka Model 2002 % % insert the common global parameters into this function global_time % This data runs the coefficients needed to run the Lateral function Magic _Formula_Coefficients(); format compact format long e dfz = ( Fz Fz0(ntire) )/Fz0(ntire); Fz0prime = lambdaFz0 Fz0(ntire); % *********************************************************** % Pure Longitudinal equation % ***************************** ****************************** SHx = (pHx1(ntire) + pHx2(ntire)*dfz)*lambdaHx; kappax = kappa + SHx; Cx = pCx1(ntire)*lambdaCx; % should be > 0 mux = (pDx1(ntire) + (pDx2(ntire)*dfz))*lammuxstar; mux = max(mux, 1*mux); Dx = mux*Fz*zeta1; Ex = (pEx1(nti re) + (pEx2(ntire)*dfz) + (pEx3(ntire)* ... dfz*dfz))*(1 (pEx4(ntire)*sign(kappax)))*lambdaEx; % should be <=1 Kxk = Fz (pKx1(ntire)+(pKx2(ntire)*dfz))*exp(pKx3(ntire)*dfz)* ... lambdaKxk; Bx = Kxk/((Cx*Dx) + etax); SVx = Fz*(pVx1(ntire) + pVx2(ntire)*dfz)*lambdaVx *lammuxprime*zeta1; Fx0 = Dx*sin(Cx*atan(Bx*kappax Ex*(Bx*kappax atan(Bx* ... kappax)))) + SVx; % *********************************************************** % Combined Longitudinal Force equation % ********************************************************** SHxa = rHx1(ntire); alphas = sangle + SHxa; Exa = rEx1(ntire) + rEx2(ntire)*dfz; Cxa = rCx1(ntire);
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65 Bxa = rBx1(ntire) cos(atan(rBx2(ntire)*kappa))*lambdaxa; Gxa0 = cos(Cxa*atan(Bxa*SHxa Exa*(Bx a*SHxa atan(Bxa*SHxa)))); Gxa = cos(Cxa*atan(Bxa*alphas Exa*(Bxa*alphas atan(Bxa*alphas))))/Gxa0; Fx = (Gxa*Fx0); % *********************************************************** % Pure Lateral equation % *************************************** ******************** SHy = (pHy1(ntire) + pHy2(ntire)*dfz)*lambdaHy + pHy3(ntire)* ... spin_camber*lambdaKyy*zeta0 + zeta4 1; alphay = sangle + SHy; % Cy = pCy1(ntire)*lambdaCy; % should be > 0 muy = (pDy1(ntire) + pDy2(ntire)*dfz)* ... (1 pDy3(ntire)*spin_camber^2)*lammuystar; Dy = muy*Fz*zeta2; Ey = (pEy1(ntire) + pEy2(ntire)*dfz)*(1 (pEy3(ntire) + ... pEy4(ntire)*spin_camber)*sign(alphay))*lambdaEy; Ey = min(Ey, 1*Ey); Kyao = pKy1(ntire)*Fz0prime*sin(2*atan(Fz/(pK y2(ntire)* ... Fz0prime)))*lambdaKya; Kya = Kyao*(1 pKy3(ntire)*spin_camber^2)*zeta3; By = Kya/((Cy*Dy)); SVy = Fz*((pVy1(ntire) + pVy2(ntire)*dfz)*lambdaVy + (pVy3(ntire)+ ... pVy4(ntire)*dfz)*spin_camber*lambdaKyy)*lammuyprime*zeta2; Fy0 = Dy*sin(Cy*atan(By*alphay ... Ey*(By*alphay atan(By*alphay)))) + SVy; % *********************************************************** % Combined Lateral Force equation % ********************************************************** DVyK = (muy*Fz)*(rVy1(ntire) + rVy2(ntire)*dfz + rVy3(ntire)*spin_camber)* ... cos(atan(rVy4(ntire)*sangle))*zeta2; SVyK = DVyK*sin(rVy5(ntire)*atan(rVy6(ntire)*kappa)); SHyK = rHy1(ntire) + rHy2(ntire)*dfz; kappas = kappa + SHyK; EyK = rEy1(nt ire) + rEy2(ntire)*dfz; EyK = min(EyK, 1*EyK); CyK = rCy1(ntire); ByK = rBy1(ntire)*cos(atan(rBy2(ntire)*(sangle rBy3(ntire))))*lambdayK; GyK0 = cos(CyK*atan(ByK*SHyK EyK*(ByK*SHyK) atan(ByK*SHyK))); GyK = cos(CyK*atan(ByK*kappas EyK*(ByK*kappas atan(ByK*kappas))))/GyK0;
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66 Fy = (GyK*Fy0) + SVyK; % *********************************************************** % Pure Aligning Torque equation % *********************************************************** Kyaprime = Kya; SHt = (qHz1(ntire) + qHz2(nt ire)*dfz + (qHz3(ntire) + qHz4(ntire)*dfz)*spin_camber); SHf = SHy + (SVy/Kyaprime); alphat = sangle + SHt; alphar = sangle + SHf; Dr = Fz*R0(ntire)*((qDz6(ntire) + (qDz7(ntire)*dfz))*lambdaMr*zeta2 + ... (qDz8(ntire)+qDz9(ntire)*dfz)*spin_camber*l ambdaKzy*zeta0) ... *cospa*lammuystar+zeta8 1; Cr = zeta7; Br = (qBz9(ntire)*(lambdaKy/lammuystar)+(qBz10(ntire)*By*Cy))*zeta6; Mzr0 = Dr cos(Cr*atan(Br*alphar)); Bt = (qBz1(ntire)+(qBz2(ntire)*dfz)+(qBz3(ntire)*dfz^2))* ... (1+(qBz5(ntire)*a bs(spin_camber))+(qBz6(ntire)*spin_camber^2)) ... *(lambdaKya/lammuystar); % should be > 0 Bt = max(Bt, 1*Bt); Ct = qCz1(ntire); % should be > 0 Dt0 = Fz*(R0(ntire)/Fz0prime)*(qDz1(ntire) + (qDz2(ntire)*dfz))*lambdat; Dt = Dt0 (1 + (qDz3(ntire)*ab s(spin_camber)) + (qDz4(ntire)*(spin_camber^2)))*zeta5; Et = (qEz1(ntire)+(qEz2(ntire)*dfz)+(qEz3(ntire)*dfz^2))* ... 1 + ((qEz4(ntire)+(qEz5(ntire)*spin_camber))*(2/pi)*atan(Bt*Ct*alphat)); % <= 1 Et = min(Et, 1*Et); t0 = Dt*cos(Ct*atan((Bt*alphat) Et*(Bt*alphat atan(Bt*alphat))))*cospa; Mz0prime = t0*Fy0; Mz0 = Mz0prime + Mzr0; % *********************************************************** % Combined Aligning Torque equation % ********************************************************** alphat_eq = (sqrt(alphat^2 + ((Kxk/Kyaprime)^2)*(kappa^2)))*sign(alphat); alphar_eq = (sqrt(alphar^2 + ((Kxk/Kyaprime)^2)*(kappa^2)))*sign(alphar); s = R0(ntire)*(sSz1(ntire) + sSz2(ntire)*(Fy/Fz0prime)+(sSz3(ntire) + ... sSz4(ntire)*dfz)*spin_camb er)*lambdas; Mzr = Dr*cos(Cr*atan(Br*alphar_eq)); Fyprime = Fy SVyK; t = Dt*cos(Ct*atan((Bt*alphat_eq) Et*(Bt*alphat_eq atan(Bt*alphat_eq))))*cospa; Mzprime = t*Fyprime; Mz = Mzprime + Mzr + s*Fx;
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67 % ************************************************ *********** % OverTurning moment equation % *********************************************************** Mx = Fz*R0(ntire)*( qSx1(ntire) qSx2(ntire)*spin_camber + qSx3(ntire)*Fy/Fz0(ntire) ); % ********************************************************* ** % Rolling Resistance equation % *********************************************************** VroverVo = 1.0; My = Fz*R0(ntire)*( qSy1(ntire)*atan(VroverVo) + qSy2(ntire)*Fx/Fz0(ntire) );
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68 REFERENCES [1] Optimum G. (2012) El ectronic References [online]. Available: http://www.optimumg.com/OptimumGWebSite/Documents/OptimumK%20Help%20File%2 0v1.1/Wheel_legacy.htm [2] Milliken, W.F. and Milliken, D.L. (2002): Chassis Design principles and Analysis. SAE 2002 51 73, 77 82, 213 221 [3] Karnopp, D. (2004): Vehicle Stability. Marcel Dekker Inc. 2004 97 145 [4] Milliken, W.F. and Milliken, D.L. (1995): Race Car Ve hicle Dynamics. SAE 1995 14 82, 125 229, 782 791 [5] Esmailzadeh, E., Goodarzi, A., Vos soughi G.R. (2003): Optimal yaw moment control law for improved vehicle handling McGraw Hill 2003. 98 100 [6] Zhu, L.D., Li,L., Xu, Y.L and Zhu,Q. (20 11 ). Wind tunnel investigations of aerodynamic coefficients of road vehicles on bridge deck [online ]. Available: http://www.sciencedirect.com/science/article/pii/S0889974612000060 [7] Honda Fit Sport. (2009). Electroni c References [online]. Available: Honda News Web si te: http://www.hondanews.com/channels/honda automobiles fit press kit/releases/2009 honda fit sport [8] Nikravesh, P.E. (2008): Planar Multib ody Dynamics. CRC press 2008 47 65, 73 83 [9] Baruh, H. (1999): Analytical Dynamics. McGraw Hill 1999 153 157 [10] Wong, J.Y. (2001): Theory of Ground Vehicles. John Wiley and Sons Inc., 2001 3 43, 335 338 [11] Jazar R.N. (2008) : Vehicle D ynamics: Theory and Application. Springer, 2008 [12] Oxford English Dictionary Oxford University Press Dash [13] Holmes, M.H., (2009): Introduction to the Foundations of Applied Mathematics Springer. 2009 [14] Pacejka, H.B. (2002). Tyre and Vehicle Dynamics. Butterworth Heinmann, 2002 1 7, 61 63 172 191 [15] Pacejka, H.B. and Besselink, I.J.M. (2007). Magic Formula with Transient Properties, Vehicle System Dynamics. Internatinal Journal of Vehicle Mechanics and Mobility, 27:S1, 234 249
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69 [16] Kasprzak, E.M. and Gentz D. (2006). The Formula SAE Tire Test Consortium Tire Test ing and Data Handling SAE 2006 [online]. Available: http://www.millikenresearch.com/TTC_SAE_paper.pdf [17] Desert Rides (2007): Electronic References [online]. Available: http://www.desrtrides .com/reference
