Citation |

- Permanent Link:
- http://digital.auraria.edu/AA00003179/00001
## Material Information- Title:
- Vehicle modeling utilizing skidpad data
- Creator:
- Garber, John Henry
- Publication Date:
- 2009
- Language:
- English
- Physical Description:
- xv, 107 leaves : ; 28 cm
## Subjects- Subjects / Keywords:
- Automobiles, Racing -- Mathematical models ( lcsh )
Automobiles, Racing -- Chassis ( lcsh ) Automobiles, Racing -- Springs and suspension ( lcsh ) Formula vee - Genre:
- bibliography ( marcgt )
theses ( marcgt ) non-fiction ( marcgt )
## Notes- Bibliography:
- Includes bibliographical references (leaves 106-107).
- General Note:
- Department of Mechanical Engineering
- Statement of Responsibility:
- by John Henry Garber.
## Record Information- Source Institution:
- |University of Colorado Denver
- Holding Location:
- Auraria Library
- Rights Management:
- All applicable rights reserved by the source institution and holding location.
- Resource Identifier:
- 515968696 ( OCLC )
ocn515968696 - Classification:
- LD1193.E55 2009m G37 ( lcc )
## Auraria Membership |

Full Text |

Vehicle Modeling Utilizing Skidpad Data
by John Henry Garber B.S. University of Colorado Denver, 2002 A thesis submitted to the University of Colorado Denver in partial fulfillment of the requirements for the degree of Master of Science Mechanical Engineering 2009 by John Henry Garber All rights Reserved. This Thesis for the Master of Science degree by John Henry Garber has been approved by Ronald Rorrer fOtfV 30 2.GG[ Date Garber, John Henry (M.S. Mechanical Engineering) Vehicle Modeling Utilizing Skidpad Data Thesis directed by Professor John Trapp ABSTRACT Formula Vee racing is one of the most popular forms of amateur racing in the United States. One of the biggest challenges in any automotive racing is the proper setup of the car. For this class of vehicle, there are only a few tire types allowed by the sanctioning bodys rules. Regardless of its popularity, there is no published manufacturers data for any of the available tires. The lack of tire data can make choosing the correct type a matter of trial and error. To compound the problem each vehicles suspension can have different setups which can change the dynamic balance of the vehicle and effect the forces on each tire. This also means that setting up a car the same as a competitor is not the best way to be competitive because every driver is different as well. The base line of a vehicle can be determined experimentally. This publication outlines the principles involved in analyzing the Formula Vee race vehicle. Areas covered include suspension analysis, derivation of tire properties, and a look at the classic vehicle roll model and its application specifically to the Formula Vee chassis. This abstract accurately represents the content of the candidates thesis. I recommend its publication. Signed DEDICATION To my family. ACKNOWLEDGEMENT My thanks to Joe Cullen for the patience and intellectual support that helped to guide me and for which this body of work would never have been possible. TABLE OF CONTENTS List of Figures....................................................xi Preface...........................................................xiv Chapter 1. Introduction....................................................16 1.1 Motivation for Studying Race Vehicle Dynamics..................16 1.2 Motivation for Research........................................16 1.3 Overview of Research.......................................... 17 1.4 Objectives of Research.........................................17 1.5 Benefits of Research...........................................18 1.6 Literature Search..............................................18 1.7 Outline and Chapter Summary....................................18 2. A Closer Look at the Problem....................................19 2.1 Slip Angle.....................................................20 2.2 Slip and Thrust................................................22 2.3 Low Speed Cornering............................................23 2.3.1 Off Tracking.................................................25 2.4 Initial cornering stiffness....................................26 2.5 High Speed Cornering...........................................28 2.6 Tire Data......................................................29 2.7 Tires..........................................................30 2.8 Tire Model.....................................................31 2.9 Inputs That Effect Slip Angle..................................34 2.10 Steering Geometry............................................34 2.10.1 Taking Advantage of Steering Geometry.......................34 2.11 The Steering Problem.........................................35 3. Classic Vehicle Model...........................................36 viii 3.1 Suspension Geometry and Roll Axis..............................36 3.1.1 Roll Center SLA..............................................37 3.1.2 Suspension Roll Center Height................................40 3.2 Roll Axis SLA................................................40 3.2.1 Roll Stiffness...............................................41 3.2.2 Roll Angle SLA...............................................43 3.3 Distribution of Forces During Turning..........................43 4. Formula Vee Chassis Specific Equations..........................45 4.1 Front Suspension...............................................45 4.1.1 Front Roll Center............................................47 4.1.2 Front Suspension Forces......................................47 4.2 Rear Suspension Roll Center..................................49 4.2.1 Rear Roll Center.............................................50 4.2.2 Rear Suspension Forces.......................................53 4.2.2.1 Jacking Effect.............................................54 4.3 Roll Axis......................................................55 4.3.1 Roll Angle.................................................56 5. Setup, Measurements and Procedure...............................61 5.1 Initial Setup and Static Measurements........................63 5.2 Testing Procedure..............................................64 5.3 Yaw Analysis...................................................65 5.3.1 Rear Suspension and Contact Patch Forces.....................72 5.3.2 Front Vertical Forces........................................77 5.3.3 Finding Tire Constants Cm and Wm.............................78 5.3.4 Yaw and Front Slip Angle Verification........................84 6. Results.........................................................87 6.1 Error Sources..................................................93 IX 7. Conclusion......................................................95 Appendix A. Table of Variables.............................................96 B. Vehicle setup pictures........................................100 C. How Tires Generate Lateral Forces.............................104 References........................................................106 x LIST OF FIGURES Figure 2.1 TURNING ANGLES..............................21 2.2 SLIP ANGLE..................................22 2.3 CAMBER ANGLE AND THRUST.....................23 2.4 LOW SPEED TURNING...........................24 2.5 FORCE VS. SLIP ANGLE........................27 2.6 HIGH SPEED TURNING..........................29 2.7 CALSPANS TIRF TESTING MACHINE..............30 2.8 PARABOLIC CURVE.............................33 3.1 SLA SUSPENSION..............................37 3.2 HALF SIDE SLA...............................38 3.3 INSTANTANEOUS CENTER........................38 3.4 CENTERLINE VIEW.............................39 3.5 ROLL CENTER.................................40 3.6 ROLL CALCULATION VARIABLES..................41 3.7 ANTI-ROLL BAR...............................42 3.8 FBD SLA SUSPENSION..........................44 4.1 PARALLEL ARM SUSPENSION SIDE VIEW...........46 4.2 PARALLEL ARM SUSPENSION TOP VIEW............46 4.3 FRONT SUSPENSION FRONT VIEW.................47 4.4 PARALLEL ARM FBD............................49 4.5 REAR SUSPENSION VIEW........................50 4.6 REAR ROLL CENTER............................51 4.7 INSIDE WHEEL AXLE COMBINATION...............52 4.8 OUTSIDE WHEEL AXLE COMBINATION..............52 xi 4.9 VEHICLE BODY.................................53 4.10 JACKING.....................................54 4.11 FRONT AND REAR ROLL CENTERS.................55 4.12 ROLL AXIS...................................56 4.13 INNER REAR SUSPENSION.......................57 4.14 SUSPENSION GEOMETRY VARIABLES...............58 4.15 VARIABLE LABELS.............................58 4.16 FLUCRUM POSITION............................59 4.17 REAR ROLL ANGLE.............................60 5.1 PROCESS FLOWCHART............................62 5.2 CENTER OF GRAVITY LOCATION...................64 5.3 YAW GEOMETRY.................................66 5.4 FRONT AXLE STEERING GEOMETRY.................67 5.5 FRONT SLIP ANGLES............................68 5.6 REAR SLIP ANGLES.............................70 5.7 REAR INNER SLIP ANGLE........................71 5.8 REAR OUTER SLIP ANGLE........................72 5.9 FBD SUSPENSION ROCKERS.......................74 5.10 FBD INSIDE WHEEL AXLE.......................75 5.11 FBD OUTSIDE WHEEL AXLE......................76 5.12 FRONT FBD...................................78 5.13 CORNERING COEF. AND STIFFNESS PLOT..........79 5.14 BASIC PARABOLA..............................81 5.15 WM VS. CM FOR AY = 0.5......................83 6.1 BASELINE CURVES..............................89 6.2 PREDICTED CURVES.............................90 xii 6.3 TEST UNDERSTEER GRADIENT....................91 6.4 PREDICTED UNDERSTEER GRADIENT...............92 6.5 TIRE PROPERTIES DATA POINTS.................93 B.l TEST VEHICLE...............................100 B.2 DATA AQUSITION UNIT........................101 B.3 FRONT LINEAR POSITION SENSORS..............101 B.4 REAR ROCKER POSITION SENSORS...............102 B.5 STEERING SENSOR............................102 B.6 REAR G SENSOR..............................103 B.7 FRONT G SENSOR.............................103 xiii PREFACE I had originally started this work as an investigation into the vehicle dynamics of the Formula Vee race car. The goal was to define the car in terms of a multi-bodied dynamic system to learn more about race car handling dynamics. Having defined the vehicle in mathematical terms, I was going to make predictions about cornering properties and compare those predictions to actual test data. The test platform was a continuation of my under graduate senior design project: Chassis Design and Evaluation. The purpose of this project was to design an ultra rigid Formula Vee chassis. After my graduation the chassis sat in a campus lab for a while. When I returned to the university to get my graduate degree, in Mechanical Engineering, I knew that I wanted to further investigate the unique properties of the rear suspension of this class of race car. During the course of this research I started finding problems with the system of equations I had created for solving this dynamics problem. Ill conditioned matrices were wreaking havoc on this research and caused delays of its presentation. I started having concerns about being able to complete my masters degree requirements on time if at all. I was forced to re-evaluate my research from the start. Searching back through my text books I discovered that the assumptions that I had made about tire models were off base and too complex. I needed to find a way to xiv determine the properties of the tires that were on the test vehicle. This was the beginning of a unique method for finding tire properties. xv 1. Introduction This chapter explains the motivation for studying race car vehicle dynamics and this particular research. A general overview of the research is contained within this chapter. Background information that is relevant to race car vehicle dynamics and research objectives are introduced. A description of how this research is intended as a tool for others is included. Finally, an outline of the thesis and a brief description about the contents of each chapter are also presented here. 1.1 Motivation for Studying Race Vehicle Dynamics The technical objective when designing or tuning a race car suspension is to create a vehicle with good handling characteristics. The configuration of a race car vehicle suspension has to be done within a practical interpretation of the rules. For each sanctioning body, the rules are different for every class of race vehicle. When the rules limit the scope of suspension modifications, finding the right tire combination becomes very important. 1.2 Motivation for Research The average Formula Vee racer typically does not have a budget for research and development. Most of the testing and tuning is done at the track. Predicting the adjustments necessary for better handling and gaining a competitive advantage is a matter of trial and error. If you dont have a background in engineering and understand the effect on the total vehicle, optimizing the handling can be very hard to accomplish. 16 One of the best ways to analyze any vehicle handling properties is to perform a series of skid pad tests. SAE has developed a set of procedures for testing transient and steady-state handling properties for cars and light trucks. These procedures are described in described in SAE J266 [1]. Major car manufactures have used these procedures for years to analyze handling characteristics. Most racers rely on information discovered from track laps. Seldom do racers ever use such simple tests to analyze their vehicles. Skid pad testing can help to quickly develop a baseline from which any changes can easily be evaluated. 1.3 Overview of Research The use of approximations is common in engineering. They provide a means to simplify a complex system and yield a useful solution to the problem. The types of approximations used are determined by the degree of accuracy required. The test platform for this research is a race prepared SCCA Formula Vee. The vehicle has been outfitted with a sensor array to measure its suspension deflections, steering angles, lateral and longitudinal accelerations. 1.4 Objectives of Research The intent of this paper is to find a straight forward method to determine the handling and tire properties for a Formula Vee that is using biased ply non-treaded tires. There is no tire model data available for any of the race car tires for the Formula Vee. For the part-timer or small circuit racer it is more or less a trial and error process to find the tires and correct chassis setup. 17 1.5 Benefits of Research The benefits of this research are to find a low budget method for evaluating handling performance. Laid out in the following pages is a tool that can be used for evaluating baseline of a vehicles performance and the effects of any suspension or tire changes made to the vehicle. Contrary to passenger cars which have suspensions that are designed for comfort, high performance race car suspensions are designed and tuned to help the car stick to the road. For racing, small changes in suspension stiffness, shock damping, roll resistance, camber and caster angle; as well as changes in center of gravity can make a measurable difference in vehicle performance. 1.6 Literature Search A search of SAE Publications electronically [2] and print [3] result in no cost effective methods in finding tire properties and using said properties to find and predict vehicle properties. 1.7 Outline and Chapter Summary Chapter 2 has an explanation of the problem of tuning a vehicle for racing including a look at tire construction and force development mechanisms. Chapter 3 contains a detailed description of the classic mathematical tools used to analyze vehicle performance. Chapter 4 will show the adaptation of the mathematical tools from Chapter 3 to be used on the unique suspension of the Formula Vee. Chapter 5 is a description of the vehicle setup, measured parameters and the methods used to find quantify baseline performance. 18 2. A Closer Look at the Problem Formula Vee is one of the most popular racing classes in the Sports Car Club of America (SCCA) [4 p. 191]. The requirements of this formula car class puts limits on engine capacity, weight, tire type, rim and size, front suspension and steering components, and rear axle and transmission type. However there are no limitations on the vehicle setup specifically steering linkage, toe, camber, ride height, and rear suspension design. Formula Vee chassis are all custom built. There are a few commercial manufacturers that sell bare frames requiring some assembly as well as complete kits. Through careful interpretation of the SCCA rules, a custom chassis may be built from scratch. In either case everyone has their own opinion on the proper vehicle setup and there is no clear opinion about what is the best setup. The Formula Vee, like other classes in SCCA racing, is called a restricted class. The restrictions limit the customizing of the vehicle in order to emphasize the preparedness and skill of the driver. The rules establish limits to adjustments made to the vehicle to ensure homogeny across the class [4 pp. 191-215]. Adjustments allowed are tire size and compound, suspension spring and damping rates, anti-roll bar size, camber, toe and caster adjustments, steering linkage and rear suspension type. All of these adjustments effect what is called the balance of the vehicle and can make the winning difference with the right driver. The balance of the vehicle is critical. Because all Formula Vee chassis are slightly different, a specific change to one may not work as well on 19 another chassis. Adjustments to the balance of the vehicle become critical during high speed flat cornering. A suspension system, in general, is a relatively easy system to analyze. When a vehicle is moving forward along straight path, on a level surface, the loads on each tire can be determined from the distribution of its mass. When a vehicle enters a flat corner the transient condition causes the vertical loads on the tires to change. The way that the suspension is tuned affects the magnitude of the change of the vertical loads on the tires. One of the most important decisions about vehicle handling is tire choice. In order to be able to choose the correct tire one must understand the relationship between slip angles and developed lateral forces. There are only two manufacturers making racing tires for the Formula Vee class with a few compound and size choices each. Neither of these manufacturers provides any tire data that would help make choosing one over the other less difficult. 2.1 Slip Angle Slip is a term that is used to describe the deformation of a tire when it changes direction. Slip does not imply that there is any sliding, skidding or loss of traction. Slip comes from an aerodynamics term used to describe when an airplanes nose is not in line with its forward velocity vector during flight. In the automotive industry slip angle a is the angle measured between the wheels heading and its velocity vector. The velocity vector is tangential to the path of the tire, see Figure 2.1. On a turning vehicle the velocity vector of the tire is different than that of the vehicle. When turning, the steering wheels (tires) are turned to 20 generate the slip angle required to make the vehicle change direction. This is called the steering angle 8 and is different for the inner and outer tires. The measurement of steering angle is made between the longitudinal axis of the vehicle and the wheel heading, see Figure 2.1. The steering angle required is dependent on the vehicles velocity and the corner radius. Heading FIGURE 2.1 TURNING ANGLES A tire develops forces in response to steering inputs. When turning, tires develop lateral forces that keep the vehicle on the path, and at the same time the tires develop a slip angle. Deformation occurs between the contact patch and the rolling plane of the tire. This deformation is measured in angular degrees. During cornering the centrifugal forces from the vehicle are counteracted by the lateral forces developed on the ground at the contact patch of the tire. The ability of the tire to distort is what makes cornering possible. The lateral force is assumed to be directly in the center of the contact patch. 21 Path of Tire Slip Angle Wheel Heading Lateral Force Velocity of Tire Contact Patch Distortion Recovery Area FIGURE 2.2 SLIP ANGLE The slip angle is important to a vehicles ability to corner, it is key to how much lateral force can be developed. There is a maximum slip angle that correlates to a maximum lateral force on a tire for a given vertical load on that tire. 2.2 Slip and Thrust Tires generate lateral forces by two mechanisms, camber thrust and deformation. Camber is the angle at which the wheel makes to the vertical when viewed from the front or rear. 22 W/cos B W tan P W FIGURE 2.3 CAMBER ANGLE AND THRUST Figure 2.3 illustrates a vehicle with a negative camber. Negative camber is a condition where the top of the tire leans towards the body of the vehicle. Camber thrust occurs when a tire rolls at a camber angle (/>. This lateral force component, camber thrust Fyc is always present when the vehicle is traveling straight or cornering. Whether the top of the wheel is tilted towards or away from the centerline of the vehicle determines if it is negative or positive camber, respectively. Camber thrust varies with suspension movement as well as vehicle roll and is a function of suspension design. Its overall contribution to the total lateral force is small if the camber angle is small. 2.3 Low Speed Cornering At low speed cornering, 5 mph maximum, tires develop negligible slip angles [5 p. 196]. Body roll and lateral load transfer are negligible as well. There are lateral forces acting on the front and the rear of the vehicle and each of the tire rolls on its own concentric circular path. The front tires 23 roll on two different radii, the front inner being smaller than the outer. Ackerman steering assumes that the center of the turn lies on the projection of the rear axle and a perpendicular line projected from both the front wheels pass through the center shown in Figure 2.4. I R CENTER OF TURN FIGURE 2.4 LOW SPEED TURNING A steering geometry is said to be Ackermann when the wheel angle of the inside turning wheel is rotated a greater angle into the turn than the outside wheel. There is a particular Ackermann angle that is dependent on the track, the wheel base of the vehicle and the radius of the turn. Proper setup for the front steering geometry, assuming small angles, is determined mathematically. 24 Steer angles: l (2.1) The average of the two steer angles do and Si is the Ackermann angle in radians. If correct Ackermann steering is not established then the tires will experience scrub. Scrub is when a tire slides in a direction that is outside of its rolling plane. 2.3.1 Off Tracking Ackerman is a simple model. At low speeds the rear experiences off tracking [5 p. 197] which is the distance off the centerline of the turn that actually locates the rear track. This is because the rear wheels cannot follow exactly in the same tracks as the front wheels. (2.2) Off tracking distance: ( l \ A = R 1 cos \R y (2.3) 25 Using a series expansion: cos(z) =1 2! 4! 6! (2.4) Now: (2.5) The larger the wheel base l of the vehicle the worse the off tracking becomes. For the purposes of this research Ackerman steering and off track calculations are do not apply to this research. The test vehicle is not making low-speed turns. 2.4 Initial cornering stiffness All tires have an inherent resistance to out of plane twisting. This resistance is called the initial cornering stiffness C measured in pounds per degree. The initial cornering stiffness is dependent on the vertical load on the tire: for each vertical load Fz there is a unique value of the initial cornering stiffness. The cornering coefficient C is the initial cornering stiffness per pound of vertical load on the tire. The units are pounds per degree per pounds which works out to be deg1. 26 Force vs. Slip Angle FIGURE 2.5 FORCE VS. SLIP ANGLE The initial cornering stiffness C is the slope of the lateral force vs. slip angle curve FY' at a 0 as shown in Figure 2.5. The value of C is determined by the design, construction, inflation pressure, and the vertical load on the tire. For low-speed turns the values of the slip angle are small the initial cornering stiffness is used to determine a tires lateral force [5 p. 198]. FY = Ca (2.6) It is possible to derive the initial cornering stiffness for the tire experimentally using the constant radius test procedure outlined in the SAE J266 standards [1]. 27 2.5 High Speed Cornering At speeds greater than 5 mph the body roll, weight transfer, lateral acceleration, and suspension and tire compliance effect handling. During cornering the lateral acceleration increases as velocity increases or as the radius of the turn decreases. Lateral Acceleration: A (2.7) In order for a vehicle to be able to perform a high speed cornering maneuver the centrifugal force created by the collective masses must be counteracted by the forces developed at the contact patch of the tire. In response to cornering, a proportional lateral force is generated. Figure 2.5 shows, in general, the relationship between the vertical loading on the tire, the slip angle and the lateral force that is generated. The lateral force increases as the slip angle increases, but without low speed assumption of a linear relationship. Instead, the slope dFy/da decreases and becomes zero when the slip angle reaches the value am. After this point Fy decreases as the slip angle increases. The value of am for a given a tire is dependent on the vertical load on the tire provided that the load is the only variable. The curves for every tire differ, and currently can only be determined experimentally. In Figure 2.6 each tire has a different vertical loading and slip angle. The lateral acceleration increases the slip angles on the rear tires, causing the rear of the vehicle to swing out. This is called yaw and is perceived by the 28 driver as if the vehicle is rotating around the inside front tire. This is because the path of the vehicle is controlled by the front inside wheel and any corrections to the course emanate from here. Vehicle Path FIGURE 2.6 HIGH SPEED TURNING 2.6 Tire Data The traditional way to get tire data is from a tire testing apparatus. The machine shown in Figure 2.7 is at the Calspan Tire Research Facility (TIRF) designed and built in 1972. TIRF can completely define and measure the forces and moments transmitted between the tire and the road. 29 FIGURE 2.7 CALSPANS TIRF TESTING MACHINE Well-funded racing like NASCAR, Formula 1, and the IRL can afford to lease time on one of these testing machines and can collaborate with the tire manufactures directly to design a tire for their needs. 2.7 Tires The most complicated part of vehicle dynamic analysis is modeling the tire. Race tire compounding and construction specifications are closely guarded of the manufacturers, as well any tire data. Only the most prestigious and big budget race teams can work with the manufacturers to design tires to fit their needs. As for the rest of the racing world it is left to experimentation. 30 All tires share similar features: a bead, sidewalls, and a tread. Aside from these similarities each series of racing tires are designed with a specific purpose. There are drag racing, slalom, dirt, road, rock, rain tires and so forth. The primary function of the tire is to provide grip. Grip is developed in a small area called the contact patch and is of major importance for the dynamic performance of a race vehicle. Longitudinal and lateral forces that enable a vehicle to accelerate, brake and corner are developed at the contact patch. These forces are the tires reaction to the loads imposed on it by the distributed masses of the vehicle and suspension. The ability and to what limits a tire can generate grip is dependent on many variables. Physical properties such as compounding, cord type and angle, height, width, and side wall thickness in part determine the initial cornering stiffness. Other variables effecting the initial cornering stiffness are determined by chassis setup and environmental conditions: tire pressure and temperature, track temperature, static and dynamic loading, and suspension setup. 2.8 Tire Model When doing a dynamic analysis of a vehicle a set of complex equations are used to simulate the tires. Tire rubber is a polymer with viscoelastic properties. Just by the nature of its characteristics the mathematics involved in modeling a tire are extensive and non-linear. Typically these equations have numerous variables many of which have little effect on the calculation of lateral force. 31 For the purposes of simulation a tire model is required. There are many of tire models available that range in complexity. Currently popular is the Magic Formula developed by Pacejka [6] which is continually undergoing refinement. The Magic Formula and others, including the Fiala tire model, SWIFT tire model, Radt curves and various parameter based models, are all designed to reproduce the laboratory curves created from real data collected on tire testing machines. For this study many of the many of these formulas are far too complex to use. For instance, the Magic Formula can have well over 50 parameters depending on its application [6 pp. 172-196]. Instead, a parabolic curve fit based on a non-dimensional representation of tire data, developed by Dr. Hugo S. Radt based on the work of Fiala, will be used. The accuracy of this method is demonstrated in Chassis Design: Principals and Analysis [7 pp. 59-71], Parabolic tire model: Fy ( = Ca 1- V 1 Ca " 4 max y (2.8) The parabolic model has been found to generate curves similar to the ones generated on tire testing apparatus [7 pp. 60 66]. 32 FIGURE 2.8 PARABOLIC CURVE In Equation (2.8) and Figure 2.8 the maximum lateral force generated is ultimately a function of friction coefficient and the vertical force on the tire: Fy max = MFz (2.9) The initial cornering stiffness C in Equation (2.10) is not only a function of vertical force Fz, but a function of the maximum initial cornering stiffness Cm and the maximum normal load Wm. The initial cornering stiffness is Cm when the normal load is Wm. The typically value for ju is 1.20 for paved roads [7 p. 69]. The values for C and /u may be determined experimentally, which make Equations (2.8) and (2.9) well suited for this study. 33 C=Func(FCm,Wm) (2.10) An important property to note when looking at Figure 2.5 and Figure 2.8, is that slip angle continues increase while the lateral force increase up to a maximum then decreases. The value of the slip angle where this occurs varies from tire to tire and depends on tire loading. 2.9 Inputs That Effect Slip Angle The force that is developed at the contact patch for a particular can be modified to peak at different load conditions. This can be done by lowering or raising the tire air pressure, or changing the amount of camber in the suspension. On cars that utilize aerodynamic downforce the amount of vertical load on the suspension maybe adjusted as well. 2.10 Steering Geometry In Section 2.3 the idea of Ackerman steering was introduced. While this is well suited for passenger vehicles there are other steering geometries possible for track racing; Parallel Steer and Reverse Ackerman. For Parallel Steer the front tires remain parallel to each other through the whole range of motion of the steering system. On Reverse Ackerman the inside tire does not rotate as far as the outside tire. Both of these other steering geometries can have an adverse effect on tire wear but may provide an advantage in high-speed cornering. 2.10.1 Taking Advantage of Steering Geometry Referring back to end of Section 2.8, there is a slip angle value am that corresponds to the maximum generated lateral force. It is possible to alter the amount of lateral force generated by the front tires by changing the 34 range of slip angles over which the tire operates. This is done by taking advantage of the different steering geometries. Turning a tire too much or too little can shift the point where the maximum lateral force peaks. Ackerman steering works well for low-speed turns, allowing the tires to track on their own paths, but does not produce the best conditions for high-speed cornering. Using a Reverse Ackerman setup over steers the outer front tire progressively as the radius of the turn decreases or as the lateral acceleration increases. This can cause the front outer tire to reach the maximum slip angle at a lower turning velocity and may cause the front to plow through the turn. The Parallel Steer setup allows the front outer tire to run at a slightly higher slip angle than the Ackerman Steer, again reaching a higher value of slip angle sooner. 2.11 The Steering Problem One of the problems encountered in high-speed cornering is that the inside tire may reach the maximum lateral force that it can produce before outside tire does. This can affect the maximum cornering speed. Beyond a specific slip angle value am a tires maximum generated lateral force starts to decrease. If aFi is too large then the tire is literally being dragged along. To alleviate this problem the amount of steer angle on the tire should be reduced. For this study the test vehicle has been setup with Parallel Steer geometry. 35 3. Classic Vehicle Model Automotive designs have advanced using basic analysis methods developed in the middle of the 20th century. Tabulating values and physical testing have driven new designs in the areas of chassis, suspension, and tire development. In the latter half of the 1980s computation methods for analyzing vehicle design started emerging which allowed engineers to further fine tune chassis, suspension, and tire designs. This was a quicker and less expensive way to prototype designs. But without an understanding of how and why the classical methods worked these new methods could not evolve. 3.1 Suspension Geometry and Roll Axis A system of forces from a vehicle suspension system, for a steady-state turn, can be resolved in to a lateral force, a vertical force and a roll moment at the suspension roll center. The roll center is a point attached to the body that is determined by the suspension geometry, and can be determined by applying a virtual displacement or rotation to the suspension links. It turns out that the roll center is the instantaneous center of rotation of the system. After locating the ICR the principle of virtual work can be used to find the forces in the system. Using these two methods in a quasi static solution, the effect of the change in vertical load on the contact patch of the tires can be calculated. 36 Each type of racing calls for specialized suspension setups. Usually the rules dictate the how exotic a suspension may be. There are several different types of vehicle suspensions. One of the most common is the SLA (Short Long Arm) suspension. As a demonstration the location of its roll center will be detailed. The front and rear roll centers will then be used to construct the roll axis. Figure 3.1 shows the SLA suspension in a basic layout. There are many different configurations possible depending on a vehicles designed application. In general the SLA suspension allows independent wheel movement up and down and little change in camber along a straight path. For turning the SLA offers good control of outside camber. 3.1.1 Roll Center SLA Finding the instantaneous center of rotation of a suspension system involves looking at each of the suspension links. To start, look at the suspension from along the centerline of the vehicle and draw the links as they exist. Consider the body as stationary and that all the connections 37 from body to the links, from the links to the wheels and from the wheels to the ground are pin connections and have one degree of freedom. Analyzing the links involves imposing a second order rotation at points A and B, see Figure 3.2, then looking at the velocity vector of points C and D. FIGURE 3.2 HALF SIDE SLA Drawing a right angle line to the velocity vector the ICR of the two links can be graphically determined by the intersection of the two lines. FIGURE 3.3 INSTANTANEOUS CENTER Since point E is on the same solid body, the wheel, as C and D the velocity vector of that point can be found by drawing a line from the ICR to E. The 38 velocity is perpendicular to the line between E and the ICR. Repeat this procedure on the other side of the vehicle. In Figure 3.4 the suspension is shown as symmetrical about the centerline, but some vehicles may have an asymmetrical design. Typically an asymmetrical suspension is found on vehicles designed to turn in one direction, but the process of finding the roll center remains the same. Combining the left and right hand sketches shows an intersection of the lines drawn from the ICRs to the tire contact patches. Where these lines meet the centerline, is the roll center of the suspension for that instance. 39 Here the analysis has been more qualitative but the mathematics is just straight forward trigonometry. 3.1.2 Suspension Roll Center Height The distance from the suspensions ICR to the road surface is called the suspension roll center height see Figure 3.6. The front and rear height, Hf and h,R respectively, can be different. Most commonly the front ICR is designed lower than the rear. 3.2 Roll Axis SLA The roll axis is determined by the front and rear suspension roll center height. A line is drawn through the front and rear suspension ICRs, this is the roll axis for the whole car. As the vehicle speed increases the centrifugal force on the sprung weight causes it to rotate about this roll axis. Dynamic calculations are made for the roll angle based on the height of the center of gravity from the roll axis. The front and rear suspension tries to resist this over turning moment via their individual roll stiffness. 40 FIGURE 3.6 ROLL CALCULATION VARIABLES 3.2.1 Roll Stiffness The roll stiffness of the suspension is a designed parameter. On a SLA suspension that has a suspension spring and damper on each tire when the vehicle corners, the body rolls towards the outside of the turn. The outside suspension springs are compress resisting roll, but the opposite happens for the inside wheels as the springs unload driving the roll. If the cornering speed is fast enough the vehicle will become unstable and spin out or flip over. In order to correct this problem an Anti-roll bar is added to the system. This is usually some sort of transverse spring linking the wheels on a single axle to each other and the chassis, and is installed on the front suspension. The spring increases stiffness as the vehicle rolls, which decreases the vehicles rate of roll and redistributes the vertical loads. The simplest of Anti-roll bars is a U-shaped piece of spring steel attached to the suspended chassis and each end to the suspension linkages. 41 FIGURE 3.7 ANTI-ROLL BAR The roll of the body increases the effective stiffness of the Anti-roll bar by two mechanisms; torsional resistance and bending resistance. Here is a typical calculation of the front suspension stiffness. Front Roll Stiffness: K = K + K &SF (3.1) The front roll stiffness is the sum of the Anti-roll Bar stiffness due to body roll and the suspension stiffness due to the springs Equations (3.2) and (3.3) K k s^ Kbar*RB (3.2) 1 ( Grl 3EI' R2 { lt ' l3 Lb y (3.3) 42 (3.4) K K t2 2 SPrinSl'F Typically the first adjustment in roll stiffness is the installation of an Anti-roll bar on the front suspension. A turning vehicle experiences the greatest change in force at the front outer tire. 3.2.2 Roll Angle SLA The roll angle is a measurement of how much the chassis rotates about the roll axis. In Figure 3.6 the center of gravity is above the roll axis a distance hi, which directly affects the amount of chassis roll. The greater this distance the more exaggerated the roll angle. Roll Angle: \\VAy ~K^F+K^R-Wh1 (3.5) 3.3 Distribution of Forces During Turning If all the right hand parameters in Equation (3.5) are known, chassis roll of a vehicle traveling around a curve can be calculated. Using the value for (j), the change in mass distribution and the amount of vertical force, the suspension exerts on the tires can be calculated. The change in vertical load effects the initial cornering stiffness and alters the tires ability to generate lateral force over a range of slip angles. 43 FIGURE 3.8 FBD SLA SUSPENSION Summing the moments around O The change in vertical loading front: AFzf - Wp-Ayflp tF Similar to the front the change in vertical loading rear: (3.6) ^ZR ~ ^4 (3.7) This is added to the static vertical load calculation for the outside front tire and subtracted from the inside front tires static vertical load calculation. The process is the same for the rear of the vehicle. Equations (3.5), (3.6) and (3.7) can be coded into a spreadsheet and tabulated for different values of lateral acceleration. 44 4. Formula Vee Chassis Specific Equations The test platform used in this study is a race prepared Formula Vee which has a parallel trailing arm front suspension and a swing axle rear suspension. One of the challenges involved in analyzing the dynamics is adapting the classic vehicle models to this chassis. Unlike most vehicles, the rear suspension linkage on a Formula Vee is setup to eliminate a problem that occurs with the swing axle suspensions called jacking. Section 4.2.2.1 covers the mechanics of jacking in more detail. The result of this special setup is that the rear suspension is uncoupled from the chassis which takes the rear suspension stiffness out of the roll angle calculation, and puts a larger requirement on the front anti-roll bar to control the entire vehicle roll. 4.1 Front Suspension The Formula Vee uses a parallel trailing arm suspension on the front, as shown in Figure 4.1 and Figure 4.2. It is characterized by the upper and lower trailing arm linkages staying parallel with one another as the wheel moves up and down. 45 Upper Trailing Arm Lower Trailing Arm V/////////7777/////////7////// FIGURE 4.1 PARALLEL ARM SUSPENSION SIDE VIEW From the top view of the chassis, the transverse tubes house torsional springs see Figure 4.2 & Figure 4.3. In the stock configuration, both upper and lower tubes house the springs and there is an anti-roll bar attached to the chassis and to each of the bottom trailing arms, left and right side. The Formula Vee is about half the mass of the standard Volkswagen Beetle, so commonly one of the torsion springs are removed and replaced with an anti-roll bar. FIGURE 4.2 PARALLEL ARM SUSPENSION TOP VIEW The spindles are attached to the trailing arms with a link bolt and the camber is adjusted by placing shimming disks between the mating surfaces. 46 4.1.1 Front Roll Center This type of suspension linkage is considered to have a roll center at the ground along the centerline of the vehicle [7 p. 332], 4.1.2 Front Suspension Forces Similar to the SLA suspension the vertical forces can be calculated. The front roll stiffness comes from two springs; a torsional suspension spring and a Anti-roll bar spring. From Section 3.2.1 K. =K + K ^SF (4.1) The Anti-roll bar is similar to the SLA calculation but there is no bending arm. 47 f t \ ka = GJ R2 RB (4.2) \ "t s 1 ( f \ (ll.2xl06psi) (0.625in)4 V 32) \ (5.98m)2 36.375in (47.24m)2 (4.3) = 287840.8 in-lb/rad For the Formula Vee front suspension the torsion spring is only as long as the roll bar. ~ SRB (4.4) K K 2 2 IYsPrinet>RB (4.5) K^f = (33.9/6/m)(47.24m)2 2 (4.6) = 37825.9 inlb/rad The total roll stiffness is =235666.7 in-lb/rad (4.7) 48 FIGURE 4.4 PARALLEL ARM FBD The front roll height is zero which changes Equation (3.6). AFzf - tp COS (j) (4.8) 4.2 Rear Suspension Roll Center The rear suspension of this vehicle is very unique it utilizes a swing axle. There are two short axles connected to the transmission housing called a transaxle. At the inboard point of attachment to the transaxle there is a pivot composed of a spade on the end of the axle, two fulcrum plates and a slotted spider gear that engages the differential. Outboard, the brake drum is keyed to a spline axle end and fixed with an axle nut; the rim and tire are bolted to the brake drum. The whole axle is one rigid assembly with a rotational axis about its length that transmits power to the tire. The axle is constrained to rotate about the fulcrum point when the suspension moves up and down. 49 Unlike most vehicles where the springs are attached to the body, here the left and right linkages are attached to one another via a common spring and damper see Figure 4.5. The primary function of the suspension, outside of keeping of the rear of the chassis off of the ground, is to allow the body to roll without affecting the camber angle of the rear tires. With the correct suspension linkage geometry the rear suspension can achieve this, in doing so the suspension becomes uncoupled from the effects of body roll. This design eliminates the jacking effect typically associated with the swing axle suspension. On the other hand now the roll stiffness of the rear has been eliminated and must be compensated by the front suspension. 4.2.1 Rear Roll Center Figure 4.6 shows a simplified view of the rear suspension of the test vehicle, the set up is shown in a zero camber orientation. The wheels and axles are rigidly connected thus a line may be drawn from the contact patch Cm and Cr0 to the fulcrum points Oi and Oo. The Line CO represents 50 a virtual link, connects the suspended body to the ground at the tire contact patch. Like the SLA suspension a second order rotation is introduced. The rear roll center location is determined by the intersection of two lines drawn from the center of the rear tire contact patch and the pivot point of the rear suspension. 51 FIGURE 4.7 INSIDE WHEEL AXLE COMBINATION Taking the inside wheel axle combination and imposing a second order rotation the direction of the velocity vector at Oi can be determined. % = rcr x % (4.9) By similar process the outside is determined. FIGURE 4.8 OUTSIDE WHEEL AXLE COMBINATION 52 ICR (4.10) (4.11) The exact location of the roll center can be calculated using the rocker angle measurements. These equations are detailed in Section 4.3.1 for the rear roll angle calculations. 4.2.2 Rear Suspension Forces The rear suspension experiences forces at the contact patch, fulcrum points and rocker these are an effect from the vehicles share the distributed masses supported by the rear axle. Because of the unique geometry of the rear suspension and the fact that there is almost zero roll stiffness. The rear roll stiffness cannot be determined using the same methods applied on a SLA suspension. FIGURE 4.9 VEHICLE BODY VOJO, voa vQi 1rcr x(ro0- r0,) = (O aRCR x rojol VO0 ~ RCr X rO / ICR 53 The special geometry of the rear suspension is to reduce the risk of the rear axle to jack under hard cornering. To understand the geometric properties of the suspension a description is required. Remove the front tires and setup the front beam to pivot about its roll center, at the ground. The chassis would roll all the way to the suspension stops, while at the same time the rear suspension camber would remain unchanged. The chassis would be free to rotate uninhibited left or right. There is absolutely no mechanism in the rear suspension geometry to stop the chassis from rolling. The unaffected camber angle prevents the outside axle from developing an overturning force that would normally cause the rear to lift. 4.2.2.1 Jacking Effect A vehicle with a swing axle rear suspension can become unstable when cornering. Both tires develop lateral forces, with the outside one having the largest value of the two. The lateral force on the outside tire tries to lift the vehicle, causing both tires to tuck under. The suspension geometry helps drive the instability because of the location of the axle pivots and raises the rear ICR and the center of gravity. FIGURE 4.10 JACKING 54 The combined effect of the wheel tuck and the rise of the rear ICR is that the lateral force on the rear axle decreases, which increases the probability of the vehicle to spin out and over turning. 4.3 Roll Axis The front suspension geometry results in a fixed roll center at the ground along the centerline of the vehicle. The rear suspension roll center floats around a stationary theoretical roll center as shown in Figure 4.11. FIGURE 4.11 FRONT AND REAR ROLL CENTERS The roll axis is defined by a line that intersects each of the roll centers defined by the front and rear suspension geometry. 55 Front ICR' FIGURE 4.12 ROLL AXIS 4.3.1 Roll Angle Laid out in the next few pages are the geometry equations used to find the roll angle using the suspension rocker angles Oik and 9r0. Inner suspension variables use the subscript i , and outer suspension variables are denoted with the subscript o. Rocker Length: In the following diagrams, physical links at represented by thick lines and virtual links are represented with dashed lines. (4.12) Pushrod Length: (4.13) 56 Rear Roll wRi/ fci A : 1 i 1 T J 7777777ZW777777777777777^777777777?. FIGURE 4.13 INNER REAR SUSPENSION Position of Di from Oi XDi ~ (Lb Lq ) + LRi cos 0Ri (4.14) yDi ~ LRi sin dR[ Hp (4.15) Enables definition of Idi tan0=^ X Di lDl = yl(Xoi)2 +(yDif (4.16) (4.17) 57 \b..LEl ,'O, Pa y'L Ci H. V///7//////////////////)//?/////////, V. FIGURE 4.14 SUSPENSION GEOMETRY VARIABLES The diagrams in Figure 4.14 and Figure 4.15 show the variables used in all the geometric relations between the suspension nodes. The virtual links were created to simplify the geometry between the axle, shock pushrod and the suspension rocker. FIGURE 4.15 VARIABLE LABELS Using the Law of Cosines. T2 T2 ^Pi ~ ^Ei + lDi 2IjEilDl cos0Oi (4.18) 58 (4.19) T2 + 72 T2 COS0Ol= El a p' Ai ~ @0i + P(a @Di sin0 = ~~ sin 8()i L'Pi (4.20) (4.21) Figure 4.16 details the variables used to find the fulcrum height which is used in calculating the rear roll angle. 7777777Z77777777777777777P77777777? FIGURE 4.16 FLUCRUM POSITION The roll angle of the rear suspension can be determined from the suspension rocker angles Ori and 9r0. Equation (4.22) reflects the usage of the angles formed between the virtual link from the contact patch center to the fulcrum pivot and a line through both the fulcrums see Figure 4.17. The vehicle is drawn in an exaggerated state of roll that is the result of the vehicle making a left hand turn and the body rolling to the outside of the turn. 59 La sinea LCo sin0Co FIGURE 4.17 REAR ROLL ANGLE Here is the specific roll angle calculation for this particular rear suspension geometry. Rear Roll Angle: tan^fl = Lasin0cl-Lcosin0co 2Lo + Lc, cos a + Lco cos Co (4.22) 60 5. Setup, Measurements and Procedure The test vehicle initial settings have been dialed in and remain unchanged during the entire testing session. No modifications were done to the steering, suspension or the tire pressures. The baseline of the vehicle and the properties of the tires can be determined experimentally using skidpad tests. This is accomplished by measuring and recording real time vehicle data with a data acquisition system, using an array of sensors to measure suspension deflections, lateral and longitudinal acceleration, and steering angles. The individual slip angles on the tires, when examined at collectively, can be used to establish the baseline measurement of chassis setup and derive the tire properties. Ideally the attitude ^of the vehicle as it moves around the circle would be good information to record using a perpendicular radar system or high speed photographic methods, but both are expensive and would not be accessible by most amateur racers. Instead of measuring y/ directly, the sensor data and tire model are used in an iterative method to find the attitude angle. The coefficient of friction is also determined using this method. When // is chosen too high Wm will not converge and when chosen too low Cm does not converge. In Figure 5.1 the steps for this iterative process is described in a flow chart. 61 FIGURE 5.1 PROCESS FLOWCHART 62 In Chapter 4 the equations for rear suspension deflection, front suspension roll stiffness, roll angle and load transfer were derived. In this chapter the accuracy of these equations will be tested. This will be done by finding the roll stiffness and roll angles experimentally, then evaluating the suspension geometry equations, mathematically, for the Formula Vee chassis and correlate those results with experimental data. The resulting data, from the suspension geometry, will be used to determine the initial cornering stiffness, build the tire model and construct a handling diagram. 5.1 Initial Setup and Static Measurements The test vehicle was set up with parallel steering geometry as described in Section 2.10. The camber angle y was set at -0.5 degree and the toe in was initially set to +0.25 for both the front and rear axles. All tires were of the same make, model, batch, and all tire pressures were set to 20 psi. All the tires were balanced were balanced on matching rims. Before the start of testing static data was taken for the test vehicle including: static loading on each tire, tire rolling radius, suspension deflection and spring rates, roll stiffness, and location of the center of gravity to mention a few. The location of the vehicles center of gravity along its longitudinal axis was determined by measuring corner weights. Next the front of the vehicle was raised to different heights with the rear wheels still on the scales. The shift in wheel loading and the angle of inclination were recorded at these different positions and used to determine the vertical location of the center of gravity. 63 When the driver sits in the vehicle it has reached its testing weight. The location of the C.G. for the vehicle changes slightly when a driver is seated. Using information compiled on human factors the C.G. is corrected for the loaded vehicle [8 pp. 705-772]. This calculation was used to find hi, the height of center of gravity above the roll axis. 5.2 Testing Procedure Skid pad testing is a practical and economical way to establish the vehicles baseline performance. The baseline of a vehicle can quickly be determined by isolating parameters. The repeatability of skidpad testing allows variation of parameters to be used to determine vehicle baseline performance. The use of a skidpad is ideal for this test. The method used was to hold the skidpad radius constant, vary the lateral acceleration and measure the steering angle and suspension deflections. The vehicle test speeds were increased by increments no greater than 0.10 g until the vehicle 64 could no longer hold the skidpad radius. The vehicle was run on a constant radius circle of 15 meters [1], and the data was collected over multiple left and right hand turns. The data acquisition system was set to collect values for the suspension deflections, lateral and longitudinal accelerations, wheel speeds and steering angles. Each data pass consisted of entering the skidpad at a predetermine speed and holding that speed for anywhere from five to seven laps. Each lap was timed and recorded. The test vehicle was stopped and the sensor data was downloaded to a laptop computer. The same test was run again but the skidpad was entered from the opposite direction so that data was collected for clockwise and counterclockwise direction of travel. Opposing runs at the same lateral acceleration kept the suspension from developing a set, ensuring even wear on the tires and averaging out any testing surface irregularities. 5.3 Yaw Analysis Finding tire properties is dependent on being able to determine the chassis attitude angle if/. This angle, also known as yaw, can be used to find the slip angle of the tires. As the vehicle travels around the skidpad the driver is controlling its trajectory by keeping the front inside tire on a predetermined path. In this test that path was an offset distance defined by the cones that outlined the perimeter of the test circle. 65 \ Vehicle Path FIGURE 5.3 YAW GEOMETRY The geometry of the steering wheels and how they rotate about the steering knuckles on either side of the vehicle are used to compute the slip angles for a specific value of the chassis attitude y/. The wheel rotates around the spindle and when the car is steered the spindle/wheel combination rotates about the king pin axis. This movement also rotates the tire contact patch fore and aft of a virtual straight line that would connect the inner and outer tire contact patches if the vehicle had no steering input. 66 p FIGURE 5.4 FRONT AXLE STEERING GEOMETRY The angles that the wheels travel through are their respective steering angles. Trigonometry is used to find the changes in the positions of the contact patches. In this system the contact patch on the inner front tire is considered to be the point about which the chassis yaws. The relative position for the outside front contact patch is calculated from point A, see Figure 5.4. The chassis yaw angle y/ and the steering angle 8 are used to calculate the front slip angles. Additionally an angle 9b is used to determine the slip angle on the outside tire. These angles will be calculated using vector mathematics. xB = 2lsp sin(Â£; +y/) + tk sin(^) yB = 2lSP cos(<5i +y/) + tk cos(y/) (5.1) (5.2) vA = Rco (5.3) CO cok (5.4) 67 (5.5) vB a>kx rB FIGURE 5.5 FRONT SLIP ANGLES rB = [/SP sin(8i + y/) + tk siny/ + lsp sin(SQ + y/)]i + [lSP cos(St + y/ ) + tk cosy/ + lsp cos(50 +w)]j vB = + lsp cos(Si +y/) + tk cosy/ + lsp cos(<70 +y/)\ +ft>[Zsp sin(J( +y/) + tk siny/ + lsp sin (8a + y/)]j (5.6) (5.7) 68 The angle Ob is the difference between the velocities uA and uB which can be found by taking the inverse tangent of the X and Y components of uB . vBX = R + lsp cos{Si +y/) + tk cosy/ + lSP cos(S0 +y/) (5.8) vBY = lSpS(8l+y/) + tksiny/ + lSPsm(S0+y/) (5.9) 0B= tan-1 {^4 (5.10) \VBX\ Now the front slip angles can be calculated. aFl=St+y/ (5.11) The slip angles for the rear are composed of the yaw plus their respective velocity angles. FIGURE 5.6 REAR SLIP ANGLES 70 Rear inner slip angle equations. |YC| = R + lSP cos(c7 +y/^ + cosy/ + Lsiny/ cosy/ 2 2 \XC\ = -lsp sin(Â£. + y/)- siny/ + Lcosy/ + siny/ 2 2 a\XC FIGURE 5.7 REAR INNER SLIP ANGLE i \Xr\ 9r = tan 11 Y rc| aRi = ys + @c (5.13) (5.14) (5.15) (5.16) 71 Rear outer slip angle equations. (5.17) (5.18) FIGURE 5.8 REAR OUTER SLIP ANGLE lY^I = i? + lSP cos(Â£ + y/} + cosi// + Lsin + cosy/ 2 2 |X01 = -lSP sin ( ix D (5.19) r d\ aRo=V/ + 0D (5.20) 5.3.1 Rear Suspension and Contact Patch Forces The data from the suspension position sensors and the accelerometers contains enough information to calculate the forces in the rear suspension and the lateral and vertical load transfers. The load transfers are calculated using the sensor data and the geometric relationships developed for the rear suspension in Chapter 4. This calculation is possible because the rear suspension is a swing axle with no roll stiffness. 72 The forces in the suspension spring are used to calculate the pushrod forces. In the static position, the force in the suspension spring needs to support the rear weight of the vehicle. The spring rate ks is known and using the geometry of the rockers the distance Ls between psi and ps0, in Figure 5.9 is calculated. The forces in the rear suspension and the distance Ls are used to compute an effective uncompressed length for the spring L0. This length L0 will be used to calculate the spring force Fs during testing. Spring force during testing Fs=ks(Lo~Ls) (5.21) The suspension rocker sensor values were used for recalculating the spring force once the chassis has rolled during the skidpad test. The rocker sensors have 0.01 degree resolution. The rocker angles completely define the rear suspension geometry. The FBD for the rockers are shown in Figure 5.9 this figure also shows angles and lengths used to complete the description of the rear suspension geometry. A spreadsheet was used to take the two rocker arm angles, from rocker sensor data, and compute the deflection of the rear suspension as shown in Figure 4.17. 73 FIGURE 5.9 FBD SUSPENSION ROCKERS When the chassis rolls, the attitude of the suspension spring changes and the change can be determined with Equation (5.22). 6S = arcsin Lr sin(8Ro + ) Lr sin[0Ri + fiR) 2Lb + Lr cos(0Ro + PR) + Lr cos(0Rl + PR) (5.22) The spring attitude and force can be used to solve for inner and outer pushrod forces by summing the moments about the pivot points Bi and B0. After determining the suspension spring and pushrod forces, the forces at the tire contact patches of the rear tires are computed. This is accomplished by summing the moments about the fulcrum points, Oi and 00 and solving for the changes in the contact patch forces AFyr and AFzr. The combined weights of the axle and wheel Wa on each side of the vehicle are included in the summation of forces. The location of this center of mass is Lw along the axle measured from the fulcrum (not shown in the figures). The lateral acceleration value from the data is used for this 74 calculation. The spreadsheet is used to calculate how the geometry and forces change when the chassis rolls. Rear Forces inside: xl= FPlcos(0Pl-^) + fw, (W 5 -fAY-AFYR v z y cos + -A F, ZR sin^ WaLw cos(0Pi (/>) yi =-FPlsin(0Pi -)- Ay A Fyr sin^ + A F, ZR cos(j> + WaAyLw cos(6pi (5.23) (5.24) 75 FIGURE 5.11 FBD OUTSIDE WHEEL AXLE Rear Forces outside: xo =-Fpocos{Qp0+ ) + W \ R AY + AFra v ^ y cos^ + W + A F ZR s\n (w } Oyo = FPo Sin {ePo + 0) + AK + hFYR J Sin wD + A F. ZR cosif> + WaAyLw cos[6P0 + (j>) (5.25) (5.26) 76 Rear moments around the fulcrum inside: AFm sin(#c; -) + AFm cos(0Cl -$) = -^-(cos(ea -(f) + AY sin(#a - -^-(WAcos(6Al -<1>) + WaAy sin(0Ai-)) Lc Rear moments around the fulcrum outside: AFyr sin(#Co +
W |