Construction of a Computational Model of a Go-Kart for Dynamic Analysis

Multibody model construction and theoretical analysis

Jonathan Mikler A.

Advisor: Andres Leonardo Gonzales Mancera Ph.D A thesis presented to apply for the degree of B.Sc Mechanical Engineer

Department of Mechanical Engineering Los Andes University Bogot´a,Colombia December 2018 Abstract

A multibody model was developed in the Msc-Adams/Car software to recreate an electric Go- Kar. The prototype vehicle was built on top of a ICE chassis, where an PMAC motor and the subsequent powertrain were installed. The model was created by examining the real prototype and modelling its components to a basic but satisfying level of details, taking into account each individual weight and position. The simulations carried out were a step steer were for most cases was ran at 30 km/h (48 mph) oﬀered results on the dynamic response of the vehicle to this simple maneuver and simple analysis of the consistency and congruence with theory was possible. Weight distribution for both axles, as well as slip angle development were observed in the simulations, and since the vehicle was cataloged as understeer, this results were consistent with the theory. lastly, the characteristic speed was validated through multiple simulation ran at diﬀerent speeds, where the maximum yaw rate was observed at a speed close to the theoretical characteristic speed.

2 Dedication

This work is completely dedicated to my family, who in diﬀerent ways contributed, promoted and celebrated my passing through engineering school. To my parents who’s trust never quivered, to my siblings, who’s support for me was felt at every step and to my grandparents, who’s love and pride played a much more subtle but equally powerful role in my life.

3 Acknowledgements

I want to thank on first account to my work advisor, Andres Gonzales Mancera, for accepting me when I struggled to find a supervisor for my bachelor thesis. This meant a big leap of faith which I worked hard to meet the expectations that came with it. Furthermore, this work could not have been done without the constant help of many parties. First and foremost, the support staff at MSC software, embodied primarily by Lucas Fazecas, signified an invaluable help, without which I would not have gotten as far as i did. The support staff at the advance simulation room also helped in some measure and therefore a warm thank you is in order. Lastly, I want to thank the mechanical engineering department of my school, specially the teachers that contributed to my formation along the 4 and a half years spent in the campus. With out their open mindedness, and their will to enable students explore a variety of different topics somehow distant from the mainstream research topics on campus, my project would not have been possible.

4 Contents

1 Introduction 8 1.1 Motivation ...... 8 1.2 Previous work ...... 8 1.3 Objectives ...... 9 1.3.1 General Objective ...... 9 1.3.2 Speciﬁc objectives ...... 9

2 Background theory 10 2.1 Vehicle Dynamics Basics ...... 10 2.1.1 Forward Vehicle Dynamics ...... 11 2.1.2 Cornering ...... 13 2.2 Tire characteristics ...... 18 2.2.1 The contact patch pressure ...... 18 2.2.2 Tire forces and moments ...... 19 2.3 Multibody System approach ...... 22 2.3.1 modelling basics ...... 23 2.3.2 Tire modelling ...... 27 2.3.3 Solving linear and non-linear equations ...... 28

3 Model development 30 3.1 The Go-Kart model ...... 30 3.1.1 The Kart’s Dynamic Characteristics ...... 31 3.2 MBS modelling ...... 34 3.2.1 Part modeling ...... 34 3.2.2 Topology ...... 37

4 Summary of achievements 39 4.1 results and interpretations ...... 39 4.2 Conclusions ...... 44 4.3 Challenges and limitations ...... 45 4.4 Future work ...... 45

5 List of Figures

2.1 SAE Vehicle axis system (Automotive Engineers 2008) and (Gillespie 1994) . . . . 10 2.2 Alternative coordinate axis system (Jazar 2014) ...... 11 2.3 Inclined scenario for CoG determination (Jazar 2014) ...... 12 2.4 Vehicle’s condition on steady state cornering(Bradley 2018) ...... 13 2.5 Lateral force’s dependency on slip angle (Gillespie 1994) ...... 14 2.6 Lateral force dependency on normal force (Jazar 2014)...... 15 2.7 Cornering of a bicycle model (Gillespie 1994) ...... 15 2.8 Change of steer angle with speed for a fixed radius turn. Taken from (Gillespie 1994) 17 2.9 Yaw gain as a function of speed. Taken from (Gillespie 1994) ...... 18 2.10 Pressure distribution on a stationary tire (Blundell and Harty 2014) ...... 19 2.11 Normal stress (pressure) on the contact patch due to pressure (Jazar 2014) . . . . 19 2.12 Vertical forces wheel model ...... 20 2.13 Tire lateral deflection ...... 20 2.14 Contact patch deflection, asymmetric for a turning tire (Jazar 2014) ...... 21 2.15 Generation of lateral force and aligning moment due to slip angle (Blundell and Harty 2014)...... 21 2.16 Development of the camber thrust lateral force ...... 22 2.17 MSC Interface for creating parts ...... 23 2.18 Information display for reviewing and editing inertial properties ...... 24 2.19 Common examples of joints Blundell and Harty 2014 ...... 25 2.20 Example of joint implementation ...... 25 2.21 Example of a couple joint implemented in this work’s model (Blundell and Harty 2014) ...... 26 2.22 Spring and damper model for a tire (Blundell and Harty 2014) ...... 27 2.23 Function shape for the ”magic formula” ...... 28

3.1 E-Kart prototype ...... 30 3.2 Basic connections map of the developed prototype ...... 31 3.3 Lever arm steering mechanism (Jazar 2014) ...... 31 3.4 Go-Karts yaw rate gain and lat. accl. gain ...... 32 3.5 Front forces developed in a 16 m corner ...... 33 3.6 Rear forces developed in a 16 m corner ...... 33 3.7 Maximum cornering speed for diﬀerent radii ...... 34 3.8 E-Kart MBS model ...... 34 3.9 Steering connection topology ...... 35 3.10 Basic connections map of the model’s parts ...... 37 3.11 Model’s detailed topology ...... 38

4.1 Steering wheel angular imposed motion ...... 39 4.2 Normal forces at the axles for a step steer maneuver ...... 40 4.3 Slip angles developed at the left side ...... 41 4.4 Slip angles at both rear and front axles ...... 41 4.5 Front wheels steered angles ...... 42 4.6 Lateral forces at both rear and front axles ...... 43 4.7 Yaw rates for diﬀerent velocities ...... 44

6 List of Tables

2.1 Common joints and DOF removed by them ...... 27

3.1 Kart’s principal characteristics ...... 30 3.2 Joint count for the steering system ...... 36 3.3 Rear axle part characteristics ...... 36 3.4 Joint count for the entire model ...... 36 3.5 Model parts and roles ...... 38

4.1 Maneuver parameters ...... 39

7 Chapter 1

Introduction

1.1 Motivation

Multibody computational simulations are one of the modern engineering tools used to improve design in different areas. In automotive design, both commercial and competitive, they are used to improve the design proposals, reducing costs due to the clear advantages offered in contrast to trial and error methodologies. Cases such as the United States National Advanced Driving Simulator (NADS) (Heydinger et al. 2001) are recurrent in several divisions of the automotive industry as well as at the level of sports competition (CNN n.d.). However, the construction of these models is a monumental challenge in both reverse engineering and direct engineering. A high fidelity model implies complex mathematical approaches of all subsystems of the vehicle, as well as an experimental validation of said models. The case of the NADSdyna model is recognized as a pioneer in the field (Pastorino et al. 2015). These models are the result of a balance between mathematical complexity (which implies an increase in computing capacity and difficulty of implementation) and reduction of complexities of the model in areas outside the focus of the particular system to be analyzed (EE and WW 2003). As a result, we have a tool that allows us to diagnose and predict behaviors and improvements in the design of vehicle prototypes in specific areas of its design. Within the context of Los Andes university, the team called Bogota Team Andes Uniandes (BTA in short), is the student group interested in automotive engineering applied to electric vehicles. Its activities revolve around an electric Kart that serves as a test and research bank. This electric kart plays a central role in the team’s ability to test new designs and their effectiveness in real use conditions. It is precisely the potential offered by the computational simulations of a virtual model, which suggests the need for a virtual model of the prototype that can be subjected to dynamic track simulations, Furthermore, it is aligned with these interests, that the opportunity to conduct a bachelor thesis which revolves around the creation of a computational model of a kart type vehicle presents itself, alongside its coupling to a simulation ecosystem for dynamic behavior analysis.

1.2 Previous work

Multibody simulation has been a research topic or a long time, even outside the automotive area. Within the university, relevant works have been carried out in this area. Among the most relevant is the dynamic modeling of a kart (Rojas 2003), which developed a highly detailed mathematical model in the Matlab - Simulink software and obtained results that allow predicting improvements in the performance of a vehicle by altering vehicle characteristics. Simulations of dynamic structural analysis were carried out in (Ardila 2014) and (Calderon 2004) where they obtained conclusions about torsional rigidity and deflection that serve as a guide for procedures to be carried out within this project. Focused directly on the real prototype belonging to the BTA group, (Sepulveda 2018) simulated static load situations presented in other publications referenced in his work that allowed to make assertions about the structural behavior and made changes in the design of the chassis, proposing a modified design of the team’s actual chassis. The result was a proposal for a modification to the electric kart framework that meets all the design restrictions imposed within the job. This work, as proposed below, will serve as the fundamental basis for the development of this work.

8 CHAPTER 1. INTRODUCTION 1.3. OBJECTIVES

Multibody models have been built in larger projects where their approach has been validated with field experimentation. Exercises such as those made by (Pastorino et al. 2015) and (Heydinger et al. 2001) offer relevant procedures and considerations when modeling. Likewise, publications such as (Sol´ısand A. Medina 2003) and (Sandu and S. Freeman 2003) offer precise modeling protocols and methodologies. Specifically (Sandu and S. Freeman 2003) is focused on the MSC Adams program, which offers an important advantage.

1.3 Objectives 1.3.1 General Objective 1. Build a computational model of an electric kart that allows to evaluate its dynamic performance in diﬀerent driving maneuvers.

1.3.2 Speciﬁc objectives 1. Develop a mathematical model encompassing the theoretical behavior of the kart to evaluate the dynamic phenomena in ”quasi-static” conditions for diﬀerent scenarios.

2. Build a digital prototype of the vehicle within the Adams / Car program that allows the simulation of the Kart in conditions similar to those posed in the quasi-static platform. 3. Contrast the dynamic results found in the quasi-static simulator with the results delivered by the simulation and evaluate their plausibility

9 Chapter 2

Background theory

2.1 Vehicle Dynamics Basics

The section of vehicle dynamics which is of interest for this work is concerned with the movements of vehicles on a road surface. The movements of interest are acceleration and braking, ride, and turning. Dynamic behavior is determined by the forces imposed on the vehicle from the tires, gravity and aerodynamics (Gillespie 1994). Despite the fact that a vehicle is formed by multiple components, all of them move conjointly in most manoeuvres the vehicle performs. For example, the entire vehicle accelerates at virtually the same rate whenever the vehicle is subjected to a straight-line acceleration. In the case for a constant speed turning, the speed of any point in the body can be described by relating both the vehicle’s yaw rate and distance of the point from the center of gravity (CoG) to it’s tangential velocity. This supports the decision to have the vehicle represented as a lumped mass located at the center of gravity, with associated inertial properties and mass. For most kinematic analyses, one mass representing the entire vehicle is sufficient (Gillespie 1994). When ride analyses are needed, the tires are treated differently because of the interface between tires and car body happening at the suspension, which directly influences car dynamics but not tire dynamics, thus differentiating the sprung mass (the lumped mass representing the body) from the un-sprung mass (The mass of the wheels)1. For single mass representation, the vehicle is regarded as shown in the figure 2.1, alongside the usual coordinate axis system used.

Figure 2.1: SAE Vehicle axis system (Automotive Engineers 2008) and (Gillespie 1994)

Axis convention Forces are said to be applied on the vehicle. In that sense, forces acting in the vehicles forward direction are said to be ”longitudinal” forces, and on the plane-perpendicular are regarded as ”lateral”. If the SAE J670e Automotive Engineers 2008 convention is followed, the vertical force

1For the Go-Kart case, there is no un-sprung mass, the entire vehicle is regarded as sprung mass; as the wheels are ﬂexible and behave as a spring.

10 CHAPTER 2. BACKGROUND THEORY 2.1. VEHICLE DYNAMICS BASICS

(a) Side view (b) Front view

Figure 2.2: Alternative coordinate axis system (Jazar 2014) on the tires would be negative. To solve this inconvenience, vertical downward forces are regarded as Normal forces, and Vertical forces as the negative of such Normal forces. Throughout this work, for presenting the mathematical model two conventions will be used, the one presented in 2.1 will be regarded as the SAE convention and the one presented in ﬁgure 2.2 as the alternative convention.

2.1.1 Forward Vehicle Dynamics A kinematic examination of a vehicle model yields, among other results, the acceleration, braking and turning capabilities. We ﬁrst start with the straight-line motion. When a vehicle is parked as in ﬁgure 2.2a the normal force Fz under each front and rear axle are: 1 a F = mg 2 (2.1) z1 2 l

1 a F = mg 1 (2.2) z2 2 l

Where l is the wheelbase of the vehicle and a1 and a2 are the distances from the front and rear axles respectively, hence l = a1 + a2. Equations 2.1 and 2.2 can be rearranged to calculate the position of the mass center on the horizontal axis.

2l a = F (2.3) 1 mg z2 2l a = F (2.4) 1 mg z1

For determining the vertical location of the CoG, the equilibrium equations need to be developed further within an inclined scenario, such as in ﬁgure 2.3. Having the force under the front wheels measured, regarded as 2Fz1 , the height of the mass center can be calculated by static equilibrium conditions:

ΣFz = 0 ΣMy = 0 (2.5)

2Fz1 + 2Fz2 − mg = 0 (2.6)

− 2Fz1(a1Cosφ + (h − R)Sinφ) + 2Fz2(a2Cosφ − (h − R)Sinφ) = 0 (2.7) Solving for h, we obtain the following expression yielding the height of the CoG:

2F h = R + a − z1 Cotφ (2.8) 2 mg

From the 3 assumptions mentioned in Jazar 2014 for the development of this model, the one regarding the wheels as rigid cylinders is the only one useful for the model developed for the Go-Kart since no suspension is present and ﬂuid shifts are non-existent.

11 2.1. VEHICLE DYNAMICS BASICS CHAPTER 2. BACKGROUND THEORY

Figure 2.3: Inclined scenario for CoG determination (Jazar 2014)

Straight acceleration When a car speeds up with constant acceleration a, on level ground, the reaction on each wheel shifts due to weight transfer. A third equation regarding equilibrium in a constant acceleration scenario must be considered alongside the ﬁrst two from the static equilibrium (eq. 2.5):

ΣFx = ma (2.9)

2Fx1 + 2Fx2 = ma

The normal forces increase or decrease depending on the direction of acceleration and it’s magnitude: 1 a 1 h F = mg 2 − ma (2.10) z1 2 l 2 l 1 a 1 h F = mg 1 + ma (2.11) z2 2 l 2 l

1 h The term 2 ma l is regarded as the longitudinal dynamic weight transfer (Bradley 2018).

Maximum acceleration attainable The maximum acceleration for a rear wheel drive (rwd) vehicle is achieved when the longitudinal force on the rear axle Fx2 reaches its maximum frictional force (Fx2 = µFz2). Since it’s a rwd vehicle, the frontal longitudinal force is practically zero (Fx1 = 0). Using these new considerations on the equilibrium on the X axis equation (eq. 2.9) , and including the weight transfer factor(eq. 2.11), we obtain:

ΣFx = ma a ha µ mg 1 + = ma x l lg rearranging for the normalized acceleration (with respect to g) we obtain: a µ a rwd = x 1 (2.12) g h l 1 − µx l If friction were strong enough, the car could spin with respect to the rear wheel, lifting from the ground the front wheel, rendering the vertical force there zero (Fz1 = 0). Replacing in eq. 2.10 F for 0 we obtain the following: z1 a a rwd = 2 (2.13) g h In any case, we can be certain that the maximum obtainable acceleration will be less than those in either eq. 2.12 or 2.13

12 CHAPTER 2. BACKGROUND THEORY 2.1. VEHICLE DYNAMICS BASICS

2.1.2 Cornering Lateral Weight Transfer2 When a vehicle negotiates a corner, due to the inertia carried by it, the vehicle will experience an acceleration outwards which will be traduced as an inertial force applied at the CoG. To maintain equilibrium, the outer wheel of the vehicle will increase the reaction force in vertical direction upwards (ﬁg 2.4).In order to also maintain equilibrium in the vertical direction, the inner wheel will reduce its magnitude by the same amount the outer wheel increased.

Figure 2.4: Vehicle’s condition on steady state cornering(Bradley 2018)

Using ﬁg 2.4 as reference, we take moments about the inner wheel: t F t = mg + ma h (2.14) zo 2 y mg h F = + ma (2.15) zo 2 y t Assuming the CoG to be located in the middle of the vehicle’s track we obtain the following weight transfer added to the outer wheel ans subtracted from the inner one: h F = F + ma (2.16) zo Szo y t h F = F + ma (2.17) zi Szi y t

FSzo and FSzi being the static normal force for the outer and inner wheel respectively. The term h may t is regarded as the total lateral weight transfer. The ”total” part is because from this weight transfer, the suspension will characterize which percentage of it will go to the front outer wheel and which will go to the outer rear. This is proportional to the so-called roll stiffness of the vehicle and it’s development is beyond the scope of this work. Instead, a simple proportion term ρwtr can be used to dictate the proportion of the total lateral weight transfer directed to the front and back. Thus the vertical forces on each wheel due to lateral weight transfer in steady-state cornering are as follows: h F = F + ρ ma (2.18) zof Szo wtr y t h F = F − ρ ma (2.19) zif Szi wtr y t h F = F + (1 − ρ )ma (2.20) zof Szo wtr y t h F = F − (1 − ρ )ma (2.21) zif Szi wtr y t The most useful effect of the normal forces on the tires, in the cornering event, is the development of lateral forces. This are in fact dependant (among other things) on the normal force. A blunt approach to the maximum lateral force to be developed, in function of the normal force is the relation between them proportional to the friction coefficient:

Fy = µFz (2.22)

2Section adapted from (Bradley 2018)

13 2.1. VEHICLE DYNAMICS BASICS CHAPTER 2. BACKGROUND THEORY

Inasmuch as the friction high school theory has it’s ﬂaws (Blundell and Harty 2014), it is useful to evaluate the maximum force obtainable for a given load. In contrast, the lateral force at each tire needed to be developed to overcome the centripetal force due to negotiating a corner of radius R is seemingly straightforward: 2 Fy = mv /R (2.23)

Hence a comparison of this two forces would yield the top conditions at which a certain corner could be negotiated.

High Speed Cornering

Because the main purpose of the Go-Kart is to be operative at mostly high speeds, and thus his maneuverability at low speeds is of no interest to this work, only high speed cornering dynamics will be addressed in this section. At high speeds, lateral acceleration due to inertia will be present. to counteract this lateral acceleration, lateral forces must develop and they will be a function of the slip angle of each wheel. This is illustrated in ﬁg 2.5

Figure 2.5: Lateral force’s dependency on slip angle (Gillespie 1994)

The slip is caused from the lateral acceleration on the wheel, this means the wheel will be moving in one direction while facing another (unlike low speed cornering or straight line motion where the direction of motion of the wheel matches the heading direction). The Slip angle is then deﬁned as: V α = arctan y (2.24) Vx

At zero Camber, the lateral force is denoted as ”cornering force”. For a given load, the cornering force grows with slip angle. At low slip angles (5 > α) the relation between cornering force and slip angle is relation, proportional to the Cornering stiﬀness:

Fy = Cαα (2.25)

Since the lateral force depends so much on the vertical load, it is common to present the tire characteristics included in Cα as the ratio between the cornering stiﬀness and the vertical force:

Cα CCα = (2.26) Fz

The lateral load is sensitive to the vertical load, so much as it is sensitive to the slip angle. Just as without slip angle lateral slip would not develop and the lateral stress would not occur, so it happens with vertical load. The variation can be appreciated in figure 2.6, which illustrates the lateral force behavior of a sample tire for different normal loads. It can be appreciated that not only the maximum lateral force changes with load but the cornering stiffness as well.

14 CHAPTER 2. BACKGROUND THEORY 2.1. VEHICLE DYNAMICS BASICS

Figure 2.6: Lateral force dependency on normal force (Jazar 2014).

Negotiating the corner: steering angle according to vehicle setup

To analyze the required forces needed for cornering, Newton’s second law comes to play. For simpliﬁcation purposes the model is better represented as ﬁgure 2.7.

Figure 2.7: Cornering of a bicycle model (Gillespie 1994)

The vehicle experiences centripetal acceleration which must be countered by lateral force at rear and front tires, additionally, Moment equilibrium in the z axis must be secured:

V 2 ΣF = F + F = M (2.27) y yf yr R ΣMz = Fyf b − Fyr = 0 (2.28)

By using this two equations we obtain values for both front and rear lateral forces:

b V 2 F = M (2.29) yf L R c V 2 F = M (2.30) yr L R

What both equations 2.29 and 2.30 show is the proportion of lateral force required at the front and b back which is directly proportional to the weight distribution in the vehicle (M L is the rear weight supported by the rear axle). Replacing these expressions for both front and rear wheel (axle) in

15 2.1. VEHICLE DYNAMICS BASICS CHAPTER 2. BACKGROUND THEORY equation 2.25 we obtain the needed slip angles for both cases:

V 2 αf = Wf (2.31) CαgR V 2 αr = Wr (2.32) CαgR

Wf and Wr being the weight proportions of both front and rear. The ultimate output for this analysis is knowing the needed steer angle for negotiating a give corner, assuming that the tire can give the needed force. For that, the relation between the steering angle δ and the slip angles both at the front and at the rear must be established. Also from ﬁgure 2.7 we obtain: L δ = + α − α (Radians) (2.33) R f r L δ = 57.3 + α − α (Degrees) (2.34) R f r (2.35)

The relation of interest is between the steering angle and the tire properties enabling the lateral force development, namely Cα. By replacing the expressions in equations 2.31 and 2.32 and reordering a little we obtain:

L W W V δ = 57.3 + f − r 2 (2.36) R Cαf Cαr gR

W The factor f − Wr is usually regarded as the Under steer coeﬃcient (Gillespie 1994), and Cαf Cαr equation 2.36 is sometime shortened to:

L δ = 57.3 + Ka (2.37) R y Three cases are possible for the Understeer coeﬃcient:

Neutral steering Wf Wr = −→ K = 0 −→ αf = αr Cαf Cαr When K is zero, the condition is called neutral steering and means the force needed to be developed at the front and rear must be the same in order for the vehicle to maintain the corner radius without slipping out.

Understeer Wf Wr > −→ K > 0 −→ αf > αr Cαf Cαr In the case of understeering the change in steering angles is proportional to the lateral acceleration by K and therefore to the square of the tangential velocity. Due to the position of the CG and/or the tire characteristics, the front wheels need to be steering to a greater angle to develop the necessary lateral forces to overcome centripetal acceleration.

Oversteer Wf Wr < −→ K < 0 −→ αf < αr Cαf Cαr In the case of Oversteering the drift in the rear wheels is greater than that at the front, this turn the front wheels inwards reducing the corner radius even more and thus increasing lateral acceleration, increasing at the same time the drift of the rear wheels. This process continues until the vehicle looses the trajectory or until the steering angle is reduced to maintain the radius. For a ﬁxed radius corner maneuver, the amount of steering correction from the Ackerman condition varies clearly with speed. This can be evidenced in ﬁgure 2.8, where for an under steered vehicle the steering angle increases with the square of the speed. A useful speed value is the so- called characteristic speed in which the steering is double it’s Ackerman condition. On the contrary, for an over steer vehicle the correction must be smaller with an increase of speed, reaching a total steer angle of zero at what’s known as the critical speed.

16 CHAPTER 2. BACKGROUND THEORY 2.1. VEHICLE DYNAMICS BASICS

Figure 2.8: Change of steer angle with speed for a ﬁxed radius turn. Taken from (Gillespie 1994)

Characteristic speed In the scenario of an understeer vehicle, it’s understeer characteristic can be measured by a parameter known as the characteristic speed. It is simply the speed at which the steering is double the Ackerman condition for it’s given radius (Gillespie 1994). It’s calculated as such: p Vchar = 57.3Lg/k (2.38)

Critical speed In the over steer case, the critical speed is the speed at which a zero-valued angle is needed to negotiate the turn. At this point the vehicle is unstable (Gillespie 1994). It is similarly calculated as the characteristic speed: p Vcrit = −57.3Lg/k (2.39) Where it must be remembered that in this case k is to be negative, indicating the oversteer condition of the vehicle.

Lateral acceleration gain and Yaw velocity gain It is of interest to evaluate how effective is such steering angle at negotiating a curve at a specific speed. For this two figures of merit are proposed by Gillespie. The lateral acceleration gain and the yaw rate gain. ay Solving equation 2.37 for the ratio δ would allow the analysis to be about how much lateral acceleration -and therefore how much lateral force is needed to negotiate the corner at some specific ay speed-corner conditions. The ratio δ is known as the lateral acceleration ratio

2 a V y = 57.3Lg (2.40) δ KV 2 1 + 57.3Lg

From 2.40 it can be seen that for a neutral steer vehicle (K zero) the lateral acceleration in determined only by the square of the speed. When K is positive (understeer) the gain is diminished by its understeer factor, being always less than the neutral condition. On the other side, when the vehicle has a oversteer configuration (K negative) the gain increases with the velocity. When the speed reaches the critical speed, the gain is mathematically infinite. In this scenario the vehicle would be unstable of spin out of control. Another analysis useful for assessing the performance of the vehicle’s setup in cornering is to evaluate it’s responsiveness in changing it’s heading direction for a given steering input at a given speed. The Yaw gain ratio is the variable useful for knowing this kinematic characteristic. The yaw rate, the rate of rotation of the heading direction of the vehicle, with the corner radius R and the vehicle speed V is defined as: V r = 57.3 (2.41) R

17 2.2. TIRE CHARACTERISTICS CHAPTER 2. BACKGROUND THEORY

r we can substitute this expression into equation 2.37 and solve for the ratio δ to obtain the following expression: r V = L (2.42) δ KV 2 1 + 57.3Lg

The implication of this analysis can be better understood by examination of figure 2.9. For an oversteer vehicle the point where the gain is infinite matches the critical speed point. beyond this point the inertial forces render unstable the vehicle and spin it virtually with an infinite gain. In reality if that the tires slip beyond their static condition and drift the vehicle inwards. For the understeer case the gain increases with speed up to the characteristic speed, and decreases thereafter. Hence the characteristic speed gains significance as the point where the vehicle is most responsive (Gillespie 1994).

Figure 2.9: Yaw gain as a function of speed. Taken from (Gillespie 1994)

2.2 Tire characteristics3

2.2.1 The contact patch pressure

The contact patch is the interface where the tire interacts with the road, which with give parameters force outputs develop. Tires may be considered as force generator with two mayor and three minor outputs, namely longitudinal force, lateral force for mayor outputs and aligning moment roll and pitch moment (Jazar 2014). These input parameters are the vertical load Fz, the slip angle α (sometime called side-slip angle), longitudinal slip s or k, and the camber angle γ. However, the interaction occurs due to the pressure distribution at the tire’s contact patch, which negotiates with frictional mechanisms to produce the force output mentioned above. Each section on the contact patch is subjected to a normal pressure P and a shear stress τ acting in the road surface (Blundell and Harty 2014). For a static tire, the pressure distribution is symmetrical and presents front and rear peaks depending on how adequately the tire has been inﬂated. Pressure distribution is not symmetrical for rolling tires, let alone driven or braked tires. When the rubber moves into the contact patch, it will be progressively loaded (compressed) until reaching the middle of the contact patch and then gradually unloaded until reaching the end of the patch. Due to the hysteresis present in the behavior of rubber, the pressure point will be present in front of the contact patch center point. We can see how the pressure distribution on the contact patch looks like in ﬁg 2.11; the pressure distribution will be asymmetrical with more pressure present at the front of the contact patch.

3Adapted from (Jazar 2014) and (Blundell and Harty 2014)

18 CHAPTER 2. BACKGROUND THEORY 2.2. TIRE CHARACTERISTICS

Figure 2.10: Pressure distribution on a stationary tire (Blundell and Harty 2014)

Figure 2.11: Normal stress (pressure) on the contact patch due to pressure (Jazar 2014)

2.2.2 Tire forces and moments

Vertical Force

It is generally sufficient to treat the tire as a linear spring and damper when calculating the vertical force (Jazar 2014). The middle of the contact patch, regarded as point P in figure 2.12a is subjected to forces due to damping and to spring stiffness of the wheel:

Fz = Fzk + Fzc (2.43)

Fzk = Kzδz (2.44)

Fzc = cvz (2.45)

√ with c = 2ζ mtKz. The model can be further developed by modelling non linear spring relations and an interpolation mechanism to relate spline speciﬁed data to the simulation conditions. It must be clariﬁed, that according to SAE Axis system the normal will always be negative, rendering the forces a little inconvenient.to overcome this, the positive magnitude of this calculations are regarded as Normal forces (More exactly the negative of the vertical for which is always negative).

19 2.2. TIRE CHARACTERISTICS CHAPTER 2. BACKGROUND THEORY

(a) Taken from (Blundell and Harty 2014) (b) Taken from (Jazar 2014)

Figure 2.12: Vertical forces wheel model

Lateral force and aligning moment

The mechanisms that generate both the lateral forces and the aligning moment are somehow similar and therefore it is useful to present them together. The first mechanism which develops the lateral forces is the deflection of the tire laterally as can be seen in figure 2.13. The tire, assumed as a linear spring, develops a lateral force proportional to the lateral stiffness:

Fy = kyδy (2.46)

This increase in lateral forces has a up limit, namely the product of the friction coeﬃcient and the vertical force: µFz.

Figure 2.13: Tire lateral deﬂection

From ﬁgure 2.14 it can be seen that the application of the lateral force is not applied at the center of the contact patch, but due to the stress distribution explained before in this section it is applied to the rear of the contact patch area, such distance is regarded as the pneumatic trail.

20 CHAPTER 2. BACKGROUND THEORY 2.2. TIRE CHARACTERISTICS

Figure 2.14: Contact patch deﬂection, asymmetric for a turning tire (Jazar 2014)

This mechanism introduces, thanks to the lateral force development location, a vertical moment regarded as the alingnig moment. Lateral distortion of the tire treads is a result of a tangential stress distribution τy over the tireprint (Jazar 2014). Assuming that the tangential stress τy proportional to the distortion, the resultant lateral force Fy over the contact patch area Ap is then: Z Fy = τydAp (2.47) Ap and the pneumatic trail is calculated as: 1 Z ax = xτydAp (2.48) Fy Ap Because of the point of application of the force being not centered, the resulting aligning moment is then:

Mz = Fyaxα (2.49) Another good illustration of the kinematic mechanism is presented by Blundell and Harty in ﬁgure 2.15. Useful for understanding the relation of the lateral force and the aligning moment to the slip angle. As mentioned in subsection 2.1.2 the lateral force is caused by the reaction of the shear stress on the patch, directed in the same slip angle. the shear stress gradually increases until reaching the aforementioned boundary limit µFz. As the slip angle increases, the rate at which lateral stress is generated as the tread rubber moves back through the contact patch increases so that the point at which slippage commences moves forward in the contact patch (Blundell and Harty 2014).

Figure 2.15: Generation of lateral force and aligning moment due to slip angle (Blundell and Harty 2014).

Camber thrust The lateral force that arises due to an inclination of the tire from the vertical is referred to as camber thrust. If the tire is inclined at a camber angle γ, then deﬂection of the tire and the associated

21 2.3. MULTIBODY SYSTEM APPROACH CHAPTER 2. BACKGROUND THEORY

radial stiﬀness will produce a resultant force, FR, acting towards the wheel center. Resolving this into components will produce the tire reaction load and the camber thrust (Blundell and Harty 2014). This is better illustrated in ﬁgure 2.16.

Figure 2.16: Development of the camber thrust lateral force

Jazar provides a good explanation for the development of this force: When a wheel is under a constant load and then a camber angle is applied on the rim, the tire will deflect laterally such that the tireprint area is longer in the cambered side and shorter in the other side... As the wheel rolls forward, undeflected treads enter the tireprint region and deflect laterally as well as longitudinally. However, because of the shape of the tireprint, the treads entering the tireprint closer to the cambered side have more time to be stretched laterally. Because the developed lateral stress is proportional to the lateral stretch, the nonuniform tread stretching generates an asymmetric stress distribution and more lateral stress will be developed on the cambered side.(Blundell and Harty 2014)

2.3 Multibody System approach to vehicle dynamics simulation

This section is intended to brieﬂy introduce the topic of multibody simulation and planning. It is inspired by the thorough explanation given by (Blundell and Harty 2014, ch. 3.2). If the reader is new to the multibody system approach to dynamic simulation, it is suggested to refer to the aforementioned author.

22 CHAPTER 2. BACKGROUND THEORY 2.3. MULTIBODY SYSTEM APPROACH

2.3.1 modelling basics

Basic model component

When developing the data set for a model in a MBS analysis program, the following can be considered to be basic model components Blundell and Harty 2014:

1. Rigid Bodies

2. Geometry

3. Constraint joints

4. Forces

5. User-Deﬁned algebraic and diﬀerential equations

Bodies

For a dynamic analysis, it is necessary that each body that makes part of the assembly has defined inertial properties, as well as defined orientation for such properties that it so requires. Such properties are mass, CoG, CoG position, mass moments of inertia and the orientation of such moments. Additionally, for some situation, the initial kinematic conditions must be also specified for each body. In the development of the model parts, a graphic approach is useful. The user interface allows for the definition of parametric coordinate to define both CoG properties and geometric characteristics of a part. This generates based on the part material and other geometric properties the rest of the inertial properties autonomously. Such is the process followed while developing this work’s model.

Figure 2.17: MSC Interface for creating parts

23 2.3. MULTIBODY SYSTEM APPROACH CHAPTER 2. BACKGROUND THEORY

Figure 2.18: Information display for reviewing and editing inertial properties

Basics constraints and standard joints

Constraints are used to restrict the motion of parts.Such restrictions can be in regards to the global reference coordinate (ground) or relative to another part. Some joints constrain individual or multiple degrees of freedom (DOF’s) between bodies which in real life would be constrained by mechanical elements such as gears or bushings. Other constrain multiple DOF’s, for example, by setting a point in one body to be placed on a specific plane on another. It is important to note that joint primitives (in-line, parallel, plane or point), but also standard joint result in a specific type of force or moment which need to be added to the general force an moment balance for each part. Joint primitives are the 4 basic constraints that together form standard joints, but which can also be implemented separately. Such primitives are present in the interaction between the model assembly and the environment where it’s being set-up (in the case of this work, whenever the environment is mentioned is a reference the the so-called TEST-RIG in Adams parley). For standard joints, the number of possible joints is large, each restricting a specific number of degrees of freedom, both for transnational and rotational possibilities. An example of the most commonly used joints is presented in figure 2.19

24 CHAPTER 2. BACKGROUND THEORY 2.3. MULTIBODY SYSTEM APPROACH

Figure 2.19: Common examples of joints Blundell and Harty 2014

Configuring the joint characteristics is joint-specific, and mean specifying (whenever the case) the orientations and location of the joint’s axes and parts. An illustrative example given in figure 2.20.

(a) Universal Joint displayed in the UI

(b) Information display of the joint

Figure 2.20: Example of joint implementation

Further modiﬁcations on the joints involve setting initial conditions. For example, in a cylindrical joint an initial translation on 100 mm can be in forced upon the joint, and even be deactivated once the model initiates the simulation (i.e. @t=0). For some joint including the common translational, revolute, and cylindrical it is possible to impose a motion on one of the DOF’s left free by the joint, it is so that some maneuvers in the simulations are performed, for example the step- steer maneuvers imposes a rotating motion on the revolute joint connecting the steering shaft of a vehicle to the suspension system (in this work’s case, the steering shaft is connected to the body since no suspension is present.

25 2.3. MULTIBODY SYSTEM APPROACH CHAPTER 2. BACKGROUND THEORY

The connection topology is discussed further in chapter 3). The imposed motion further constrains the model reducing the DOF’s since, in spite of movement being involved, it is enforced and cannot be altered by external forces or inertial properties change. Another constraint method are the so-called couple, or coupling equation, which restricts the motion of a speciﬁc degree of freedom on a joint by connecting it to another motion output on a diﬀerent joint. A common example is a reduction gear connecting the rotational degree of freedom of the steering shaft and the translational degree of freedom of the steering rack in a Rack-n-Pinion steering system; the rotation of the shaft in real life would signify a translation of the rack in it’s longitudinal axis. Such movements are represented by the aforementioned couples or gears. As with the imposed motions, a degree of freedom is lost to the system, as the kinematic relationship cannot be altered by external factors (Blundell and Harty 2014).

Figure 2.21: Example of a couple joint implemented in this work’s model (Blundell and Harty 2014)

In the example given in ﬁgure 2.21, the joint that enables the wheel to rotate in it’s own axis in coupled to the revolute joint allowing the rotation of the steering shaft on it’s own axis as well. Here the reduction ratio is one because it belong to the steering system of the Go-Kart model developed for this work. However, typical reduction ratios could vary from model to model.

Degrees of freedom A degree of freedom is an independent parameter with which the mechanical system can be described in term of his kinematic characteristics. A free floating object with no restriction in movement or rotation is said to have six degrees of freedom. ”They are brought into being by the declaration of bodies and taken out of existence by constraint equations” Blundell and Harty 2014. One early task for any MBS analysis is for the analyst to be able to identify and understand the total DOF in the system. This can be done using the Chebychev-Grübler-Kutzbach criterion (Grüblercount): Total DOF = 6 × (N − 1) − S (2.50) Where N is the number of bodies including the ground and S is the number of constraints all the bodies have together. Some typical joints (Constraints) and the amount and type of degrees of freedom they removed is presented in table 2.1 .The DOF number must never be negative in a model development, not only because it has no real meaning but also because the system would be over constrained. In term of the solution of the equations of motion to be solved by the software, it renders impossible to come up with a solution. Thus, most software (MSC-Adams among them) have some algorithm to eliminate redundant constrains (those rendering the DOF count negative)Blundell and Harty 2014. although a solution, it is not a good one since the forces and moments developed at those joints will be defined to zero and if they are of interest it can affect the analysis.

Forces For an MBS model, forces can be classiﬁed into two types: internal and external. Internal forces come from the interaction between bodies, either through spring and dampers or through joint restrictions between them. External forces are applied to the body. they are regarded as ”action only” forces, since the reaction on the counter body (ground, air, etc.) is not important for the analysis or solution. The forces generated by a tire model and input through the wheel centres into a full vehicle model can also be considered to be this type of force Blundell and Harty 2014.Both type of forces are formulated through the use of system variable equations.

26 CHAPTER 2. BACKGROUND THEORY 2.3. MULTIBODY SYSTEM APPROACH

Constraint element Translational constraints Rotational constraints Total Cylindrical 2 2 4 Fixed 3 3 6 Planar 1 2 3 Revolute 3 2 5 Spherical 3 0 3 Translational 2 3 5 Universal (Hooke) 3 1 4 ConVel 3 1 4 At-point primitive 3 0 3 Inline primitive 2 0 2 In-plane primitive 1 0 1 parallel primitive 0 2 2 perpendicular primitive 0 1 1 motion (trans) 1 0 1 motion (rotation) 0 1 1

Table 2.1: Common joints and DOF removed by them

2.3.2 Tire modelling

As stated by Blundell and Harty, in relation to the purpose of the tire model: ”Disregarding the approach of the wheel model, the function of the Tyre model is to establish the forces and moments occurring at the Tyre to road contact patch and resolve these to the wheel centre and hence into the vehicle”(Blundell and Harty 2014). For handling studies derived from vehicle dynamics, the point follower approach is used, where a single point at the contact patch center is used to calculate all kinetic outcomes (Forces and momenta). The interaction of the vehicle body with the ground is non-existent, it is through the tires ant the bodies connecting the body to them that the kinematic behavior occurs (ﬁgure 2.22).

Figure 2.22: Spring and damper model for a tire (Blundell and Harty 2014)

Within the handling analysis, the most important variable to be analyzed and therefore calculated by the tire model are (presented in dependency hierarchy): The vertical or normal Force Fz Longitudinal or tractive force Fx, the lateral or cornering force Fy and the aligning moment Mz. geometric and kinematic information at the wheel center is taken as input and the forces and moments will be the output, back-fed to the wheel center to calculate the vehicle’s response in that speciﬁc time step. The outcome will be new conditions for the wheel center to be inputted in the model once again, and on.

27 2.3. MULTIBODY SYSTEM APPROACH CHAPTER 2. BACKGROUND THEORY

The magic formula Blundell and Harty state that the tire model that is now most well established is based on the work by Pacejka and is referred to as the ‘Magic Formula’. The Magic Formula is a set of mathematical functions that relate:

1. The lateral force Fy as a function of slip angle a.

2. The aligning moment Mz as a function of slip angle a.

3. The longitudinal force Fx as a function of longitudinal slip k.

The general formula, represented in ﬁgure 2.23 is a sine function enlarged in the x-axis thanks to two arc-tangent functions, one within the other:

Y (x) = Dsin [Carctan (Bx − E (Bx − arctan(Bx)))] (2.51)

In this case Y is either the side force Fy, the aligning moment Mz or the longitudinal force Fx and X is either the slip angle α or the longitudinal slip. The details of how this formula works

Figure 2.23: Function shape for the ”magic formula” for developing the kinetic outcomes is beyond the scope of this document, however it needs to be mentioned, that this formula is universal, meaning that it’s the same for calculating the lateral, longitudinal force and aligning moment. To diﬀerentiate for each, a set of numerical parameters are formulated for each scenario. These take into account the studied geometrical properties that inﬂuence each magnitude (for example, the camber value is taken into account for the aligning moment but not as much for the longitudinal force). For further information it is recommended hat the reader explores the work done by Blundell and Harty and Pacejka and Besselink.

2.3.3 Solving linear and non-linear equations4 Two different setups can be ran. For static analysis, the velocities and accelerations are set to zero and the loads are balanced with the reactions until an equilibrium (iterative convergence) is achieved, such process is better described by Blundell and Harty: ...This may involve the system moving through large displacements between the initial definition and the equilibrium position and therefore requires a number of iterations before convergence on the solution closest to the initial configuration. Usually static analysis is a precursors to dynamic simulations. It defines the ”kerb height” of the vehicle before moving it forward. Some dynamic simulation involved what regarded as quasi.static

4Adapted from (Blundell and Harty 2014)

28 CHAPTER 2. BACKGROUND THEORY 2.3. MULTIBODY SYSTEM APPROACH simulations; which are static simulation with non-zero initial parameters calculated one after the other. The system can be considered to be moving, although the dynamic response is not required. Dynamic analysis is performed on systems with one or more DOF. The diﬀerential equations representing the system are automatically formulated and then numerically integrated to provide the position, velocities, accelerations and forces at successively later times, each one referred to as time step.

Solving linear equations During the solution phase of the simulation, the MBS software solves both linear and non-linear equations. For the linear case, all the equation can be assembled in matrix form:

[A][x] = [b] where [A] is a square matrix of constants, [x] is a column matrix of unknowns and [b] is a columns of constants. usually, matrix A is a sparse matrix, and MBS programs take advantage of that to solve them more easily. Such solution starts by splitting matrix A into two triangular matrices, namely L (a lower triangular matrix) and U (an upper triangular solution):

[A] = [L][U]

Each matrix is solved separetly by introfucing a new set of variables, namely [y]:

[L][U][x] = [b] [U][x] = [y] [L][y] = [b]

Although not apparent here, in order to obtain a solution, the intermediate variable [y] are obtained by dividing some linear configuration of [b] and [y] by the diagonal terms of [L] (called pivots, therefore this cannot be zero or (for computational reasons) approach very closely zero. this conditions can occur from poor systems modelling and/or mechanism configuration at a bifurcation. Changing the configuration will change the order of the equation, thus altering the value of the ”pivots”.

Solving linear equations An arrange of non-linear equations may be described in matrix form like this:

[G][x] = 0 with [x] a set of unknown variables and [G] a set of implicit functions of [x]. The solution may be obtained using Newton - Raphson iteration based on the assumption that very close to the solution the curve may be approximated to a straight line where it crosses the x-axis. A detailed explanation of this process is presented in the works of Blundell and Harty and the reader is encouraged to refer to this work for extended explanation.

29 Chapter 3

Model development

3.1 The Go-Kart model

Figure 3.1: E-Kart prototype

The model developed for this work involves the basic components for a multibody system for dynamic simulation mentioned in chapter 2. Inspired in the examination of a real prototype belonging to the mechanical engineering department at Los Andes University.

The prototype is built upon a Birel Cr-32// X ICC chassis. It’s powertrain was changed to become an electric driven vehicle, with a 48V battery bank distributed throughout the chassis’s body, a motor controller and a PM-AC motor. The Kart main parameters are presented in table 3.1.

Wheelbase [mm] Track [mm] Mass [Kg] Front weight 1040 870 160 45%

Camber [◦] Castor [◦] Toe [◦] CoG Height [mm] -1.5 -15 0 100

Table 3.1: Kart’s principal characteristics

The kart’s dimensions and weights were measured to set a baseline to be used as reference for the Go-Kart modeling. Because of time restrictions, the kart could not be separated into it’s most elemental pieces, and therefore, the car’s body was assembled with some minor accessories when measured. The assumption that the actual chassis weight does not distance itself too far from the measured value was taken. It must be noted that the additional equipment as well as the replacement of ICC specific parts to convert the prototype into an electric Go-kart heavily influenced the weight distribution for the kart both longitudinally and laterally. For this reason, these masses were also included in the final model as well as the pilot.

30 CHAPTER 3. MODEL DEVELOPMENT 3.1. THE GO-KART MODEL

Figure 3.2: Basic connections map of the developed prototype

The Kart prototype is composed by 7 different subsystems each with a specific task to perform, each being very distinct and very easy to identify on general terms. The seven subsystems are the body, the steering system, the front and rear wheels, the rear axle, the braking system and the powertrain. A basic diagram of the physical connection which are not for support is illustrated in fig 3.2. The steering system implemented is often called a lever arm steering system illustrated in figure 3.3. it is often used where large wheel steering angles are needed, because it renders them possible. Being simply a pair of four link mechanisms sharing one mobile link (the lever arm), the geometric parameters enable the range of angular movement to be wide at the output of the mechanism, namely the wheel carrier connected to the hub.

Figure 3.3: Lever arm steering mechanism (Jazar 2014)

The wheels used in the Go-Kart are a model YZ from MG-Tires racing, which are speciﬁc for competition Go-Karts. The general dimensions are 10 x 4.60 - 5 inches, referring to the nominal diameter, the tread width and the rim diameter for the front tires and 11 x 7.10 - 5 for the rear ones (YZ tire model n.d.). The maximum load was unavailable for the speciﬁc tires, however for analysis and modelling reason a similar tire maximum load, taken from (Corporation n.d.) was taken as reference.

3.1.1 The Kart’s Dynamic Characteristics

A simple analysis on the dynamic characteristics of the Go-kart’s prototype utilizing some of the equation given in section 2.1 yields some important results about the vehicle’s performance. As mentioned before, the Understeer gradient is calculated using the tire cornering stiﬀness

31 3.1. THE GO-KART MODEL CHAPTER 3. MODEL DEVELOPMENT and the weight distribution as explained before:

W W f − r (3.1) Cαf Cαr

The cornering stiﬀness value used was taken from the work of Mirone and the weight distribution was studied in the faculty’s laboratories:

(0.45)160 N/g (0.55)160 N/g K = − = 5.27 (deg/g) (3.2) 69.8 N/deg 174.5 N/deg

Given the positive value of the understeer coeﬃcient the vehicle catalogs as of understeer conﬁgu- ration. It is useful to evaluate and compare it’s characteristic speed (equation 2.38), the yaw rate gain (Equation 2.42) and the lateral acceleration gain (equation 2.40) —Both mentioned in section section 2.1. The characteristic speed is then:

r 1040 ∗ 9.81 V = 57.3 ∗ char 5.27 = 10.5 m/s = 37.93 km/h

The response of the vehicle’s turning is evaluated with both gains mentioned before. Ranging from standstill to 100 Km/h, both values are evaluated and presented in ﬁgure 3.4 .

Figure 3.4: Go-Karts yaw rate gain and lat. accl. gain

It is thus, that can be asserted that the best speed in terms of its yaw rate response is close to 40 km/h. This is not to say that all corner should be taken at this constant speed, but it’s a baseline value to asses how eﬀective the negotiation of a curve might be. To further understand of the Go-Kart’s needs in terms of cornering it is useful to relate the possible lateral forces to be developed in a corner and compare them to the needed lateral forces to overcome lateral centripetal force. If for example, a corner of 16 m is studied, we can use equations 2.22 and 2.23 to calculate top speed for each wheel (Bradley 2018):

32 CHAPTER 3. MODEL DEVELOPMENT 3.1. THE GO-KART MODEL

Figure 3.5: Front forces developed in a 16 m corner

Figure 3.6: Rear forces developed in a 16 m corner

In the case of the front wheels, it can be seen that the maximum speed at which the Go-Kartcan go is limited by the right tire. This is congruent with the fact that the right tire —for a right hand turn— get progressively unloaded at the speed increases. This in contrast to the need to overcome an ever-growing lateral inertial force deﬁnes the limit.

In relation to the characteristic speed, when comparing all four wheels at the same time, it is clear that even if the characteristic speed is surpassed, the maximum speed is limited by the front right left tire but still yield a good yaw rate response. The longitudinal weight distribution plays an important role on deﬁning the speed limit tire —In the case of the kart, since the weight distribution is 45-55 the limiting tire would be the front one.

The Go-Kart activities are usually performed in the ”Juan Pablo Montoya” Go-Kart racetrack in Tocancipa, Colombia. The corners’ radii in this track range between 12 and 25 meters. There- fore, it is of interest to evaluate the karts cornering performance in this range of radius. Equaling equations 2.22 and 2.23 and solving for the velocity we obtain the maximum speed for each radius, which can be compared to the characteristic speed:

33 3.2. MBS MODELLING CHAPTER 3. MODEL DEVELOPMENT

Figure 3.7: Maximum cornering speed for diﬀerent radii

From ﬁgure 3.7 it can be seen that for every radius, the limiting speed is on the inside of the vehicle (right in this case) and from the longitudinal weight transfer ratio (0.5 in this case) the limit would be at the front or rear.

3.2 MBS modelling

Figure 3.8: E-Kart MBS model

3.2.1 Part modeling The Msc-Adams/Car subsystem modeling process was based on the explanations given in (Msc- Software 2016). It starts with specifying location in the 3-dimensional space from where the part would be constructed followed by the creating of the part entity and finally, adherent to that part (if necessary) geometries are generated associated with that part, defining the inertial properties of it. Such geometries must be specified with a material in order to calculate its inertial properties. Although steel is the predetermined material for all parts, one can specify not a material but a density, with which the mass and rest of the inertial properties will be calculated. This process is followed for every part in the subsystem. Then, joints are used to relate the kinetics of one part to another. It is obvious that some part in one subsystem connects to other. The connection between the must be specified as well. Msc-Adams/Car has a special way of doing this through the so-called

34 CHAPTER 3. MODEL DEVELOPMENT 3.2. MBS MODELLING mount communicators which must be speciﬁed in the pertaining parts1. Here follows a description of the modelling of the involved subsystems for the Go-Kart model.

Body

Past work done in the faculty involved generating a series of coordinates for the Go-kart’s chassis intersection of circular profiles (Sepulveda 2018). The geometry adopted from the work of Sepul- veda 2018 was selected to implement in the model’s development, both for coherence reason within the research work done on the Go-Kart prototype and for the assumption that given the weight distribution within the body’s chassis, the remaining not modelled sections (the two bars holding the steering shaft in its place) have little effect on the overall dynamic response of the model. The entire chassis is assumed to be made of circular tubular sections of 32 mm of diameter with a thickness of 1 mm. Since it was easier to model the part as solid cylinders, the density of the part was adjusted so the weight of the part would match that one of the real chassis, which was already measured previously. The value for this density was 3.36 e−6kg/mm3. The Center of gravity was obtained from modelling with basic geometric features the different elements pertaining the powertrain and locating them as geometric features of the Kart’s body. The density of each element was then altered to match it’s real mass value.

Steering system

Figure 3.9: Steering connection topology

For the case of the steering system, no coordinates were available in any previous work that would reﬂect the actual geometry of the prototype’s setup. This led to begin the modeling by measuring the dimension of all the steering components and specifying their key locations. This task was somehow daunting, since a global reference coordinate system must have been used and the geometric derivation of its geometry involved specifying orientations diﬃcult to assess rapidly. However, the points’ locations were established and followed by the creation of the parts, their geometry and the joints binding the together.

1This is merely a summary of the process followed, for a detailed instruction on Msc-Adams/Car please refrain to (Msc-Software 2016)

35 3.2. MBS MODELLING CHAPTER 3. MODEL DEVELOPMENT

Element Translational Rotational Total Number of constrains Fixed joint 3 3 6 1 Revolute joint 3 2 5 4 Spherical joint 3 0 3 2 Universal joint 3 1 4 2 Motion (rotational) 0 1 1 0 Red. gear 0 1 1 1

Table 3.2: Joint count for the steering system

Using the table 3.2, derived from ﬁgure 3.9, the degrees of freedom of the steering subsystem can be calculated using equation 2.50 as follows:

DOF = 6 × (10 − 1) − 50 = 3

The subsystem has three degrees of freedom. Two of them, must be noted, are the rotating hubs a the end of each wheel carrier. I we took those out with the hubs, to analyze only the steering motion, we would end up with 1 degree of freedom —The calculation would be then with 8 parts and four revolute joints instead of six. In other words, with an input value for the rotational constrain at the steering wheel, the entire system can be characterized. It must be noted here, that for the maneuvers performed in the analysis of this model, the simulation had an initial condition for the revolute joint at the steering wheel, thus eliminating the single degree of freedom of the subsystem.

Rear axle, front and rear tires The rear axle was modeled as a rigid beam of dimension 1040 mm x 40 mm with a thickness of just 3 mm. However, the density of the part was altered to match the weight of the actual piece, which encompassed also the rear brake disk and three ball bearings. The subsystem has a single degree of freedom, speciﬁcally the rotational longitudinal DOF. All the rest are restricted by the ball bearing, in reality, and by the revolute joint, in the model. The full description of the rear axle part are presented in table 3.3.

Outer Diameter [mm] Thickness [mm] 40 3 Length [mm] Density [kg/mm3] 1040 4.82 e−6

Table 3.3: Rear axle part characteristics

For the front and rear tires, the model was adapted from the geometry of the actual tires references earlier in 3.1. The property file for the Pacejka model was adapted, using the same parameters as the default file version since such detailed property data for competition Go-Kart tires was difficult to come by.

Element Translational Rotational Total Number of constrains Fixed joint 3 3 6 5 Revolute joint 3 2 5 7 Spherical joint 3 0 3 2 Universal joint 3 1 4 2 Motion (rotational) 0 1 1 0 Red. gear 0 1 1 1

Table 3.4: Joint count for the entire model

In general the entire model would need to have ten degrees of freedom: The free movement of the Go-kart’s body delivers six, the single degree of freedom from the steering system adds two

36 CHAPTER 3. MODEL DEVELOPMENT 3.2. MBS MODELLING more, each of the front wheels fixed at their respective hubs rotate around the left and right wheel- carrier meaning two more and finally, the rear axle rotates with it’s wheels fixed to him around it’s longitudinal axis providing the final degree of freedom. Once again, referring to the model joints in table–, equation 2.50 is used to evaluate this.

DOF = 6 × (16 − 1) − 80 = 10

3.2.2 Topology

Figure 3.10: Basic connections map of the model’s parts

The basic description of the model topology can be found in ﬁgure 3.10. The model has 6 subsystems, necessary in Msc-Adams/Car to perform full vehicle analyses. The subsystems and their role are identiﬁed in table 3.5. As stated previously, to build a full vehicle model in Msc- Adams/Car a minimum of six subsystems must be created, namely the body, the front and rear suspension, the front and rear tires and a steering system.

Since the Go-Kart has by default no suspension system, the role of front suspension was given to a part-less subsystem, with the only purpose to communicate alignment parameters to the front wheels. Similarly, the role of rear suspension subsystem was given to the rear axle, performing the same job as the front suspension subsystem along with connecting the rear wheel to the rest of the vehicle.

Joining the bodies together and restricting their movement are 16 joints and 1 reduction gear, which in turn impose 79 constraints. The detailed connection topology is presented in ﬁgure 3.11. Based on those constrains the model has – degrees of Freedom.

37 3.2. MBS MODELLING CHAPTER 3. MODEL DEVELOPMENT

Subsystem name Part Part ID Role Body Chassis 1 Body Steering wheel 2.1 Steering shaft 2.2 Lever Arm 2.3 Left Pitman Link 2.4 Steering Right Pitman Link 2.5 Steering system Left Wheel carrier 2.6 Right Wheel carrier 2.7 Left Hub 2.8 Right Hub 2.9 Rear Axle Rear Axle 3 Rear suspension Left front tire 4.1 Front tires Front tires Right front tire 4.2 Left rear tire 4.1 Rear tires Rear tires Right rear tire 4.2

Table 3.5: Model parts and roles

Figure 3.11: Model’s detailed topology

38 Chapter 4

Summary of achievements

4.1 results and interpretations

The MBS model developed was tested to evaluate (either quantitatively or qualitatively) it’s accu- racy and sensitivity. A simple step steer maneuver was simulated to evaluate the vehicle dynamics phenomena discussed previously. Beforehand, a sort of ”convergence” test was carried out to identify the amount of time steps minimum at which the solution for a certain parameter would not vary too much (less than 5%). At the end a time step of 1000 steps/s was selected.

The Step steer maneuver To evaluate the vehicle’s response a step steer maneuver was performed applying at the steering shaft a right hand turn of 15◦degrees1. The maneuver works as follow, after the initial parameters as logged in, the assembly starts by running a static simulation. If and only the simulation is successful, it passes onto running the dynamic simulation. An initial speed is initialized at the CoG after the car is ”settled” on the road interface. Then, the maneuver occurs at the time indicated. The speed of the maneuver progressively decays until the vehicle stops in a standstill. For this work’s case, the basic parameters of the maneuver are presented in table 4.1 and the kinetic imposed motion on the steering wheel is better illustrated by ﬁg 4.1.

Time Steps Initial velocity Steering angle Starting time 10 s 1000/s 30 km/h 15◦(for 0.5s) 1st s

Table 4.1: Maneuver parameters

Figure 4.1: Steering wheel angular imposed motion

1The maneuver is selected from a previously conﬁgured list of open-loop maneuvers, the direct motion conﬁgu- ration is done by Msc-Adams/Car internally

39 4.1. RESULTS AND INTERPRETATIONS CHAPTER 4. SUMMARY OF ACHIEVEMENTS

A ﬁrst approach to present the validity of the model is to evidence typical dynamic behavior of any four wheel vehicle. As explained in chapter chapter 2 along the negotiation of a corner of a given radius, a lateral acceleration occurs to the body. The lateral weight transfer due to this eﬀect can be better observed in the shift of normal forces at the wheels.

(a) Front normal forces

(b) Rear normal forces

Figure 4.2: Normal forces at the axles for a step steer maneuver

From figures 4.2a and 4.2b the weight transfer can be made evident at both the front and rear forces. The oscillation of the forces is supposed to be due to vibrations in the whole of the Go-Kart, which affect the contact patch geometry and therefore the magnitudes derived from it. Because the velocity decreases with time, the analysis is pertinent initially just the the very first seconds of the simulation. For this purpose the span of the conclusions are limited to the results up to the 3rd second where the forces have fully developed and the speed is not too different from the target speed. Successively and due to the steered wheels, slip angles develops on the wheels. The decline on the values accounts for the decrease in speed as well. The values reached on both the front and rear wheels are within what would be considered low (Gillespie 1994) which allows for the analysis of the force development due to them to be thought about in the lineal region of the curve represented in 2.5 and are consistent with the steered input based on the cornering explanation on chapter chapter 2. Observable in figure 4.3, the magnitude of the front slip angle is higher than the one developed on the back. The slip angle is dependant entirely on the lateral and heading speeds of the tire. For the Go-Kart’s case, the wheels at the front are less stiff than the rear ones which, despite the rear axle being slightly more loaded than the front one, causes the lateral acceleration at the front to be higher, which means a higher lateral velocity. Another characteristic that catches the attention is the asynchronous development of the peak slip angle between the front and rear. To understand why this is consistent with the theory, a further explanation on the order of the slip angle development is in order; As the front wheels turn, the inertia carried by them develops into a lateral acceleration. The slip of the wheels

40 CHAPTER 4. SUMMARY OF ACHIEVEMENTS 4.1. RESULTS AND INTERPRETATIONS

Figure 4.3: Slip angles developed at the left side produces a lateral speed on them after overcoming the initial friction that stops them initially from moving —this happens very fast though being dependent on the vehicle’s velocity and the wheels stiﬀness. Just after this, the car, now subjected to a lateral force on the front turns. With the turn, the car’s inertia produces a lateral acceleration (centripetal) on it’s CoG and impulses both the body and the rear wheels outwards, thus a lateral speed on the rear wheels appears. The latency between the appearance of front and rear lateral velocities is the reason behind the phase shift which can be evidenced also in ﬁgure 4.3.

(a) Front slip angles

(b) Rear slip angles

Figure 4.4: Slip angles at both rear and front axles

41 4.1. RESULTS AND INTERPRETATIONS CHAPTER 4. SUMMARY OF ACHIEVEMENTS

A difference on the left and right slip angles is also evident on both axles, illustrated in figures 4.4a and 4.4b. The reason for this comes from the geometry of the steering system. While at the rear axle the wheels are connected only by the solid rear axle, the movement of each wheel at the front depends on the steering system which does not produce the same wheel steering angle for each wheel as can be seen in figure 4.5.

The direct consequence of a presence of slip angle and a vertical load is of course the appearance of lateral forces on he contact patch of each wheel, as a reaction consequence for the shear stress on it. The magnitude of the forces at the front and rear axles is proportional with the magnitude of the slip angles (front and rear ones) as indicated in ﬁgure 2.5.Furthermore, As the lateral forces are a function of slip angles and vertical load, it is only logical that both for the front and rear axle, a diﬀerence between lateral forces exists.

Figure 4.5: Front wheels steered angles

The same accounts which affect the slip angle, affect also the lateral force development, for it is dependable on the slip angle. Additionally, the weight transfer and the asymmetry in the weight static distribution (however small) both contribute to a difference in the force values observed in figure 4.6a and 4.6b.

42 CHAPTER 4. SUMMARY OF ACHIEVEMENTS 4.1. RESULTS AND INTERPRETATIONS

(a) Front lateral forces

(b) Rear lateral forces

Figure 4.6: Lateral forces at both rear and front axles

Consistent with the bigger transfer of weight at the front and the smaller difference in slip angles for left and right wheels at the rear, a smaller difference on the lateral forces is present in the rear axle. It must be noted that similar forces are not a highly possible scenario, for the outside wheel (in this case the left one) is evermore loaded and this a difference should most of times be present. Furthermore, the response in turning for the vehicle was evaluated while performing the same maneuver at different speed. As mentioned in table 4.1, the speed for all the previous analyses was 30 km/h. For this comparison the same maneuver was simulated at speed in around the theoretical critical speed (37.9 km/h) to asses its response. For purposes of visualization, an averaged values plot was elaborated, taking the average of the values in a range of 1/14Th of a second for this is the frequency of oscillation for the higher mode observed in all the results. The results are presented in figure 4.7.

As seen in ﬁgure 4.7, the maximum value for the yaw rate is achieved at 40 km/h, very close to the critical speed. Furthermore, the results observed for the simulations of 50 and 60 km/h show that at higher speeds a higher yaw rate is not obtained, also being the case for the lower speed of 30 km/h. This indicates that the theoretical yaw rate gain matches the simulated scenarios and that the maximum yaw rate also for the simulation is reached at around 40 km/h.

43 4.2. CONCLUSIONS CHAPTER 4. SUMMARY OF ACHIEVEMENTS

Figure 4.7: Yaw rates for diﬀerent velocities

4.2 Conclusions

A multibody model was developed in the Msc-Adams/Car software, to recreate an electric prototype developed at university grounds. The model resembles very closely the actual prototype as it has similar inertial properties; come of which can be easily validated in the real vehicle as to serve as feedback for the model creation. Weight distribution, total mass, individual part’s weight and location as well as pilot location resemble closely the real Go-kart. How ever, some other more detailed aspects of the model were not taken into account. This details include geometric characteristics of the chassis, which is tubular instead of solid cylindrical. Other parts of the vehicle were constructed similarly. This difference, need to be revised as it affects the dynamic result as a consequence of the alteration in the basic inertial properties of the different parts, although it is believed that it does not affect on a very high degree the altogether behavior of the vehicle. Dynamic simulation were conducted, proposing a simple step steer maneuver to evaluate the basic notions of the vehicle’s dynamic behavior. Since no powertrain was modeled, the vehicle was initialized with a constant speed (usually 30 km/h). The result output for the vehicle’s behavior was qualitatively within what was expected. For both axles, the weight distribution resembled the theoretical values, although at the front axle the differences were bigger, reaching differences of 11.5% whereas a the front the maximum difference noted was 4%. The magnitude of the slip angles developed is consistent with the fact that the speed developed in the kart is not extremely high, and thus the lateral forces at a curve are also relatively small. the maximum slip angle noted was about 1◦at the front and around 0.2◦. This difference in magnitude, as well as a recorded offset in the reaching of such maximum values is consistent with the theoretical analysis of the kart being of understeer fashion. In further regard to the understeer condition, it must be noted that it is not a desirable condition, and in turn an oversteer condition is more common on competition vehicles for it’s handling advantages. The vehicle has an understeer condition due to the relatively balanced weight distribution (45% front) being boosted by the very different stiffness on front and rear tires. Lastly, the analysis for different speeds of the yaw rate developed in the vehicle, shows a much more strong consistency between theoretical analysis and the model of the vehicle. Being an understeer vehicle, the quantity known as ”characteristic speed” yields the speed at which the vehicle develops a higher yaw rate per degree of turn, hence the speed at which the go.kart is most responsive. In theoretical calculations, such speed was found to be around 40 km/h (specifically 37,9 km/h). Simulations at different speed ranging from 30 km/h up to 60 km/h showed, yielded different yaw rates, from which the one at 40 km/h (the simulation closest to the theoretical characteristic value) was the highest, as can be seen in figure 4.7. This confirms the close relation between the theoretical and multi-body models.

44 CHAPTER 4. SUMMARY OF ACHIEVEMENTS 4.3. CHALLENGES AND LIMITATIONS

4.3 Challenges and limitations

The understanding of multiple aspects of vehicle dynamics is of paramount necessity, specially if a model is to be built to behave, within reasonable tolerances, as the complex theory proposes. For this reason, the first challenge was to relate to the best possible extent, vehicle dynamic theory to the specifics of a suspension-less small vehicle such as a Go-Kart. Literature and published work studying the dynamic behavior of the Go-Kart, in relation to the amount of work carried out in regards to bigger vehicles is virtually non-existent. Several of the sources consulted for this work stated similar difficulties. However, multibody modelling as a whole is a very widespread line of research and this was very helpful. Another challenge presented itself at the moment of dealing with the software itself. Msc- Adams/Car is undoubtedly a very powerful tool for dynamic and dynamic-mechanical simulation, but it is equally daunting for un-instructed persons who wish to perform activities at topics beyond basic analysis for pre-conceived systems in the software. Since the project involved creating from scratch the subsystems instead of adapting the existing ones (which is common for normal vehicle analysis since all major four wheelers have similar components), the tutorials fell short for some tasks, which in turn consumed a considerable amount of time performing.

4.4 Future work

The model needs to have several areas studied further in order to perfect the representation of the electric prototype. Firstly, further refinement of the actual model is in order. Introduce actual tubular structures and replace the existing cylindrical ones is a first step. This need to be not only in the chassis but in other parts as well. furthermore, a powertrain, even a basic one, needs to be included in the model. Further more simulations can be achieved if a powertrain is included. In relation to the simulations performed, two main issues need to be addressed; the high frequency variation of the results, which is though to be a consequence of some modelling error, has an important impact on the results, because it difficulties the reading and interpretation of the results. It must be noted that such variation appears in two different modes, one with a frequency of about 1/14th of a second and the second of about 1 second. The second issue is the deceleration of the vehicle in the maneuvers simulated for no known reason. The simulations are set to maintain a constant speed, but this does not happen as the vehicles slows down to a stand still after some simulation run time. This also makes the interpretation of the result more difficult and impoverishes them altogether. Further efforts can be made into incorporating mechanical analysis to the model, since it is known that for this type of vehicles who have no dynamic suspension, the elastic characteristic of it’s chassis has a bigger influence on the cornering behavior of the vehicle. Implementing the possibility of evaluating such effects could offer the possibility of proposing design changes on the chassis for different track configurations.

45 Bibliography

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