Wheel and Tire Customization Influence On

Vehicle Dynamics Performance

Xianjie Zhou

Dr. Paul Venhovens

04/08/2014

Disclaimer

The data provided in this status report are based on best-practice

simulation-based engineering methods where assumptions and

simplifications were made. The outcome may vary due to different

physical conditions of the vehicle and its components.

This is supplemental information only for manufacturers and

customers considering customization of wheels and tires. All

modifications to the vehicle should comply with national/state

regulations and product manuals.

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1 Introduction The purpose of the study is to quantify the vehicle dynamics performance impact

of customized wheels/tires for a chosen reference vehicle using simulation tools. Automobile product is designed as an integrated system. The vehicle’s suspension, wheels, tires, etc., are defined to meet the vehicle dynamics targets. Any change of the components will potentially change the characteristics of the system. In this study, this change will be quantified in terms of several fundamental engineering metrics. And then they are translated into consumer languages as a supplemental material for customers who are looking to customize wheels and tires.

2 Qualitative Metrics for Vehicle Dynamics Performance Similar to human beings, vehicles are different from brand to brand because

they are characterized by their own “DNAs.” The vehicle’s DNA is composed with many metrics in different aspects. For example (see figure 1): The Cadillac is luxurious, expensive and exclusive, while the MINI is youthful, bold and distinctive.

Figure 1. Vehicle DNAs

These DNAs consist of different engineering performance metrics, such as vehicle dynamics; noise, vibration, and harshness (NVH); safety; comfort; and so on. Under each of them, there are even more sub catalogs. In our study vehicle dynamics is the main topic. The study consists of bunches of engineering sub catalogs, such as steady-state handling, transient state handling, on-center steering, emergency handling, etc. And under each sub catalog, there are specific data to

-30%

-20%

-10%

0%

10%

20%

30%

40%Sporty

Youthful

Fun to Drive

Family-Oriented

Distinctive

Luxurious

Eco-Friendly

Economical

UpscaleExpensive

Exclusive

Simple

Affordable

Good value

Powerful

Bold

Conservative

Comforting Cadillac Chevrolet Lexus MINI

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describe the level of the performance. The vehicle dynamics metrics are listed in table 1. A sample vehicle dynamics spider chart (figure 2) is used to graph the general profile of the performance.

Table 1. Vehicle Dynamics Metrics Steady-State Handling • Understeer Gradient • Yaw Rate Gain • Lateral Acceleration

Gain • Yaw Rate Linearity • Roll Gain • Sideslip Angle Gain

Transient Handling Yaw Rate Response Time Yaw Rate Overshoot Roll Rate Response Time Roll Angle Overshoot Lateral Acceleration Phase Lag

Steering Feedback Steering Torque Gradient Steering Torque Linearity Steering Torque Feel

On-Center Handling Steering Torque Steering Stiffness-Gradient Yaw Rate Velocity Gain Response Dead Band Steering Sensitivity Steering Hysteresis

Emergency Handling Yaw Stability - FMVSS 126 Roll Stability - NHTSA Fishhook

Parking Static Steering Torque

Road Adaptability Rough Road Cornering Index Wet Road Cornering Index

Coupled Dynamics Cornering Understeer Angle Increment

Straight-Line Stability Pull and Drift

Disturbance Sensitivity Understeer Rate

Figure 2. Vehicle Dynamics DNAs

0

1

2

3Steady State

Transient

On-Center

Emergency

Parking

Steering Feedback

Disturbance Sensitivity

RoadAdaptability

Straight LineStability

CoupledDynamics

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3 SEMA Product Performance Index Label In order to make the numbers more meaningful and readable, a color scale bar is

designed to provide the quantification of product performance. It allows a side-by-side product comparison and differentiation with respect to the reference component performance. It benefits in supporting product development such as, “How much performance gain or loss ”; Supporting marketing and sales: “These are the benefits: this product is % better”; Supporting brand development: “This sets our company apart from our competitor.”

Below is an example of the color scale bar. It shows the understeer gradient of different tire setups with the same reference vehicle. The results are positioned according to their values. We consider the understeer gradient as the smaller the better as long as it is positive (negative means oversteer—that is not a design preference). So we mark “excellent” (neutral steer) in the left end and “fair” (understeer) in the right end. The reference baseline product is marked with red triangle. This is more user-friendly to help people understand which product will improve this specific performance, and which product might not have as much benefit as baseline setup.

Figure 3. Color Scale Bar Sample

In our study, there will be nine performance index labels. The “excellent” and “fair” implications are explained as follows: a) Understeer gradient, tagged as “Directional response.”

It means the vehicle cornering directional response to the input of steering wheel. • “Excellent ” (neutral):

The vehicle can maintain the same cornering responses level as the lateral acceleration increase—that is neutral steer trend. Neutral steer gives more steady response through different cornering intensities (low speed to high speed, large angle to sharp angle). The driver will have more confidence.

• “Fair” (understeer): Vehicle responses decrease as the lateral acceleration increases. So the driver might not be so confident to make a sharp turn or high-speed turn as those driving a neutral steer vehicle.

b) Sideslip-lateral acceleration gain, tagged as “road grip ability.” Sideslip is the ratio of lateral velocity and longitudinal velocity. It describes whether the vehicle moves along the heading direction or with lateral sliding. • “Excellent” (less slip):

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The vehicle can follow the heading direction with less slip and hold tightly on the expected path in the driver’s mind. The cornering action is more predictive and accurate.

• “Fair” (more slip): The vehicle has more lateral slip, and the passenger might feel transversal sliding. The vehicle does not have as great a grip as the reference product when taking turn.

c) Roll gain, tagged as “body rolling level.” It describes the vehicle body roll level under certain lateral acceleration. • “Excellent” (roll less):

The vehicle body rolls less than the baseline vehicle. • “Fair” (roll more):

The vehicle body rolls more than the baseline vehicle. d) Lateral acceleration time lag, tagged as “lateral agility.”

It describes the vehicle lateral acceleration buildup delay with respect to the steering input. Lateral acceleration is one of the most evident metrics that easily be sensed by passengers. It gives the feeling of the so-called “centrifugal force.” • “Excellent” (more quick):

The “centrifugal force” responds to the steering wheel much quicker. • “Fair” (less quick):

The “centrifugal force” takes longer time to respond to the steering wheel. e) Roll over shoot, tagged as “body oscillation.”

It describes the vehicle body rolling overshoot during transient handling. It is usually a trade off for quick lateral response. • “Excellent” (more tight):

The vehicle rolling is damped within a limited level so the passenger will not experience intense oscillation during transient handling.

• “Fair” (less tight): The vehicle will behave with sensible roll overshoot and feel less tight.

f) Steering torque gain, tagged as “steering feedback response level.” It describes the firmness of the steering wheel when the driver steers. The steering wheel is the vital component to transfer the man-machine communication. And the torque gain is the metric to quantify the intensity of the feedback. • “Excellent” (firmer):

The steering wheel gives more feedback to the driver, thus, it creates more fun driving. The driver can have a comprehensive understanding of how the vehicle behaves in cornering.

• “Fair” (lighter): The steering wheel does not produce so much torque feedback. The steering wheel will feel lighter for the driver but there is a lack of feedback.

g) Steering torque lag, tagged as “steering feedback response speed.” It describes the feedback speed of the steering wheel when the driver steers. • “Excellent” (more quick):

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It takes less time for the steering wheel torque to build up. In other words, once the driver turns the steering wheel he will sense the torque feedback.

• “Fair” (less quick): It takes longer time for the steering wheel torque to build up. The driver might not feel the anticipated torque feedback. It is kind of loose feeling.

h) Lateral deviation (split-mu braking), tagged as “brake pull tendency.” It describes the tendency of brake-pull when driver brakes on split-mu surface, such as asphalt/icy surface. Due to the difference of left and right longitudinal forces in the hard braking, there will be yaw moment acting on the vehicle and cause brake pull. It happens in emergency situations. So it is regarded as an emergency handling case. This maneuver is studied also because it is one of the most impacted metrics caused by wheel-offset changes. • “Excellent” (fewer pull tendency):

The vehicle has fewer tendencies to laterally deviate in hard braking maneuvers on split-mu surfaces.

• “Fair” (More pull tendency): The vehicle has the tendency to laterally deviate in hard braking maneuvers on split-mu surfaces.

i) CG vertical acceleration RMS (root of mean square), tagged as “ride comfort.” It describes the harshness of the vehicle ride on a typical uneven road surface. • “Excellent” (softer ride):

The passenger feels a softer ride and it is more comfortable. The tires will largely filter the unevenness of the road surface.

• “Fair” (stiffer ride): The passenger feels a harsher ride compared to the reference product.

4 Customer Domain Transformation The engineering metric is not a comprehensive language to customer. To

determine the customer metrics there are three aspects to consider. Firstly, the commonly used metric by the American publication Consumer Reports is a good starting point. Consumer Reports is one of the most well-known rating publications for all kinds of products, including automobile products. There are three metrics related to vehicle dynamics: routine handling, emergency handling and ride (They are located in Performance subdirectory and comfort/convenience subdirectory). (See figure below)

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Figure 4. Example of Consumer Report for Cars

Secondly, the steering feel is one of the most dominant feedbacks of a vehicle for a customer; it tells customers the different characteristics of vehicles, so it is considered as an independent catalog. Last but not least, the metrics that will be impacted from the modification of wheels and tires will be the priorities. There are numerous metrics, and some of them are not affected by the wheel/tire change, so they are not considered in the current study.

In other words, we determine four main catalogs of customer metrics: routine handling, emergency handling, steering feel and ride comfort.

Table 2. Handling Metrics

Customer Handling Requirements

Engineering Domain

Routine Handling

Steady State Handling Understeer Gradient, Yaw Rate Gain,

Roll Gain, Side Slip Angle Gain Transient Handling Yaw Rate Time Constant, Lateral

Acceleration Phase Lag, Yaw Rate Damping Ratio, Roll Angle

Overshoot Emergency Handling Roll over* (For wheel

offset study only) Static stability factor

Split-mu Braking (Pull/Drift)

Lateral Pulling Ratio

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Steering Feel Steering Feedback (Off-Center)

Steering Torque Feel (Torque vs. Angle, Torque vs.

Lateral Acceleration Gradient) On-Center Steering Steering Torque Time Lag

(vs. Steering Angle) at Low Lateral Accelerations

Comfort

Primary Ride Vertical Acceleration of C.G.

• Routine handling: It describes the vehicle handling behavior in daily-use

situations. It consists of normal maneuver such as low-speed large angle cornering, mid-speed small angle cornering and normal lane change. The steady state handling and transient state handling in engineering metrics are used to describe the routine handling.

• Emergency handling: It describes the maneuver in emergency situation such as obstacle avoidance, high-speed lane change and hard braking. In emergency handling the key factor is the vehicle stability, which includes rolling stability and directional stability. The static stability factor and split-mu braking are two stability intrinsic metrics because they are determined by the vehicle physical parameters without any control program engaged. However more and more vehicles are equipped with stability controller. They help stabilize the vehicle in emergency handling maneuver. In this case the controller take a determinant role of the vehicle stability. In the study the controller is not involve.

• Steering feel: It consists of two parts—the off-center steering and on-center steering. Off-center steering describes the steering torque feedback in general steering operation. On-center handling describes the steering “feel” and precision of a vehicle during nominally straight-line driving and while negotiating large-radius, low lateral acceleration bends at high speeds. It is within the most important metrics to indicate steering quality.

• Ride comfort: It has a tight connection to the wheel and tires. And it is also one of the most important customer requirements for a vehicle. In this study, the RMS of vertical acceleration in the center of gravity of the vehicle is used to indicate the trend of wheel and tire impact to ride comfort.

In the engineering to customer translation, the engineering metrics are not rated

with absolute performance level because first, there are no absolute good or bad evaluations. Different product segments have different definition of “good,” and different customer groups have different expectations from their products. Second, this study focuses on the wheel and tire change impacts. So it is better to show the relative level with respect to the baseline vehicle so the customer will have better idea of what kind of improvement they will have with the new wheel and tire.

In this study there four steps in translating the engineering metrics to customer metrics.

Steps 1: The vehicle dynamics engineering values are achieved from simulation, respectively. Then, they are positioned in the color scale bar.

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Step 2: The relative performance is rated with five different levels based on the position with respect to the reference product:

• Level 1: The performance of the new wheel and tire is greatly improved compared with the baseline product.

• Level 2: The performance of the new wheel and tire is a little better than the baseline product.

• Level 3: The new wheels and tires provide performance above the level of the baseline.

• Level 4: The products’ modification can maintain same-level performance as the baseline.

• Level 5: The modification performance does not generate as good a performance as the baseline.

Figure 4. Rating Interpretation

Step 3: All the metrics rated value of each group are weight-averaged to generate the final rate of the catalog. For example, in table 3, the steady-state handling data of 275/45R22 is compared with reference tire 275/65R18 and gets a rate for each item. Then all of the items accounted for steady state are averaged to get a final rate. Any score between 1-2 is rated as level 1; a score between 2-3 is rated as level 2; and so on.

Table.3 Engineering metrics to consumer metrics transfer example Steady state handling 275/65R18(Ref) 275/45R22 Rate

Kus deg/g 1.88 1.93 3 Side slip gain deg/g 0.29 0.37 3

Roll gain deg/g 5.69 5.56 1 LatAcc lag ms 0.96 132 3

Roll Overshoot % 0.11 10.37 3 Average 2.6 Final rate 2

Step 4: The product specification and performance rates are listed in a table to provide guidance for wheel and tire customers. Difference symbols will be assigned to the different performance levels of the products. The levels are: excellent, improved, above baseline and maintained performance (see figure 5).

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Figure 5. Example of Tire Product Rating Cards.

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5 Product Analysis and Assessment Process The process of study is a little bit different from OE development process in that it starts from reverse engineering. Then it comes to the normal way that includes baseline vehicle mathematical model setup, vehicle dynamics DOE simulation and objective quantification and evaluation.

Figure 6. Product Analysis Process

5.1 Reverse engineering Reverse engineering is used to achieve critical data of reference vehicles, which includes dimensional data, mass and inertia data, geometrical hardpoints data, suspension kinematics and compliance (K&C) data, tire data and necessary component properties (bushing property, spring stiffness, damper characteristics, etc.). To achieve all the above data, several tests are necessary.

a) Suspension kinematics and compliances test. Such test provides data of suspension characteristics that is the determined factors of vehicle dynamics performance.

Figure 7. K&C Test Rig

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Figure 8. K&C result example (bumpsteer test)

b) Even though the K&C test has achieved most of the property information of

the suspension, we are still not able to cover all kinds of situations where there are different wheel offsets. In this case, we will need the multi-body simulation tools to analyze the impact of geometrical changes. If there is no CAD data available, suspension and steering system kinematics hardpoints measurement (CMM) will be the backup solution for the information of the suspension multi-body simulation model. In the F-150 project, the CAD models of the front and rear suspension are already available (as shown below), so no measurement is required.

Figure 9. CAD Data of the Reference Vehicle

c) Tire property test. Tire test measures the tire longitudinal, lateral and

aligning moment and other characteristics and generate an empirical function model (Pacejka model) for all kinds of simulation. In figure 10, typical data is achieved from the tire test. This study focuses on tire change impact to vehicle dynamic performance, so tire data is one of the most critical data needed to perform a good fidelity analysis. While not all the tires data are available, we will estimate the value according to some well-developed research.

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Figure 10. Tire Property Test

(a) Longitudinal force; (b) Lateral force; (c) Aligning moment

d) Bushing property test. The bush property includes static stiffness, dynamic stiffness and damping. They are essential data for a precise suspension compliance simulation. Suspension is architecture of the linkage joint with rubber bushings and ball joints. So there is an inherent flexibility of the suspension; we call it suspension compliance. It is determined by the bushing axial and radial stiffness, which can be measured with a tensile test machine. In figure 11, shown are the F-150 lower control arm bushing test setup and results from the test carried out in CU-ICAR.

Figure 11. Lower Control Arm Bushing Test

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e) Vehicle ride and handling test. It provides valuable data for the correlation of

the simulation model in the system level so that the analysis will be much more credible. However, this study is mainly used to examine the function variation with change of variables, subsystem model correlation already meet the required precision.

5.2 Simulation-based Product Verification After the data is achieved, the simulation model is built and studied in the following process. Step 1: The vehicle multi-body model is built based on hardpoints and component property data in SIMPACK. The F-150 featured has a front double wishbone suspension and rear leaf spring suspension (see figure 12). They are simulated in terms of kinematics and compliance respectively.

Figure 12. F-150 Simpacks Simulation Model

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Step 2: The suspension K&C simulation result is compared with measured data. Slight adjustment to the model is needed in order to correlate the simulation to the measured data. The correlated model becomes the baseline suspension model. Step 3: All the parameters, which include the K&C data, will be input to the Matlab or Carsim simulation model for their fast simulation speed compared to multi-body simulation. If the vehicle ride and handling test data is available, the simulation model will be correlated again and generate a full vehicle baseline model. Step 4: Customize components will replace the baseline model parameters and update the K&C result and full vehicle simulation result. The simulation program is well developed with a user-friendly interface. The program is capable of outputting all the metrics values of the set of tires and generating all the color scale bars in the same time, which make it a very efficient tool in this project.

Figure 13. Matlab Simulation Program

Step 5: The final result will be examined and evaluated with respect to the target settings or rating system, which has been introduced in section 4.

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6 Ford F-150 Case Study and Tire Customization

6.1 Vehicle specification The first case study project is based on 2013 Ford F-150. The vehicle specifications are listed in the table below:

Table.4 F150 reference vehicle specifications Specifications Horsepower 365 hp @ 5,000 rpm Torque 420 lb.-ft. @ 2,500 rpm Length 231.9" 5,890 mm Width Without Mirrors 79.2" 2,012 mm Height 76.7" 1,948 mm Wheelbase 144.5” 3,670 mm Track width front 67.0" 1,702 mm Track Width Rear 67.0" 1,702 mm Curb Weight 5,615 lbs. 2,547 kg Axle Load at Curb Weight 57% Front / 43% Rear GVWR 7,200 lbs. 3,266 kg Maximum Payload 1,520 lbs. 689 kg Tires P275/65R18C OWL all-terrain MSRP $43,690

6.2 Vehicle Wheel and Tire Portfolio There are series of options of wheel and tire customization for the F-150 (see table 5), some of which are from OE recommendations while others from aftermarket products. OE recommendations cover a range of rim sizes from 17 to 22 inches. But there are also 24-inch and 26-inch wheels and tires available in aftermarket products. The OE recommendation product line does not have any wheel offset changes while there is a range of +44 to -44 wheel offsets in the aftermarket products. And there are also passenger tires and light-truck tires. All of this wheel and tire information will be analyzed for the impact to the vehicle dynamics metrics.

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Table 5. Wheel and Tire Portfolio

6.3 Performance Index Label of Ford F-150 Wheels and Tires By using the process introduced in section 5, we have achieved following results of the performance index label (See figure 14-18). There are several observations from the results. a) The cornering capability mainly depends on the tire aspect ratio and section

width. Generally, wider sections and lower aspect ratios will have better handling performance.

b) The load index also influences the cornering capability. The higher loads index the better performance. The baseline tire 275/65R18 has a load index of 114, which has made it better performance than most of the others.

c) Good cornering capability also means good steering feels, quick steering response and less brake pull tendency in hard braking on a split-mu road.

d) However, quick cornering response might lead to large roll overshoot. They are also trade-offs. But roll gain depends on tire vertical stiffness. High-pressure causes lower roll gain. It is a trade-off for ride comfort. For example, the tire 245/75R17 is inflated in 55 psi, which is much higher than normal passenger

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tires. And 295/30R26 has a very small aspect ratio. So we can notice both of these two tires are not showing excellent ride performance.

e) Light-truck tires are not indicating a lot of excellent performances in this study. This is because the items considered here are for normal driving conditions, not off-road or heavy-duty loading conditions.

f) The evaluation of “excellent” and “fair” only indicate the variation of different tire/wheel adoptions for the same reference vehicle (F-150). It does not necessarily tell the performance variation with respect to general vehicle products in the markets.

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Figure 14. Steady-State Handling.

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Figure 15. Transient-State Handling.

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Figure 16. Steering Dynamics

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Figure 17. Emergency Handling (Split Mu Braking)

Figure 18. Ride Comfort

6.4 Customer Guidance Report Card The tires report cards are grouped by the rim diameters. The outstanding performances of each of the products are shown below. Also the wheels are studied regarding the offset changes from baseline to negative. The result showed there are always trade-offs when considering difference product options. A comfortable tire may not have good routine handling; a tire with negative offset can improve routine handling but not the emergency handling simultaneously.

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Figure 19. Customer Report Card

(*The tires are both OE recommend and aftermarket available.)

7 Summary This study introduced a process of product analysis in system integration level.

The process is efficient and can be extended to other kinds of applications regarding vehicle dynamics issues. It creates a method to translate the engineering metrics to easy-to-understand language for consumers, which makes the study more meaningful to support product development and marketing.

The study also generates instructive data showing the potential affects of the customization of wheels and tires. They can be useful supplement materials for the customization of wheel and tire products.

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8 Appendix 8.1 Understeer Gradient

𝐾𝐾𝑈𝑈𝑈𝑈 = 𝐾𝐾𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 + 𝐾𝐾𝐿𝐿𝐿𝐿𝑇𝑇 + 𝐾𝐾𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑇𝑇𝑇𝑇 + 𝐾𝐾𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑈𝑈𝑅𝑅𝑇𝑇𝑇𝑇𝑇𝑇 + 𝐾𝐾𝐿𝐿𝐿𝐿𝑅𝑅𝑈𝑈 + 𝐾𝐾𝐴𝐴𝑇𝑇𝑅𝑅𝑈𝑈 + 𝐾𝐾𝐴𝐴𝑇𝑇𝐴𝐴𝑇𝑇 KUS : Total Understeer Gradient, deg/g

KTires : Understeer Gradient due to Tires (Static Cornering Stiffness), deg/g

KLLT: Understeer Gradient due to Weight Transfer, deg/g

KRollCamber: Understeer Gradient due to Suspension Roll Camber, deg/g,

KRollSteer: Understeer Gradient due to Suspension Roll Steer, deg/g

KLFCS: Understeer Gradient due to Lateral Force Compliance Steer, deg/g

KATCS: Understeer Gradient due to Aligning Torque Compliance Steer, deg/g

KATPT: Understeer Gradient due to Aligning Torque caused by Pneumatic Trail, deg/g

a) Understeer gradient due to tires

𝐾𝐾𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 =𝑊𝑊𝑓𝑓

𝐶𝐶𝛼𝛼𝑓𝑓−𝑊𝑊𝑓𝑓

𝐶𝐶𝛼𝛼𝑇𝑇

Where:

𝑊𝑊𝑓𝑓 ,𝑊𝑊𝑇𝑇: Front and rear tire normal load

𝐶𝐶𝛼𝛼𝑓𝑓 ,𝐶𝐶𝛼𝛼𝑅𝑅: Front and rear tires cornering stiffness

b) Understeer gradient due to lateral load transfer

𝐾𝐾𝐿𝐿𝐿𝐿𝑇𝑇 =𝑊𝑊𝑓𝑓

𝐶𝐶𝛼𝛼𝑓𝑓�𝐶𝐶2Δ𝐹𝐹𝑧𝑧𝑓𝑓2

𝐶𝐶𝛼𝛼𝐿𝐿� −

𝑊𝑊𝑇𝑇

𝐶𝐶𝛼𝛼𝑇𝑇�𝐶𝐶2Δ𝐹𝐹𝑧𝑧𝑇𝑇2

𝐶𝐶𝛼𝛼𝑇𝑇�

Where:

𝐶𝐶2: The coefficient to describe the quadratic relationship between cornering stiffness and normal force. See below equation:

Δ𝐶𝐶𝛼𝛼 = 𝐶𝐶1Δ𝐹𝐹𝑧𝑧 + 𝐶𝐶2Δ𝐹𝐹𝑧𝑧2

Δ𝐹𝐹𝑍𝑍𝑓𝑓 ,Δ𝐹𝐹𝑍𝑍𝑇𝑇: Front and rear lateral load transfer

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c) Understeer gradient due to suspension roll camber

𝐾𝐾𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑇𝑇𝑇𝑇 = �−�𝐶𝐶𝛾𝛾𝑓𝑓𝐺𝐺𝛾𝛾𝑓𝑓𝐶𝐶𝛼𝛼𝑓𝑓

� + �𝐶𝐶𝛾𝛾𝑇𝑇𝐺𝐺𝛾𝛾𝑇𝑇𝐶𝐶𝛼𝛼𝑇𝑇

�� 𝐺𝐺𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅

Where: 𝐶𝐶𝛾𝛾𝑓𝑓 ,𝐶𝐶𝛾𝛾𝑇𝑇: Additional tire cornering stiffness due to camber 𝐺𝐺𝛾𝛾𝑓𝑓 ,𝐺𝐺𝛾𝛾𝑇𝑇: Roll camber gradient due to roll angle, deg/deg 𝐺𝐺𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅: Roll gain due to lateral acceleration, deg/g

d) Understeer gradient due to roll steer

𝐾𝐾𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑈𝑈𝑅𝑅𝑇𝑇𝑇𝑇𝑇𝑇 = (𝑒𝑒𝑓𝑓 − 𝑒𝑒𝑇𝑇)𝐺𝐺𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 Where: 𝑒𝑒𝑓𝑓 , 𝑒𝑒𝑇𝑇: Roll steer gradient, deg/deg

e) Lateral force compliance steer effect 𝐾𝐾𝐿𝐿𝐿𝐿𝑅𝑅𝑈𝑈 = 2(−𝑊𝑊𝑓𝑓𝐺𝐺𝐿𝐿𝐿𝐿𝑈𝑈𝑓𝑓 + 𝑊𝑊𝑇𝑇𝐺𝐺𝐿𝐿𝐿𝐿𝑈𝑈𝑇𝑇)

Where: 𝐺𝐺𝐿𝐿𝐿𝐿𝑈𝑈𝑓𝑓, 𝐺𝐺𝐿𝐿𝐿𝐿𝑈𝑈𝑇𝑇: Front and rear lateral force compliance steer gain, deg/N

f) Aligning torque compliance steer effect 𝐾𝐾𝐴𝐴𝑇𝑇𝑅𝑅𝑈𝑈 = 2[𝑊𝑊𝐿𝐿𝐺𝐺𝐴𝐴𝑇𝑇𝑈𝑈𝐿𝐿�𝐿𝐿𝐴𝐴𝑇𝑇𝑓𝑓 + 𝐿𝐿𝑀𝑀𝑇𝑇𝑓𝑓� −𝑊𝑊𝑇𝑇𝐺𝐺𝐴𝐴𝑇𝑇𝑈𝑈𝑇𝑇(𝐿𝐿𝐴𝐴𝑇𝑇𝑇𝑇 + 𝐿𝐿𝑀𝑀𝑇𝑇𝑇𝑇)]

Where: 𝐺𝐺𝐴𝐴𝑇𝑇𝑈𝑈𝑓𝑓 ,𝐺𝐺𝐴𝐴𝑇𝑇𝑈𝑈𝑇𝑇: Aligning torques compliance steer gain, deg/Nm 𝐿𝐿𝐴𝐴𝑇𝑇𝑓𝑓, 𝐿𝐿𝐴𝐴𝑇𝑇𝑇𝑇: Pneumatic trail 𝐿𝐿𝑀𝑀𝑇𝑇𝑓𝑓 , 𝐿𝐿𝑀𝑀𝑇𝑇𝑇𝑇: Mechanical trail

g) Yaw angle due to aligning torque

𝐾𝐾𝐴𝐴𝑇𝑇𝐴𝐴𝑇𝑇 = �𝑊𝑊𝐿𝐿𝐴𝐴𝑇𝑇

2𝐿𝐿� �𝐶𝐶𝛼𝛼𝑓𝑓 + 𝐶𝐶𝛼𝛼𝑇𝑇𝐶𝐶𝛼𝛼𝑓𝑓𝐶𝐶𝛼𝛼𝑇𝑇

�

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8.2 Vehicle dynamics bicycle model

Fig. A1 Vehicle dynamics bicycle model

The lateral dynamics model is a 2 DOF (yaw rate and side slip angle as the) bicycle model. It is used to calculate the steady and transient handling metrics. It is described with following equations:

𝑚𝑚𝑚𝑚��̇�𝛽 + 𝑟𝑟� + 𝑚𝑚�̇�𝑚𝛽𝛽 = �−�𝐶𝐶𝛼𝛼𝑇𝑇𝑇𝑇

� 𝛽𝛽 + �−1𝑚𝑚�𝐶𝐶𝛼𝛼𝑇𝑇𝑥𝑥𝑇𝑇𝑇𝑇

� 𝑟𝑟 + �(𝐶𝐶𝛼𝛼𝑇𝑇 + 𝐹𝐹𝑥𝑥𝑇𝑇)𝛿𝛿𝑇𝑇𝑇𝑇

𝐽𝐽𝑧𝑧�̇�𝑟 = 𝛽𝛽�(−𝑥𝑥𝑇𝑇𝐶𝐶𝛼𝛼𝑇𝑇 + 𝐶𝐶𝑀𝑀𝑧𝑧𝑇𝑇 + ℎ(𝑓𝑓0 + 𝑓𝑓1𝑚𝑚2)𝐶𝐶𝛼𝛼𝑇𝑇𝑇𝑇

+𝑟𝑟𝑚𝑚�(−𝑥𝑥𝑇𝑇2𝐶𝐶𝛼𝛼𝑇𝑇 + 𝐶𝐶𝑀𝑀𝑧𝑧𝑇𝑇𝑥𝑥𝑇𝑇 + ℎ(𝑓𝑓0 + 𝑓𝑓1𝑚𝑚2)𝐶𝐶𝛼𝛼𝑇𝑇𝑥𝑥𝑇𝑇)𝑇𝑇

+ 𝛿𝛿𝑇𝑇�(𝑥𝑥𝑇𝑇𝐶𝐶𝛼𝛼𝑇𝑇 − 𝐶𝐶𝑀𝑀𝑧𝑧𝑇𝑇 − ℎ(𝑓𝑓0 + 𝑓𝑓1𝑚𝑚2)𝐶𝐶𝛼𝛼𝑇𝑇 + 𝐹𝐹𝑥𝑥𝑇𝑇𝑥𝑥𝑇𝑇)𝑇𝑇

−�𝐹𝐹𝑥𝑥𝑇𝑇𝑦𝑦𝑇𝑇𝑇𝑇

Where: m: Vehicle mass 𝐽𝐽𝑧𝑧: Vehicle moment of inertia V: Longitudinal speed 𝛽𝛽: Sideslip angle at center of gravity 𝑟𝑟: Yaw rate of the vehicle 𝛿𝛿𝑇𝑇: Steer angle of ith wheel 𝐶𝐶𝛼𝛼𝑇𝑇: Cornering stiffness at ith wheel 𝐶𝐶𝑀𝑀𝑧𝑧𝑇𝑇: Aligning moment stiffness at ith wheel 𝑥𝑥𝑇𝑇 ,𝑦𝑦𝑇𝑇: The coordinate of ith wheel center in vehicle coordinate system. h: Height of the center of gravity 𝑓𝑓0, 𝑓𝑓1: Coefficient of rolling resistance

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8.3 Rolling dynamics model To indicate the effect of tire to the roll dynamics, the steady state roll gain is modeled with the consideration of unsprung mass rolling. It is meaning full especially in the case of trucks with high ground clearance and large wheels. The model is shown in figure A.2 and following equation.

Fig. A2 Rolling dynamics model

𝐺𝐺𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 = �1 +𝐾𝐾𝜙𝜙𝑈𝑈𝐾𝐾𝜙𝜙𝑈𝑈

�𝑚𝑚𝑈𝑈ℎ𝑐𝑐𝑐𝑐2𝑇𝑇𝑐𝑐𝑈𝑈

𝐾𝐾𝜙𝜙𝑈𝑈 − 𝑚𝑚𝑈𝑈𝑔𝑔ℎ𝑐𝑐𝑐𝑐2𝑇𝑇𝑐𝑐𝑈𝑈+

1𝐾𝐾𝜙𝜙𝑈𝑈

(𝑚𝑚𝑈𝑈ℎ𝑐𝑐𝑐𝑐𝑈𝑈 + 𝑚𝑚𝑈𝑈(𝑎𝑎𝐿𝐿ℎ𝑇𝑇𝑐𝑐𝑇𝑇 +

𝑏𝑏𝐿𝐿ℎ𝑇𝑇𝑐𝑐𝑓𝑓))

Where: 𝐺𝐺𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅: Roll gain, deg/g 𝐾𝐾𝜙𝜙𝑈𝑈: Sprung mass roll stiffness 𝐾𝐾𝜙𝜙𝑈𝑈: Unsprung mass roll stiffness 𝑚𝑚𝑈𝑈: Sprung mass 𝑚𝑚𝑈𝑈: Unsprung mass ℎ𝑐𝑐𝑐𝑐2𝑇𝑇𝑐𝑐𝑈𝑈: Distance between sprung mass CG and rolling center ℎ𝑐𝑐𝑐𝑐𝑈𝑈: Unsprung mass CG 𝑎𝑎: Distance from front axle to CG 𝑏𝑏: Distance from rear axle to CG 𝐿𝐿: Wheelbase ℎ𝑇𝑇𝑐𝑐𝑓𝑓: Rolling center height of front axle ℎ𝑇𝑇𝑐𝑐𝑇𝑇: Rolling center height of rear axle

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8.4 Steering dynamics modeling The steering system is modeled based on conventional power assist rack and pinion architecture.

Fig. A3 Steering system model

𝐽𝐽𝑇𝑇𝑠𝑠�̈�𝜃𝑇𝑇𝑠𝑠 + 𝐵𝐵𝑇𝑇 ��̇�𝜃𝑇𝑇𝑠𝑠 −�̇�𝑥𝑇𝑇𝑅𝑅𝑝𝑝𝑇𝑇𝑝𝑝

� + 𝐾𝐾𝑇𝑇 �𝜃𝜃𝑇𝑇𝑠𝑠 −𝑥𝑥𝑇𝑇𝑅𝑅𝑝𝑝𝑇𝑇𝑝𝑝

� = 𝑇𝑇𝑇𝑇𝑠𝑠 − 𝐹𝐹𝑝𝑝𝑇𝑇𝑝𝑝𝑅𝑅𝑝𝑝𝑇𝑇𝑝𝑝

𝑚𝑚𝑇𝑇�̈�𝑥𝑇𝑇 + 𝐵𝐵𝑇𝑇�̇�𝑥𝑇𝑇 = 𝐹𝐹𝑝𝑝𝑇𝑇𝑝𝑝 + 𝐹𝐹𝑅𝑅𝑇𝑇𝑇𝑇𝑅𝑅 − 𝐹𝐹𝑇𝑇

𝐹𝐹𝑇𝑇 =𝑀𝑀𝐾𝐾

𝐿𝐿𝑅𝑅𝑇𝑇𝑅𝑅

𝑀𝑀𝐾𝐾 = 𝑀𝑀𝐾𝐾𝑅𝑅 + 𝑀𝑀𝐾𝐾𝑇𝑇

𝑀𝑀𝐾𝐾𝑅𝑅 = 𝑀𝑀𝑧𝑧𝑓𝑓𝑅𝑅 cos��Φ𝐾𝐾𝐴𝐴𝐾𝐾2 + Φ𝑅𝑅𝑅𝑅𝑇𝑇𝑅𝑅

2 � − 𝐹𝐹𝑦𝑦𝑓𝑓𝑅𝑅𝐿𝐿𝑀𝑀𝑇𝑇 − 𝑑𝑑(𝐹𝐹𝑥𝑥𝑓𝑓𝑅𝑅𝑐𝑐𝑐𝑐𝑐𝑐Φ𝐾𝐾𝐴𝐴𝐾𝐾 + 𝐹𝐹𝑧𝑧𝑓𝑓𝑅𝑅(𝑐𝑐𝑠𝑠𝑠𝑠Φ𝐾𝐾𝐴𝐴𝐾𝐾𝑐𝑐𝑠𝑠𝑠𝑠𝛿𝛿

+ 𝑐𝑐𝑠𝑠𝑠𝑠Φ𝑅𝑅𝑅𝑅𝑇𝑇𝑅𝑅𝑐𝑐𝑐𝑐𝑐𝑐𝛿𝛿))

𝑀𝑀𝐾𝐾𝑇𝑇 = 𝑀𝑀𝑧𝑧𝑓𝑓𝑇𝑇 cos��Φ𝐾𝐾𝐴𝐴𝐾𝐾2 + Φ𝑅𝑅𝑅𝑅𝑇𝑇𝑅𝑅

2 � − 𝐹𝐹𝑦𝑦𝑓𝑓𝑇𝑇𝐿𝐿𝑀𝑀𝑇𝑇 − 𝑑𝑑(−𝐹𝐹𝑥𝑥𝑓𝑓𝑇𝑇𝑐𝑐𝑐𝑐𝑐𝑐Φ𝐾𝐾𝐴𝐴𝐾𝐾

+ 𝐹𝐹𝑧𝑧𝑓𝑓𝑇𝑇(𝑐𝑐𝑠𝑠𝑠𝑠Φ𝐾𝐾𝐴𝐴𝐾𝐾𝑐𝑐𝑠𝑠𝑠𝑠𝛿𝛿 − 𝑐𝑐𝑠𝑠𝑠𝑠Φ𝑅𝑅𝑅𝑅𝑇𝑇𝑅𝑅𝑐𝑐𝑐𝑐𝑐𝑐𝛿𝛿))

𝛿𝛿 =𝑥𝑥𝑇𝑇𝐿𝐿𝑅𝑅𝑇𝑇𝑅𝑅

Where: 𝐽𝐽𝑇𝑇𝑠𝑠: Steering wheel/column equivalent moment of inertia 𝑚𝑚𝑇𝑇: Rack equivalent mass 𝜃𝜃𝑇𝑇𝑠𝑠: Steering wheel angle 𝐵𝐵𝑇𝑇: Steering column damping 𝐵𝐵𝑇𝑇: Rack damping 𝐾𝐾𝑇𝑇: Steering column stiffness 𝑥𝑥𝑇𝑇: Rack displacement 𝑅𝑅𝑝𝑝𝑇𝑇𝑝𝑝: Pinion gear radius

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𝑇𝑇𝑇𝑇𝑠𝑠: Input steering torque 𝐹𝐹𝑝𝑝𝑇𝑇𝑝𝑝: Force between pinion and rack 𝐹𝐹𝑅𝑅𝑇𝑇𝑇𝑇𝑅𝑅: Power assistant force 𝐹𝐹𝑇𝑇: Counter acted force from tie rod 𝑀𝑀𝐾𝐾 ,𝑀𝑀𝐾𝐾𝑅𝑅 ,𝑀𝑀𝐾𝐾𝑇𝑇: Wheel counter acted moment about king pin, total, left and

right. 𝑀𝑀𝑧𝑧𝑓𝑓𝑅𝑅, 𝑀𝑀𝑧𝑧𝑓𝑓𝑇𝑇: Tire aligning moment, left and right. Φ𝐾𝐾𝐴𝐴𝐾𝐾: King ping inclination angle Φ𝑅𝑅𝑅𝑅𝑇𝑇𝑅𝑅: Caster angle 𝐹𝐹𝑦𝑦𝑓𝑓𝑅𝑅,𝐹𝐹𝑦𝑦𝑓𝑓𝑇𝑇: Tire lateral force, left and right 𝐹𝐹𝑥𝑥𝑓𝑓𝑅𝑅,𝐹𝐹𝑥𝑥𝑓𝑓𝑇𝑇: Tire longitudinal force, left and right 𝐹𝐹𝑧𝑧𝑓𝑓𝑅𝑅,𝐹𝐹𝑧𝑧𝑓𝑓𝑇𝑇: Tire normal force, left and right 𝐿𝐿𝑀𝑀𝑇𝑇: Mechanical trail 𝐿𝐿𝑅𝑅𝑇𝑇𝑅𝑅: Steering arm length 𝑑𝑑: Scrub radius 𝛿𝛿: Wheel steer angle

8.5 Tire property estimation functions a) Cornering stiffness regression function[R. Wade Allen 2002]

CSC: Cornering stiffness Asp Rt: Aspect ratio Sec Size: Section width Press: Tire pressure FZ: Tire normal load DL: Tire design load a1, b1, c1, d1, e1, A1: Regression coefficients

b) Vertical stiffness estimation [Rhyne 2005]

KZ: Tire vertical stiffness Pi: Pressure W: Section width OD: Outer diameter

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9 Reference 1. Genta, Giancarlo (1997), “Motor Vehicle Dynamics”, World Scientific,

Singapore 2. Gillespie, Thomas D. (1992), “Fundamentals of Vehicle Dynamics”, Society of

Automotive Engineers, PA, US 3. R. Wade Allen, Thomas T. Myers, Theodore J. Rosenthal and David H. Klyde,

(2000), “The Effect of Tire Characteristics on Vehicle Handling and Stability”, SAE 2000 World Congress, 2000-01-0698

4. R. Wade Allen, Theodore J. Rosenthal and David H. Klyde (2002), “The Relative Sensitivity of Size and Operational Conditions on Basic Tire Maneuvering Properties”, SAE 2000 World Congress, 2002-01-1182

5. Rhyne, Timothy (2005), “Development of a Vertical Stiffness Relationship for Belted Radial Tires”, Tire science and technology, Vol. 33, No. 3, PP. 136-155

6. Hazare, Mandar (2013) “An Integrated Systems Engineering Methodology for Design of Vehicle Handling Dynamics”, Clemson University PhD Research Proposal.

7. Ishio, Jun, Ichikawa, Hiroki, Kano, Yoshio and Abe, Masato (2008) “Vehicle-handling quality evaluation through model-based driver steering behavior”, Vehicle System Dynamics, Vol. 46, No. 1, PP. 549-560

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