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High fidelity quasi steady state aerodynamic model development and effects on race vehicle performance predictions.

ABSTRACT

Presented in this paper is a procedure to develop a high fidelity quasi steady state aerodynamic model for use in race car vehicle dynamic simulations and its application in a race vehicle multi-body full lap simulation. Developed to fit quasi steady state (QSS) wind tunnel data, the aerodynamic model is regressed against three independent variables: front ground clearance, rear ride height, and yaw angle. An initial dual range model is presented and then further refined to reduce the model complexity while maintaining a high level of predictive accuracy. The model complexity reduction decreases the required amount of wind tunnel data thereby reducing wind tunnel testing time and cost. The quasi steady state aerodynamic model for the pitch moment degree of freedom is systematically developed in this paper. This procedure is extended to the other five aerodynamic degrees of freedom to develop a complete, high fidelity, six degree of freedom quasi steady state aerodynamic model. This high fidelity model reduces the QSS aerodynamic fit error compared to conventional aerodynamic model development. Both the newly developed high fidelity aerodynamic model and a conventionally derived aerodynamic model are implemented in a NASCAR Truck multi-body, full lap QSS simulation to determine the effects of the high fidelity QSS aerodynamic model on the simulation results. Performance metrics are calculated from simulation results and compared to assess the effects of the aerodynamic models on the performance predictions. The increased accuracy of the high fidelity aerodynamic model is found to have discernable effects on the vehicle performance predictions resulting from the QSS simulation.

CITATION: Mohrfeld-Halterman, J. and Uddin, M., "High Fidelity Quasi Steady State Aerodynamic Model Development and Effects on Race Vehicle Performance Predictions," SAE Int. J. Passeng. Cars - Mech. Syst. 9(2):2016.

INTRODUCTION

Race vehicle simulation is broadly and frequently utilized in the motorsports industry to predict vehicle handling and performance across every race track. Given the high stakes for race vehicle performance, the accuracy of the vehicle simulation is paramount, requiring accurate modelling of all the vehicle subsystems, i.e. chassis, powertrain, tires, and aerodynamic subsystems. This paper focuses on the development of the latter, an accurate QSS aerodynamic model, composed of a mathematical model of the aerodynamic forces and moments acting on the vehicle as a result of the vehicle orientation which includes the front ground clearance, rear ride height, and yaw. In the motorsports industry, vehicle simulations are conducted prior to race events to select the starting vehicle setup for each track event, and then trackside simulations are utilized during race events to adjust the vehicle setup for race day track and weather conditions. Pre-race and trackside simulations focus on analyzing the interchangeable components of the vehicle which affect the suspension characteristics, i.e suspension geometry, spring and sway bar rates, damping characteristics, etc. Modifying the suspension characteristics will result in vehicle orientation changes that will affect the predicted aerodynamic forces and moments acting on the vehicle. The change in the aerodynamic forces and moments will in-turn affect the vehicle orientation. To understand the effect the suspension changes will have on the aerodynamic forces and moments, and vice versa, both an accurate vehicle model and accurate aerodynamic model are required in the simulation.

Historically, aerodynamic models have been developed and applied to vehicle models with increasing degrees of complexity. Original steady state models used constant aero forces or coefficients with no vehicle orientation dependencies [1][2]. These aerodynamic models only included three degrees of freedom: front lift, rear lift, and drag. Additional degrees of freedom were then added to these models to include side force and yaw moment [3]. As wind tunnel testing of vehicles furthered, the effect of yaw on the aerodynamic forces and moments was documented [4, 5, 6] and incorporated into aerodynamic modelling of vehicle simulations [7][8]. The final degree of freedom, the roll moment, was finally added to vehicle models for complete modelling of all aerodynamic forces and moments acing on the vehicle [7].

The next step in quasi steady state model development came with the inclusion of front and rear vehicle travels as dependencies of the aerodynamic model [9]. More recently, the effect of individual race vehicle body components, such as wings, have begun to be characterized independently and modeled in vehicle dynamic simulations [10].

Limited testing at each race event leads the race engineers to rely on vehicle simulations to help analyze additional vehicle setups that cannot be physically tested during the event. The utilization of simulation in pre-race and especially track side analysis demands minimal run time of accurate simulations. Quick simulation with high levels of fidelity can be utilized by race engineers to run through a large number of potential setup variations to help optimize their vehicle setup for race day. Implementing an aerodynamic model of an empirical nature within the simulation can meet the track side run time requirements as opposed to a solution utilizing computational fluid dynamics, CFD. Generating a CFD solution is slow and does not lend itself to the quick turnaround time required of race vehicle simulations; thus, industry practice has trended toward incorporating empirical, mathematical aerodynamic models.

Mathematical models used to calculate the QSS aerodynamic forces and moments were first developed in the Aerospace industry [11]. Second order polynomial fits were applied to the model's independent variables to compute the aerodynamic forces applied to an aircraft. This concept was later applied to automobiles by modelling the aerodynamic forces as second order fits of front and rear ride heights [12]. To the authors' knowledge, little other documentation exists on the evolution of this modelling technique; however, industry practice now develops QSS aerodynamic models using full or partial second order fits of the independent variables, typically front travel, rear travel, and yaw.

Recent race vehicle setup trends travel the front of the vehicle close to the ground to capitalize upon aero advantages of maintaining a vehicle at this orientation. For instance, in all of the three official racing series, viz. Sprint Cup, XFINITY and Camping World Truck Series, managed by America's National Association for Stock Car Auto Racing, popularly known as NASCAR, the current setup trends attempt to minimize the front ground clearance of the vehicles such that the leading edge of the vehicle can intentionally make contact with the track. In these conditions, as the authors' have observed, the QSS aerodynamic forces and moments become highly nonlinear and can no longer accurately be modeled by unified second order fits applicable over the entire range of front travel. A new mathematical model is proposed in this paper to accurately model the QSS aerodynamic data.

Creating QSS aerodynamic models for force and moment data requires an adequate amount of test data spanning the full range of the independent variables. Wind tunnel testing and CFD are both methods that can be employed to create the test data, but the processing time and cost required by CFD, about 100 hours on a 96-Core computing cluster for a single vehicle orientation will make the CFD approach a more costly option. Computing the forces and moments at 40 vehicle orientations, a standard number with which to fit a conventional QSS aerodynamic model, 400,000 CPU hours at a rate of $0.10 per CPU hour, results in a $40,000 CFD cost.

Wind tunnel testing is less expensive than CFD but is still a costly alternative. Acquiring data for aerodynamic mapping can take up to five hours of wind tunnel time. Of the two wind tunnels readily available to the NASCAR industry, AeroDyn charges $1895 per hour [13], costing almost $10000 for a five hour shift, and Windshear charges new customers $35000 for a one-off 10-hour shift [14]. Decreasing test time has large cost advantages, so developing a methodology to limit the required amount of the wind tunnel testing is desirable. In addition to developing a high fidelity QSS aerodynamic model, an aerodynamic model term reduction procedure is also presented in this paper to minimize the amount of test data required to develop the QSS aerodynamic model. This will minimize the cost of physical testing while maintain a high accuracy of the aerodynamic model fit. To facilitate the forthcoming analysis and discussion, the final reduced parameter high fidelity QSS aerodynamic model is abbreviated as MUAM. In addition to the development of a high fidelity aerodynamic model, the MUAM aero model is applied to a standard race vehicle full lap QSS simulation to quantify the results of the MUAM compared to conventional aerodynamic models on the predicted performance metrics resulting from the simulation.

Past research has shown a significant influence of the aerodynamic model on vehicle simulation results [10][15, 16, 17, 18]. The effect of lift and front-to-rear lift distribution has been studied for high speed and braking conditions [15]. Initial limit handling analysis calculated the effects of aerodynamics on additional lateral acceleration potential and the resulting vehicle balance [16], and the effect of aerodynamics on transient vehicle handling has been analyzed with smoothed step steer simulations [17][18]. More in depth studies have analyzed the effects of aerodynamic and active aerodynamic models on the vehicle performance potential, as well as analyzed the vehicle model in a steady state constant radius turn simulation and analyzed the transient response in a step steer maneuver [10]. In all studies, the influence of the aerodynamic forces and moments applied to the vehicle model resulted in predicted handling changes in the simulation results.

No standard simulation experiments are documented in literature to quantify the predicted vehicle handling and performance results, especially for race vehicle analysis. However, a standard simulation experiment does exist within the NASCAR industry. A full lap QSS simulation is commonly used to predict the vehicle performance around the race track. A full lap QSS simulation is independently implemented with two vehicle models, one modeled with the MUAM and the other modeled with the conventional aerodynamic model. This simulation inputs are generated from a representative NASCAR 1.5-mile oval track, Homestead-Miami Speedway, and performance metrics calculated from the simulation results are compared.

AERODYNAMIC MODEL THEORY AND SIMULATION METHODOLOGY

Aerodynamic Model Theory

The conventional QSS aerodynamic model used in the NASCAR industry is a full second order polynomial fit of three independent variables, front ground clearance, [z.sub.f], rear travel, [z.sub.r], and yaw angle, [beta], described by the equation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where [C'.sub.a] is the predicted aerodynamic coefficient (force or moment), and [D.sub.i,j,k] is the parametric fit coefficient for each term of the model. The high fidelity aerodynamic model is developed to capture the highly nonlinear tends observed in the wind tunnel data when the leading edge of the vehicle is traveled close to the ground. Figure 1 contains the plots of the aerodynamic pitch moment coefficient, [C.sub.Mx], vs. the front ground clearance of the race vehicle at a constant rear travel and yaw angle that illustrates the nonlinearity of this dependence. This data was collected as part of wind tunnel QSS aerodynamic model development test conducted at AeroDyn using a 2007 NASCAR Truck.

To address the nonlinearity that occurs at the lower front ground clearance, a more complex aerodynamic model is proposed. It is hypothesized, that utilizing an additional mathematical model term applied to one section of the front ground clearance data will capture the additional nonlinearities unable to be accurately modeled using the conventional approach. Thus, an aerodynamic model composed of two mathematical model terms is developed. One math model term applies to the entire range of the front ground clearance input and the other math model term is weighted to be applied to only the high ground clearance data. The proposed aerodynamic model equation is:

[C'.sub.a] ([z.sub.f],[z.sub.T],[beta]) = [f.sub.1]([z.sub.f].[Z.sub.r].[beta]) + W([z.sub.f]) ([f.sub.2]([z.sub.f][z.sub.T],[beta]) (2)

where [f.sub.1] is the unweighted mathematical model, [f.sub.2] is the weighted mathematical model, and W is the weighting function that is a function of the front ground clearance. The weighting function has the form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

and is displayed in Figure 2 with three regions, A, B, and C, illustrated to describe the effect of this function. This weighting function drives the weighting of [f.sub.2] to 0 at low front ground clearance values, Region A, below a cut-off value of [z.sub.fc], and maintains the weighting of [f.sub.2] at 1 for the higher ground clearance values, Region C. The other variable in the equation, [w.sub.1], influences the width of the transition section, Region B. The weighting function is designed to allow for [f.sub.2] to operate as a high ground clearance deviation function. The unweighted math model, [f.sub.1], is effective over the entire range of the front ground clearance, but the weighted math model, [f.sub.2] is only effective when the weighting function becomes non zero. The weighting function increases the influence of the [f.sub.2] throughout Region B until the [f.sub.2] function has the maximum influence on the aerodynamic model through the high ground clearance range of Region C.

Each mathematical model term in Equation 2 is proposed to be a function of front ground clearance, rear ride height, and yaw. The first math model term, [f.sub.1], has no weighting and thus must remain valid over the entire range of [z.sub.f] inputs, so only a linear dependency of [z.sub.f] will be included in this math model. The other independent variables will be implemented with a second order dependency consistent with finding in literature [4, 5, 6, 7, 8, 9] and current industry modelling practices. The function [f.sub.1] is thus defined as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

where [A.sub.i,j,k] is the parametric fit coefficient for each term, and 18 terms are present in this equation. This will result in 18 model coefficients in the [f.sub.1] fit.

For the higher ground clearance [f.sub.2] fit, the [z.sub.f] input range is limited and the front ground clearance input can thus be modeled with more nonlinearity, so [f.sub.2] is modeled as a second order function of each independent variable and can be defined as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

where [B.sub.i,j,k] is the parametric fit coefficient for each term of the model, and 27 terms are present in this equation. This will result in 27 model coefficients in the [f.sub.2] fit.

In total, including the weighting function, there are 47 parameters that must be determined to generate the aerodynamic model given in Equation 2. An iterative least squares approach is applied to calculate these parameters. First, a reasonable starting range for [w.sub.1] and [z.sub.fc] is evaluated for the fitting process. For each of n combinations, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], of [w.sub.1] and [z.sub.fc] a linear least squares fit is applied to calculate the values of [A.sub.i,j,k] and [B.sub.i,j,k]. This is represented in the following equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

where P = [[[A.sub.n], [B.sub.n]].sup.T], [C.sub.a] is a 1 x p matrix of one degree of freedom of the measured aerodynamic coefficients, and [X.sub.n] is a p x m matrix of each model parameter from Equation 2 evaluated using [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [Z.sub.fc] where p is the sample size of the data and m is the number of parameters in the fit.

For each n, the sum square error is calculated as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

where h is the number of each test data point. The final set of coefficients, P, equals the [P.sub.n] which minimizes [S.sub.n].

The proposed aerodynamic model consists of two math model functions with a combined total of 45 individual terms. Not all of the terms, however, have a significant impact on the accuracy of the model. Decreasing the number of terms in the model will also decrease the amount of test data required to fit the model, thus reducing wind tunnel test time and cost.

A parameter reduction scheme is developed to maintain the model fidelity while reducing the model complexity. The first step of the parameter reduction scheme is to determine the effect of each parameter on the accuracy of the model fit. The high impact parameters are kept in the model, and the low impact parameters are removed. The parameter effectiveness is determined by analyzing the sensitivity of the model error to a change in each model parameter. This is achieved by numerically determining the partial derivative of the modelling error with respect to each parameter as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

where [P.sub.m0] is the original coefficient of each parameter, [P'.sub.m] is a small deviation from the original coefficient value, and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

The parameter with the smallest partial derivative [partial derivative]S/[partial derivative][P.sub.m] value, is initially removed from the model, and new coefficients are calculated for the remaining parameters using the procedure outlined above. This model reduction process is repeated for each least significant model parameter of the aerodynamic mathematical model until the error of the model exceeds a set threshold.

This procedure can be applied to each aerodynamic degree of freedom, both forces and moments, to generate a full six degree of freedom aerodynamic model.

Simulation Methodology

To determine if the improved fidelity of the MUAM has a significant impact on the predicted performance metrics from race vehicle simulation, each aerodynamic model is independently applied to a multi-body NASCAR Truck vehicle model used in a full lap QSS simulation. The QSS simulation is conducted for a 1.5 mile oval track, Homestead Miami Speedway.

The full lap QSS simulation is a collection of quasi static simulations at individual states around the race track. At each state, a set of simulation inputs, vehicle speed, V, longitudinal acceleration A lateral acceleration, A, and vertical acceleration, A, are used to control the steer, roll, pitch, yaw, and heave of the chassis model as well as the drive and brake torques applied to the wheels. The QSS simulation is run dynamically using the quasi steady state controller QuasiStatics, from the Modelon Vehicle Dynamics Library in Dymola. For each input state, the simulation is run for a set length of time allowing the simulation controller and vehicle model to converge on the final internal vehicle model states that meet the demanded inputs.

The vehicle model used in the QSS simulation is an 84 degree of freedom multi-body chassis model. This model is built from the base NASCAR chassis model in the Claytex VDL Motorsports Library by populating the model with the proper suspension geometries and stiffness, damping, and mass elements.

The chassis model consists of an asymmetric double wishbone front suspension and a trailing arm rear suspension model. The steering system is modeled as a draglink system, and the tires are modeled using the MF-Swift 6.1 tire model [19] populated to fit test data from fat track tire testing. The aerodynamic model is developed as previously described for both the MUAM and conventional aerodynamic model, and applied to the vehicle model in the ground plane at the centerline of the vehicle track and wheel base - the same location used to report the force and moment data from the wind tunnel measurements. As a multi-body model, the chassis and individual suspension components are modeled as separate parts that have unique mass and inertia properties. Notably removed from the vehicle model in the quasi steady state simulation are the brake system and powertrain models. These systems are not necessary in this simulation because the drive and brake torques at the wheels are controlled by the quasi steady state simulation controller to achieve the longitudinal acceleration targets.

Prior to running a QSS simulation, a setup simulation is run to determine the spring and sway bar preloads and sprung mass cg location required to achieve the static chassis heights and vertical corner loads. The setup results, along with the other chassis parameters emulating a NASCAR Truck Homestead setup, are input into the QSS simulation to properly model the vehicle.

To analyze the results of the QSS simulation, performance metrics are calculated from the simulation results. The performance metrics are applied to select segments of the track that are applicable to the performance metric being analyzed. The track is broken up into nine segments to support this analysis. Figure 4 illustrates the division of the track in the analysis segments.

Segments [S.sub.1], [S.sub.5], and [S.sub.9] are all straight away segments. Segments [S.sub.2] and [S.sub.6] are turn entry segments, segments [S.sub.3] and [S.sub.7] are the mid-corner segments, and segments, [S.sub.4] and [S.sub.8] are the turn exit segments.

Three performance metrics were chosen to analyze the QSS simulation results in this study. One metric quantifies the vehicle handling, or balance, in cornering conditions, and the other two metrics quantify the vehicle's lateral performance potential while cornering and the longitudinal performance while on throttle.

The vehicle handling metric, static margin, SM, is a classical vehicle dynamics handling metric [4, 20], and is defined as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

where [l.sub.f] and [l.sub.r] are the distances front the front and rear axles, respectively, to the vehicle cg, l is the wheelbase of the vehicle, and [K.sub.f] and K are the front and rear axle cornering stiffness.

A vehicle with a static margin greater than zero will exhibit understeer, while a vehicle with a static margin less than zero will exhibit oversteer Vehicle handling is important to race vehicle performance, as a vehicle that exhibits excessive understeer or oversteer will not corner at speeds as high as a vehicle that exhibits more neutral steer characteristics. The static margin metric is only analyzed for the corning segments of the track, [S.sub.2] - [S.sub.4] and [S.sub.6] - [S.sub.8], as vehicle balance will only affect the performance in the corners. The lateral performance potential is calculated by post processing the QSS simulation results in conjunction with the tire models to determine the maximum corner speed, [V.sub.max], possible if all four tires are operating at their maximum lateral capacity. Thus [V.sub.max] is a result of the total maximum lateral force acting on the vehicle, from both tire and aerodynamics forces, as well as the corner radius, R, and vehicle mass, m, and is calculated as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], ana [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are the maximum lateral orce capacities of each tire, and [F.sub.y] is the aerodynamic side force. The prefixes LF, RF, LR, and RR in the subscripts in Equation 12 represent left-front, right-front, left-rear and right-rear respectively, and this convention is used throughout this paper. The minimum mid corner speed for the entire corner is experienced during the end of the turn entry segment or the beginning of the mid corner segment therefore, this metric is analyzed for the turn entry and mid turn cornering segments, [S.sub.2] - [S.sub.3] and [S.sub.6] - [S.sub.7].

The drive force demand, [F.sub.dx], metric quantifies the total drive force needed to achieve the longitudinal acceleration target. This is calculated as

[F.sub.dx] = [F.sub.LRX] + [F.sub.RRX] (13)

for a rear wheel drive race vehicle, where [F.sub.LRx] is the longitudinal force of the left rear wheel and [F.sub.RRx] is the longitudinal force of the right rear wheel required to meet the vehicle speed and longitudinal acceleration targets. This drive force demand can change based on the variation of the longitudinal forces acting on the vehicle, such as a change in the aerodynamic drag force. The less drive force demand that is required to meet the acceleration target will result in additional drive force that will be available to accelerate the vehicle thereby improving performance. The drive force demand is only critical, and thus analyzed, for vehicle performance when the vehicle is operating at wide open throttle conditions, which occurs in the turn exit and the straightaway segments, [S.sub.1], [S.sub.4] - [S.sub.5], and [S.sub.8] - [S.sub.9].

Each of these performance metrics is compared to determine the effects of each aerodynamic model on the full lap QSS simulation results.

AERODYNAMIC MODEL DEVELOPMENT AND COMPARISON

High Fidelity Aerodynamic Model Development

The wind tunnel data used to create the QSS aerodynamic model was collected at AeroDyn using a test plan designed to collect enough data points over the range of independent variables tested, front ground clearance, rear ride height, and yaw, to adequately develop the aerodynamic model. For each yaw angle tested, 0, -2, and -3 degrees, the same front and rear ride height matrix was tested. The front and rear ride height matrix was designed to encompass the range of travel of the race vehicle, sampling at intervals of 12.5 mm. The front ground clearance ranges from the vehicle position at inspection heights to the minimum achievable ground clearance at the wind tunnel. The rear travel of the vehicle ranges from inspection heights to 76 mm of jounce.

Initially, the model in Equation 2 is applied to the wind tunnel pitch moment coefficient data. Then the parameter reduction scheme is applied and the modelling error, [S.sub.n], is plotted against the number of removed parameters in Figure 4. An expected increase in the modelling error is seen as the number of parameters is reduced.

There is, however, only a small decrease in the modelling error by reducing the model by nine or less parameters. A plot of the model error for each test data point is plotted in Figure 5 for both the initial model with the full 47 terms, and for the nine term reduced model with 38 terms. For both the full term fit and the nine term reduced fit, the average absolute fit error is less than 1 count or 0.001, and the maximum fit error is less than 3 counts.

Another jump in the modelling error can be seen after reducing the model by 17 terms. This modelling error for each test point is plotted in Figure 5 overlaid with the modelling error for the full 47 term model. For the 17 term reduced fit, the average absolute error is slightly over 1 count and the maximum fit error is less than 4 counts.

There is minimal sacrifice in the modelling error for the nine term reduction where the 17 term reduction results in larger error. Given the desire for the most accurate aero model to be applied to the simulation, the nine term reduction model is chosen for the simulation evaluation against the conventional aerodynamic model. If wind tunnel testing cost is of more significance than model accuracy for other applications, then the 17 term model reduction could be implemented with only a small increase in the modelling error.

The pitch moment fit of the MUAM, including the nine term reduction, is displayed in Figure 6 and Figure 7 for 0[degrees] and -3[degrees] of yaw, respectively. The markers in the figures represent the raw test data and the lines represent the results of the model fit at constant rear ride height values.

A more in depth description of this model development can be found in another paper dedicated solely to the development of the MUAM model [21].

Aerodynamic Model Comparison

Similar to the development of the pitch moment component of the MUAM, the completed MUAM is developed for the other five aerodynamic degrees of freedom with the nine term reduced model. The average absolute fit error for each degree of freedom of the MUAM is presented in Table 1, and compared to the average absolute error from the conventional aerodynamic model.

In comparing these results, the MUAM reduces the model error for all the degrees of freedom, most significantly for the pitch moment coefficient, [C.sub.Mx], and the lift coefficient, [C.sub.z]. The pitch moment error is reduced from 5.35 counts in the conventional model to less than one count in the MUAM, and the lift coefficient error is reduced from 2.52 counts in the conventional model to 1.42 counts. The comparison of the model error at each test data point for these two degrees of freedom is included in Figure 8. The error reduction for each data point is noticeable, and the reduction in the maximum error can be seen to reduce from almost 10 counts for both the lift and pitch moment coefficients to just over 6 counts for the lift coefficient and less than 3 counts for the pitch moment coefficient.

For the other aerodynamic coefficients, drag, [C.sub.z], side force, [C.sub.y], roll moment, [C.sub.x], and yaw moment, [C.sub.z], the conventional model already results in a small error, and the increased accuracy of the MUAM only results in small additional error reductions.

QSS Simulation Comparison

The input states used in the QSS simulation are extracted from measured NASCAR Truck vehicle data at Homestead Miami Speedway. Each data set is first filtered using a one Hz, four pole Butterworth filter and then sampled at every second to gather the collection of input states. Plots of the input states used for the Homestead simulation are in Figure 9. For proprietary reasons, the accelerations states are presented as normalized accelerations.

The results of the full lap QSS simulation are compared to first determine the difference in the aerodynamic forces and moments calculated from each aerodynamic model. Then the QSS simulation results are further analyzed to compare the effect of these aerodynamic differences on the QSS simulation performance metrics.

The difference in the aerodynamic forces and moments are calculated by subtracting the conventional aerodynamic results from the MUAM results. The differences in these results for the full lap QSS simulation are plotted in Figure 10, and the average difference across the entire lap is included in Table 2.

The aerodynamic moment comparison shows that the pitch moment, [M.sub.y], is affected the most by the different QSS aerodynamic models. The pitch moment difference peaks over 300 Nm, and the average pitch moment difference throughout the lap is 138 Nm. Similar to the lift force, the difference in the pitch moment varies throughout the lap as the vehicle orientation and speed change. The other aerodynamic moments, roll moment, [M.sub.x], and yaw moment, [M.sub.z], exhibit smaller differences between the two aerodynamic models. The roll moment has an average difference of 15 Nm, and the yaw moment has the least difference with an average difference of less than 2 Nm.

The comparison of the aerodynamic forces reveals that the lift force, [F.sub.z], is shown to have the largest difference between the two aerodynamic models. While the difference in the lift force varies as the vehicle orientation and speed varies throughout the lap, the average difference of the lift force between the two QSS aerodynamic models is 40 N with a max difference of almost 100 N. The drag force, [F.sub.x], and the side force, [F.sub.y], both result in minimal difference between the two models with an average difference of less than 5 N.

In the full lap QSS, the aerodynamic degrees of freedom resulting in the largest difference between the two aerodynamic models are the lift force and the pitch moment. This is consistent with the QSS aerodynamic model error results, as these were the two aerodynamic degrees of freedom with the most reduced modelling error from the MUAM.

The variations in the lift and pitch moment resulting from the aerodynamic models is expected to have an impact on the vehicle performance metrics calculated from the simulation results. The differences in the performance metrics resulting from each aerodynamic model are included in Figure 11. The average change of each performance metric across the states of interest is summarized in Table 3.

A difference in the static margin balance metric does not intuitively translate into the magnitude of the handling change. Therefore, the full lap QSS simulations were also conducted for a common vehicle setup change, a half percent nose weight increase, applied to the conventional aerodynamic model simulation. The nose weight of the race vehicle can be adjusted by moving ballast located in vehicle frame rails from the back of the frame rails toward the front of the frame rails. This change will typically result in more understeer, thus a higher static margin, of the race vehicle through the corners.

The predicted static margin results reveal that implementing the more accurate MUAM has a significant effect on the prediction of the vehicle handling, especially in turn entry segments, [S.sub.2] and [S.sub.6]. It is shown to have as significant of an effect as a half percent nose weight change throughout the corners at Homestead.

The difference in this balance metric can be attributed to both the difference in the calculated lift force and pitch moment. Modifying the lift force and the front-to-rear distribution of the lift force (the effect of the pitch moment) will change the vertical load on the front and rear axles thereby modifying the lateral capacity of each axle. Decreasing the lateral capability of the front axle relative to the rear axle will result in more understeer of the vehicle as seen in the static margin results. This is consistent with the large pitch moment difference shifting the vertical load distribution to the rear axle with the MUAM implementation.

A small effect on the difference in the maximum potential corner speed is seen from the implementation of the MUAM compared to the conventional aerodynamic model. The change in maximum potential corner speed is attributed primarily to the change in the lift forces calculated from each aerodynamic model. Through the track segments with the lowest corner speed, the MUAM changes the prediction of the maximum potential corner speed by -0.055 m/s on average. While this is small change in predicted performance, in the competitive motorsports industry, being able to predict and make decision on even small gains is still important.

The difference in predicted drive force demand resulting from the simulation comparison is minimal, with the MUAM only resulting in a -0.41N average difference. This is expected as the difference in the drag force between the two aero models is also minimal. A change in the drive force demand is mostly attributed to a change in the drag force difference since additional drive force is required to maintain the same vehicle speed with increased drag force.

The MUAM is shown to primarily affect the vehicle handling metric resulting from the full lap QSS simulation when compared to the conventional aerodynamic model. The MUAM implementation has a small effect on the prediction of mid corner speed, but has minimal effect on the drive force demand.

If the conventional aerodynamic model is applied to race vehicle simulation in place of the MUAM, the vehicle handling predictions will not be as accurate. When used to make setup decision at the race track, this could result in an improper decision resulting in a lower performing race vehicle. Thus for accurate handling predictions, the authors recommend the application of the high fidelity MUAM to race vehicle simulations.

CONCLUSION

The high fidelity MUAM proposed, developed, and applied in this paper is shown to more accurately model the quasi steady state wind tunnel test data than the conventional aerodynamic model over the range of vehicle orientations tested. The model is developed to utilize the minimum amount of test data required to maintain the highest model accuracy thus reducing the wind tunnel test time and cost required to generate a high fidelity aerodynamic model.

When implemented in a full-lap QSS simulation, the increased accuracy of the MUAM changes the aerodynamic forces and moments, primarily the lift force and pitch moment, applied to the vehicle model when compared to the implementation of the conventional aerodynamic model. This aerodynamic force and moment difference results in changes in the performance metrics calculated from the QSS simulation results. While the maximum potential cornering speed and the drive force demand metrics were minimally affected, the vehicle handling metric was significantly affected. Thus, for accurate vehicle balance prediction, the high fidelity MUAM is recommended for use in the vehicle simulations.

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ACKNOWLEDGMENTS

Thanks to Chevy Performance and Pratt & Miller Engineering for supporting these research efforts, and special thanks to Chris Gilligan, NASCAR Program Manager at Pratt & Miller Engineering, for enabling this project.

Jackie A. Mohrfeld-Halterman

Pratt & Miller Engineering

Mesbah Uddin

UNC Charlotte Motorsports Engineering

Table 1. Average absolute fit error for each aerodynamic coefficient
model.

Aerodynamic  Conventional Model  MUAM Error (count)
Coefficient  Error (count)

[C.sub.x]    0.72                0.53
[C.sub.y]    0.71                0.60
[C.sub.z]    2.52                1.42
[C.sub.mx]   0.55                0.20
[C.sub.my]   5.35                0.89
[C.sub.mz]   0.60                0.25

Table 2. Average change in aerodynamic forces and moments from the
Homestead full lap QSS simulation.

              Average Force and
              Moment Difference

[F.sub.x],N     -4.51
[F.sub.Y],N     -3.74
[F.sub.z],W     40.30
[M.sub.x],Nm   -15.45
[F.sub.y],Nm  -138.28
[F.sub.z],Nm    -1.89

Table 3. Average change in performance metrics.

Metric             Average Performance
                   Metric Change

Static Margin       0.019
Corner Speed, m/s  -0.055
Drive Force, N     -0.41
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Author:Mohrfeld-Halterman, Jackie A.; Uddin, Mesbah
Publication:SAE International Journal of Passenger Cars - Mechanical Systems
Article Type:Report
Date:Jun 1, 2016
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