Experimental Determination of Aerodynamic Downforce with Pseudo-Inverse Method Variation

The effective evaluation of loads acting on a vehicle is a fundamental challenge in several engineering applications. Where possible, instruments such as load cells are used for a direct measurement of these loads, otherwise local strain measurements can provide an indirect determination of forces or pressures reaction. In this case, an adequate number of strain measurements in proper location is required to achieve correct load measurements, using a pseudo inverse method. In the present work, a variation of the classical method of applied forces calculation from the deformation data is presented for automotive applications, in particular in the case of reduced detected deformations number, lower than necessary for loads determined by static equilibrium. Finally, the method is tested on a real case study, where the aerodynamic downforce generated by the rear wing of a high-performance car is measured.


Introduction
In structural mechanics, a case of considerable interest is to determine the loads to which a structure is really subjected during its service life.When a designer is called to develop a new structure shape, not always is perfectly aware of the loads that will stress the structure and therefore extensive use is made of both experimental experience and the socalled safety coefficients, allowing the designer a certain confidence on the calculations outcome.However, an experimental test for certification of sustained loads is often desirable and, in several cases required.
The exact determination of the loads applied to a structure or vehicle frame is not an easy problem to solve: especially when the structure has a complex geometry, elementary geometrical simplifications that could allow assimilating the structural elements to a system of links or beams may introduce unacceptable approximations.In addition, the direct measurement of the loads acting in an experimental environment could be not easily applicable too.A further complication is represented by the circumstance that the stress-strain state of structures is almost never generated by the application of a single load but by the combined loads acting in different directions or generated by complex pressure fields.Finally, these actions could also interact among them or could be influenced by the deformation of the structure itself.
In the literature, there are several examples about tracing techniques of a complex load condition capable of generating a certain deformation field.Among them, several studies introduced some methodology for the reconstruction of the load conditions, associated to certain state of deformation.A classical and very useful method is the pseudo-inverse method, discussed in their works by Lee [1] and Masroor et al. [2].This method provides the definition of a system of equations, from which it is possible to estimate the complex load conditions, capable of generating a certain state of deformation that has been measured experimentally with strain gages.Other works of Khoo et al. [3] applied the pseudo-inverse method, together with the Operating Deflection Shape (ODS) analysis and FRF (Frequency Response Function) methodology to evaluate the dynamic loads that presumably occur on a vehicle, schematized as a block of suitable masses.With this approach, the cases of underdetermined, fully determined, and over-determined systems have been analysed, experimentally highlighting how the over-determined case is the one that offers greater accuracy in determining the loads.In another study, Leclère et al. [4] determined the internal loads of a diesel engine during operation by evaluating the dynamic deformations of the engine, given that the transfer functions between excitations and measured responses is correctly known.In this case, the pseudo-inverse method is accompanied by a weighing phase of the various contributions, which allow the optimization and regularization for the values of the estimated loads.From the perspective of Structural Health Monitoring (SHM), Shakarayev et al. [5] on the other hand developed a technique based on finite elements, involving also the pseudo-inverse method to determine loads, stresses, and displacements of an aircraft structure in real time.By the combined use of numerical and experimental techniques, the quantities of interest and related to structural health needs are determined with a good approximation.A further work was proposed by Möller [6], who introduces a first evaluation of the load system acting on a static structure using Betti's theorem; in this way, he could test a series of complex load conditions without particular computational effort.Subsequently, the load conditions that produce the same strain field are studied, according to an optimization criterion that aims to determine the required load condition.Miller et al. [7] also used the approach offered by the inverse Finite Element Method (iFEM) in the aerospace field to determine the deformations and structural applied loads in real time on a test wing.Avilasha et al. [8] used the same method but for a completely different application, represented by a hacksaw machine.Furthermore, the estimation loads acting on a structure also plays an important role in the study of fatigue related problem, as done by Pasquier et al. [9] for the study of the fatigue loads acting on a full-scale bridge from a numerical and an experimental point of view.
In the present work, a modification of the pseudo-inverse method for the determination of the load conditions that generate a certain deformation field is proposed to overcome the limitations, which arise in the case of under-determined systems.In particular, the proposed method could be useful when an insufficient number of strain gauges is applied on the structure, due to inaccessibility for instance, aesthetic needs or other relevant reasons.The approach here presented is based on the possibility to turn an underdetermined system in a better-determined system adding a certain number of relationships derived by physical and data considerations within the system of equations.This approach shows the possibility to correlate insufficient experimental data and add specific conditions in order to derive the loading conditions.
The modified method was developed and applied to measure the loads acting on the rear wing of a high-performance car, for which the installation areas of the strain gauges on the supports were limited and insufficient to obtain a complete description of the problem.The deformation measurements were performed specifically on wing supports, which were transformed into improper load cells.The instrumented supports were calibrated in laboratory environment and the missing parameters for solving the problem were numerically determined using a finite element correlated model.
The novelty of the paper is also represented by the experimental evaluation of the loads acting in service condition, since the experimental deformation measurements have been taken directly on the car engaged in standard track tests.The measurements considered the most common driving conditions of the car, at different speed levels and for the different aerodynamic configurations that the wing can assume, according to simple registration mechanism, identifying the presumed aerodynamic downforce developed by wing and highlighting how this force varies in the different configurations.

Application of the Pseudo-Inverse Method for Underdetermined System
The possibility to determine the unknown load condition on structures through the knowledge of the deformation state could be non-trivial problem in the case of structures that have complex geometries and complex load conditions, where there is a certain difficulty in separating the directional load -deformation relationships.Therefore, the need arises to break down the load condition into load components orthogonal to each other, or in any case according to appropriate directions, in order to allow an easier examination of the relationships existing between the applied loads and the strain measurements.The pseudo-inverse method (also known as D-optimal technique) is a useful method in this case [1].The basic concept is that each deformation in the linear elastic range can be expressed as a linear combination of the various load contributions.By extending the concept to multiple deformation values, the elastic relationship assumed between the n-th deformation vector {S} n×1 and the p-th vector of the applied loads {L} p×1 can be expressed: where [A] is called the matrix of the sensitivity coefficients.
The number of strain gauges needed, their position and their orientation are chosen to maximize the determinant of the product [A] T [A] and therefore to reduce the variance of the estimated load.The n elements of the deformation matrix {S} are given by deformation readings (in mV/V) at certain chosen points of the structure under study.In the case of an iso-determined system ( n = p ), the matrix [A] is square; moreover, if the n relationships are independent, the matrix (1) [A] is invertible and it is possible to solve the system and to obtain the vector of the unknown loads as follows: In the case of an under-determined or over-determined system, the matrix A is no longer square and therefore it is not invertible; the pseudo-inverse method, as described in detail by Moore -Penrose [10], allows proceeding with the system resolution anyway, providing as output an estimated load vector {L} .In fact, with the data {S} n×1 and {L} p×1 , the best estimate of {L} p×1 is found to be given by: The variance-covariance matrix of the estimated load is given by equation (4): where s is the error measurement for strains, [A] the matrix of the sensitivity coefficients of L and the standard deviation.
In order to reduce the error as much as possible, this method assumes that the number of deformations, and consequently the number of strain gauges used, are sufficient and equal to or greater than the number of the load components.
If the number of strain gages is not sufficient to evaluate all the load components correctly, there should be no possibility to evaluate the vector load only based on the experimental measurements.To overcome this difficulty, the idea is to complete the equation linear system adding a certain number of conditions, exploiting physical and structural considerations, to arise from experimental data analysis.The purpose of the introduced variation is to achieve the number of unknowns equal to the number of system conditions, from under-determined to even-determined problem, consequently having the possibility to evaluate all loads.However, these further equations must be determined for each case based on reasonable hypothesis and subjected to a posteriori verification, in order to avoid the risk of significant errors.By introducing relations between the various load components, if physically possible and in the case these relations are invariant with respect to loads variations, the number of unknowns may be reduced, bringing the system determined and thus guaranteeing its resolution.
In detail, supposing to have a structure, for which the deformation vector {S} n×1 and the number of components of the load vector {L} p×1 are known, as reported in equa- tion (1), and let p − n = q .For the system to be determined, q indicates the degree of indeterminacy of the system, hence a number of q new relations must be conceived.Since a (2) load acting on a structural system is in most cases characterized by a constant direction, this implies that the ratios between its components along three orthogonal directions is quite constant for relatively stiff structures, independently from the loading amount.To clarify what has been stated, it is useful to consider a simple system, for which a single uniaxial strain is the result of the plain load acting on the structure.Assuming that the two load components are F 1 and F 2 , with a 1 and a 2 the respective sensitivity coefficients producing deformation 1 , it can be written that: The system is apparently under-determined and clearly the equation has ∞ 1 solutions.Nevertheless, if the direction of the force is defined and constant over time and load conditions, the following simple relationship between the two loads can be introduced: By inserting this relationship in equation ( 5), it is immediate to simplify the relationship between measured strain and acting load as follows: In this way, it is possible uniquely to trace the loads F 1 and F 2 which generated the deformation 1 .However, the validity of equation ( 7) is guaranteed only if it is possible to find on a physical basis a verified and constant relationship between the loads.

Aerodynamic Down Force Measurement of a Car Rear Aero-Profile
In the case of under-determined systems, the case study of downforce developed by the rear wing of a high-performance car during track tests, hence in real operating conditions, is here studied.In fact, vehicle dynamics depend by many different factors: the key-factor is the tire capability to exchange forces with the road surface.These forces, due to the friction between the tires and road surface, in turn depend on the vertical load acting on each one.Therefore, aero-profiles are commonly used on high performance vehicles in order to increase the capacity of the tires to exchange relevant forces with the road plane and to improve road holding and stability at high speed, hence safety.From these general considerations, it arises the demand to investigate the load on the axles, especially on the rear one, in order to understand the effects of the aerodynamic wings on the vehicle dynamics, such as high-speed stability.There are some experiences in the literature regarding this aspect. (5) Cheng et al. [11] investigated the vertical load on a car model equipped with different types of rear spoilers and as a function of the yaw angle.The experiments were done in the wind tunnel on a simplified model of a body hatchback, positioned on a base equipped with 6 load cells in order to measure the forces and moments.Other types of measurement also include the use of CFD/FEM numerical simulations that evaluate the load acting on the air foils and on the vehicle itself.In fact, Kajiwara [12] conducted an experience on a rear wing prototype equipped with three air foils and installed on an experimental vehicle with the aim of evaluating the vertical downforce developed by different configurations of the three profiles on the overall wing.Wing supports were instrumented with strain gauges, of simple geometry and optimally positioned to evaluate the forces that arise from the action of air.The experiments took place both in the wind tunnel and on the road, with the support of CFD analyses that validated results.On the other hand, in this work the authors make use of complex wing supports instrumented with strain gauges and load for down force evaluation were investigated only with experimental track tests, without the aid of wind tunnel simulations.

The Aero-Profile Supports
The aerodynamic profile under investigation is mounted on the rear axle of a sports car and is hold by two symmetrical supports (Fig. 1).The aim is to evaluate the aerodynamic downforce during track tests, measuring the deformation of the two supports, so to introduce minimal changes of the aerodynamic behaviour of whole system.
The analysed component is an aluminium alloy support, of suitable shape and equipped with 5 holes: three lower holes (holes 3, 4 and 5 in Fig. 1(b)) are used to fix the support to the rear compartment cover of the car, while the two upper holes (holes 1 and 2 in Fig. 1(b)) are used to fix the wing to the support.In this way, the loads received by the wing due to the aerodynamic action of the air are transferred through the supports onto the rear axle of the car.
Given the geometric and load symmetry of the system (Fig. 1(c)), a FEM model of the single support was conceived to understand its structural behaviour and to identify the best areas where install strain gages needed for the track test measurements and then load estimation task with pseudo-inverse method.

Numerical Model of the Support
The experimental study involves the use of strain gauge techniques, therefore is fundamental to know the area where the strain gauges will be glued, making the load calculation easier, also considering the assembly constraints and the shape of the components.Therefore, a numerical model of the support was developed to identify which areas were suitable to host the strain gauges for reliable strain analysis, i.e., the areas that show significant deformations.
Given the complex geometry, a reverse engineering phase was preliminarily done by laser scanning in order to reconstruct its geometry (Fig. 2).The point cloud representing the external surface of the component was used to generate a 3D cad solid model in a neutral import format (file format .stp).The model thus generated was imported into the Salome-Meca FEM pre-processing environment correcting the geometry and inserting some elements necessary for the correct application of the load and constraints that reproduce the real operating conditions.
In detail, hole axes have been added and used to correctly apply loads and constraints.The volume was meshed with 10 nodes tetrahedral solid elements having parabolic shape functions and beam elements with Euler-Bernoulli The static analysis was performed using Code -Aster (https:// www.code-aster.org/) applying a representative vertical load equal to 1000 N, equally distributed between the two holes used to join the wing to the support (holes 1 and 2 in Fig. 1(b)); all DOF of the holes 3, 4 were blocked (see Fig. 1(b)).The results of this calculation showed that the deformation is uniaxial distributed along the direction of the inclined connections of the support and uniform in its central area (Fig. 3), which are the preferable conditions for deformation measurement.The directions of maximum deformation are parallel to the axis of the two slender sides of the support and correspond to 18° and 40° with respect to the vertical direction along which the loads were applied (Fig. 4(a)).Consequently, the strain gauges were applied in these directions and locations.
Starting from these considerations, the supports were instrumented in four points.A half-bridge configuration with three-wire connection (Fig. 4(b) and (c)) was opted to compensate, respectively, the deformations due to temperature variations and the resistance readings due to the connecting cables [13].
Each strain gage reading was identified with a unique code for easy and correct identification.The strain gage nomenclature includes three groups of letters (Fig. 4(d The scheme of Fig. 4(d) shows the different names used later for the analysis of results.Since the strain gages were put on both supports, the load is distributed almost uniformly, and the supports are symmetrically loaded, the average strain of opposite strain gages have been considered, as: In this way, four experimental strain data for entire structure are available for the evaluation of the loads acting on the wing.

Determination of the Sensitivity Coefficients
The application of the pseudo-inverse method requires the knowledge of the sensitivity coefficients of the structure (the two supports in this case) for each load direction.
These loads are the aerodynamic vertical load, drag load, due to the horizontal action of the air on the wing, and the centrifugal force, due to traveling through high-speed bends.The classic reference system used in the case of vehicles associated to the cash desk has been adopted for the correct identification of the loads; the x axis is parallel to the direction of road and directed towards the front of the vehicle, the y axis is directed to the left of the vehicle and the z axis is directed upwards [14], as shown in the following Fig. 5.
Aerodynamics loads on the wing are distributed loads and they are transferred as resultant forces to analysed supports through the two connection holes (numbered 1 and 2 in Fig. 1).
For what concerns the vertical load components, analysing the wing structure, the supports, and the cover of the rear compartment of the car, it can be observed that the constraints on the interface of the various components are similar to fixed joints.Consequently, assuming that the pin is infinitely rigid and responsible of structure constraints, the assembly can be structurally modelled by a simple 2D beam structure (Fig. 6).
Since connection between support and wing can be assumed as an internal fixed joint, the actions that are transmitted on the single support will be two forces, one vertical and one horizontal, and a bending moment, whose amount depends not only by the geometry but also by the stiffness of wing and supports.The supports stiffness has been determined by FEM analysis, while a composite beam has been assumed for the wing, using the geometric and material characteristics based on experimental surveys and comparisons with similar structures.In this way, it was possible to reconstruct the resulting loads acting on the support, reproducing the same deformation field that occurs when the support is mounted on vehicle.The correlation between experimental strain measurements on the real part and the simplified 2D model allowed to estimate the wing stiffness and loads and moments transferred to each support in the case of the vertical load condition.These simulations and comparisons were analysed for all the positions that the wing can assume on the car, because the position of the wing centre of pressure changes and load is differently distributed.
Similar considerations were made for horizontal drag loading; in this case, the single components acting on each single fixing holes 1 and 2 of the support were considered (Fig. 1).Therefore, the concentrated load calculated by the 2D model must be referred to the connections wing/support (holes 1 and 2 in Fig. 1); the local load components must be distributed between these two locations, as displayed in Fig. 7.In conclusion, all loading forces to be determined are the vertical load ( F z ), the drag load ( F X1 and F X2 ) and the transverse load ( F Y1 and F Y2 ), applied into the two connection holes, for a total of five unknowns.
Once the local deformations in the strain gauge positions for each load and for each wing position were determined, the sensitivity coefficients were calculated according to the following criterion: where x i is the generic coefficient related to the deformation i , generated by the application of load F i .
For the calculation of aerodynamic downforce acting on the aero-profile, the proposed method was applied.Referring to the vehicle reference system previously indicated, the known deformations are related to 8 strain gauges, (9) Fig. 5 DIN 70,000 Axes system Fig. 6 Wing and support 2D schematization averaged between the two wing supports for a total of four known deformations, the system of equations cab be written, in terms of deformations: In matrix form, it can be written: This system of equations is under determined and presents ∞ 1 solutions.To get out of this condition, the pseudo- inverse method need to be completed as discussed in paragraph 2. It would be preferable for the system of equations to be overdetermined.In this case, it helps how forces are transmitted to the support.Since the structure is in the elastic field, it is possible to define the ratios (see Fig. 7): Consequently, the method could be applied by introducing these two ratios between the forces.In this way, the system is even determinate and it can be stated as: and in matrix form: where: is the full matrix of the sensitivity coefficients (see equation (1)) with the introduced variation.This system of equations (equation ( 13)) allows calculating the various force contributions, associated to specific deformation field inside the component, experimentally determined.The next step is the determination of the sensitivity coefficient x i defined in equation ( 9) and which constitute the sensitivity coefficients matrix [A] (equation ( 15)).For the geometry of the support, it was decided to proceed as follows: the coefficient relating to the vertical load F z was determined experimentally with calibration tests in the laboratory.As an experimental evaluation of the remaining coefficients would be difficult, due to the complication related to the correct application of a uniform horizontal load to the system, the remaining coefficients were determined numerically using the numerical model of the support, after having validated it with the result of the experimental calibration for vertical load.

Experimental Calibration for the Evaluation of Vertical Sensitivity Coefficient
In order to correlate the deformations detected by the strain gauges to the vertical load values acting on the wing, it ( 14) Steps for the determination of the ratios of forces generated by distributed horizontal load was necessary to carry out an experimental calibration of the supports, effectively transforming the wing supports into improper load cells.In this way, the sensitivity coefficients relating to the vertical load were experimentally determined.
The mounting configuration that normally exists on the car was used to carry out the calibration.Given that the supports of the wing are mounted on the rear hood of the car, this was taken from the car and placed on a suitably constructed wooden structure that reproduces the actual position of the hood -supports -wing assembly on the car itself.
The wing was loaded with calibrated weights to reproduce the action of a vertical load using a second structure in wood and polyurethane foam to have a uniform force distribution over the entire length of the wing (Fig. 8).
The calibration process was repeated for all four wing positions (Fig. 9); this is because small variations in the angle of inclination of the wing vary the position of the pressure center of the airfoil and therefore also the distribution that the load has on the two wing -support connection holes.Figure 10 shows the calibration curves performed for position 3 of the wing, which were obtained for each strain gauge.
The load, as reported in Fig. 10, is applied in several load step.For each load step, the strain, after an initial oscillation, determined by the physical application of the load, is stabilized to a mean value (for example the values in the blue ovals in Fig. 10), which has been considered for each applied load.
Consequently, the calibration curves reported in Fig. 11 are obtained plotting the average strain values for each applied load.The experimental data are referred to the unloading phase, in which any clearance has been recovered, and were accurately interpolated by linear curves, confirming the linear elastic behaviour of the support.The calibration curves of the symmetric strain gage are practically coincident, due to the geometry and load symmetry of the uniform vertical load.From the knowledge of these curves, the relationship between the load and the deformation for each strain gauge used and for each position of the wing have been expressed through the angular coefficient of the calibration curves.In detail, these coefficients were identified with the following relationship (according with equation ( 9) an equation ( 14)): where a i is the coefficient relating to the deformation i gen- erated by the application of the vertical load F Z .These coef- ficients were determined in all four operating positions of the wing, because the distribution of the load varies as the position varies.The following Table 1 shows the values of the coefficients a i ( ∕N ) determined experimentally.

Numerical Model for the Evaluation of the Other Sensitivity Coefficients
The numerical model of the support was validated using the experimental data obtained by the static calibration reported before.However, the vertical load applied cannot be applied directly to the single support, since several preliminary considerations are needed to evaluate how the vertical load is distributed on the supports.( 16) Assuming that the wing is constrained to the support by means of a rigid joint, when a uniform load acts on the wing, the constraint acts with two forces and a moment (in-plane analysis).These identical and opposite actions act on each support, determining a specific deformation state.Thanks to the symmetry of the problem, it was possible to reduce the hyper static degree of the structure, passing from a three times hyper static structure to a structure two times hyper static (Fig. 12).By applying to the numerical model, the same loads applied to the real structure in the experimental phase, it was possible to compare the results obtained in terms of deformation.The loads applied to the support are divided between the two mounting holes according to the position assumed by the wing.The distribution percentages were evaluated with a numerical model of the wing to take into account its stiffness.The following table shows the deviation of the numerical strains respect to the experimental strains measured during calibration.These values are calculated as below: where exp is the experimental strain and num is the numerical strain.The comparison was made for all wing positions (Table 2).
The presence of percentage differences is due to the hypothesis of considering the internal constraint between the support and the wing as a perfect joint, neglecting its compliance.Given the low values of these differences, the hypothesis was considered acceptable.In this way the numerical model was validated.
Finally, it was possible to proceed to identify the sensitivity coefficients for the loads that were assumed to act on  the wing, and which cannot be replicated in the laboratory.These were determined by applying one force at a time to the numerical model and for each position taken by the wing.

Track Tests
After the first laboratory and modelling/simulation phase, the track tests followed in order to obtain the data for the calculation of vertical load acting on the wing as a function of speed.In this phase, in addition to the instrumentation necessary for recording the strain gauge data (HBM Quantum X strain gauge control unit), the car was instrumented with a series of typical instruments for vehicle dynamics characterization on the track, in particular: -IMU (High-Grade Inertial Measurement Unit), necessary for the measurement of acceleration components (lateral, longitudinal and vertical), rotational rates along the three main axes (yaw, roll and pitch) and velocity components trough Differential-GPS system; -Height from ground laser sensors at each corner for the direct measurement of the vehicle body roll and pitch.-Sensorized steering wheel to measure the steering steer angle and steer rate.-Additional signals from the vehicle CAN-bus such as accelerator pedal position, engine speed, etc.
The signals sampling rate was fixed to 100 Hz, which is typical for vehicle dynamics data gathering.The acquisition systems from strain gauges and vehicle dynamics (Fig. 13) were synchronized with a trigger activated directly by the driver.
The track tests allowed to study the aerodynamic contribution of the wing (downforce and drag) and to correlate it with the dynamic performance of the vehicle and were  • Car Circular Track (Fig. 14(a)): It is a perfect circle of 12.6 km and a diameter of 4 km.The centrifugal force is compensated by the track banking.These unique features allow high speed testing with speed compensation until 240 km/h.A high-speed test for each position of the wing was performed on the circular track on the fastest lane in exclusive use and at the vehicle's maximum speed (this test was executed two times for each position; in the first repetition two laps were made, while in the second only one).This test is named test A. • Car Dynamic Platform (Fig. 14(b)): 2 straight lanes (700 m long and 20 m wide each) and a square (regular sides of 280 m).Tests were conducted for positions 1 and 4 on the car dynamic platform at intermediate speeds, keeping them constant for a certain time to understand the downforce available at these reference speeds.This test is named test B. • Handling Track (Fig. 14(c)) with a length of 6.2 km, it counts 16 main curves with very different radii.The track was developed to achieve comprehensive characteristics that might include some of the most particular features of the European circuits commonly adopted for high performance vehicles development and tuning.A test for position 1 on the handling was performed here.This test is named test C.
The strain gauges data that were extracted at the end of the track tests were previously filtered with the moving average method to reduce the scatter of the experimental data (Fig. 15).
By examining the strain gauge data obtained in the circular track of the car, a certain difference in the recorded strain value was observed between the left and right symmetrical strain gauges (for example, R_F_OUT and L_F_OUT).This effect is due to the presence of centrifugal force acting on the body and wing.Considering the arithmetic mean of the homologous strain gauges, it was possible to exclude this effect, which is symmetrical between the right and left side of the car.The wing, however, undergoes the action of other side components that are not quantifiable, such as crosswind or other environmental causes.For this reason, the term relating to the presence of lateral loads is still present in the equations system for determining the vertical load.
From the graph in Fig. 15, it can be seen that in the time interval in which the car is at maximum speed (in the interval 60 − 300 s) the deformations have significant oscillations.This is due to the surface of the track which is not perfectly flat which causes jolts which are recorded by the strain gauges.For this reason, it was decided to use the deformation values detected for a limited test interval corresponding to the flattest section of the circular track.This interval is between km 6 and km 8 of the track (in the interval between about 200 and 250 s).
The strain course for the tests on the dynamic platform does not have particular characteristics, as shown in Fig. 16 for the test B in position 1.What it may be observed in this case is that the strains do not start from zero, as in the previous case, because the test has been started at 80 km/h.
From these first two tests, it is evident that the strain values of the strain gauges indicated with OUT are positive and those indicated with IN negative, as it is expected.This means that the aerodynamic load tends to apply a bending stress to the support, inducing traction on the outside and compression on the inside of the support.The strain gauge data of the tests on the handling track are of a completely different nature, as shown by the trend as in Fig. 17(a).
Due to the conformation of the track, deformation trends reflect the various trajectories that the vehicle takes on the track.Increases and decreases in deformation when the vehicle is in a straight line and, respectively, in an acceleration and deceleration phase; asymmetries in the deformation values in the travel of curves at high speed, with a variation in the sign of the deformations as a function of the direction of the curve, whether right or left.It can also be seen how the same block of deformations (black rectangle) is repeated six times, as six laps have been made.Figure 17(b) focuses on one of the highest peaks (blue rectangle in Fig. 17(a)).No abnormal trends are observed, but all strains grow and decline together, with minimal differences between the left and right symmetrical strain gauges.
In the handling circuit, the centrifugal effect was very evident, especially when traveling through curves (Fig. 18).For this reason, an arithmetic average was carried out between the homologous deformations of the two supports to eliminate the centrifugal contribution.
A further consideration led us to exclude F_IN strain gages from the analysis.These data, in fact, are affected by anomalies and irregularities that have not obtained in the other strain gage data.The unreliability of F_IN strain gage data is originated by the fact that F_IN strains gauges are exposed to a further effect due to their positions (Fig. 19).They are located on the inside of the supports, facing towards and this part of the support on which they are glued is slightly diverging forward and therefore it is more exposed to airflow.
Since strain gages are sensitive to the pressure, it was assumed that strains recorded by the front-side gauges are influenced by this hydrostatic pressure of airflow.For this reason, in the final analysis, the strain gages F_IN were excluded by the analysis.The exclusion of these strain gauges results in the elimination from the system of equations of the relation describing Fin .Since the system of equations is overdetermined, this elimination does not affect the resolution of the problem.The system of equations goes from overdetermined to determined, making it easier to solve.This system of equations is now:

Results
The implementation of the strain gauge data in the resolution method made it possible to derive the downforce and aerodynamic drag values on the rear wing of the analysed vehicle.
In this section, we will refer to the high-speed test on circular track and we will focus on results in the section included between km 7 and km 8.5, which has no longitudinal slope.Load values are shown with a positive sign only to simplify visualization (in the adopted reference system the z-axis is pointing upwards whereas the downforce values are negative).
Tests were performed for all positions in the profile.From the dynamic data, it is observed that the maximum speed decreases from position 1 to position 4 due to the drag action of the profile, which increases its braking component (Fig. 20).In all tests, the surrounding conditions are obviously the same with, for example, gas pedal at 100% for tests in road cars, highest gear, etc.
The increase in aerodynamic load passing from position 1 to 4 is also confirmed by the value of the distance read by the rear lasers of the vehicle: this distance decreases  From these preliminary observations on the dynamic data of the tests carried out on the circular track, it can be stated that the trend of aerodynamic load and aerodynamic resistance is in line with expectations, given that a greater incidence angle of the profile leads to a greater load on the rear axle (which is more flattened downwards) and to a reduction in the maximum speed of the car (due to the greater frontal area that the profile exhibits by increasing the attack angle).
By applying the proposed calculation method to the strain gauge data detected during the tests, various behaviours were observed, some in line with what was expected and some inconsistent and in opposition to what was detected by the dynamic data of the car.In fact, the processing of the data relating to positions 1, 2 and 3 are consistent with what was expected, while the processing of the data relating to the tests carried out in position 4 of the profile in some tests returned inconsistent results.
The results reported in the previous graphs show how the loads determined for positions 1 to 3 are increasing passing from position 1 to position 3 and therefore in line with expectations (Fig. 22).Instead, the comparison between the first three positions and the position 4 (Fig. 22) and positions 1 and 4 (Fig. 23(b)) showed that the loads determined in position 4 are not in line with what was expected.The downforce and drag determined for position 1 in two different tests show the same trend as a function of speed (Fig. 23(a)), while the same cannot be said of what is determined in position 4.In fact, the downforce turns out to be higher in the test carried out on the circular track compared to the test carried out on the dynamic platform and the drag calculated for the circular test is lower.
This behavior in position 4 is probably due to an excessive deformation of the surface of the strain gauges due to the aerodynamic pressure component that is generated.While in the other positions the shielding was sufficient to protect the strain gauges from these non-evaluable loads, in position 4 this protection was not sufficient.This consideration arises from the fact that the loads calculated in this position do not respect the physics of the problem.In fact, an increase in downforce and drag should be observed, respectively, due to the greater inclination of the wing and the greater frontal surface of the wing.The results of the tests on the handling track are now analysed.Figure 24 shows the values of downforce, drag and speed for a single repetition of the lap of the track (for the other laps the analysis will be the same).The very high correlation between the vehicle speed and the wing measured downforce is evident.This validates the close relation between downforce and vehicle speed and, more generally, the reliability of strain gage measured data.
In a straight line, the maximum test speed is achieved.Being on a straight path, the influence of lateral disturbances is minimal, and both the aerodynamic load and the aerodynamic drag follow the increase (or decrease) of the speed (yellow rectangle in Fig. 24).It can be seen that 25% of the maximum downforce is reached.This 75% loss is caused by two causes: • this graph relates to position 1 of the wing and this position is characterized by the lowest aerodynamic load; • this test reaches a speed lower than the maximum speed (compared to the maximum speed of all tests).
In the remaining part of the test there are curves with very low radius of curvature travelled at low speed.As regards the deformations, also in this case aerodynamic load and aerodynamic resistance assume opposite trends (for example blue rectangles in Fig. 24).In this case, the strains have very low values, and the associated error is certainly greater.Furthermore, below a certain speed the deformation measured by the strain gauges is probably due to torsional effects rather than to the aerodynamic forces.

Conclusion
The measurement of unknown loads acting on a structure is a non-trivial problem, in particular when it is not possible to apply specific sensors such as load cells capable of performing a direct measurement.In this case, it is necessary to carry out an indirect measurement, calculating the loads once the structure deformation state in some points is known.In this way, it is possible to apply the pseudo-inverse method and calculate the load condition that produces that specific structure deformation state.
If the number of deformation measurement points is insufficient, the system of equations describing the problem seems to be underdetermined.To eliminate this problem, in this work a modified pseudo-inverse method was proposed, which consists in adding to the system a certain number of equations exploiting physical and structural considerations.In this way, an even over determined or determined system is obtained and the resolution is possible.
The proposed modified method was applied to a case study, in which it was necessary to estimate the aerodynamic load developed by the rear wing of a high-performance car with track tests.The indirect measurement was conducted by measuring the deformations of the wing supports.A numerical model was developed to identify the installation positions of the strain gauges and the sensitivity coefficients of the loads that could not be determined experimentally.An experimental calibration of the supports was performed, replicating the assembly scheme that exists on the car, to determine the sensitivity coefficients relating to the vertical load and to validate the results of the numerical model.Finally, tests were performed on the track, in different conditions of use, applying the modified calculation method in order to determine the aerodynamic load in the different configurations that the wing can assume.
The proposed calculation method was applied to the results of the tests on the track, obtaining results that proved the existence of a consistent aerodynamic load, which is a function of the speed and position assumed by the wing.
Finally, applying the proposed modified calculation method, it was observed how the aero-profile can produce a vertical load and how this increases with the angle of the wing and as the speed increases.This result was also confirmed by the reading of the height of the four corners of the car: by increasing the angle of attack, the car also lowered the rear of the car, due to the presence of a greater vertical load.The processing of the strain gauge data produced consistent results for positions 1 to 3, while inconsistent results were obtained for position 4 due to unpredictable aerodynamic effects.Experimental results for this last position require further investigation.
In conclusion, it has been demonstrated how, by making a modification to an analytical method of the pseudo-inverse for the determination of a particular load condition, it is possible to trace a particular load.This modified method was applied to determine the loads developed by an aerodynamic appendage of a high-performance car, demonstrating it the effectiveness for different values of the angle of attack.Furthermore, it was demonstrated that it is possible to run combined deformation/load and vehicle dynamics data acquisition in complex and real operation conditions as track testing at high speed.This method, after further refinement, hopefully will support the specific analysis of aero profiles contribution to vehicle dynamics characteristics in sports cars.

Fig. 1
Fig. 1 Aerodynamic wing installed on the car (a), Cad model of the support (b) and rear (c) and side (d) view of the assembly

Fig. 2 Fig. 3
Fig. 2 Scanning process step (a) and reverse engineering process (b); mesh of component: outside view (left) e inside view (right) (c) )): • The first letter indicates which support the strain gage has been mounted (L = left and R = right relative to the driver position); • The second letter indicates if the strain gage has been positioned at the front or rear of the support, where the front is the one closest to the driver (F = front and R = rear); • The end letters indicate if the strain gage has been fixed in the interior or exterior part of the support, where the exterior part is the exterior of the vehicle (IN = inside and OUT = outside).

2 Fig. 4
Fig. 4 Support reference system and strain gages orientation (a); detail of the strain gauges (b) and instrumented support mounted on the rear bonnet of the car (c); strain gage position and nomenclature in a wing axonometric frontal (d)

a
Fout b Fout c Fout d Fout e Fout a Rout b Rout c Rout d Rout e Rout a Rin b Rin c Rin d Rin e Rin a Fin b Fin c Fin d Fin e Fin 0 1 −h 0

a
Fout b Fout c Fout d Fout e Fout a Rout b Rout c Rout d Rout e Rout a Rin b Rin c Rin d Rin e Rin a Fin b Fin c Fin d Fin e Fin 0 1 −h 0

Fig. 8 Fig. 9 Fig. 10
Fig. 8 The calibration test in the laboratory with calibrated loads

Fig. 12
Fig. 12 Distributed load acting on half structure and equivalent support scheme

Fig. 14
Fig. 14 Nardò Technical Center track's top view (a) and test tracks (b)

Fig. 17
Fig. 17(a) Normalized strain on the handling track (position 1) and (b) Normalized strains in the straight path of the handling track Fig. 17(a) Normalized strain on the handling track (position 1) and (b) Normalized strains in the straight path of the handling track

Fig. 21
Fig. 21 Rear left laser (in the yellow box the interval of track distances considered)

Fig. 24
Fig. 24 Normalized dynamic track load data (Pos.1) Fout c Fout d Fout e Fout a Rout b Rout c Rout d Rout e Rout a Rin b Rin c Rin d Rin e Rin a Fin b Fin c Fin d Fin e Fin

Table 1
Values of the coefficients a i( ∕N)

Table 2
Percentage deviation between experimental and numerical strains