Analysis and evaluation of a tyre model through test data obtained using the IMMa tyre test bench

Vehicle System Dynamics Vol. 43, Supplement, 2005, 241 252 Analysis and evaluation of a tyre model through test data obtained using the IMMa tyre test bench A. ORTIZ*, J.A. CABRERA, J. CASTILLO and A. SIMÓN University of Málaga, Málaga, Spain Nowadays, obtaining results on the forces and torques in a tyre, in controllable and repetitive working conditions in a laboratory, is fundamental for research groups in vehicle dynamics simulations. There is a need, more than a convenience, to own a facility to test tyres of a wide range of dimensions and with versatile and non-restricted operating conditions of functioning. In this paper a group of experimental data obtained from tests on a tyre in the IMMa tyre test bench is presented. With the goals of determining the accuracy of these experimental test data and also the operating performance of the test bench in the steady state, the analytical model of pneumatic tyres for vehicle dynamics simulations reported by Gim and Nikravesh is employed. Keywords: Tyre test bench; Tyre model; Tyre test data 1. Introduction In this paper, we make use of the IMMa tyre test bench [1] to obtain the values of the empirical characteristic variables C z, C s, C α and C γ of the tyre model and the test data needed to contrast these with the forces and moment obtained with the analytical expressions of the tyre model validated by Gim and Nikravesh [2, 3]. This bench, which has been entirely developed by the IMMa research group, enables the analysis and study of the dynamic properties of a tyre in the laboratory. The operating conditions can be controlled and repeated. Basically, the tyre-testing machine consists of a high-stiffness tyre loading and positioning assembly, a flexible closed-loop flat track system and a specifically designed data acquisition and control system. It has the capacity to monitor and control a diversity of operative parameters, each varying within a wide range of values. The process of data acquisition was carried out in steady-state conditions; we control an input variable so that it remains constant and we measure the output variable during a period of time, picking up data per second at a chosen frequency. Finally, every test is repeated a determined number of times and the average and standard deviation of the output variables are obtained. We proceeded as follows with the objective of analysing the analytical model of pneumatic tyres for vehicle dynamics simulations reported by the University of Arizona [2, 3]. We are *Corresponding author. Email: aortizf@uwe.es Vehicle System Dynamics ISSN 0042-3114 print/issn 1744-5159 online 2005 Taylor & Francis http://www.tandf.co.uk/journals DOI: 10.1080/00423110500140096

242 A. Ortiz et al. also interested in using it as a tool to compare with the experimental data obtained on the IMMa tyre test bench in the steady state. (i) Initially, the value of the empirical variables of the model for each tyre is calculated by tests. Basically, we evaluate by testing the elastic properties that consist of radial stiffness, longitudinal stiffness, cornering or lateral stiffness and camber stiffness. Here, the testing methodology will be presented to obtain C z, C s, C α and C γ. The influence of radial stiffness on the inflation pressure and the dependence of the longitudinal, lateral and camber stiffnesses on the vertical load will be demonstrated. These were obtained by tests on the IMMa tyre test bench. (ii) Camber stiffness will also be analysed as a function of the cornering stiffness of equation (1) and validated by means of the camber stiffness experimentally obtained on the IMMa tyre test bench. The length l of the contact patch was evaluated on a test bench using the method of total internal reflection of light [4]; the tyre radius ρ 1 is a geometric value given by C γ = C αl. (1) 6ρ 1 (iii) Test data on the forces F x and F y and the autoaligning torque M z on a tyre for pure slips are obtained by means of the IMMa test bench, for different vertical loads F z. With the purpose of estimatinge the accuracy of these test data, the validated analytical model of pneumatic tyres reported by Gim and Nikravesh [5] of the University of Arizona is used. 2. Testing oriented to obtain the values of the elastic properties of a tyre by means of the IMMa tyre test bench Basically, the IMMa tyre test bench, on which the tests have been performed, consists of a high-stiffness tyre loading and positioning assembly, a flexible closed-loop flat track system and a specifically designed data acquisition and control system. The bench is able to test solid rubber or pneumatic tyres with radii in the range 10 60 cm at maximum road speeds of up to 100 km h 1. It has the capacity to monitor and control a diversity of operative parameters, each varying within a wide range of values. This is accomplished by means of a set of appropriately placed sensors and actuators, sending and receiving control signals through a specifically designed data acquisition and control system. The software required for these tasks has been developed by the research group. A wheel-sensor (quartz-based instrument) for passenger cars was used to obtain test data on forces and moment in this paper. 2.1 Radial stiffness C z The aim of the tests was to obtain the radial deformation δ produced in the tyre for every vertical force F z exerted. The tyre was an INSA TURBO 185/60 R14 type and the test was performed with three different inflation pressures p i equal to 1.8, 2.2 and 2.7 bar. The test results are given in figure 1. The test data on radial deformation versus vertical load obtained on the IMMa tyre test bench show the well-known features that radial stiffness is affected by the vertical load and inflation pressure. The radial stiffnesses C z evaluated from the previous test data in figure 1 for every vertical load and inflation pressure are presented in table 1. The radial stiffness C z is defined as the

Analysis and evaluation of a tyre model using the IMMa tyre test bench 243 Figure 1. Radial deformation produced in the tyre with three inflation pressures. slope of the vertical force versus the radial deformation δ, that is C z = F z δ. (2) 2.2 Longitudinal stiffness C s A testing procedure was performed to evaluate the longitudinal force against the longitudinal slip ratio. The tests were carried out at a constant speed with three vertical loads for an INSA TURBO 185/60 R14 tyre with an inflation pressure of 2.2 bar. The methodology of the testing process consists in exerting a vertical load on the tyre and a constant controlled braking force, without slip and a camber angle, while the tyre rolls at a constant speed. At the same time the longitudinal force and longitudinal slip ratio are measured. The process is repeated to cover the longitudinal slip ratio range from 0 (free rolling) to 1 (total sliding). Table 1. Radial stiffnesses of an INSA TURBO 185/60R14 tyre for different inflation pressures p i. Radial stiffness C z (N m 1 ) for the following p Fz i (N) 1.8 bar 2.2 bar 2.7 bar 1100 10 4924 11 0860 12 4632 1500 12 3270 12 8050 14 4758 1650 12 7857 13 4310 15 1467 2000 13 7030 14 7330 16 4884 2100 14 1616 15 0460 17 1592 2500 15 5375 16 4010 18 5010

244 A. Ortiz et al. F z Table 2. Test results of the longitudinal force F x against the longitudinal slip ratio κ. F x (N) for the following κ (N) 0 0.02 0.04 0.06 0.08 0.1 0.2 0.4 0.6 0.8 1 1100 0 900.5 1030 1070 1088 1097 1106 1088 1071 1060 1051 1650 0 645 1083 1332 1472 1555 1695 1715 1696 1676 1659 2000 0 1207 1858 1946 1969 1979 1990 1994 1994 1995 1995 The results of the tests are presented in table 2. The longitudinal stiffness is defined as the slope at the origin of the curve of the longitudinal force against the longitudinal slip ratio without slip and a camber angle, that is C s = df x ds. (3) s=0 The calculated longitudinal stiffness of the tyre is presented in figure 2(a). In this figure, the known dependence of the longitudinal stiffness C s on the vertical load is noted. In figure 2(b), the variation in the slope of this longitudinal force against longitudinal slip ratio with three vertical loads is drawn. Also, the value of the slope of the longitudinal force F x with respect to the longitudinal slip ratio, at κ = 0, is observed. 2.3 Lateral stiffness C α A testing procedure was performed to evaluate the lateral force against the slip angle. The tests were carried out at a constant speed with three vertical loads for an INSA TURBO 185/60 R14 tyre with an inflation pressure of 2.2 bar. The methodology of the testing process consists in exerting a vertical load on the tyre and a constant controlled slip angle, without camber angle and slip ratio, while the tyre rolls at a constant speed. At the same time the lateral force and slip angle are measured. The process is repeated to cover the slip angle range from 0 to 10. The results of the test are presented in table 3. Figure 2. force. (a) Influence of the vertical load on the longitudinal stiffness; (b) variation in the slope of the longitudinal

Analysis and evaluation of a tyre model using the IMMa tyre test bench 245 Table 3. The lateral force F y versus the slip angle α for different vertical loads. F z F y (N) for the following α (N) 0 0.5 1 1.5 2 2.5 3 4 6 8 10 1650 0 310 646 950 1129 1323 1464 1704 1982 2139 2060 2100 0 439 792 1215 1456 1653 1852 2101 2451 2630 2636 2500 0 463 845 1308 1624 1868 2081 2462 2859 2970 3011 The lateral stiffness is defined as the slope at the origin of the curve of the lateral force against the slip angle without a camber angle and slip ratio, that is C α = df y dα. (4) α=0 The calculated lateral stiffness of the tyre is presented in figure 3(a). In this figure the known dependence of the lateral stiffness C α on the vertical load is noted. In figure 3(b), the variation in the slope of this lateral force against the slip angle with three vertical loads is drawn. Also, the value of the slope of lateral force F y with respect to slip angle, at α = 0, is observed. 2.4 Camber stiffness C γ A testing procedure was performed to evaluate the lateral force against the camber angle. The tests were carried out at a constant speed with three vertical loads for an INSA TURBO 185/60 R14 tyre with an inflation pressure of 2.2 bar. The methodology of the testing process consists in exerting a vertical load on the tyre and a constant controlled camber angle, without a slip angle and slip ratio, while the tyre rolls at a constant speed. At the same time the lateral force and camber angle are measured. The process is repeated to cover the camber angle range from 0 to 16. The camber stiffness C γ evaluated from the previous test data in figure 4 for every vertical load and an inflation pressure of 2.2 bar are presented in table 4. The camber stiffness C γ is Figure 3. (a) Influence of the vertical load on the lateral stiffness; (b) variation in the slope of the lateral force.

246 A. Ortiz et al. Figure 4. The camber stiffness as a function of the vertical load. Table 4. Camber stiffness of an INSA TURBO 185/60R14 tyre for different vertical loads. F z Camber stiffness C γ (N) (N deg 1 ) 1700 12.3833 2300 27.9547 2600 39.6499 defined as the slope at the origin of the lateral force versus the camber angle γ, that is C γ = df y dγ. (5) γ =0 In figure 4, the known dependence of the camber stiffness on the vertical load from test data obtained on the IMMa tyre test bench is drawn. 3. Comparison between the analytical and the experimental values of camber stiffness Here, we compare the values of camber stiffness obtained by means of test data collected on the IMMa tyre test bench and determined using the analytical expression propsed by Gim and Nikravesh [2, 3]. The method of total internal reflection used for pneumatic tyres by Pérez et al. [4] is employed to evaluate the length of the contact patch. Assuming a rectangular contact area, the effective length is evaluated from the effective area divided by the width. The experimental data on the camber stiffness, lateral stiffness, effective area, width and length of the contact patch and the analytical value of the camber stiffness are compiled in table 5. The test data in table 5 correspond to an inflation pressure of 2.2 bar. The INSA TURBO 185/60 R 14 tyre has a geometric radius of 288.8 mm.

Analysis and evaluation of a tyre model using the IMMa tyre test bench 247 Table 5. Comparison between the experimental and analytical values of the camber stiffness (p i = 2.2 bar). Camber Lateral Effective F z stiffness stiffness Effective Effective length C γ = C α l/6ρ 1 (N) C γ (N deg 1 ) C α (N rad 1 ) area (mm 2 ) width (mm) l (mm) (N deg 1 ) 1700 12.3833 41248 5034 109.1 46.14 19.17 2300 27.9547 50960 6767 119.1 56.82 29.16 2600 39.6499 52047 7633 122.1 62.51 32.77 The analytical and experimental values of camber stiffness show a certain level of similarity. The analytical expression (1) gives values for the camber stiffness that agree fairly well with the experimental camber stiffness obtained by means of testing in the above-mentioned setup. 4. Experimental data versus analytical tyre model Here, we use the analytical tyre model previously validated by Gim and Nikravesh [4] with the aim of verifying the test data obtained on the IMMa tyre test bench. Here, we make use of the radial, longitudinal and cornering stiffnesses presented above in section 2. The tyre radius is 0.2888 m. We used an elliptical friction concept with µ x = 1 and µ y = 1.2 as the semiaxes of the ellipse, owing to the evidence obtained in our test. 4.1 Longitudinal force due to a pure slip ratio The friction parameters used were µ 0 = 1 for s = 0, and µ 1 = 1 for s 1 = 1, for the longitudinal characteristic and were obtained from previous experimental results about the behaviour of the adimensional longitudinal force versus the slip ratio. The evolution of the friction coefficient is very similar to a horizontal straight line, once it passes the maximum, because of the frictional force existing in the tyre steel belt contact on the bench. Therefore, a linear dependence of the friction coefficient on the sliding speed is considered [6]. Using the friction value mentioned above and the stiffness presented in section 2.2, the longitudinal characteristic of the Gim Nikravesh tyre model is evaluated and employed to compare with the experimental data in table 2. The result of this comparison can be observed in figure 5. There is a fair amount of disagreement between the experimental results and the analytical expression for the longitudinal characteristic of the tyre model, in the same way as in braking and traction situations with different vertical loads. We do not use the radial stiffnesses presented in table 1 because we found the values of vertical loads from tests. At any rate, we propose to use a modified expression for the radial stiffness, that is F z = C z (δ δ 0 ). (6) With this expression it is possible to calculate the vertical load for every radial deformation δ when the radial stiffness and the slope of the tangent of the curve of the vertical load versus the radial deformation δ 0 with the x axis are known (table 6). This can be observed in figure 6.

248 A. Ortiz et al. Figure 5. Comparison between the experimental data on the longitudinal force and the analytical tyre model for three vertical loads. 4.2 Lateral force and autoaligning torque due to a pure slip angle The friction parameters used were µ 0 = 1.2 for s α = 0, and µ 1 = 1.2 for s α = 1, for the lateral characteristic and were obtained from previous experimental results about the behaviour of the adimensional lateral force versus the slip angle. Using the friction coefficient value mentioned above and the stiffness presented in section 2.3, the lateral characteristic of the Gim Nikravesh tyre model is evaluated and employed to compare the experimental data in tables 3 and 7. In this section, we use the lateral force test data in table 3 and the autoaligning torque test data in table 7 with the objective of making a comparison between the experimental data and the analytical expression for the lateral force and autoaligning torque. In figures 7(a) and (b) the results of the analytical tyre model for lateral force and autoaligning torque respectively are drawn. Also a comparison with the experimental results is made. Table 6. Radial deformation δ and initial lineal radial deformation δ 0 for every vertical load (p i = 2.2 bar). Radial deformation Initial lineal radial deformation F z δ δ 0 (N) (m) (m) 1100 0.0195 0.0095 1500 0.0228 0.0111 1650 0.0240 0.0117 2000 0.0265 0.0129 2100 0.0271 0.0132 2500 0.0297 0.0144

Analysis and evaluation of a tyre model using the IMMa tyre test bench 249 Figure 6. Values of δ 0 obtained from experimental data tests. Table 7. The autoaligning torque M z versus the slip angle α for different vertical loads. M z (N m) for the following α F z (N) 0 0.5 1 1.5 2 2.5 3 4 6 8 10 1650 0 7.48 15.3 18.43 17.77 18.68 16.26 13.99 11.46 8.88 8.74 2100 0 13.13 22.22 27.15 28.61 27.59 26.62 24.38 19.15 14.04 12.69 2500 0 17.41 28.97 37.06 38.81 39.01 38.33 34.43 25.70 18.31 15.49 Figure 7. Comparison of the experimental data on (a) the lateral force and (b) the autoaligning torque with the analytical tyre model for three vertical loads.

250 A. Ortiz et al. There are few differences between the tyre model lateral characteristic and our lateral force test data. Their values increase nonlinearly with a slope equal to the cornering stiffness, then reach their maximum and remain at this peak value for larger slip angles. The comparison between analytical and experimental autoaligning torque values shows a higher level of disagreement. The reasons for this, which are basically due to the approximations used in the tyre model for the characteristic of the contact patch and the pressure distribution on the contact, has been commented on by Gim and Nikravesh [2, 3]. 4.3 Lateral force and autoaligning torque due to a pure camber angle In this section we make use of the experimental data in table 8. These data were obtained by measuring the lateral force generated in the tyre for every camber angle exerted; three vertical loads are used. The camber stiffness was obtained previously, as reported in section 2. In figure 8 it is possible to observe the good level of agreement between the experimental data on the lateral force obtained on the IMMa tyre test bench and the analytical result of the Gim Nikravesh tyre model. The experimental data only disagree for camber angles smaller than the critical camber angle. Table 8. Experimental data on the lateral force F y due to the camber angle γ for three load cases. F y (N) for the following γ F z (N) 0 2 4 6 8 12 16 1700 0 33.5053 45.2362 86.2684 97.7633 103.5924 2300 0 105.2903 190.0109 349.3369 410.6189 2600 0 176.4610 327.6812 480.7365 579.9085 Figure 8. Comparison between experimental and analytical data on the lateral force versus the camber angle for three vertical loads.

Analysis and evaluation of a tyre model using the IMMa tyre test bench 251 Figure 9. The autoaligning torque versus the camber angle for three vertical loads. For motorcycle tyres it is possible to perform experiments to obtain the lateral force versus the camber angle for larger camber angles than those in figure 8. Then it will be possible to compare the experimental and analytical results for camber angles larger than the critical camber angle. The test data on the autoaligning torque versus the camber angle are shown in figure 9. The existence of a non-zero autoaligning torque suggests that the pneumatic trail is not zero for a pure camber angle. Hence we did not obtain evidence that the lateral force acts in the geometric centre of the patch. 5. Conclusions We have used the IMMa tyre test bench to obtain the elastic properties of the tyre and the friction coefficient so that they can be used in a known analytical tyre model. We have checked the experimental data with the longitudinal, lateral and autoaligning torque characteristics obtained by means of the analytical tyre model expressions. We have also proposed a new expression to obtain the vertical load with the radial stiffness, radial deformation and initial lineal radial deformation. We have used the method of total internal reflection of light to obtain the correct contact area and we have used the patch contact length to verify the camber stiffness expression described by the Gim Nikravesh tyre model. A good level of agreement between the test data and the analytical expressions employed in the tyre model has been obtained. Hence, the IMMa tyre test bench has proved to be a proper facility for tyre tests in the steady state although the comparison between experimental and analytical autoaligning torque show a higher level of disagreement for the reasons mentioned by Gim and Nikravesh with respect to their tyre model. Also the test data on the autoaligning torque versus the camber angle have been given. The existence of a non-zero autoaligning torque suggests that the pneumatic trail is not zero

252 A. Ortiz et al. for a pure camber angle. Hence we did not obtain evidence that the lateral force acts in the geometric centre of the patch. Acknowledgements We thank the Instituto Universitario de Investigación del Automóvil, Madrid, for lending us the quartz-based instrument used to obtain the test results. References [1] Cabrera, J.A., Ortiz, A., Simón, A. and García, F., 2003, A versatile flat track tyre testing machine. Vehicle System Dynamics, 40, 271 284. [2] Gim, G. and Nikravesh, P.E., 1990, An analytical model of pneumatic tyres for vehicle dynamic simulations. Part 1: pure slips. International Journal of Vehicle Design, 11, 589 618. [3] Gim, G. and Nikravesh, P.E., 1990, An analytical model of pneumatic tyres for vehicle dynamic simulations. Part 2: comprehensive slips. International Journal of Vehicle Design, 12, 19 39. [4] Pérez, A., Prado, M. and Simón, A., 2000, Sistema de alta resolución para la medida de presiones de contacto de neumáticos comerciales. Anales de Ingeniería Mecánica, 17, 367 376. [5] Gim, G. and Nikravesh, P.E., 1991, An analytical model of pneumatic tyres for vehicle dynamic simulations. Part 3: validation against experimental data. International Journal of Vehicle Design, 12, 217 228. [6] Dugoff, H., Fancher, P.S. and Segel, L., 1970, An analysis of tyre traction properties and their influence on vehicle dynamic performance. SAE Paper 700377, Society of Automotive Engineers, New York.