Lu et al. Chin. J. Mech. Eng. (2020) 33:85 https://doi.org/10.1186/s10033-020-00513-8 Chinese Journal of Mechanical Engineering

ORIGINAL ARTICLE Open Access Dynamics Modeling Method Based on Rapid Test Method Dang Lu1, Lei Lu1, Haidong Wu1*, Wei Wang2 and Manyi Lv2

Abstract Combined with the tire dynamics theoretical model, a rapid test method to obtain tire lateral and longitudinal both steady-state and transient characteristics only based on the tire quasi-steady-state test results is proposed. For steady state data extraction, the test time of the rapid test method is half that of the conventional test method. For transient tire characteristics the rapid test method omits the traditional tire test totally. At the mean time the accuracy of the two method is much closed. The rapid test method is explained theoretically and the test process is designed. The key parameters of tire are extracted and the comparison is made between rapid test and traditional test method. The result show that the identifcation accuracy based on the rapid test method is almost equal to the accuracy of the conventional one. Then, the heat generated during the rapid test method and that generated during the conven- tional test are calculated separately. The comparison shows that the heat generated during the rapid test is much smaller than the heat generated during the conventional test process. This benefts to the reduction of tire wear and the consistency of test results. Finally, it can be concluded that the fast test method can efciently, accurately and energy-efciently measure the steady-state and transient characteristics of the tire. Keywords: Tire dynamics modeling, Rapid test method, Tire steady-state and transient characteristics, Identifcation of tire characteristics parameters

1 Introduction method, the combined brake and side test method by Te tire is the only part of the vehicle that is in contact using the sweeping method of the outdoor trailer. Tey with the road surface. Studying the dynamic characteris- found that the mechanical properties of the are tics of the tire is the basis of [1]. Tire quite diferent at diferent loading and speeds. dynamics research is based on tire dynamics testing. Te Janowski [3] studied the test method of tire driving accuracy of the tire dynamics model depends to a large and braking in the winter environment. Te longitu- extent on the data quality of the tire modeling test. Te dinal force of the tire under diferent slip ratios was tire modeling test is infuenced by many factors, such as collected with the method loading diferent longitu- temperature and wear. Tese factors afect the dinal slip rate by the outdoor trailer. However, due to quality of the test data, so the consistency of the tire state the limitations of the test equipment and the uncon- and environmental state during the test must be ensured. trollability of environmental factors at that time, the However, diferent test methods will also make the tire reliability of the data was relatively poor. Sui et al. [4] state consistency diferent. studied and developed the tire mechanical character- Beauregard et al. [2] studied the linear brake test istic test rig. Tire cornering characteristics test and method, the steady-state test method, transient test longitudinal dynamics characteristics test were car- ried out using this test rig with the stepped steady state test methods. Pottinger et al. [5] measured *Correspondence: [email protected] 1 State Key Laboratory of Automotive Simulation and Control, Jilin the cornering characteristics of heavy truck tire dur- University, Changchun 130025, China ing free rolling by trailer test, at the same time, they Full list of author information is available at the end of the article also studied the infuence of tire infation pressure and

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tire surface properties on the mechanical properties steady-state characteristics and friction work are com- of tires. Liu et al. [6] analyzed the test methods of the pared with the results obtained by the conventional tests. non-steady state side slip characteristics of tires, and they also designed the steering and pure lateral motion test using the tire test rig. Sommerfeld et al. [7] pointed 2 Tire Longitudinal Transient Model out that tire constraints, unevenness and other fac- Te longitudinal characteristics of the tire include lon- tors have infuence on the vertical load control accu- gitudinal steady state characteristics and longitudinal racy of tire dynamics testing. By improving the design transient characteristics. In this paper, based on the lon- of the tire force and moments sensor, the infuence of gitudinal quasi-steady-state test results, the longitudinal the above factors on the test accuracy of the tire sen- relaxation length which represent the transient char- sor is reduced. Sui et al. [8] built a virtual tire testing acteristics of the tire is extracted. Te transient charac- platform through a fnite element method. Some spe- teristics extraction method is based on the longitudinal cial tire testing which can’t completed in conventional transient model of the tire. tire testing platform can be completed in the virtual tire testing platform. Tuononen et al. [9] used trailer equip- ment to measure the lateral stifness on ice and snow 2.1 Tire Longitudinal Transient Modeling by a new method. Li et al. [10] obtained the lateral Te longitudinal transient characteristics of the tire is relaxation length of the tire through the t method of the mainly caused by the elasticity of the tire carcass, which tire side angle step test. Te accuracy of the model is is refected by the hysteresis of the longitudinal force of verifed by comparing the relaxation lengths obtained the tire relative to the driving torque of the tire [12–16]. by the rate of stifness method and the slip angle step Te parameter refecting longitudinal transient charac- method. Qiu et al. [11] studied the diferent measure- teristics of the tire is the longitudinal relaxation length. ment methods to obtain tire relaxation length at difer- Te longitudinal transient model of the tire is shown in ent vertical loads. Figure 1 [14, 17, 18], where u is the longitudinal defor- In summary, steady-state modeling in tire dynamics mation of the carcass, u˙ is the longitudinal deformation studies is generally based on a quasi-steady-state test rate of the carcass, K is the longitudinal stifness of the method or the stepped steady state test method. Te carcass, C is the longitudinal damping of the tread, and Vx quasi-steady-state data of the tire contains not only the is the longitudinal speed of the tire, Vsx is the longitudinal steady-state characteristics but also the transient charac- slip speed of the tire. Ω is the tire rolling rotational veloc- teristics. Although the quasi-steady-state characteristic ity, and Re is the efective rolling radius of the tire. of the tire is approximately equivalent to the steady-state Assuming that the longitudinal force of the tire is Fx at characteristic, the test error will be inevitably introduced this time, based on the principle of force balance, Eq. (1) by directly using quasi-steady state data for steady-state can be obtained: modeling. Using the stepped steady state test method, F =−C · (V −˙u) = C · V , the tread wear is severe before and after tire testing, and x x sx F =−K · u. (1) the result data consistency is poor. Tire dynamics mod- x eling methods based on quasi-steady-state data to build Taking the derivative of second formula in Eq. (1) with tire transient model and extract steady state data are respect to time: rarely reported. Terefore, the rapid test method which is based on the quasi-steady-state test data of tires to extract steady-state data and transient parameters is a key technical research work to solve the modeling of tire dynamics test. Tis paper frst introduces the longitudinal transient model of the tire and the method of extracting the lon- gitudinal transient parameters from the quasi-steady state test results, and then introduces the lateral transient model without considering the turn-slip properties and the method of extracting the lateral transient parameters. Based on the above theory and method, a tire rapid quasi- steady-state test is designed, and the parameters of the transient characteristics and steady-state characteristics are extracted. Finally, the tire transient characteristics, Figure 1 Schematic diagram of longitudinal transient model Lu et al. Chin. J. Mech. Eng. (2020) 33:85 Page 3 of 10

F˙ the longitudinal force and the derivative of the longitu- u˙ =− x . K (2) dinal force versus time when the longitudinal slip ratio is zero, and the rolling speed of the tire. In the tire dynamics model, the longitudinal force and the longitudinal slip ratio of the tire are linearly changed 2.2 Key Parameters Extraction Method in a small slip state, so there is the following equation: When the tire is sinusoidally moved within a small lon- gitudinal slip ratio, the relationship between the longi-

Vsx Kx tudinal force of the tire and the longitudinal slip ratio is Fx = Sx · Kx =− · Kx =− · Vsx, approximate as shown in Figure 2. Due to the transient Vr Vr (3) characteristic of the tire, the longitudinal force of the tire where Sx is the longitudinal slip ratio of the tire; Kx is the forms a circular curve with the slip ratio. S = 0 F 01 F 02 longitudinal slip stifness of the tire; Vr is the tire rolling When x . x and x are extracted from the speed. above fgure. And the longitudinal force with the time According to the tire steady characteristic at small slip derivation of zero point can be obtained from these two F˙ 01 F˙ 02 S = 0 ratio, the expression of the tread damping coefcient is: points: x and x at x . K |Fx01|+|Fx02| x F 0 = , C =− . x 2 (11) Vr (4)

Substituting Eq. (2) into Eq. (1), Eqs. (5) and (6) can be F˙x01 + F˙x02 F˙ 0 = . obtained: x 2 (12) F˙x F =−C · (V + ), Te longitudinal relaxation length of the tire can be x sx K (5) solved by bringing the values obtained according to Eqs. (11) and (12) into Eq. (10). F˙ · C −C · V = F + x . sx x K (6) 3 Tire Lateral Transient Model without Considering the Impact of Turn‑slip Bring Eq. (4) into Eq. (6), Eq. (7) can be obtained: Lateral dynamics properties of the tire include lateral Vsx Kx Kx transient characteristics and lateral steady state charac- · Kx = Fx − F˙x · = Fx − F˙x · · Vr. Vr Vr · K K (7) teristics. In this paper, the quasi-steady-state data of the tire is obtained based on the tire slip angle sweep test, According to the defnition of the longitudinal relaxa- the lateral relaxation length which represent the tire tion length lx , Eq. (8) can be obtained: lateral transient characteristic is extracted from quasi- K steady-state data. Te lateral transient characteristics l = x . x K (8)

Bring Eq. (8) into Eq. (7), Eq. (9) can be obtained: l ˙ x . −Sx · Kx = Fx − Fx · (9) Vr

When Sx = 0 , the longitudinal relaxation length expression of the tire can be obtained by transforming Eq. (9):

F 0 = x · , lx0 Vr (10) F˙x0 ˙ where Fx0 is the longitudinal force at Sx = 0 ; Fx0 is the derivative of the longitudinal force versus time at Sx = 0. Te longitudinal relaxation length of the tire can be solved by Eq. (10). It is necessary to obtain the value of Figure 2 Schematic diagram of longitudinal transient model result Lu et al. Chin. J. Mech. Eng. (2020) 33:85 Page 4 of 10

parameters are extracted based on the tire lateral tran- Yt (X, x) = Yt (X − (a − x), a), sient model [16, 18, 19]. Xt (X, x) = Xt (X − (a − x), a). (14)

3.1 Tire Lateral Transient Model without Considering By making a diference between the ordinate of the the Impact of Turn‑slip point on the tread and the ordinate of the point on Te tire lateral transient model is shown in Figure 3 [20, the carcass, the lateral deformation of the tread can be 21]. obtained, and the following equation is obtained:

First, the following assumptions are proposed: �y(X, x) = Yt (X, x) − Yc(X, x), 1) Te carcass has only lateral translational  �y(X, x) = Y (X − (a − x)) − Y (X) + yc0(X − (a − x)), deformation;  − yc0(X) + a · ψ(X − (a − x)) − x · ψ(X), 2) Te tire only makes a small movement, and there  (15) is no slide between the tire and the road surface in the ψ entire footprint; where is the steering angle. 3) ignore the infuence of the tire width. Te uniform lateral force expression of the tire in the First the mathematical relationship between the posi- footprint is: tion of the tread points and the corresponding position of the carcass in the footprint of the tire is analyzed. d = d , Fy kty y x (16) Pt (Xt , Yt ) is the point on the tread of the tire in the foot- , print, and Pc(Xc Yc) is the corresponding point on the where kty is the bristles lateral distributed stifness. tire carcass in the footprint. Te point on the carcass Te expression of the lateral force of the tire in the and the point on the tread can be represented in global entire grounding footprint can be obtained by combin- coordinates XOY. Since these points are also in the local ing Eqs. (15) and (16): coordinate system on the tire tread, the tread point y-axis a Yt = Yt (X, x) , d . coordinate can be expressed as . Te y-axis Fy(X) = kty �y(X x) x (17) coordinate of the point on the same carcass can also be −a expressed as Yc = Yc(X, x) . When the tire tread is at the As can be seen from the above Eq. (17), the lateral front end of the grounding footprint, the point Pt on the force Fy(X) of the tire is a function of the distance X . tread and the point Pc on the carcass coincide with each According to the Laplace transform, the Laplace trans- other, and the following relationship exists: formation of the lateral force Fy(X) with respect to the Yt (X, a) = Yc(X, a), travel distance X can be obtained as: Xt (X, a) = Xc(X, a). (13) +∞ −sX F(s) = F(X) · e dX. (18) After entering the footprint, since the tire tread does 0 not slip relative to the road surface, the point coordinate Bringing Eq. (17) into Eq. (18), the Laplace transform position on the tread can be converted to the coordinate expression of the lateral force can be obtained: position just entering the grounding footprint, the fol- a lowing expression exists: , d . Fy(s) = kty �y(s x) x (19) −a

It is known from the static mechanical properties of the tire:

Fy(X) Fy(X) yc(X) = = . Kcy 2akcy (20)

Based on Laplace transformation Eq. (21) can be obtained:

Fy(s) Fy(s) yc(s) = = . Kcy 2akcy (21)

By taking a Laplace transformation on the second Figure 3 Schematic diagram of lateral transient modelling formula in Eq. (15), the following equation can be obtained: Lu et al. Chin. J. Mech. Eng. (2020) 33:85 Page 5 of 10

−(a−x)s l V �y(s, x) = (a − x) · ψ(s) − aQ(s) · (1 − e ). y ˙ y . Fy + Fy = Ky · ϕ − (32) (22) Vr Vr  Among them When the steering angle is 0, according to the corre- Y (s) yc(s) spondence of tire kinematics, the following equation can be Q(s) = ψ(s) + + . a a (23) obtained.

A zero-order E function is defned: Vy a tan , 1 (α) = (33) 1 −(a−x)s d . Vr E(s) = ( − e ) x (24) 2a −a where α is the side slip angle. Based on Eq. (22), bring Eq. (20) into Eq. (16), Eq. So Eq. (29) becomes: (25) can be obtained. Fy0 2 0 =− · V . ( ) = 2 (ψ( ) − ( ) · ( )). ly ˙ r Fy s a kty s Q s E s (25) Fy0 (34)

Based on Eq. (23) and Eq. (25), Eq. (26) can be Te lateral relaxation length of the tire can be solved by obtained: Eq. (34). It is necessary to obtain the value of the lateral 2 force and the derivative of the lateral force versus time Fy(s) = 2a kty − ε0E(s) · Fy(s), (26) when the steering angle is zero, and the rolling speed of the where ε0 (the translation feature ratio) is the ratio of tread tire. translational distribution stifness to carcass translational 3.2 Lateral Feature Parameters Extraction Method distribution stifness, and When the steering angle move sinusoidally in a small range, k = ty . the approximate relationship between the lateral force and ε0 (27) kcy the steering angle of the tire is as shown in Figure 4. Due to the transient characteristic of the tire, the lateral force of Ignoring the high-order terms the one-order approxi- the tire forms a circular curve with the steering angle. mation of Taylor expansion of zero-order E function When Sx = 0 . Fy01 and Fy02 are extracted from the above can be obtained: fgure. And the lateral force and the time derivation of zero ˙ ˙ . point can be obtained from these two points: Fy01 and Fy02 E(s) ≈ as (28) at Sx = 0: According to the defnition of lateral relaxation Fy01 + Fy02 length ly: F 0 = , y 2 (35) 2 Ky 2 · a · kty kty l = = = a = a · ε0. y K 2 · a · k k (29) cy cy cy F˙y01 + F˙y02 F˙ 0 = . y 2 (36) Te expression of the lateral force of the tire can be obtained based on Eqs. (25), (26) and (28):

dFy dY dψ F +l · = K · ψ − − a · . y y dX y dX dX  (30)

Eq. (30) refects the relationship between the lateral force of the tire and the lateral input in the domain of rolling distance. Te relationship converted to the time domain can be obtained as follows:

ly Vy ϕ˙ Fy + F˙y = Ky ϕ − − a . Vr Vr Vr  (31)

Since the infuence of the turn-slip properties of the tire is neglected, Eq. (31) becomes: Figure 4 Schematic diagram of lateral transient model result Lu et al. Chin. J. Mech. Eng. (2020) 33:85 Page 6 of 10

Table 1 Test conditions of longitudinal quasi-steady state slip test Vertical load Side Camber Longitudinal Loading ratio (N) slip angle slip ratio of longitudinal 1 angle slip ratio (s­ − )

2390 0 0 0.3~0.3 0.1 − 4791 0 0 0.3~0.3 0.1 − 7193 0 0 0.3~0.3 0.1 −

4 Figure 5 MTS CT III tire six component test rig x 10 1

0.5 Te lateral relaxation length of the tire can be solved by bringing the values obtained according to Eqs. (35) and ) (36) into Eq. (34). (N 0 Fx

4 Quasi‑Steady State Rapid Test Design -0.5 Fz = 2390N Fz = 4791N Te longitudinal relaxation length and the lateral Fz = 7193N relaxation length of the tire are calculated based on -1 -0.4 -0.2 0 0.2 0.4 the extracted parameters. Due to the relative motion Sx (-) between the tire and the road during the test, the tire Figure 6 Result of longitudinal quasi-steady-state test tread will inevitably wear to a certain extent. In order to reduce wear, the relative sliding distance between the tire and the road surface should be minimized. Since the speed during the test is constant, the test time should be Table 2 Transient parameters of tire reduced. Terefore, the steering angle loading rate of the N m/s ˙ m Vertical load (N) Fx0 ( ) Vr ( ) Fx lx0 ( ) tire should be improved during the test of the tire lateral mechanical properties. Te longitudinal slip loading rate 2390 144.0 16.67 9333.9 0.353 of the tire should be increased during the test of the lon- 4791 724.7 16.67 21470 0.700 gitudinal mechanical properties of the tire. Te MTS test 7193 1956.1 16.67 28233 1.111 rig is shown in Figure 5, that is one of the best test rig with high stability and accuracy, which can minimize the efect of the test rig on the test results. 4.2 Lateral Rapid Test Method According to Eq. (34), the lateral relaxation length of 4.1 Longitudinal Rapid Test Method the tire side slip is solved, and it can be obtained from According to Eq. (11), the longitudinal relaxation length the value of the lateral force and the derivative of the of the tire is solved, and it can be obtained from the value lateral force versus time when the steering angle is zero, of the longitudinal force and the derivative of the longitu- and the rolling speed of the tire. In order to reduce the dinal force versus time when the longitudinal slip ratio is wear of the tire during the test, a larger steering angle zero, and the rolling speed of the tire. loading rate is employed. Te tire longitudinal quasi-steady-state slip rate sweep Te tire lateral quasi-steady-state steering angle test is carried out using a tire of the type HANKOOK sweep test is carried out using a tire of the type LIN- 195/ 55R16. Te specifc test conditions are shown in GLONG 205/ 55 R16. Te specifc test conditions are Table 1. shown in Table 3. Te longitudinal quasi-steady-state slip rate sweep test Te results of the lateral quasi-steady-state steering result is shown in Figure 6. angle sweep test are shown in Figure 7. Based on the test result and proposed method, the Te parameters refecting the transient characteris- parameters characterizing the transient properties of the tics of the tire are extracted from the above-mentioned tire under each load are respectively obtained. Te spe- steering angle sweep test results, as shown in Table 4. cifc parameters are shown in Table 2. Lu et al. Chin. J. Mech. Eng. (2020) 33:85 Page 7 of 10

Table 3 Test conditions for steering angle sweep test side slip angle nonlinear region, and the above three parts Vertical load Longitudinal Camber Steering Side slip of data are extracted separately, and polynomial ftting is (N) slip ratio angle angle (°) angle cycle performed respectively, and the obtained ftting data rep- rate ((°)/s) resents the steady state data of the tire. 2041 0 0 16~16 4 Te tires of the type ATLAS 235 55R18 are used for − 4070 0 0 16~16 4 the conventional steering angle sweep test and the rapid − 6100 0 0 16~16 4 test method of steering angle sweep test. Te specifc − test contents are shown in Table 5. Te test comparison results under diferent loads and the comparison of the steady-state data are extracted. As shown in Figures 8, 9 6000 Fz = 2030N and 10. Te test time of the rapid test method is half that 4000 Fz = 4058N of the conventional test method. Fz = 6087N Te error calculation formula is as shown in Eq. (37): 2000 ) (N

0 2 − Fy ytrad ynew dev = , -2000   2  (37) ytrad -4000    y -6000 where trad is the steady-state extraction result of con- -15 -10 -5 0 5 10 15 ventional test method; ynew is the steady-state extraction SA (deg) result of the rapid test method. Figure 7 Result of lateral quasi-steady-state test Te error calculation results under each vertical load are shown in Table 6. It can be seen from the above results that the steady Table 4 Parameters for steering angle sweep test state test results extracted by the rapid test method are very close to the results extracted by the conventional Vertical load (N) F 0 (N) V (m/s) ˙ l 0 (m) y r Fy0 y test method, and the error is under 0.5%. 2041 160.1 16.67 6796 0.3737 4070 452.5 16.67 11735 0.6428 5 Comparison Results of Rapid Test Method 6102 590.9 16.67 13732 0.7173 and Conventional Test Method For the longitudinal relaxation length, the stifness ratio method is generally used to solve the problem. For the Table 5 Parameters for steering angle sweep test Test methods Vertical load (N) Longitudinal slip Camber angle Side slip angle (°) Side slip angle ratio cycle rate ((°)/s)

Conventional test method 3100 0 0 16~16 2 6100 − 9200 Rapid test method 3100 0 0 16~16 4 6100 − 9200

4.3 Steady State Data Extraction Method lateral relaxation length, the slip angle step method is According to the quasi-steady-state test data, the piece- generally used. In this paper, the relevant tests are carried wise function ftting method is used to extract the steady- out respectively, and the longitudinal relaxation length state data from the quasi-steady-state data. Te specifc and the lateral relaxation length obtained by the conven- piecewise function ftting method which is based on the tional method and compared with the results of the rapid steering angle sweep test data is an example. Te quasi- test method. steady-state data is divided into three parts, the frst part Te friction work generated during the test of the con- is the linear area, and the second part is the positive side ventional test method and the rapid test method is calcu- slip angle nonlinear area. Te third step is the negative lated and compared. Lu et al. Chin. J. Mech. Eng. (2020) 33:85 Page 8 of 10

4000 4000 SArate = 2deg·s-1 SArate = 4deg·s-1 2000 2000 ) ) 0 0 (N (N Fy Fy -2000 -2000 SArate = 2deg·s-1 SArate = 4deg·s-1 -4000 -4000 -20 -10 0 10 20 -20 -10 0 10 20 SA (deg) SA (deg) Figure 8 Lateral force test results and steady-state data extraction results under 3100 N

SArate = 2deg·s-1 SArate = 2deg·s-1 5000 SArate = 4deg·s-1 5000 SArate = 4deg·s-1 ) ) (N 0 (N 0 Fy Fy

-5000 -5000

-20 -10 0 10 20 -20 -10 0 10 20 SA (deg) SA (deg) Figure 9 Lateral force test results and steady-state data extraction results under 6100 N

4 4 x 10 x 10 1 1 -1 SArate = 2deg·s-1 SArate = 2deg·s -1 SArate = 4deg·s-1 SArate = 4deg·s 0.5 0.5 ) ) (N (N 0 0 Fy Fy -0.5 -0.5

-1 -1 -20 -10 0 10 20 -20 -10 0 10 20 SA (deg) SA (deg) Figure 10 Lateral force test results and steady-state data extraction results under 9200 N

5.1 Longitudinal Transient Test Method Comparison Table 6 Result error of rapid test method relative Te conventional longitudinal transient test method of to conventional test method in steady state data tires is the stifness ratio method [22, 23]. Te longitu- extraction dinal stifness of the tire is obtained by the static tire Vertical load F (N) 3100 6200 9200 longitudinal stifness test. Te longitudinal slip stifness z of the tire is obtained by the longitudinal slip steady Error (%) 0.0318 0.073 0.126 state test. Lu et al. Chin. J. Mech. Eng. (2020) 33:85 Page 9 of 10

5000 2000

) 4000 1500 ) (N e (N rc 3000 1000 Fo

rce Fy nal Fo 2000 500 udi ra l it te La ng Fz = 2411N Fz=2050N 1000 0 Lo Fz = 4822N Fz=4077N Fz = 7232N Fz=6100N 0 -500 0 5 10 15 20 25 30 0 5 10 15 20 25 30 Longitudinal Deformation (mm) Rolling Distance D (m) Figure 11 Result of longitudinal translational stifness test Figure 12 Result of steering angle step test

Table 7 Result of stifness ratio method Table 8 Result of steering angle step test

− − Vertical load Longitudinal Longitudinal lxc(m) lx0 lxc · 100% Vertical load (N) lyc(m) ly0 lyc · 100% (N) translation slip stifness lx0 ly0 stifness 2041 0.3779 1.11% 2411 79541.4 144800 0.342 3.216 4070 0.6492 0.99% − 4822 176749 252500 0.700 0 6102 0.7282 1.49% 7232 275576 251300 1.097 1.276 −

Te static test results of the tire are shown in Figure 11, Table 9 Tire lateral friction work calculation comparison and the longitudinal translational stifness of the tire is Vertical load (N) Conventional test (J) Rapid test (J) calculated. Ten, the tire longitudinal slip stifness is extracted 2041 1339.9 839.8 from the tire quasi-steady-state data, which is used to 4070 2500.6 1588.3 solve the longitudinal relaxation length of the tire. Te 6102 3475.5 2181.2 comparison between the longitudinal relaxation length lxc solved by the stifness ratio method and lx0 solved by longitudinal rapid test method is shown in Table 7. It can be seen from the above test results that error tire obtained by the side slip angle step measurement between the longitudinal relaxation length of the tire method and the rapid test method proposed in this obtained by the stifness method and the fast test method paper is less than 2%. proposed in this paper are less than 4%, which meets the requirements of dynamic modeling. 5.3 Friction Work Comparison 5.2 Lateral Transient Test Method Comparison Tire friction work is done by the tangential force Te conventional tire lateral transient modeling test is between the tire and the road surface on the tire tread based on the slip angle step test method [24–26]. Te [27, 28], the main energy source of the tire tread heat specifc test method is as follows: First, the tire slip angle generation and the cause of tire tread wear. is set to 1° under 0 N, and then the tire load is adjusted to Te calculation results of friction work are shown in the test value, the data should be measured until the force Table 9. is not change, the test results are shown in Figure 12. From the calculation results of the friction work Ten, the comparison between the lateral relaxation under the above diferent loads, the friction work gen- length ly solved by lateral rapid test method and the lat- erated in the conventional test process is greater than eral relaxation length lyc of the tire solved by side slip that generated in the test process of the rapid test angle step method are shown in Table 8. method. It can be seen that the rapid test method has It can be seen from the above test results that the less heat generation, less tread wear, higher data con- deviation between the lateral relaxation length of the sistency, and reliable data quality. Lu et al. Chin. J. Mech. Eng. (2020) 33:85 Page 10 of 10

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