DYNAMIC FRICTION IN THE BRAKING, TIRE

International Conference on Traffic and Transport Engineering - Belgrade, November 27-28, 2014

DYNAMIC FRICTION IN THE BRAKING, TIRE – ROAD CONTACT - Ivan Ivković, Željko Janjoš, Gradimir Danon, Srećko Žeželj Ivan Ivković1, Željko Janjoš2, Gradimir Danon3, Srećko Žeželj4 1,4

University of Belgrade, Faculty of Transport and Traffic Engineering, Belgrade, Serbia SR Center City Net Ltd., Belgrade, Serbia 3 University of Belgrade, Faculty of Forestry, Belgrade, Serbia 2

Abstract: The differential equation of the tire-road contact sliding speed is presented in the paper. It is based on the equations of: normal pressure distribution in contact, available friction and tangential stress within the tire-road contact. Solving this equation makes it possible to calculate the sliding speed and available friction within the contact path. Some numerical examples are presented in the paper. Indoor testing of radial tires was performed and the results of the measurement were used for these calculation s. The formulated equation may be incorporated in the friction model of tires, for the purpose of making a more exact prediction ab out the friction behaviour of the braking tire on the roadway of known characteristics. Keywords: tire-road contact, adhesion, braking, tire sliding.

Introduction The forces required to support and control a vehicle all appear in the tire contact area. In contact, an element of a tire tread is acted upon by a force vector which can be expressed in terms of two components. One is perpendicular to the contact surface and one tangential to the contact surface. The transfer of forces in the tire and road contact is limited by available friction in contact, and it is inevitably followed by a slip. The slip phenomenon is manifested as the vector difference of the translator tire speed and longitudinal speed of the tread part in contact. The compensation of this difference is possible by tread deformation or by sliding of the tread part in the plane of the contact path. The contact path of a braking tire is separated in two regions. In the first, named “adherence” region, the dominant is a deformation component of the slip and there is no move between the tire and road surface. Available friction (µo), in this region, is nearly constant, because the sliding of the tread pattern against road surface is negligible. Sliding begins when the value of shear stress, which is proportional to the strain of a small piece of tire tread in contact (x=Kx·Sslip), reaches available unit friction in contact (µo ·p). From this moment on, sliding begins between a piece of the tire tread and surface. Sliding speed value duly depends on exterior efforts and contact pressure distribution, and it has a great effect on available friction. By formulating the differential equation of sliding speed and solving it, available and obtained friction within the contact path may be calculated. The equation of sliding speed, available and obtained friction, which is presented, may be incorporated as useful in the tire friction models. This paper outlines the equation of the sliding speed in the tire and road contact and calculations of the sliding speed distribution that are made for the straight line motion of the tire under braking effort. The equation for the sliding speed in contact is formulated on the basis of corresponding assumptions of simplification, as the outcome of the investigation of the results (Danon, 1988; Clark, 1981; Pillai and Fielding-Russell, 1986; Janicijević and Danon, 1987; Danon and Markun, n.p.; Zeng-Xin et al., 2001; Pottinger, 2005; Burke and Olatunbosun, 1997; Grigolyuk, 2004). The equations for calculating tire dimensions in the function of air pressure, radial deformation, pressure in contact and dimensions of contact are stated in the paper. Dimensions of an inflated pneumatic tire On the basis of research results presented in (Clark, 1981; Pillai and Fielding-Russell, 1986), it is assumed that the circumference, diameter and width of an unloaded tire are not significantly modified regardless of compressed air it is filled with. The assumption is tested by means of corresponding laboratory measurements (Danon, 1988). The measurements are carried out on four 145 R 13 Best tires and mean results are shown in Table 1. Table 1 145 R 13 Best radial tire dimensions measurements (Danon, 1988)* Tire pressure [bar] Tire dimensions Unit of Measure 0.5 1.3 1.5

1.9

2.1

2.3

Tire free diameter

mm

565.60

565.45

565.52

565.52

565.52

565.60

565.66

Total tire width

mm

144.3

145.5

145.5

145.6

145.6

145.7

145.9

* Tire was fitted around a 4Jx13 wheel rim

1

1.7

Corresponding author: [email protected]

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International Conference on Traffic and Transport Engineering - Belgrade, November 27-28, 2014 The results of the measurement have corroborated the stated assumption. A possible explanation for this kind of tire behaviour is belt, which all radial tires have and which is practically inelastic, i.e. its circumference, regardless of the internal air pressure variation, does not alter. Dimensions of tire contact The vertical load acting on the tire makes it deformed and changed in shape. The deformation and change in shape are mostly evident in contact with the road surface. The contact of a stationary wheel with a rigid road surface is most frequently of a rounded rectangular shape (Ivanov, 2010). The radial deformation, area and dimensions of contact of the stationary wheel and the surface greatly depend on air pressure in the tire and vertical load. Here the following rule applies-the larger the load on the tire, the larger the contact patch, and the larger the inflation pressure, the smaller the contact patch (Taylor et al., 2010). In Fig. 1, the results of the measurement of change in radial deformation of the tire are given in the function of the vertical load for a 145R13 Best tire (Danon, 1988). Different vertical forces acted on the examined tires and in addition radial deformation, area and contact dimensions were measured. 35

Tire radial deflection [mm]

30

y = 0,0072x + 2,0442 R2 = 0,9952

25

20

0.5 bar 1.3 bar

15

1.5 bar 1.7 bar

10

1.9 bar

5

2.1 bar 2.3 bar Linear (1.9 bar)

0 0

500

1000

1500

2000

2500

3000

3500

4000

4500

Vertical force [N]

Fig. 1. Influence of air pressure and normal load on radial deformation of the 145R13 Best tire It is established that the dependence between vertical force and radial deformation, for all examined pressure pumping, is approximately linear. In the mentioned measurements, a linear trend line illustrates that the radial deformation dR= f(Fz)p=1,9 bar has consistently risen over the observed load range. Notice that the R2 value is 0.9952 (see Fig. 1), which is a good fit of the line to the data. The radial tire deflection dR consists of two components. One of them is tread deflection and the other is carcass deflection, Eq. (1): dR  dR t  dR c

(1)

The value of the tread deflection (dRt) depends on the vertical load (Z) and total tire deflection (dR). The tire changes its shape and volume on deformation. Deformation energy is partially consumed by the deformation of the tire structure and partially by interior air compressing. The carcass deformation (dRc) is proportional to the vertical load and tire inflation pressure. In Fig. 2, the schematic representation of the tire road contact is given, where the tire tread, carcass and air pressure are symbolically represented by springs.

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Fig. 2. Scheme of the tire tread and tire carcass behaviour undergoing a vertical force; Springs of equivalent stiffness: 1-tire carcass; 2-tire air pressure; 3-tire tread. If it is assumed that we are really dealing with springs, it is possible, with an appropriate simplification, instead of a bunch of curves shown in Fig. 1, to formulate the universal dependence of the vertical load and radial deformation for any tire, Eq. (2) (Janićijević and Danon, 1987; Danon and Markun, n.p., Grečenko, 1995; Knoroz, 1963; Biderman, 1963): dR 

C2  FZ  2   pa  p o 

2

 C2  FZ  ,    C1  FZ  2   pa  p o  

(2)

where: FZ is the vertical load; pa is the tire air pressure; po is the measure of structural rigidity of the tire carcass; According to (Danon and Markun, n.p) po is usually between 0 and 0.3 bar. Higher values are characteristic of truck and off-road tires. For radial tires for passenger cars po can be neglected; the parameter C1 depends on the tire radius, tread section radius, applied material characteristics, net/gross tread pattern ratio and its thickness; the parameter C 2 depends on the tire radius, tread section radius and tire construction; the C1 and C2 parameters are determined experimentally.

Fig. 3. Universal characteristic of tires The calculated constants C1 and C2 for examined tires were: Tire 145R13 Best C1= 0.027, C2= 0.011, Tire 155R13 Best C1=.0.0495, C2= 0.0119. At load variation, apart from radial deformation, the length of contact and the shape of contact are partially changed while the width of contact (wc) remains approximately constant. The results of the measurement for 145 R 13 Best tires are shown in Table 2.

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International Conference on Traffic and Transport Engineering - Belgrade, November 27-28, 2014 Table 2 Effect of air pressure and vertical load on the contact area and size of 145 R 13 Best tire deflection Pressure Vertical load Tire dimensions [mm] bar N Width (wc) Length (lc) Deflection (dR) 1.3 2,400 92 166 24,74 1.5 2,700 92 167 25,52 1.7 3,000 92 168 26,21 1.9 3,300 92 170 25,8 2.1 3,600 93 175 26,32 2.3 3,900 94 174 27,95 The same data but for the 155 R 13 tire are given in Table 3. Table 3 Effect of air pressure and vertical load on the dimensions of contact and size of 155 R 13 Best tire deflection Pressure Vertical load Tire dimensions [mm] bar N Width (wc) Length (lc) Deflection (dR) 1.4 3,300 100 207 32.8 1.7 3,000 99 171 26.6 2.1 3,600 100 175 27.0 2.4 3,300 97 150 23.5 1.9 2,300 96 144 20.1 1.9 2,800 99 150 23.5 1.9 3,300 98 170 26.5 1.9 3,800 99 184 30.5 1.9 4,300 102 207 33.2 1.9 4,800 102 210 36.5 The data shown in Table 3 indicate that the variation in the contact length for a large range of pressures and load is marginal. This can be seen in Fig. 4 where the variation in the contact dimensions of the 155R13 Best radial tire (inflated at a pressure of 1.9 bar and subjected to different vertical loads) is given.

[mm]

220 200 180

y = 0,0294x + 73,029 R2 = 0,9706 Tire width

160

Contact length 140

Linear (Contact length)

120 100 80 1800

2300

2800

3300

3800

4300

4800

5300

Vertical load [N]

Fig. 4. Influence of vertical load on 155 R 13 Best tire footprint dimensions tire pressure was 1.9 bar The variation in the contact length is approximately linear and directly proportional to the variation in the vertical load. This holds if air pressure does not change. The high R2 value which amounts to 0.9706 confirms it. The equation which suits the data (the contact length and the respective tire deflection) is Eq. (3):

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International Conference on Traffic and Transport Engineering - Belgrade, November 27-28, 2014 lc  2 

k  R o   R o  dR  2

2

(3)

The coefficient k depends on tire deflection. The increase of the radial tire deflection entails the decrease of the coefficient k. The linear dependence is presumed Eq. (4):

k  A  B  dR

(4)

The coefficients of the Eq. (4) have been determined for all values given in Table 3 by means of the method of least squares Eq. (5) (Danon and Markun, n.p):

k  0,9766  0,00078  dR

(5)

The measured and calculated values of the contact length as well as the determined difference are given abreast in Table 4. Table 4 Calculated versus measured static footprint lengths of the 155 R 13 Best tire Pressure Tire length [mm]

The difference

bar

Measured

Calculated

%

1.4

207

139

3.59%

1.7

171

158

-5.03%

2.1

175

158

-5.03%

2.4

150

172

-1.28%

1.9

144

173

-0.96%

1.9

150

174

0.29%

1.9

170

190

-3.08%

1.9

184

199

3.92%

1.9

207

200

3.17%

1.9

210

213

-1.28%

The determined differences are between -4% and +5%, which might be considered satisfactory. The influence of the vertical load on the variation in the contact width is irrelevant. According to the data set, shown in Table 3, it has been stated that the contact width increased by 6 mm in length change by 50 %. Therefore, it is safe to assume that the width of contact remains constant regardless of the level of vertical load. During the wheel’s rotation, the variation in radial deflection occurs. In order to determine the effect of the circumferential speed on radial deformation, the corresponding investigations on a tire test drum have been conducted (Danon, 1988). The tests for different inflation pressures and different vertical loads have been carried out. Table 5 shows the measured results for the 155R13 Best tire for air pressure to be equal to 1.9 bar and vertical load of 3,300 N. The values of static measurements are given in the column marked with V = 0. Table 5 Measurements of radial tire deflection of the 155R13 Best tire on test drum. Tire air pressure was 1.9 bar and vertical load 3,300 N Drum speed

km/h

0

40

60

80

100

120

140

Tire deflection

mm

29.5

29.5

28.5

28.5

28.5

28.5

27.5

The variation of radial deformation is very low around 2 mm within the speed range of up to 140 km/h. The tire deflection of the stationary tire (29.5 mm) mounted on the test drum is somewhat higher in relation to the result presented in Table 3 (26.5 mm). The difference is 3 mm and it is the result of the variation in the contact surface. The measurements presented in Table 3 are carried out on a flat surface, and the surface of the test drum is curved. It can be assumed that the variation in radial deformation could have been smaller if the dynamic investigations of tires had been carried out on a flat surface.

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International Conference on Traffic and Transport Engineering - Belgrade, November 27-28, 2014 For calculating the dimensions of contact and speeds in contact it has been accepted that the statistic and dynamic deformation are equal ( dR  dR d ) (Springer and Weisz, 1975). Speeds in tire contact Let us assume that a vertically loaded tire moves rectilinearly at a constant speed (V), i.e. it is rotated at an angular speed (ω). Let us also assume that the tire deformation is limited to the contact area and that it is very small, i.e. negligible outside it. Then we can say that the circumferential speed of a point on the surface of the tire tread not subjected to contact is likewise constant and equal to Vc   R o and that it is approximately equal to the longitudinal speed of the wheel (V). The circumferential speed remains constant until the beginning of contact. From the beginning of contact the situation keeps changing. The longitudinal speed (V1) in contact is not constant but varied, due to radial and tangential deformations of the tire tread, along the contact (Uljanov, 1982). The deformation speed (V2) and the circumferential speed (V3) are components of the speed in the tire and road contact, Eqs. (6), (7) and (8):

V1 

 h cos 2 ()

(6)

V2 

 h  sin() cos 2 ()

(7)

V3 

 h cos()

(8)

where: h is the distance between the wheel axle and road surface of the rotating tire, Eq. (9): h  R o  dR d

(9)

Where: Ro is the free radius of the wheel and dRd is the tire radial deformation, and it is given in the equation stated in (Janjićijević and Danon, 1987):

Fig. 5. Tire and road contact With a braking wheel the situation is somewhat different. The slipping speed of the tire and road contact, in the case of straight line braking, can be expressed as an ordinary difference of the translator speed of the tire (V) and longitudinal speed in contact (V1) , Eq. (10):

Vslip  V  V1

(10)

Slipping speed can be divided in two components:

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 deformation speed (Vdef), which is dominant in the adhesion part of the tire and road contact and  sliding speed (Vslide), which is characteristic of the sliding part of contact, Eq. (11): Vslip    V  Vdef  Vslide ,

(11)

Where the slip coefficient is, Eq. (12)



V  V1 , V

(12)

Since there is a variation in speed (V1) along the trajectory the slip coefficient (λ) is variable even when the longitudinal speed of the wheel is constant (V=const). In the first part of contact (A-C) the slip path is approximately equal to the tangential deformation of the tread part, Eq. (13):  h  a  x   (13) sslip   a  x    atn    atn     , 1  o   h   h    where: a is half of the contact length, x is a coordinate in the contact path and λ0 is the slip coefficient in the centre of contact for x=0. Due to this, the deformation speed can be expressed as the first derivative of (τ) tangential stress (within contact path) per time, Eq. (14): Vdef 

dSslip dt



1 d  K x dt

(14)

In the second part (C-E), after the friction limit has been reached, the tangential deformation of the tread part in contact changes only due to normal pressure and obtained friction. The difference between a growing slip path and decreasing tangential deformation is compensated by sliding of the tread part over pavement (Fig. 6).

Fig. 6. Slip path in the tire-road contact Sliding speed can be expressed in the equation of available friction in the tire and road contact (μ) (Moore, 1975). If the linear dependence between friction coefficient and sliding speed is assumed, the equation for sliding friction is, Eq. (15):    o    Vslide

(15)

where: μ 0 is the “static” coefficient of friction in contact, while Vslide= 0; β is the coefficient of proportionality.

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International Conference on Traffic and Transport Engineering - Belgrade, November 27-28, 2014 A linear function is assumed for two reasons: It is much easier to solve final differential equation and Linear function is not too far from reality, especially for the speeds among 10 and 100 km per hour. The realized coefficient of friction (μ) varied along the contact path in the function of the value of sliding speed (refer to Eq. (15)). Therefore, we can say that obtained friction is equal to Eq. (16):



 px, y 

(16)

where: p(x,y) is the normal contact pressure in the point with coordinates x and y. In our example, we have assumed that the pressure distribution along the contact length of tire is parabolic, and along the contact width uniform, i.e. the dependency applies Eq. (17):

 x2  p  p max  1  2   a 

(17)

Where the maximum value of pressure is calculated as, Eq. (18):

p max 

3  Fz 2  lc  w c

(18)

If by increasing the vertical force (Fz), pressure reaches the limit value (plim), in some segment of contact, the form of distribution is changed into trapezoidal and the contact length is increased. With the tire that rolls, the pressure distribution in contact varies due to hysteresis and becomes asymmetric against vertical axis. Part of energy that the tire absorbs in the load area remains captured within the tire’s structure and it turns into heat. For the assumed friction-Vslide relation Eq. (15) it is easy to solve the equation Eq. (16) as follows, and this equation is valid from x= xa to the end of contact, Eq. (19):

Vslide

  K x t      Vslip  1  e p    

(19)

In other cases, if direct solving is not possible, the Runge-Kutta method can be applied. The calculated variations of slip and sliding speeds and the friction coefficient (μ) are shown in Fig. 7. The shown distributions of velocities within the tire and road contact have been calculated for the input values given in Table 6. Table 6 Input values for contact speed calculations Input values

Symbol

Measure

Value

Inflation pressure

pv

bar

2

Vertical load

Z

N

3000

Radial deformation coefficients

C1 C2

m2/N m-1

0.0447 0.0116

Slip coefficient

λ0

%

15

Longitudinal stiffness

Kx

N/mm2

400

Translator speed

V

m/s

40

Static friction coefficient

μ0

-

1

Friction correction coefficient

β

s/m2

0.005

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International Conference on Traffic and Transport Engineering - Belgrade, November 27-28, 2014 By looking at the Eq. (19) we can note that the magnitude of sliding speed depends primarily on the value of the slip speed and it depends on the slip coefficient (λ) and on the translation speed (V). Fig. 7 shows the variation of the friction coefficient (μ), the slipping and sliding speed in the tire-road contact.

Fig. 7. Speed and friction distribution within the contact The form of the sliding speed is not usual and differs from those usually found in the literature (Clark, 1981; Ivanov, 2010). In the cited references, it is assumed that the longitudinal speed in contact is constant which makes the change of sliding speed in contact to increase monotonously. In our case, we have taken the speed variability (V 1) in contact into consideration. At the beginning of contact, the speed (V1) is as follows, Eq. (20): V1 

h

2

 a 2     1  

for x  a

(20)

where: ρ1 – the shortest distance from point (A) at the beginning of contact to the axis of rotation, usually taken to be equal to Ro. After the beginning of contact this speed decreases and falls onto minimum in the middle of contact, Eq. (21): V1  h  

for x  0

(21)

After completing half of contact (V1) the speed increases and at the end of contact it has nearly the same value as at the beginning of contact. The circumferential speed of the wheel at the beginning of contact is different from the longitudinal speed of the wheel (V). With the braking wheel, the circumferential speed is lower than the longitudinal speed of the wheel. The amount of variation depends on the slip coefficient (λ). From the beginning of contact (point A) the difference between these two speeds is compensated by the tangential deformation of the tire tread. Sliding begins at point (C) and it lasts until the end of contact. Conclusions The presented procedure for the calculation of adherence is simple and has quite a few advantages compared to the procedures which have been applied so far. The first advantage appears at the calculation of the tire radial deformation. Instead of a bunch of curves for the dependence of radial deformation and vertical load, the so-called carpet diagram, we only applied the Eq. (2) for which we only had to know the coefficients C1 and C2. The tests have shown that the deviations, for one dimension of the tire of the same construction, are not considerable (Pottinger, 2005; Taylor et al., 2010). If the data on the inflation pressure and vertical load of the tire are available, it is possible to apply the presented formulas to accurately calculate radial deformation, footprint area and distance between the wheel axle and road surface. By applying these data, it is possible to calculate the rolling radius of a free wheel.

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International Conference on Traffic and Transport Engineering - Belgrade, November 27-28, 2014 This paper assumes the parabolic distribution of normal pressure in contact. The formula can include some other distribution if it is more convenient for the actual conditions. The longitudinal tangential stresses (τ) acting in the plane of contact of the braking wheel were determined by a sum of stresses generated by the rolling of the free wheel (τ0) and additional stresses (τβ) obtained by the application of a braking torque. Using the Eqs. (16) and (17) it is possible to calculate the obtained force in contact and achieved adherence. The formula for the calculation of achieved adherence includes the influence of sliding speed. The equation was formulated for this purpose, for the sliding speed in the tire and road contact. Sliding speed, shown in Fig. 5, includes the assumption that the speed in contact (V1) is variable. The shown variation differs from those shown in (Clark, 1981; Ivanov, 2010) where sliding speed is presented as an exponential curve. The friction in contact depends on the instantaneous sliding speed. Although we applied linear dependence here, this procedure does not exclude an application of some other dependence. The presented equations may be incorporated in the frictional model of the tire. The purpose is to improve the prediction of the friction behaviour of the braking tire. For the application of the presented procedure it is necessary to know a number of input data on the tire and exterior wheel load. Acknowledgements This paper is based on the project TR36027: "Software development and national database for strategic management and development of transportation means and infrastructure in road, rail, air and inland waterways transport using the European transport network models" which is supported by the Ministry of science and technological development of Republic of Serbia (2011-2014).

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International Conference on Traffic and Transport Engineering - Belgrade, November 27-28, 2014 References Biderman, V. 1963. Avtomobilnije šini. Moskva :GHI, 378 p. Burke, A., Olatunbosun, O. 1997. Static Tyre/Road Interaction Modelling, Meccanica 32(5), 473-479.. Clark, S.(1981). Mechanics of Pneumatic Tires. USA: National Bureau of Standard, 931 p. Danon, G. 1988. Dynamic adhesion during straight line car braking. Doctoral Thesis: Univeristy of Belgrade, Faculty of Mechanical Engineering, 232 p. Danon, G. 1990. Matematički model elastičnog točka putničkog vozila, u Zborniku VI Međunarodni simpozijum “Motorna vozila i motori 90”, 7 p. Grečenko, A. 1995. Tyre Footprint Area on Hard Ground Computed from Catalogue Values, Journal of Terramechanics 32(6),325-333. Grigolyuk, E., Kulikov, G., Plotnikova, S. 2004. Contact Problem For a Pneumatic Tire Interacting with a Rigid Foundation”, Mechanics of Composite Materials, 40(5), 427-436. Ivanov, V. 2010. Analysis of Tire Contact Parameters Using Visual Processing, Advances in Tribology 2010, 11 p. Janjićijević, N., Danon, G. 1987. Comparison of Experimental and Calculated Dynamic and Steady-state Tire Adhesion during Braking, in Proceeding of EAEC Conference, 135-142. Knoroz, V., Klenikov, E. 1975. Šini i kolesa. Moskva: Mašinostroenie, 220 p. Lenasi, J., Žeželj, S., Danon, G. 1995. Motor Vehicles. Belgrade: Faculty of Transport and Traffic Engineering, 433 p. Moore, H. 1975. Friction of the Pneumatic Tires. Amsterdam: Elsevier Scientific Publishing Company, 220 p. Pillai, P.,bFielding-Russell, G. 1986. Empirical Equations for Tire Footprint Area, Rubber Chemistry and Technology 59(1),155-159. Pottinger, M. 2005.Contact Patch (Footprint) Phenomena in The Pneumatic Tire, Edited by Gent, A. and Walter, J. USA: National Highway Traffic Safety Administration, 48 p. Springer, H., Weisz, G. 1977. Theoretical Investigation on Steady-State Rolling of Steel Belted Tire, AutomobileIndustry 3, 87-94. Taylor, R., Bashford, L., Schrock, M. 2000. Methods for Measuring Vertical Tire Stiffness, Transactions of the ASAE 43(6),1415-1419. Uljanov, N. 1982. Kolesnie dvigitelji. Moskva: Masinostroenie, 325 p. Zeng-Xin, Y., Hui-Feng, T., Xing-Wen, D., Li, S. 2001. A Simple Analysis Method for Contact Deformation of Rolling Tire, Vehicle System Dynamics, 36(6), 435-443.

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