子午线轮胎设计-RCOT

Koichu Yamagishi, ~ Minora Togashi, ~ Shin'ichi Furuya, 1 Kazumi Tsukahara, 1 and Nobuya Yoshimura 1 A Study on the Contour of the Radial Tire: Rolling Contour Optimization Theory RCOT 2 REFERENCE: Yamagishi, K., Togashi, M., Furuya, S., Tsukahara, K., and Yoshimura, N., "A Study on the Contour of the Radial Tire: Rolling Contour Optimization Theory - - RCOT, "Tire Science and Technology, TSTCA, Vol. 15, No. 1, January-March 1987, pp. 3-29. ABSTRACT: The Rolling Contour Optimization Theory (RCOT) can lead to improved steering, fuel efficiency, riding comfort, and braking performance of tires relative to those of conventional shape. The conventional shape has been guided by natural equilibrium profiles, while the RCOT technology shape is guided by that of the tire in motion. This reduces useless distortions caused by running the tire under load. The RCOT design focuses on the distribution of belt and sidewall tension in the tire. Controlling tension in the belt and carcass area while the tire is in motion was the key to creating this new tire shape. KEY WORDS: tire contour, tension distribution, inflation pressure, rolling resistance, riding comfort, braking efficiency, tread buckling A tire must perform four fundamental functions: 1. sustain vehicle load 2. transmit driving and braking forces to the road surface 3. change and/or maintain vehicle direction 4. absorb shock from the road surface. Fuel consumption, wear resistance, and high speed safety are also important factors. The improvement of a single characteristic is easy, but this may damage the quality of another characteristic. For example, the drive to improve fuel consumption of automobiles has required lower rolling resistance of tires. However, this may damage other characteristics such as rolling performance or riding comfort. 1 Bridgestone Corporation, Ogawa Higashi-cho, Kodaira-shi, Tokyo 187, Japan. 2 Presented at a meeting of The Tire Society, March 26, 1985, at the University of Akron, Akron, Ohio. 3 4 T IRE SC IENCE TECHNOLOGY Solution of this "incompatibility dilemma" should optimize all important tire characteristics simultaneously. Previous efforts have been only partially successful because they dealt with only tire construction and material. The present theory offers the additional possibility of considering tire shape. The Theory of Equilibrium Shape Conventional shape-design of radial tires has used the "theory of equilibrium shape" as a base. This theory implies that inflation of a tire causes no carcass bending. It was originally developed for bias fires but has since been used, with some modifications, for the shape-design of radial tires. The radius of curvature of the natural equilibrium shape is [1] rl = (ro 2 - rm2)/2r (1) and the calculated line is used as a base for the contour design, as shown in Fig. 1. The theory of equilibrium shape, which states that a tire takes on a uniform carcass shape upon inflation may not be the ideal theory for providing the optimum shape for roging performance, which is the most important challenge. 2 2 r 1 = r o - r m 2r i m ro r r i Belt ( ii!i+ I Natural Equilibrium Shape as a base for the tire design rl = meridian radius of curvature (in the sidewall) FIG. 1 -- Natural equilibrium shape of tire cross section. YAMAGISHI ET AL ON TIRE CONTOUR 5 Shape of the RCOT Tire The RCOT tire shape shown in Fig. 2 has a sidewall radius of curvature which, especially near the belt, is smaller than that of a conventional tire. Figure 2 also shows that the radius of curvature of the belt area in the RCOT shape is larger than that in the conventional shape. The RCOT shape somewhat resembles that of the rolling conventional tire shown in Fig. 3. By using a tire in motion, rather than in the conventional static state, as a guide for the shape-design, RCOT technology minimizes useless distortions produced by rolling under load. Furthermore, they should have the following advantages over conventional tires: 1. reduced buckling, giving better cornering and braking performance 2. better fuel economy due to a decrease in useless distortions in the sidewalls 3. improved riding comfort due to the ability of the tire to adapt to the road surface. These points illustrate how the RCOT technology conquers the incompatibility dilemma. Tension Characteristics of the RCOT Axisymmetric membrane shell theory for structures implies that equilibrium for a normal to the surface gives [2] RCOT shape . . . . . . . . Conventional tire FIG. 2 - - Comparison of RCOT shape with that of conventional tire. 6 TIRE SCIENCE TECHNOLOGY f e, ~ tt l I I I 711 igl I r i I ['gt r l l" / /~ I ',,,, ,,,' ,; I I / s l i i it s~1 J ss I i ..... / :55 / Shape of conventional rolling tire (RCOT developed from this shape) . . . . . . . . Shape of inflated conventional tire FIG. 3 - - Shapes of conventional tire when unloaded and when rolling under load. .mjr~ + N,,/r2 = p, (2) where N~ = meridian membrane force (sidewall tension) N. = circumferential membrane force r~ = meridian radius of curvature (in the sidewall) r2 = circumferential radius of curvature p = inflation pressure. For radial tires we may assume that N ,=0 (3) in the sidewall area. Therefore, the sidewall membrane force (tension) is N~ = rip, (4) where Ne is the side tension per unit width of the sidewall. Total belt tension To is obtained from Fig. 4 as To = (ap/2 )( b - 2r~ sin 0), (5) where a, b, and 0 are as shown in Fig. 4 YAMAGISHI ET AL ON TIRE CONTOUR 7 / To w I Equation of the b I relationship of the tire's ~ _ shape and tension t. . ~ : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . " ~ 1 ap(b- 2rlsine) -, i To - 2 IF1_ ~ - __ / . i P i i N~ = r lp ! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1_~ a = diameter of the belt . . . . . C- - b = width of the belt c = diameter of the rim d d = rim width �9 N(~ = meridian membrane force (sidewall tension) p = inflation pressure r~ = meridian radius of curvature (in the sidewall) I ~ T O = belt tension "~ .,.,-' w = tire width �9 --- p ---- FIG. 4 - - Relation of a tire's shape to tensions in its structure [3, 4]. 8 TIRE SCIENCE TECHNOLOGY a = belt diameter b = belt width 0 = angle between carcass and belt. Equation 4 shows that the carcass ply tension is proportional to the radius of curvature. Conversely, Eq 5 shows that the total belt tension decreases when the carcass ply radius rt and angle 0 are larger than in the conventionally- shaped tire for a given belt diameter a and belt width b. This means that the tire's tension is directly related to its shape. The RCOT shape, as shown in Fig. 2, has a sidewall radius of curvature which, especially near the belt, is smaller than that in the conventional tire shape (0 is smaller). The radius of curvature at the bead area, however, is larger than that in the conventional shape. Therefore, from Eqs 4 and 5, we can assume that the circumferential belt tension is higher for the RCOT tire. However, the carcass tension near the belt is smaller, and the carcass tension near the bead is higher than those in a conventional tire. Detailed tension distribution in belt and carcass plies was obtained by using the "Finite Element Method" (FEM), with the result roughly resembling a simple model prediction as shown in Fig. 5. Analys is of Road Contact The first step in analysis of road contact, because of its direct relation to rolling performance, was a detailed consideration of tread buckling tendency. Carcass tension RCOT shape distribution ~ tension .~. ,~,~ distribution \ \ \ \ ' ~ Bead area \\~,'~, "~.,,-~.~ tension ~, ~ (stronger) . . . . c~e~t~o~, .tJl ~--/i Sid~.~, shape ,'~}'] ,~ i tension S/ /~/ (smaller) ~ / /'-,~' ," Conventional ~ J /~'~/,?"- shape tension . . . . . . '~,~,,' . distribution Belt tension distribution (stronger) il I 11 / I L .CO-bapetens,on "/I LI-J.J~ distribution ~Conventional shape tension distribution Tension (kg/mm) 185/70SR14 5 ~ p: 1.7 kg/cm 2 Nn NN \ 4 ~ jRCOT k~Oonvent iona l 3 Belt\\ RCOT\ \1 Conventional \ '~ 2 1 - ~ i 0 I Belt Center Belt Edge Bead N~/ = circumferentialmembrane force Nr = meridian membrane force (sidewall tension) FIG. 5 -- Efficiency of the RCOT shape as calculated by FEM. F IG . 6 - - A n al ys is o f b u ck lin g be ha vi or . 10 TIRE SCIENCE TECHNOLOGY Occurrence of Buckling When a car turns a corner very severely, certain parts of the contact patch separate from the road surface because of buckling. This damages rolling performance, so its occurrence is critical. Figure 6 shows the principal char- acteristics of buckling. The black areas on the photograph (taken from a contact patch color data system developed by Bridgestone) are those areas which tend to separate from the road surface. Analysis of Buckling Mechanism Circumferential belt buckling is caused by circumferential compression during severe cornering. The FEM analysis is used to calculate belt tension for an analysis of buckling mechanism, with separate consideration for the effects of inflation, loading and cornering. The distribution of circumferential belt tension caused by inflation pressure is shown in Fig. 7. Maximum tension occurs at the center of the belt and minimum tension occurs at the belt edge. Because the tire crown has two radii of curvature, loading against a fiat surface may cause the belt tension to become negative, i.e., go into compres- sion, near the center of the contact patch, leaving the maximum tension near the outer edge. This is shown in Fig. 8. When a tire turns a corner, the side force causes in-plane bending of the belt; outside bending produces tension and inside bending produces compres- sion. Belt tension distribution caused by the side force is obtained through a superposition of the loading and cornering effects. Figure 9 shows how belt tension distribution is affected by side force. As it increases, the point of maximum compression moves from the center to the 4.0 . . . . . . . . . . . . . . . . . . . . Belt Tension (kg/mm) 201 Belt Belt Center Beltl t Edge / Edgel O.0L ~ q 175/70SR13 p = 1.7 kg/cm 2 / - . . . . . 1,5 . . . . . ~. N~ = sidewall tension T O = belt tension p = inflation pressure FIG. 7 - - Belt tension induced by inflation pressure. YAMAGISHI ET AL ON TIRE CONTOUR 11 12.0 8.0 Belt Tension (kghnm) 4.0 0.0 - 4.0 Free Rolling E ge Belt Center • + ) ( + ~ (--) J 175/70SR13 p = 1.7 kg/cm 2 p = inflation pressure W = loaded weight w = 270kg F IG . 8 - - Distribution of circumferential belt tension under the loaded area of a tire that is rolling in a straight line. outer edge, with the circumferential compression of the belt increasing and finally overcoming the critical buckling load. Buckling of a belt was originally modeled by T. Akasaka as a beam on an elastic foundation with an axial compression force similar to that shown in 12.0 8.0 Belt Tension (kg/mm) 4.0 0.0 - 4.0 In f la t ion - -7~- - ,.,,~ Free Rolling Pressure / , , \ - - , . . " \ \ Belt Center/ / ~-,,, i p d jb I ~'~ Edge " - - - / Cornering Edge 175/70SR13 p = inflation pressure W = loaded weight SF = side force SA = slip angle In-Plane Side Force l ~ p = 1.7 kglcm 2 W = 270kg = SF = 110 kg In-Plane Bending I F IG . 9 - - Distribution of cireurnferential belt tension under the loaded area of a tire that is rolling around a curve. 12 TIRE SCIENCE TECHNOLOGY Fig. 10. The belt and tread are modeled as a beam with bending stiffness D. The tread is modeled on an elastic foundation with a tread spring rate K. The critical buckling axial load is [5] Nor ~ 2(DK) 1/2 (6) and the wave length is ~cr ~ 2r (D/K) 1/4. (7) Critical belt compression is often caused by severe cornering or braking which causes buckling. Prevention of Buckling Buckling may be prevented in either of two ways. One is to increase the critical buckling load so that the belt will resist any expected compression load. A stiffened belt and harder tread rubber combined with a stiff tread pattern yields a higher bending stiffness D and a higher spring rate K. Although this prevents buckling, it may also damage other rolling performance char- acteristics. J § N N + loooloooLoo~ 12, ooooooc oooo / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / Critical buckl ing force Ncr ~ 2~-DK b = tread width M = bending moment D = bending stiffness(belt) N = actualcompression force K = tread spring rate p = inflation pressure F IG . 10 - - Analytical model of belt buckling. YAMAGISHI ET AL ON TIRE CONTOUR 13 A second way to prevent buckling is to reduce the belt compression by increasing the inflation pressure. This will increase the initial belt tension and thus reduce the belt compression when the tire is rotated under load. Higher inflation pressure, however, stiffens the tire; this causes a smaller contact patch, which again damages other rolling performance characteristics. The RCOT shape solves this dilemma by having high belt tension without having high inflation pressure. This occurs because the buttress radius and angle 0 of the RCOT-shaped tire are smaller than in the conventionally-shaped tire (Fig. 4). Figure 11 shows that belt tension of the RCOT-shaped tire is about 15% higher than that of the conventionally-shaped tire at the point where buckling is most likely to occur. Figure 12 shows the belt tension distribution in the RCOT-shaped tire and in a conventionally-shaped tire during loaded rolling in a straight line. The maximum belt tension is near the belt edge, while the maximum belt compression is at the center of the belt. Therefore, because the initial belt tension of the RCOT-shaped tire is higher than that of the conventionally- shaped tire, the maximum belt compression of the RCOT-shaped tire is lower. Figure 13 shows the belt tension distribution in each of the two tire shapes when they are being cornered severely enough to produce a side force equal to 40% of the vertical tire load. The maximum belt compression is increased by the cornering, moving the peak compression level from the center of the belt, as in Fig. 12, toward the outer edge of the curve being negotiated. Since 6.0 4.0 i . . . . . 2.0 0.0 Belt Tension (kg/mm) RCOT Conventional ~ - \~ Belt Belt Edge Belt Center Edge 175/70SR13 p = 1.7 kg/cm 2 p = inflation pressure FIG. 11 - - Comparison of belt tension in the RCOT tire with that in a conventionally shaped tire when each is inflated. 14 TIRE SCIENCE TECHNOLOGY 12.0 8.0 Belt Tension 4.0 (kg/mm) 0.0 - 4.0 .q Belt Edge RCOT ~ " t Conventional "~._ ~ j~ ' Belt Belt Center Edge 175/70SR13 p = 1.7 kg/cm 2 W = 270 kg p = inflation pressure W = Ioaded weight F IG . 12 - - Comparison of belt tension in the RCOT tire with that in a conventionally shaped tire when each is inflated and rolling in a straight line under load. 12.0 8.0 m_ &/' Be ienter RCOT Edge \ / / , / Edge_ Conventional -.. - - - / - Belt Tension 4.0 (kg/mm) 0.0 - 4.0 175/70SR13 p = 1,7 kg/cm 2 W = 270 kg SF = 110 kg p inflation pressure W = loaded weight SF = side force F IG , 13 - - Comparison of belt tension in the RCOT tire with that in a conventionally shaped tire when each is inflated and rolling around a curve under load. YAMAGISHI ET AL ON TIRE CONTOUR 15 the belt compression of the RCOT-shaped tire is the lower of the two, the initial high tension effect remains. This means that the RCOT shape lowers the probability of belt buckling, not by adding extra materials but by simply using air pressure effectively as a design element. Validation of RCOT as a Preventive of Belt Buckling In comparing the RCOT shape to the conventional shape, using the same materials and construction, we observed buckling on a road surface. The contact patch of the RCOT shape had fewer black areas in color photographs like that in Fig. 6, so it gripped the road better. Also, fewer black areas mean a decrease in occurrence of buckling. (This figure isn't shown because its interpretation depends on color.) It follows that the tread friction is improved, which in turn improves both wet and dry braking performances as well as maneuverability. Furthermore, decreasing the occurrence of buckling de- creases uneven treadwear and lowers squealing noise in hard cornering. These observations demonstrate an improvement of road contact by using RCOT, as shown by the diagram in Fig. 14. Analysis of Sidewall Rolling Resistance Finite Element Method. A Finite Element Method (FEM) was used to an- alyze rolling resistance. In order to calculate a reduction in rolling resistance of less than 5%, a very precise three-dimensional non-linear FEM analysis was required. Figure 15 shows the model used. Contact problems and non- linear material properties, as well as other detailed properties, were considered I Decreases buckling 1 Uniform contact pressure 1 Increased friction Improves maneuverability I ~ Decreases uneven }___....~ Resists uneven I deformation treadwear Increases dry and wet friction I _1 -I Resists I squealing noise FIG. 14 -- Improvements in tire performance that follow from decreased buckling. 16 TIRE SCIENCE TECHNOLOGY FIG. 15 -- Three-dimensional non-linear finite element model for analysis of rolling res&tance. during the analysis of rolling resistance. Figure 16 shows the distribution of strain energy loss of each design element. Total energy loss is directly related to rolling resistance. Strain energy loss is expressed as Etoss = fv V6e tan 6dV. (8) Reduction of energy loss by the RCOT-shaped tire is mainly in the sidewall and bead areas. YAMAGISHI ET AL ON TIRE CONTOUR 17 Energy Loss 500 1000 1500 I ] t Tread Gum ~ 1 Belt Coating Gum ~ 1 Ply CoatingGum ~ S i d e G u m ~ I ~'1~-1 Gum Chafer ~ I Bead Filler ~ < I kg x meters per revolution 4500 5000 / / / / I I Conventional[ I RCOT ~-,,\\\~\\\\\\"~ /( , U Total ~ ~ I / / FIG. 16- Distribution of energy loss of an RCOT tire compared with that of a conventionally shaped tire when each is rolling under load. Figure 17 shows the strain distribution of an RCOT-shaped and a conven- tionally-shaped tire in the buttress and bead areas along a circumference of the tire. Although out-plane bending is larger in the sidewall of the RCOT Bead lO 5 "tST(~) o ~ . -120* -600 ~t -10 . "7" - - . - -7 - - 07 60* 0 120* I Buttress - -RCOT - - Convent iona l 10 3' RS (*~) 5 -150" -90* -30 ~ ,I I| U I l I t t ; , 30* 90~ 150" I FIG. I7- Strain distribution in sidewall and bead areas of an RCOT tire compared with that of a conventionally shaped tire by using a finite element method. 18 TIRE SCIENCE TECHNOLOGY tire, in-plane shear strain is higher in the conventional tire. The process of this phenomenon is discussed in the following analytical model. Ply tension of the RCOT tire is higher than that of the conventional tire in the bead area, so the bending stiffness is also higher. Therefore, the shear strain of the RCOT tire is lower than that of the conventional tire in the bead area. Model Analysis. As shown in Fig. 18, a curved-beam-model sample cut from a tire's bead to its tread, and having a deflection at the leading or trailing edge, was used for analysis in dec