Abstract:
In a computerized method for generating a finite element model of a pneumatic tire provided in the tread portion with a zero-degree band, a 2D tire model is firstly generated based on the finished tire shape in a tire vulcanization mold, where the 2D tire model includes a 2D model of a band ply. Secondary, a residual stress is computed and defined on each element of the band ply model. Then, a deformation calculation is made on the 2D tire model to obtain the deformed 2D tire model. Finally, a 3D tire model is generated by duplicating the 2D tire model around the tire rotational axis.

Description:
BACKGROUND OF THE INVENTION 
     The present invention relates to a computerized method for generating a finite element model of a pneumatic tire capable of improving simulation accuracy. 
     In general, a pneumatic tire is manufactured by first building a raw tire by assembling rubber members, cord reinforced layers and the like, and then vulcanizing the raw tire in a mold by pressurizing the inside of the raw tire. Therefore, the cords embedded in the vulcanized tire usually have residual stress. Due to the residual stress, accordingly, there is a tendency such that the shape of the demolded tire becomes smaller than the finished tire shape in the vulcanization mold. 
     In Japanese Patent Application Publication No. 2010-191612, a computerized method for simulating a rolling tire is disclosed. In this method, a finite element model of the tire is generated, based on the finished shape of a tire in a vulcanization mold, namely, the shape of the mold cavity. 
     As a result, there is a tendency that the shape of the tire model generated based on the finished tire shape in the vulcanization mold differs from (usually, becomes smaller than) the shape of the tire during use. 
     Thus, it is difficult to fully improve the accuracy of simulation using such a tire model. 
     The present inventor, therefore, made researches and found that simulation errors are large when the tire is provided in the tread portion with so called zero-degree band (namely, a cord reinforced rubber layer wherein the cord angle is about 5 degrees or less with respect to the tire circumferential direction). In such a tire, the residual stress becomes relatively large in the band when compared with other cord reinforced layers, and the band has much effect on the tire shape. 
     SUMMARY OF THE INVENTION 
     It is therefore, an object of the present invention to provide a method for generating a finite element model of a pneumatic tire provided in the tread portion with a band, in which, a residual stress is defined on a band model of a tire model to approximate the shape of the tire model to the shape of the tire during use, and thereby it becomes possible to improve the accuracy of simulation using the tire model. 
     A computerized method for generating a finite element model of a pneumatic tire according to the present invention is for the tire manufactured through a step of building a raw tire provided in a tread portion with a band ply composed of a band cord having a cord angle of not more than 5 degrees with respect to the tire circumferential direction, and a step of vulcanizing the raw tire in a vulcanization mold, and the method comprises 
     a step in which, based on a cross-sectional shape of the tire in the vulcanization mold, a tire model which is a finite element model of the tire is generated, wherein the tire model includes a band ply model of the band ply made up of elements, and 
     a residual stress defining step in which a residual stress Si caused by the vulcanizing step is computed and defined on each of the elements of the band ply model of the tire model, and a deformation calculation is made on the tire model to obtain the deformed tire model. 
     The residual stress Si is computed by the following manner (I), (II) or (III). 
     (I) For each of the elements of the band ply model, the residual stress Si is computed by the following expression
 
 Si=E ×{( Ra−ra )/ ra}×α 
 
wherein
     E: the Young&#39;s modulus of the band cord,   Ra: the outside diameter of the element,   ra: the outside diameter of the element before vulcanization   α: a coefficient.
 
(II) For all of the elements of the band ply model, the residual stress Si is computed collectively by the following expression
 
 Si=E ×{( Ra−ra )/ ra}×α 
 
wherein
   E: the Young&#39;s modulus of the band cord,   Ra: the mean value of the outside diameters of the elements belonging to the group   ra: the mean value of the outside diameters of the elements belonging to the group before vulcanization   α: a coefficient.
 
(III) The elements of the band ply model are grouped into a plurality of groups each consisting of axially abutting elements, and for each of the groups, the residual stress Si is computed collectively by the following expression
 
 Si=E ×{( Ra−ra )/ ra}×α 
 
wherein
   E: the Young&#39;s modulus of the band cord,   Ra: the mean value of the outside diameters of the elements belonging to the group   ra: the mean value of the outside diameters of the elements belonging to the group before vulcanization   α: a coefficient.   

     In the case of (III), it is preferable that the groups include: a central group consisting of the elements existing in a central region of the tread portion; and two lateral groups consisting of the elements existing in shoulder regions of the tread portion. 
     The method may further comprises, before the residual stress defining step, a step in which a deformation calculation is made on the tire model on which a condition of a tire pressure is defined. 
     The above-mentioned tire model is two-dimensional, and the method may further comprises a step in which a three-dimensional tire model is generated by circumferentially duplicating the two-dimensional tire model around the tire rotational axis at intervals of a small angle. 
     Therefore, due to the residual stress Si defined on each element of the band ply model, the tire model is deformed similarly to the actual tire. In other words, it is possible to generate the tire model having a shape approximate to that of the tire during use. As a result, simulation accuracy can be improved. 
     In the case of (I), it is possible to generate the tire model having a shape most approximate to that of the tire during use. In the case of (II) or (III), the computational time can be reduced in comparison with (I). 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  shows an example of the computer system implementing a method according to the present invention. 
         FIG. 2  is a cross sectional view of an example of the pneumatic tire. 
         FIGS. 3( a ) and 3( b )  are schematic partial cross sectional views for explaining steps of building the raw tire. 
         FIG. 4  is a cross sectional view for explaining a step of vulcanizing the raw tire. 
         FIG. 5  is a flow chart of the method as an embodiment of the present invention. 
         FIG. 6  is a cross sectional view of a two-dimensional tire model. 
         FIG. 7  is a closeup of the tire model shown in  FIG. 6 . 
         FIG. 8  is a perspective partial view of a three-dimensional tire model generated from the two-dimensional tire model shown in  FIG. 6 . 
         FIG. 9( a )  is a cross sectional view of the band ply model of which elements are grouped into five groups. 
         FIG. 9( b )  is a cross sectional view of the band ply model of which elements are grouped into a single group. 
         FIG. 10  is a cross sectional view of the band ply model of which elements are not grouped. 
         FIG. 11  is a cross sectional view of another example of the tire model. 
     
    
    
     DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     Embodiments of the present invention will now be described in detail in conjunction with accompanying drawings. 
     According to the present invention, a tire model  3  which is a finite element model of a pneumatic tire  2  is generated, wherein the tire model  3  is made up of a finite number of elements so that numerical analyses are possible by the use of a computer for example as shown in  FIG. 1 . 
     The expression “numerical analyses are possible” means that it is possible to deal with the tire model  3  in a numerical analysis method such as finite element method, finite volume method, finite difference method and boundary element method. In this embodiment, a finite element method is employed. 
     The computer  1  comprises a main body  1   a , a keyboard  1   b , a mouse  1   c  and a display  1   d . The main body  1   a  comprises an arithmetic processing unit (CPU), ROM, work memory, storage devices such as magnetic disk, disk drives  1   a   1  and  1   a   2  and the like. In the storage device, programs/software for carrying out a generating method in this embodiment are stored. 
     As shown in  FIG. 2  for example, the pneumatic tire  2  is a passenger car radial tire comprising a tread portion  2   a , a pair of sidewall portions  2   b , a pair of bead portions  2   c  with a bead core  5  therein, a carcass  6  extending between the bead portions through the tread portion and sidewall portions, a belt  7  disposed radially outside the carcass  6  in the tread portion  2   a  and a band  9  disposed radially outside the belt  7 . 
     The carcass  6  is composed of at least one, in this embodiment only one ply  6 A of carcass cords arranged at an angle of from 75 to 90 degrees with respect to the tire equator C and extending between the bead portions  2   c  through the tread portion  2   a  and sidewall portions  2   b , and turned up around the bead core  5  in each bead portions from the axially inside to the outside of the tire to form a pair of turnup portions  6   b  and a main portion  6   a  therebetween. 
     Between the main portion  6   a  and each turnup portion  6   b , a bead apex rubber  8  is disposed to extend radially outwardly from the bead core  5 . 
     The belt  7  is composed of at least two cross plies  7 A and  7 B of parallel cords laid at an angle of from 10 to 35 degrees with respect to the tire circumferential direction. 
     The band  9  is composed of one ply  9 A of at least one cord  12  (e.g. organic fiber cord) and the angle of the cord or cords  12  is almost zero or not more than 5 degrees with respect to the tire circumferential direction. 
     The ply  9 A has a width capable of covering the entire width of the belt  7 . 
     In this embodiment, the ply  9 A is formed by spirally winding one or more cords  12  around the outside of the belt  7 . 
     Next, a typical method for manufacturing the pneumatic tire  2  in this embodiment will be described. 
     Firstly, a raw tire T is built. (raw tire building step) 
     In this raw tire building step, as shown in  FIG. 3( a ) , unvulcanized inner liner rubber  17  and sidewall rubber  18  and carcass ply  6 A are wound sequentially on the outside of the cylindrical tire building drum  15 . Then, rubber bead apexes  8  and bead cores  5  are disposed on the outside of the carcass ply  6 A from the axially outside, and a tubular main body  20  is formed. Then, the axially outside portions  20   b  of the tubular main body  20  protruding axially outward from the bead cores  5  are turned. 
     Next, the tubular main body  20  is inflated into a toroidal shape as shown in  FIG. 3( b ) , while reducing the axial distance between the bead cores  5  by moving bead lock devices G for holding the bead cores  5  closer to each other. Thereby, the outer peripheral surface of the carcass ply  6 A is pressed onto the inner peripheral surface of an annular tread ring  22 . The annular tread ring  22  is an assembly of the belt  7 , the band  9  and a tread rubber  21  prepared in advance. 
     Thus, the raw tire T is built. 
     Next, as shown in  FIG. 4  for example, the raw tire T is put in a cavity  23  of a vulcanization mold  24  and vulcanized. 
     (Vulcanization Step) 
     The vulcanization mold  24  comprises a pair of bead rings  24   c  for holding the bead portions  2   c  of the raw tire T, a pair of sidewall dies  24   a  having die surfaces  23   a  for molding the sidewall portions of the raw tire T, and tread dies  24   b  having a die surface  23   b  for molding the tread portion of the raw tire T. These parts are combined to form the cavity  23 .
 
In the tire cavity of the raw tire T put in the mold, a bladder  25  is set and inflated with a high temperature and pressure fluid so that the raw tire T is pressed onto the inner surface of the cavity  23  and vulcanized.
 
Thus, the tire  2  as shown in  FIG. 2  is manufactured.
 
       FIG. 5  shows a flow chart of a method for generating the tire model  3  of the above-mentioned tire  2 . 
     In this example, according to the cross-sectional shape of the tire molded in the vulcanization mold  24  in the vulcanization step (namely, the shape of the cavity of the vulcanization mold), a two-dimensional contour of the tire model  3  is defined. (step P 1 ) 
       FIG. 6  shows an example of the tire model  3 . 
     In the tire model  3 , the band ply  9 A is discretized into two-dimensional elements Fi (i=1, 2 - - - ). 
     Other reinforcing cord members, e.g. the carcass ply  6 A and belt plies  7 A and  7 B and the like and the rubber members  26 , e.g. the tread rubber  21  and the like, are discretized into two-dimensional elements Gi (i=1, 2 - - - ). 
     Thus, the tire model  3  in this embodiment is a two-dimensional tire model which includes 
     a band ply model  30  made up of the elements Fi, and 
     a carcass ply model  28  and belt ply models  29  and rubber member models  27  each made up of the elements Gi. 
     Incidentally, generation of such models can be readily made by the use of a mesh-generating software and geometrical data (for example, CAD data) of the vulcanization mold  24 . 
     As to the above-mentioned two-dimensional elements Gi and Fi, quadrilateral elements are preferably used, but elements having other polygonal shapes can be used as well. 
     For each of the elements Gi and Fi, numerical data—for example, an element number, node point numbers, coordinates of the node points, and material characteristics (for example density, Young&#39;s modulus and/or damping coefficient and the like)—are defined and stored in the computer  1 . 
     In the band ply model  30  in this embodiment, the elements Fi are grouped as follows: 
     a central group  33 A consisting of axially abutting elements Fi positioned in a central region of the tread portion  2   a , and two lateral groups  33 B each consisting of axially abutting elements Fi positioned in tread shoulder regions of the tread portion  2   a.    
     Next, the tire pressure is applied to the tire model  3 , and a deformation calculation of the tire model  3  is carried out. (Step P 2 ) 
     In order to apply the tire pressure, the following conditions are defined: 
     the rim contacting parts  3   r  of the tire model  3  are not deformable; 
     the width w between the bead portions  2   c  of the tire model  3  is equal to the width of the wheel rim (if necessary, the bead portions  2   c  are so displaced); 
     the radial distance Rs of the rim contacting parts  3   r  from the rotation axis of the tire model  3  is equal to the radius of the wheel rim; and 
     a uniformly-distributed load corresponding to the tire pressure is applied to the inner surface of the tire cavity of the tire model  3 . 
     under such conditions, the computer  1  make an equilibrium calculation of the tire model  3  to obtain displacements of the node points of the elements of the tire model  3 . Thereby, the tire model  3  after the dilation deformation where the rubber member models  27 , carcass ply model  28 , belt ply models  29  and band ply model  30  are expanded is obtained. 
     In the meantime, in the case of the actual tire  2 , the belt cords and band cords  12  are subjected to large tension during manufacturing the tire, especially during vulcanizing the tire, and residual stress remains in the finished tire. There is a tendency that the shape of the tire  2  during use becomes smaller than the cross-sectional shape in the vulcanization mold  24 , and the shape of the tire model  3  based on the cross-sectional shape in the vulcanization mold  24  differs from the shape of the tire  2  during use. Thus, it is difficult to fully improve the simulation accuracy. 
     As explained above, the inclination angle with respect to the tire circumferential direction is smaller in the band cord than the belt cords. As a result, the residual stress of the band ply  9 A tends to become larger than the residual stress of the belt plies  7 A and  7 B. 
     Therefore, the method according to the present invention includes a residual stress defining step P 3 , in which on each element Fi of the band ply model  30 , a residual stress Si (i=1, 2 - - - ) caused in the vulcanization step is defined, and then a deformation calculation is carried out. 
     In this embodiment, on each of the elements Fi belonging to the central group  33 A, a residual stress Si obtained by the following computational expression is defined:
 
 Si=E ×{( Ra−ra )/ ra}×α   (1)
 
wherein
     E: the Young&#39;s modulus of the band cord,   Ra: the mean value of outside diameters Di of all of the elements Fi belonging to the central group  33 A,   ra: the mean value of outside diameters ri of all of the elements Fi belonging to the central group  33 A before vulcanization,   α: a coefficient.   

     Further, on each of the elements Fi belonging to the lateral group  33 B, a residual stress Si obtained by the following computational expression is defined:
 
 Si=E ×{( Ra−ra )/ ra}×α   (1)
 
wherein
     E: the Young&#39;s modulus of the band cord,   Ra: the mean value of outside diameters Di of all of the elements Fi belonging to the lateral group  33 B,   ra: the mean value of outside diameters ri of all of the elements Fi belonging to the lateral group  33 B before vulcanization,   α: a coefficient.   

     By calculating the computational expression (1), the residual stress Si can be obtained on the assumption that the stress of the band ply  9 A before vulcanization is zero. In either group, the outside diameters ri before vulcanization are defined on the elements Fi, respectively, based on those in the band ply  9 A before vulcanization. 
     The term (Ra−ra)/ra represents the expansion rate of the band ply model  30  before and after vulcanization in each group  33 A, 33 B. 
     There is a tendency that the residual stress in the band ply  9 A is reduced by a creep deformation of the band cord  12  during vulcanization. Therefore, the coefficient α is used to adjust the residual stress Si. 
     The value of the coefficient α is predetermined according to the material of the band cord  12 . For example, if the material is nylon, a value between 0.5 and 1.0 is preferred. If the material is aramid, a value between 0.75 and 1.0 is preferred. 
     There is a tendency that the temperature of the tread dies  24   b  during vulcanization becomes lower in the tread shoulder portions than in the tread crown portion. Accordingly, hot stretching of the band cord  12  during vulcanization becomes less in the tread shoulder portions in comparison with the tread crown portion. As a result, there is a tendency that the residual stress Si becomes larger in axially outer parts of the band ply  9 A than in the central part. 
     Therefore, it is preferable that, in order to incorporate such variation of the residual stress Si, a value in a range of from about 0.05 to about 0.10 is added to the above-mentioned value of the coefficient α in the case of the two lateral groups  33 B. and such modified coefficient α is used in the above-mentioned computational expression (1). 
     By multiplying each expansion rate (Ra−ra)/ra by the Young&#39;s modulus E of the band cord  12  and the coefficient α as in the expression (1), the residual stress Si of each element Fi of the band ply model  30  can be obtained. 
     By the residual stress Si, the tire model  3  is deformed so that the contour becomes smaller similarly to the actual tire  2 . Accordingly, the tire model  3  can simulate the tire  2  during use more accurately to improve the simulation accuracy. 
     In this embodiment, residual stress is not defined on the belt ply model  29 , therefore, an increase in the computational cost can be avoided. 
     In this embodiment, further, since the residual stress Si is computed collectively for the central group  33 A and also for the two lateral groups  33 B, the computational time can be greatly reduced in comparison with a method in which the residual stress Si is computed for each element Fi. 
     In this embodiment, when the outside diameters Di of the elements Fi is gradually decreased from the tire equator side to the tread edge side and the outside diameters ri of the elements Fi before vulcanization have a constant value, 
     it is possible to set the residual stress Si for the elements Fi belonging to the central group  33 A more than the residual stress Si for the elements Fi belonging to the two lateral groups  33 B in order to approximate the residual stress Si to the actual residual stress of the band ply  9 A of the tire  2 . 
     The axial width L 2  of the central group  33 A is preferably set in a range of from 30 to 60% of the axial width L 1  of the band ply model  30  as shown in  FIG. 6 . 
     If the width L 2  exceeds 60% of the width L 1 , there is a possibility that the contour of the tire model  3  becomes excessively shrunk in the tread crown portion. If the width L 2  is less than 30% of the width L 1 , there is a possibility that the contour of the tire model  3  becomes excessively expanded in the tread crown portion. 
     In this embodiment, before implementing the residual stress defining step P 3 , the above-mentioned step P 2  in which the tire pressure is applied and a deformation calculation is carried out, is implemented. Therefore, it is prevented that the size of the tire model  3  deformed by the residual stress Si becomes smaller than the size of the tire model  3  before the dilation deformation. As a result, it is possible to avoid a compression calculation of a topping rubber model (not shown) of the topping rubber of the band ply included in the band ply model  30 . Thereby, the stability of the computation can be improved. 
     Next, a step P 4  of generating a three-dimensional model  31  is implemented. 
     In this step P 4 , as shown in  FIG. 8 , the node points  3   t  of the two-dimensional tire model  3  generated as above are circumferentially duplicate around the tire rotational axis at intervals of a small angle θ, and the node points  3   t  are mutually linked. Thereby, a three-dimensional model  31  of the tire inflated to the tire pressure can be generated readily in a short time. 
     In the above described embodiment, the elements of the band ply model  30  are grouped into three groups  33 A,  33 B and  33 B. But, it is also possible to group into four or more groups  33 , for example, five groups  33 A,  33 B,  33 B,  33 C and  33 C as shown in  FIG. 9( a ) , and the residual stress Si is computed collectively for each group  33  in order to approximate the residual stress Si to the actual residual stress of the band ply  9 A. 
     Further, it is also possible to group the elements Fi of the band ply model  30  into only one group  33  as shown in  FIG. 9( b ) , and the residual stress is computed collectively for the one group  33 , namely, all of the elements Fi have uniform residual stress. In this case, the computational time can be greatly reduced. 
     Furthermore, it is also possible that the elements Fi of the band ply model  30  are not grouped as shown in  FIG. 10 , and the residual stress Si is computed for each of the elements Fi. In this case, the above-mentioned “Ra” and “ra” in the computational expression (1) are as follows:
     Ra is the outside diameter of each element Fi,   ra is the outside diameter of the element Fi before vulcanization.   (Ra−ra)/ra is the expansion rate of the element Fi before and after vulcanization.   

     As shown in  FIG. 11 , if a shrunk region  32  where the outside diameter Ra becomes less than the outside diameter ra exists in the tire  2 , a tension (residual stress) does not occur in the band cord  12  existing in the shrunk region  32 . 
     In this case, it is desirable that, in order to make the shape of the tire model  3  approximate to the shape of the tire  2  during use, the residual stress Si of zero value is defined on the elements Fi belonging to the shrunk region  32 . 
     More specifically, in the calculation of the residual stress Si using the expression (1), zero is set to the coefficient α for the elements Fi belonging to the shrunk region  32 . 
     Comparison Test 
     Twenty one pneumatic tires different from each other in tire size and/or internal structure were prepared, and under the normally inflated and loaded states of the respective tires, their footprints were recorded. 
     The tire models  3  of the above-mentioned tires were generated according to the flow chart shown in  FIG. 5 , and their footprints under the normally inflated and loaded states of the respective tires were computed, wherein the residual stress Si was computed collectively for each of the central group  33 A and lateral groups  33 B. 
     For comparison, according to the flow chart shown in  FIG. 5  from which the residual stress defining step was omitted, the tire models of the above-mentioned tires were generated, and their footprints under the normally inflated and loaded states of the respective tires were computed. 
     Then, a center rib and shoulder rib appearing in each of the footprints were measured for their circumferential lengths under the same conditions. 
     With respect to the measured circumferential lengths, the coefficient of correlation between the tire models  3  and the actual pneumatic tires, and the coefficient of correlation between the comparative tire models and the actual pneumatic tires were calculated. In the result, the former became 0.93, whereas the latter was 0.57. Thus, it was confirmed that a tire model according to the present invention has a high correlativity to the actual tire. 
     Further, it was also confirmed by visual comparison that the footprints of the tire models  3  were approximate to those of the actual pneumatic tires than the comparative tire models. 
     In the method according to the present invention, therefore, the simulation accuracy can be improved.