考虑高阶剪切的轮胎Gough刚度模型
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摘要
轮胎是一种帘线增强橡胶复合材料结构,Gough刚度是表征轮胎侧向变形与磨耗的一个重要参数,但缺乏理论推导。本文基于高阶剪切复合材料梁理论,研究子午线轮胎带束层角度对轮胎侧向变形的影响及Gough刚度的理论基础。在Fiala横梁模型的基础上,将轮胎简化为弹性基础上的复合材料梁。胎侧及胎面胶简化为弹性基础,带束和胎体简化为复合材料梁。考虑高阶剪切位移场及帘线-橡胶材料的各向异性,利用虚功原理建立模型方程,得到了复合材料梁在侧向载荷作用下的小挠度理论解,与有限元结果和已有文献结果对比验证了其合理性。算例分析表明,当胎体等效梁结构的高宽比固定时,对应Gough刚度极大值带束角度是固定的;不同的帘线模量和橡胶模量也会对Gough刚度极大值所对应的带束角度产生不同的影响;带束层厚度对Gough刚度极大值对应的带束角度影响较小。根据Gough刚度极大值准则得到了不同高宽比下的轮胎最佳带束角度,利用实际轮胎解剖结构进行了验证。本文提出的模型为子午线轮胎带束角度优选和磨耗设计提供了理论指导。
Tire is a kind of rubber-cord reinforced composite structure.Gough stiffness derived by using a beam model is a parameter of tire which is related to the transverse deflections and wear of tires.In the present study,the theoretical explanation of Gough stiffness was proposed to analyze the lateral bending deformation of the tire under different belt angles.By revising the Fiala tire model,the tire was simplified as composite beam on elastic foundation.The sidewall and tire tread pattern blocks were treated as an elastic foundation.The belt and tire body were simplified as a composite beam.Considering the high order shear deformation theory and the anisotropy of the cordrubber material,the equilibrium equations were derived by using the principle of virtual work.The deformations under the transverse load with small deflections were derived for the first time.To verify the accuracy of the model,the theoretical results were compared with the results of finite element analysis,and the good agreement could be found.The effect of length,height,thickness and modulus of cord and rubber were analyzed.It is indicated that if the length-height ratio of the equivalent beam is unchanged,the corresponding belt angle of the maximum value of Gough stiffness will not be changed.According to maximum Gough stiffness,the optimal belt angle of the tire with different high width ratio was obtained.The results also agree with those results of the actual tire structure.The proposed model can provide theoretical direction for the structure and durability design of tires.
Tire is a kind of rubber-cord reinforced composite structure.Gough stiffness derived by using a beam model is a parameter of tire which is related to the transverse deflections and wear of tires.In the present study,the theoretical explanation of Gough stiffness was proposed to analyze the lateral bending deformation of the tire under different belt angles.By revising the Fiala tire model,the tire was simplified as composite beam on elastic foundation.The sidewall and tire tread pattern blocks were treated as an elastic foundation.The belt and tire body were simplified as a composite beam.Considering the high order shear deformation theory and the anisotropy of the cordrubber material,the equilibrium equations were derived by using the principle of virtual work.The deformations under the transverse load with small deflections were derived for the first time.To verify the accuracy of the model,the theoretical results were compared with the results of finite element analysis,and the good agreement could be found.The effect of length,height,thickness and modulus of cord and rubber were analyzed.It is indicated that if the length-height ratio of the equivalent beam is unchanged,the corresponding belt angle of the maximum value of Gough stiffness will not be changed.According to maximum Gough stiffness,the optimal belt angle of the tire with different high width ratio was obtained.The results also agree with those results of the actual tire structure.The proposed model can provide theoretical direction for the structure and durability design of tires.
引文
[1] GOUGH V E.Stiffness of cord and rubber constructions[J].Rubber Chemistry&Technology,1968,41(4):988-1021.
[2] WALTER J D.Advances in tire composite theory[J].Tire Science&Technology,1973,1(1):210-250.
[3] WALTER J D.Cord-rubber tire composites:Theory and applications[J].Rubber Chemistry&Technology,1978,51(3):524-576.
[4]危银涛,杜星文,王友善,等.橡胶复合材料结构非线性有限元分析[J].复合材料学报,1998,15(4):81-88.WEI Yintao,DU Xingwen,WANG Youshan,et al.Nonlinear FE analysis for rubber composite structures[J].Acta Materiae Compositae Sinica,1998,15(4):81-88(in Chinese).
[5] FIALA E.Lateral forces on rolling pneumatic tires[J].Zeitschrift VDI,1954,96(29):973-979.
[6] DANIELS B K.A note on Gough stiffness and tread life[J].Tire Science&Technology,1977,5(4):226-231.
[7] FORD J L,PATEL H P,TURNER J L.Interlaminar shear effects in cord-rubber composites[J].Fibre Science&Technology,1982,17(4):255-271.
[8] PARHIZGAR S.Determination of stiffness properties of multi-ply cord-rubber composites[J].Tire Science&Technology,1989,17(17):201-216.
[9] PIDAPARTI R M V.Stiffness characteristics of twisted cords for cord-rubber composites[J].Composite Structures,1993,24(4):291-298.
[10] PIDAPARTI R M V.Analysis of cord-rubber composite laminates under combined tension and torsion loading[J].Composites Part B:Engineering,1997,28(4):433-438.
[11] KOCAK S,PIDAPARTI R M V.Three-dimensional micromechanical modeling of cord-rubber composites[J].Mechanics of Composite Materials&Structures,2000,7(1):19-34.
[12] RAO S,DANIEL I M,GDOUTOS E E.Mechanical properties and failure behavior of cord/rubber composites[J].Applied Composite Materials,2004,11(6):353-375.
[13] HASSIS H,AREM S E,PIDAPARTI R.Modeling coupling effects in cord-rubber composite structures[J].Jp Journal of Solids&Structures,2007,1(3):217-236.
[14] REDDY J N.A simple higher-order theory for laminated composite plates[J].Journal of Applied Mechanics,1984,51(4):745-752.
[15] REDDY J N,LIU C F.A higher-order shear deformation theory of laminated elastic shells[J].International Journal of Engineering Science,1985,23(3):319-330.
[16] MARUR S R,KANT T.Free vibration analysis of fiber reinforced composite beams using higher order theories and finite element modelling[J].Journal of Sound&Vibration,1996,194(3):337-351.
[17] SHI G,LAM K Y.Finite element vibration analysis of composite beams based on higher-order beam theory[J].Journal of Sound&Vibration,1999,219(4):707-721.
[18] HASSIS H,MIRAOUI I,PIDAPARTI R.Analysis of cordcomposite plates using a higher order formulation[J].European Journal of Mechanics A:Solids,2006,25(3):437-446.
[19] CHEN W,XU M,LI L.A model of composite laminated Reddy plate based on new modified couple stress theory[J].Composite Structures,2012,94(8):2599-2609.
[20] THAI C H,TRAN L V,TRAN D T,et al.Analysis of laminated composite plates using higher-order shear deformation plate theory and node-based smoothed discrete shear gap method[J].Applied Mathematical Modelling,2012,36(11):5657-5677.
[21] POLIZZOTTO C.From the Euler-Bernoulli beam to the Timoshenko one through a sequence of Reddy-type shear deformable beam models of increasing order[J].European Journal of Mechanics-A/Solids,2015,53:62-74.
[22] FROSTIG Y,BARUCH M,VILNAY O,et al.High-order theory for sandwich-beam behavior with transversely flexible core[J].Journal of Engineering Mechanics,1992,118(5):1026-1043.
[23] FROSTIG Y,SHENHAR Y.High-order bending of sandwich beams with a transversely flexible core and unsymmetrical laminated composite skins[J].Composites Engineering,1995,5(4):405-414.
[24] FROSTIG Y.Buckling of sandwich panels with a flexible core-high-order theory[J].International Journal of Solids&Structures,1998,35(3):183-204.
[25] PAVEL S,RAINER B,GNTHER P,et al.Optical measurement:Assessment of tire deformation with the focus on its transient handling properties[R].Hannover:Tire Technology International,2017(Annual review):26-30.
[2] WALTER J D.Advances in tire composite theory[J].Tire Science&Technology,1973,1(1):210-250.
[3] WALTER J D.Cord-rubber tire composites:Theory and applications[J].Rubber Chemistry&Technology,1978,51(3):524-576.
[4]危银涛,杜星文,王友善,等.橡胶复合材料结构非线性有限元分析[J].复合材料学报,1998,15(4):81-88.WEI Yintao,DU Xingwen,WANG Youshan,et al.Nonlinear FE analysis for rubber composite structures[J].Acta Materiae Compositae Sinica,1998,15(4):81-88(in Chinese).
[5] FIALA E.Lateral forces on rolling pneumatic tires[J].Zeitschrift VDI,1954,96(29):973-979.
[6] DANIELS B K.A note on Gough stiffness and tread life[J].Tire Science&Technology,1977,5(4):226-231.
[7] FORD J L,PATEL H P,TURNER J L.Interlaminar shear effects in cord-rubber composites[J].Fibre Science&Technology,1982,17(4):255-271.
[8] PARHIZGAR S.Determination of stiffness properties of multi-ply cord-rubber composites[J].Tire Science&Technology,1989,17(17):201-216.
[9] PIDAPARTI R M V.Stiffness characteristics of twisted cords for cord-rubber composites[J].Composite Structures,1993,24(4):291-298.
[10] PIDAPARTI R M V.Analysis of cord-rubber composite laminates under combined tension and torsion loading[J].Composites Part B:Engineering,1997,28(4):433-438.
[11] KOCAK S,PIDAPARTI R M V.Three-dimensional micromechanical modeling of cord-rubber composites[J].Mechanics of Composite Materials&Structures,2000,7(1):19-34.
[12] RAO S,DANIEL I M,GDOUTOS E E.Mechanical properties and failure behavior of cord/rubber composites[J].Applied Composite Materials,2004,11(6):353-375.
[13] HASSIS H,AREM S E,PIDAPARTI R.Modeling coupling effects in cord-rubber composite structures[J].Jp Journal of Solids&Structures,2007,1(3):217-236.
[14] REDDY J N.A simple higher-order theory for laminated composite plates[J].Journal of Applied Mechanics,1984,51(4):745-752.
[15] REDDY J N,LIU C F.A higher-order shear deformation theory of laminated elastic shells[J].International Journal of Engineering Science,1985,23(3):319-330.
[16] MARUR S R,KANT T.Free vibration analysis of fiber reinforced composite beams using higher order theories and finite element modelling[J].Journal of Sound&Vibration,1996,194(3):337-351.
[17] SHI G,LAM K Y.Finite element vibration analysis of composite beams based on higher-order beam theory[J].Journal of Sound&Vibration,1999,219(4):707-721.
[18] HASSIS H,MIRAOUI I,PIDAPARTI R.Analysis of cordcomposite plates using a higher order formulation[J].European Journal of Mechanics A:Solids,2006,25(3):437-446.
[19] CHEN W,XU M,LI L.A model of composite laminated Reddy plate based on new modified couple stress theory[J].Composite Structures,2012,94(8):2599-2609.
[20] THAI C H,TRAN L V,TRAN D T,et al.Analysis of laminated composite plates using higher-order shear deformation plate theory and node-based smoothed discrete shear gap method[J].Applied Mathematical Modelling,2012,36(11):5657-5677.
[21] POLIZZOTTO C.From the Euler-Bernoulli beam to the Timoshenko one through a sequence of Reddy-type shear deformable beam models of increasing order[J].European Journal of Mechanics-A/Solids,2015,53:62-74.
[22] FROSTIG Y,BARUCH M,VILNAY O,et al.High-order theory for sandwich-beam behavior with transversely flexible core[J].Journal of Engineering Mechanics,1992,118(5):1026-1043.
[23] FROSTIG Y,SHENHAR Y.High-order bending of sandwich beams with a transversely flexible core and unsymmetrical laminated composite skins[J].Composites Engineering,1995,5(4):405-414.
[24] FROSTIG Y.Buckling of sandwich panels with a flexible core-high-order theory[J].International Journal of Solids&Structures,1998,35(3):183-204.
[25] PAVEL S,RAINER B,GNTHER P,et al.Optical measurement:Assessment of tire deformation with the focus on its transient handling properties[R].Hannover:Tire Technology International,2017(Annual review):26-30.