Shape design of a tire contour based on approximation model

Transcription

1 Journal of Mechanical Science and Technology 25 (1) (2011) 149~155 wwwspringerlinkcom/content/ x DOI /s Shape design of a tire contour based on approximation model Dong Woo Lee 1, Jong Kyu Kim 2, Sung Rae Kim 3 and Kwon-Hee Lee 1,* 1 Department of Mechanical Engineering, Dong-A University, Busan, , Korea 2 Graduate School of Mechanical Engineering, Dong-A University, Busan, , Korea 3 R&D Center, Nexen Tire Company, 30 Yusan-dong Yangsan city, Gyeongsangnamdo , Korea (Manuscript Received October 29, 2009; Revised June 7, 2010; Accepted September 13, 2010) Abstract The basic purpose of a tire is to enhance vehicle performances such as driving performance, rolling resistance, durability, ride comfort, noise, wear resistance, etc by acting as a flexible cushion To meet the demand for increased vehicle performances, the design method of a tire has advanced This study proposes a structural design method for tire contour by considering both the tread contour and the sidewall contour, simultaneously Existing studies of tire contour optimization have focused on the tread contour and the sidewall, separately Durability, maneuverability and ride comfort are performances that are commonly investigated in tire contour design Durability, maneuverability and ride comfort can be measured by the values of the strain energy density, tension and vertical stiffness, respectively The optimization technique using a metamodel is introduced to maximize durability while satisfying the imposed constraints of tension and ride comfort To achieve this, the responses defined in the optimization formulation are expressed mathematically in explicit form with respect to the design variables by using the kriging surrogate model, resulting in a simple optimization problem Then, the simulated annealing algorithm is utilized to find the global optimum Keywords: Contour design; Kriging; Optimization; Simulated annealing Introduction A tire acts as a damper against shocks from the ground, a deliverer of the braking force and driving force to the road surface, a guide to set the direction of the vehicle and supporter of the vehicle load Recently, many diverse tire designs according to the performance enhancement of a vehicle have been proposed The tire contour is an important design factor having a direct influence on the main vehicle performances such as driving performance, rolling resistance, durability, ride comfort, noise and wear resistance [1] Tire contour design is largely divided into tread contour design and sidewall contour design, which have been investigated extensively in the past The equilibrium contour method that designs the contour such that the carcass has uniform tensile distribution was suggested in 1928 With this method, a designer can suggest the same contour because only one procedure exists, but performance becomes difficult to enhance when the vehicle is driven at high speeds The most widely method used today is the nonequilibrium contour method, which was proposed in 1970s With this This paper was recommended for publication in revised form by Associate Editor Jeonghoon Yoo * Corresponding author Tel: , Fax: address: leekh@dauackr KSME & Springer 2011 method, vehicle performance can be improved by maximizing the tensile force of the carcass adjacent to the bead and minimizing it near the belt But because no uniform procedure exists in this method, the carcass contour varies according to the designer Accordingly, only a highly skilled designer can realize a good design by applying this method [2-4] Recently, an optimization method has been introduced in tire contour design to meet the various requirements of a tire [5-8] But because existing studies on tire contour optimization were interested in either the tread contour or the sidewall contour, they neglected the interaction between the two contours, so they have been unable to find the true optimum shape [9, 10] Accordingly, the present research proposes design variables considering both the tread contour and the sidewall contour of a tire And in order to investigate the responses of durability, maneuverability and ride comfort, which are the performances of a tire, the strain energy density, the tension at the bead region and the vertical stiffness of a tire are calculated by nonlinear FE analysis of a tire The optimization of a tire contour is classified as shape optimization among structural optimization classifications But when the gradient based optimization is applied to the contour design of a tire, it has difficulty in calculating the sensitivity of each response and it generates mesh distortions of the finite elements during an optimization process Therefore, in this

2 150 D W Lee et al / Journal of Mechanical Science and Technology 25 (1) (2011) 149~155 18kgf/cm 2 (a) Two-dimensional FE model Fig 2 Design variables for a tire contour 515 kgf (b) Three-dimensional FE model Fig 1 Boundary and loading conditions for FE analysis study the optimization method using the kriging metamodel [11-13] is applied to overcome these difficulties A kriging metamodel is built to approximate the true function on the basis of the deterministic response values Then, an optimization algorithm is applied on the explicit metamodel expression Even though there is no guarantee that the obtained solution is the true optimum, it can give a valuable engineering solution for a design including nonlinear analysis or a robust design The present research first builds the kriging models for the strain energy density, tension at the bead region and tire s vertical stiffness Next, the tension at the bead region and the tire s vertical stiffness are set as constraints Then the shape design variables minimizing strain energy density and satisfying the constraints are determined Since each response is replaced with a metamodel, it is expressed as a function of the design variables, facilitating the shape optimization The computer time for solving this formulation is not greatly restricted Accordingly, even if the algorithm for searching the global optimum, such as the tabu search, a genetic algorithm and simulated annealing algorithm, is introduced, it does not become a burden on the computer time The present research utilizes a simulated annealing algorithm to solve the approximated formulation 2 Structural analysis of a tire 21 Tire performance Because the durability of a tire is the most important performance with respect to passenger safety and vehicle protection, it should be foremost considered in a design process Especially, the rotary motion of a tire translates to loading and unloading in a continuous periodic form Therefore, a tire is a structure that is vulnerable to fatigue In terms of durability, failure most likely initiates when the separation occurs at both steelbelt edge and shoulder zone [14-16] Therefore, a quantitative analysis based on strain energy density is suggested in which both stress and strain can be considered simultaneously in an effort to reflect actual displacement on the interface [17, 18] Hence, the durability of a tire can be relatively evaluated by the strain energy density Namely, a tire with low strain energy density for a load can be considered to have relatively high durability The tension at the bead region that contributes to the mounting of the tire in the rim is an important factor influencing maneuverability [19] The tension at the bead region influences the force restoration of the steering wheel during longitudinal driving or vehicle turns The tire is the first vehicle part to absorb load from the road surface during driving Also, the tire has decisive influence on ride comfort by its harmony with the suspension parts of the vehicle The influence of the tire on ride comfort can be assessed by evaluation of the vertical stiffness The present research calculates three kinds of responses, namely, the strain energy density, tension at the bead region and vertical stiffness through finite element analysis 22 Finite element analysis of tire The tire model used in the present research is P195/65R15, and the finite element analysis is carried out by using Abaqus [20, 21] For the finite element analysis of the tire, the material nonlinearity is considered, using Mooney-Rivlin model and rebar element [16, 20] The tire analysis simulates the behavior of a tire with a specified air pressure in contact with the ground, when the tire receives a load The two-dimensional and three-dimensional finite element models as well as the respective load and boundary conditions are represented in Fig 1 The three-dimensional model is constructed by performing a two-dimensional finite element analysis since a tire can be regarded as an axisymmetric model That is, the three-

3 D W Lee et al / Journal of Mechanical Science and Technology 25 (1) (2011) 149~ Fig 3 Design procedures Define the design variables Generate sample points using Latin hypercube design Perform the FEA using ABAQUS Construct the surrogate models using Kriging Solve the optimization problems using Kriging-SA algorithms dimensional model is defined based on the result of the twodimensional finite element analysis First, the tire contour is modeled for the two-dimensional analysis, as shown as Fig 1(a) A tire can show the imposed function after it is mounted in the rim and is injected with air to a specified pressure In order to simulate this situation, the node of the portion where the tire and the rim comes into actual contact is moved to the position where it is mounted in the rim by displacement control Then, uniform air pressure is applied inside the tire The applied pressure is 18 kg f /cm 2 for the analysis of the P195/65R15 tire Second, the deformed shape of the two-dimensional model becomes the primitive model for the three-dimensional finite element model shown as Fig 1(b) Because the tire contacts the ground, the contact condition between the tire and ground is given as the boundary condition The rigid area acting as the ground is fixed and comes in contact with the finite element model while the rigid area ascends to fixed distance Afterward, the finite element analysis is carried out by applying a load of 515 kg f, which is used for the analysis of the P195/65R15 tire 3 Optimization of a tire contour 31 Definition of design variables The existing researches of tire contour optimization [9-11] considered the tread contour and the sidewall contour independently The interaction between the tread contour and the sidewall contour is neglected The present research selects design variables considering both the tread contour and the sidewall contour at the same time; this is indicated in Fig 2 In the process of tire contour design, the dimensions of OD, SW and RW in Fig 2 can be regarded as constants for the given size of a tire Usually, design variables are set up as the radii R 1 and R 2, lengths L 1 and L 2 and height H in Fig 2 Then, the contour of tread can be determined by R 1, R 2, L 1 and L 2, while the contour of sidewall can be determined by L 1, L 2 and H 32 Formulation of design problem Tire manufacturers set their target values for the tension of the bead region and vertical stiffness in their design process Thus, the tension of the bead region and the vertical stiffness can be set as constraint functions On the other hand, the strain energy density of an end of the belt region can be set as an objective function for durability improvement The design problem can be formulated as follows: Find R 1, R 2, L 1, L 2 and H to minimize S E (1) subject to σ T σ a K V K a 4000 mm R mm 1800 mm R mm 300 mm L mm 700 mm L mm 574 mm H 765 mm, where S E, σ T, K V, σ a and K a mean the strain energy density of an end of the belt region, tensile stress of the belt region, vertical stiffness, allowable stress of tensile stress of the belt region and allowable vertical stiffness, respectively The use of the gradient based optimization method to determine the optimum of Eq (1) leads to the following problems First, shape variables are difficult to define and nonlinear analysis lead to excessive calculation time Also, if the shape variables defined in Eq (1) move, the finite elements can become distorted in the optimization process In order to solve these problems, the objective and constraint functions in Eq (1) can be replaced as follows: minimize S (2) subject to σ σ a K K, E T V a where means an estimated value of each response The present research builds a surrogate model of each response by using the kriging interpolation method 33 Kriging interpolation method Kriging is an interpolation method named after an engineer named D G Krige, who developed the technique while trying to increase the accuracy in predicting ore reserves In the kriging model [11-13], the approximation model y (x) for a response y (x) is represented as T 1 y( x) = β + c ( x) C ( y β i ) (3) where x = [R 1, R 2, L 1, L 2, H] T, β is a known function, c is the correlation vector, C is the correlation matrix, y is the observed data, and i is the unit vector In this study, y (x) can be replaced by S E, σ T and K v, respectively The correlation matrix and the correlation vector are defined as Eqs (4) and (5) [11-13], respectively

4 152 D W Lee et al / Journal of Mechanical Science and Technology 25 (1) (2011) 149~155 Table 1 Design of experiments and FE analysis results No Design variables Responses R 1 R 2 L 1 L 2 H S E (J/mm 3 ) σ T (kg f /mm 2 ) K V (kg f /mm) Table 2 Validation of for Kriging models Design variables S E ( J/mm 3 ) σ T (kg f /mm 2 ) K V ( kg f /mm) R 1 R 2 L 1 L 2 H Kriging FEA Kriging FEA Kriging FEA Table 3 Optimization results Average % error (%) R 1 R 2 L 1 L 2 H S E (J/mm 3 ) σ T (kg f /mm 2 ) K V (kg f /mm) Initial value Optimal value C( x, x ) Exp x x (4) n j k j k 2 = θi i i i = 1, (j=1,, n s ), (k=1,, n s ) c (x)= [C (x,x (1) ),C (x,x (2) ) C (x,x (ns) )] T (5) where n is the number of design variables, θ i is the i-th correlation parameter, and n s is the number of observed data In this study, n=5 and n s =50 To determine a kriging model expressed as Eq (3), the parameters θ should be defined The unknown correlation parameters are calculated from Maximize n s 2 ln( σ ) + ln C 2 where σ2 is the estimated variance of a metamodel, and θ i should be greater than 0 To solve Eq (6), the modified feasible direction method is utilized (6) The validity of an approximated kriging model can be evaluated as several indexes In this study, the error percentage are defined as t i= 1 n 1 t yi yi Average % error = 100 (7) n y where n t is the number of sample points for validation, which is set to 10 in this study As the kriging model is a metamodel, it always accompanies an error Thus, the calculated optimum may violate the imposed constraint In order to prevent this, the constraints of the Eq (2) are revised as follows: T i σ σ a - ε σ (8) V K K a - ε K,

5 D W Lee et al / Journal of Mechanical Science and Technology 25 (1) (2011) 149~ Table 4 Kriging and FEA results at optimum design S E (J/mm 3 ) σ T (kg f /mm 2 ) K V (kg f /mm) Kriging FEA Error (%) (a) Initial design Fig 4 Comparison between initial and optimum designs where ε σ and ε K are the penalty terms of tensile stress and vertical stiffness, respectively, considering the errors of the kriging model The approximate design process is summarized in Fig 3 First, design variables are defined As the second process, sample points are generated by using the latin hypercube design In this study, the latin hypercube design of Eq (9) is utilized to define the sample points (b) Optimum design Fig 5 Strain energy density at the initial and optimum designs (a) Initial design Minimize ns ns 1 d (9) i= 1 j= i+ 1 ij where d ij is the distance between points i and j After the finite element analyses on the generated sample points are executed, the kriging metamodels of strain energy density, tension at the bead region and vertical stiffness of the tire are constructed Then, these models are optimized by considering Eqs (1), (2) and (8) The simulated annealing algorithm is applied in this process, and the pseudo objective function is defined as follows: Minimize φ = SE + (10) α Max 0, ( σt σa + εσ ),( KL Ka + εk) where α is an arbitrary large number 4 Results and discussions The finite element analyses were carried out by generating sample points of n s =50 to generate the kriging metamodel of the strain energy density, tension of the bead region and vertical stiffness The DOE results are indicated in Table 1 Table 2 gives the kriging error of each response calculated with respect to test points of n t =10 The test points to assess the metamodel are selected by random sampling It shows an error of 37 to 50% for each response Accordingly, the allowable (b) Optimum design Fig 6 Tensile stress at the initial and optimum designs Displacement Initial Optimum Load (kgf) Fig 7 Vertical stiffness at the initial and optimum designs values of ε σ and ε K are respectively set to 5% in Eq (8) The optimum values of the design variables determined from Eq (10) are indicated in Table 3 and Fig 4 The strain energy density improved by about 298 % than the existing value was obtained The responses predicted by the kriging

6 154 D W Lee et al / Journal of Mechanical Science and Technology 25 (1) (2011) 149~155 models and the true responses by the finite element analysis in the optimum are compared in the Table 4 The responses were in good agreement within an error of about 5 % In order to increase the durability while satisfying the constraints related to maneuverability and ride comfort, the design variables should be set such that the dimension of L 1 is increasing and that the dimensions of radii R 2 and L 2 are decreasing, as shown in Table 3 and Fig 4 The strain energy density, tensile stress and vertical stiffness of the initial and optimum designs are shown as Figs 5-7, respectively Fig 5 shows the result of the strain energy density: the strain energy of the belt edge region was reduced Fig 6 shows the tension of the bead region: the tension of the bead region was increased And Fig 7 shows the displacement of the tire according to load change for measurement of the vertical stiffness of the tire: the vertical stiffness of the optimum design was reduced from that of the initial design 5 Conclusions (1) The present research proposes design variables considering both the tread design and the sidewall design of a tire, and the optimum shape of the tire contour is suggested to obtain the design that maximizes the durability while satisfying the constraints related to maneuverability and ride comfort (2) The shape optimization of a tire, which is difficult due to excessive computer calculation time, irregular finite elements and difficulty in the setting of shape design variables, was solved by applying the kriging metamodel technique (3) Because durability is very important among the performances of a tire with respect to passenger safety and vehicle protection, it should be considered foremost in the tire design process Thus, the present research set the tension of the bead region and vertical stiffness as constraints and the strain energy density as the objective function before the optimization was carried out As a result, the strain energy density was improved by about 298 % from the existing value Acknowledgment This work supported by the Korea Research Foundation Grant funded by the Korea Government (KRF D00073) References [1] S J Lee and C Y Shon, Navigating of EU Environment Regulation for Tire, Auto journal, Journal of KSAE, 31 (6) (2009) [2] H W Kim, T Kim, J G Cho and S S Mun, Tire Engineering, Goldenbell (2004) [3] R B Day and S D Gehman, Theory for the meridian Section of Inflated Cord Tires, Rubber Chemistry and Technology, 36 (1963) 11 [4] K Yamagishi, M Togashi, S Furuya, K Tsukahara and N Yoshimura, A Study on the Contour of the Radial Tire:Rolling Contour Optimization Theory- (RCOT), Tire Science and Technology, TSTCA 15 (1) (1987) 3-29 [5] J R Cho, S W Shin and W S Yoo, Crown Shape Optimization for Enhancing Tire Wear Performance by ANN, Computers & Structures, 83 (12-13) (2005) [6] G Unnithan, R KrishanKumar and A Prasad, Application of a 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7 D W Lee et al / Journal of Mechanical Science and Technology 25 (1) (2011) 149~ [20] Abaqus Standard User's Manual, Version 68, Abaqus Inc [21] H G Kwak and H J Kim, An Introduction to Computer Aided Engineering of Tire, Auto Journal of KSAE, 31 (3) (2009) D W Lee is a visiting professor of Mechanical Engineering at Dong-A University, Busan, South Korea He earned his BS in Mechanical Engineering degree in 1995, and his PhD in 2003, from Dong-A University His research interest lies in optimum design and fatigue K H Lee is an associate professor of Mechanical Engineering at Dong-A University, Busan, South Korea, where he has taught and conducted numerous research studies since 2002 He earned his BS in Mechanical Engineering degree in 1989, and his PhD in 1997, from Hanyang University He is currently working on developing a robust design methodology