Robust design of an automobile front bumper using design of experiments

Transcription

1 1199 Robust design of an automobile front bumper using design of experiments Kwon-Hee Lee* and Il-Kwon Bang Department of Mechanical Engineering, Dong-A University, Busan, Republic of Korea The manuscript was received on 9 March 2006 and was accepted after revision for publication on 12 May DOI: / JAUTO311 Abstract: Bumpers are structural components installed to reduce physical damage to the front and rear ends of a passenger motor vehicle from low-speed collisions. Damage and protection assessments are the commonly used design criteria in bumper design. For damage assessment, the relative displacements representing stiffness performance are defined and examined. At the protodesign stage for a new car, finite element (FE) analysis is often utilized to predict the stiffness of a bumper. However, conventional bumper analysis through FEM outputs a constant stiffness even though the stiffness has some distribution due to uncertainties. In this research, the uncertainties are assumed to be the tolerances of thicknesses. Under this uncertain condition, the displacements representing stiffness are calculated by approximate statistics and by worst-case analysis. Then, a robust design is determined by design of experiments (DOE) using the orthogonal array strategy to find the design having a minimum weight of bumper within the stiffness constraints. In this research, the thicknesses of the inner beam, outer beam, and stay are treated as design variables. The robust design procedure for a bumper, considering the uncertain thicknesses, is presented. Keywords: front bumper, finite element method, robust design, design of experiments 1 INTRODUCTION Bumper design has a limitation in the protodesign stage: it must contribute to the overall weight Bumpers are structural components installed at the reduction of the vehicle to enhance vehicle mileage front and rear ends of a vehicle to protect the car- but meet safety standards. The test criteria for a car body, accessories, and passengers during low-speed bumper have been enacted under various national collisions [1, 2]. Bumpers are generally composed of requirements. The test conditions for evaluating beams, stays, shock absorber, and fascia. Bumper the performance of a bumper are specified in 49 beams, composed of inner and outer beams, are CFR (Code of Federal Regulations) Part 581 of mainly formed by pressing or rolling two plates and the National Highway Traffic Safety Administration then welding them. Stays connect the bumper and (NHTSA) in the United States [1], CMVSS (Canada carbody. The shock absorber installed between the Motor Vehicle Safety Standard) 215 in Canada [2], beam and fascia absorbs the impact energy. When it and ECE regulation 42 in Europe [6]. is installed in a structure to be protected from shock, After setting the external appearance, called styling, its weight is not a problem but its volume may be. and finishing the concept design of the bumper, Bumper beams are usually made of steel, aluminium, computational simulation analysis of the bumper plastic, or composite material [3 5]. By constrast, for can be carried out to evaluate bumper safety. In this the shock absorber, low-density foam material is research, finite element (FE) analysis was carried used. out, for which the pendulum and barrier test conditions * Corresponding author: Department of Mechanical Engineering, specified in CMVSS 215 were applied. The Dong-A University, 840 Hadan 2-Dong, Busan, , Republic test bumper was the front bumper of an arbitrary of Korea. leekh@donga.ac.kr passenger car.

2 1200 Kwon-Hee Lee and Il-Kwon Bang For each test condition, two responses are defined gram PAM-CRASH [10] is used to calculate the to investigate the bumper stiffness. The first response displacements generated in the bumper structure. of whether the pendulum or barrier hits the grille or not during collisions is determined. The second response of whether the inner beam of the bumper comes into contact with the cooling system or not 2 FE ANALYSIS FOR BUMPER DESIGN during collisions is determined. The two responses 2.1 Loading conditions and FE analysis are obtained by calculating the displacements The CMVSS 215 stipulates the longitudinal impacts to generated in the bumper structure. The allowable the front of the vehicle at 5 mile/h and the pendulum value of each response is the magnitude determined impacts on the corner of the vehicle at 3 mile/h. at the initial stage of the development process. The detailed test conditions are given in CMVSS Satisfaction of such stiffness criteria minimizes the 215 [2]. The loading conditions for FE analysis are repair costs associated with low-speed collisions. summarized in Table 1. Each loading condition in A conventional bumper analysis through FEM Table 1 corresponds to each FE model in Fig. 1. outputs a constant stiffness because it does not con- In Table 1 and Fig. 1, x is the longitudinal direction sider any uncertainty [4, 7]. However, the stiffness, of the carbody coordinate system, y is the width measured as displacement, is very sensitive to the direction, and z is the height direction. In Table 1, h thickness of each part of the bumper. That is, the y is the angle between the y axis of the test vehicle and variation in thickness of a bumper part cannot the plane of the pendulum. be neglected as it induces a variation in stiffness. For FE analysis of the front bumper, FE half- Thus, stiffness should be considered as a distribution models are utilized in loading cases 1, 2, and 7 and not a constant value. This research proposes because these loading cases apply symmetric loada method for evaluating the two displacements ing. By contrast, full FE models are utilized for the related to bumper stiffness and a method for select- remaining loading cases. The pendulum and barrier ing the optimum thicknesses with account taken are modelled as rigid bodies. As shown in Fig. 1, the of the tolerances of design variables. The design foam, inner and outer beams, stays, and longitudinal variables are the thicknesses of the inner beam, members are included in the FE model, and the outer beam, and stay. To calculate the statistics of remaining curb vehicle weight (CVW) is considered the displacements, the first-order statistical approximation as a lumped mass at point P. method and orthogonal array, called the outer array, are adopted. 2.2 Responses and initial design The design of experiments (DOE) is a statistical The front bumper should be designed to protect the technique used to study the effects of multiple varihood, grille, cooling system, and various lights during ables simultaneously [8, 9]. In the present research, a low-speed collision [1, 2]. Two responses for the extent of the DOE is confined to conducting the bumper stiffness are defined to investigate whether orthogonal array experiments and selecting optimum the protection requirements are met or not. The first levels. The orthogonal array strategy is adopted response investigated for each loading condition is to determine the robust optimum satisfying the whether or not the pendulum or barrier comes into stiffness requirements while minimizing the weight. contact with the other parts of the carbody when the The thickness set considered as a design variable for each member is not arbitrarily determined but selected from standard products, and thus it is a discrete set. Because of such design characteristics Table 1 Loading conditions for front bumper analysis and the calculation time of non-linear FE analysis, Velocity of the optimization algorithm for continuous space Position of impact line pendulum or Loading case x, y, z (mm) h (deg) vehicle (mile/h) could not be applied. Therefore, the orthogonal array y is adopted for the discrete design as a superior Pendulum impact 1 y=0, z= method. As the conventional DOE does not consider 2 y=0, z= the stiffness constraints, a characteristic function is 3 y=300, z= y=300, z= defined that accounts for the effect of constraint 5 h =30, z= feasibility. y 6 h =30, z= y The present research sets out a structural design Barrier impact procedure for a car bumper. The commercial pro-

3 Robust design of an automobile front bumper using design of experiments 1201 Fig. 1 Finite element model for front bumper analysis bumper collides with the pendulum or barrier. The is calculated. Then, the second relative displacement grille could be damaged if the impact of the collision along the centre-line is the greatest one, except those goes beyond the bumper. The second response of loading cases 5 and 6. For loading cases 5 and 6, investigated for each loading condition is whether only the first response is investigated since any part or not the inner beam of the bumper comes into of the inner beam at which the maximum second contact with the cooling system. relative displacement is generated cannot come into To consider the first response in the pendulum contact with the cooling system. test, the relative displacement in the x direction An allowable value of each response is determined between point P and the pendulum in Fig. 1 is calcu- during the fixing of styling and concept design. When lated. On the other hand, in the barrier case, the each relative displacement is less than the allowable displacement of point P in the x direction is investi- value, the components mentioned can survive in gated because the barrier does not move. To consider low-speed collisions. The allowable values for the the second response, the relative displacement first and second relative displacements are presented between point P and the inner beam of the bumper in Fig. 2. Fig. 2 Allowable values for relative displacements

4 1202 Kwon-Hee Lee and Il-Kwon Bang In the early protodesign stage of a new car, the 3 ANALYSIS CONSIDERING THE TOLERANCES thickness of the inner beam t, the thickness of OF THE DESIGN VARIABLES 1 the outer beam t, and the thickness of the stay t 2 3 are 1.2, 1.2, and 2.0 mm respectively. The relative 3.1 Definition of design variables, design displacements at the initial design are given in parameters, and uncertainties Table 2. For the nominal case of the initial design, In this research, the design variables are the thickthe resulting deformations with the maximum first nesses of the inner beam, outer beam, and stay, relative displacement are as shown in Fig. 3. In all which are represented in Fig. 1(g). In general, they cases, the maximum relative displacements take are allowed a 10 per cent tolerance with respect to place between 47 and 53 ms. the nominal thickness in carbody design. The thick- In the applied vehicle, the allowable displacements ness of a panel has a normal distribution when it is with respect to h and h are set up as d =80 mm 1 produced at a steel company. The distribution of the and d =60 mm respectively. Table 2 shows that the 2 panel thickness is not considered in a conventional relative displacements satisfy the constraints related FE analysis for bumper design, which produces a to stiffness. However, for loading case 7, the second constant relative displacement. Since the relative relative displacement h is very close to the allowable 2 displacements of the bumper are very sensitive to value, while the first relative displacement h is far 1 the thicknesses of its parts, the relative displacefrom its allowable value: the second requirement ments determined from conventional analysis do may not be met when the perturbations of design not offer a reliable design recommendation. Actually, variables are considered. Thus, a robust design is the relative displacement varies according to the strongly recommended when trying to calculate the variation in thickness. This research presents a optimum thicknesses of bumper parts. method for investigating the stiffness of a bumper Table 2 by considering the tolerances of the parts in the Relative displacements at bumper structure. the initial design (mm) The barrier test condition is the worst loading condition in conventional bumper analysis. Two relative Loading case h h displacements, h and h, are expressed as h (t, p)=d (t, p) d (t, p) (1) h (t, p)=d (t, p) d (t, p) (2) where t=[t,t,t ], p=[t,t ], and t and t are the thicknesses of the inner and outer longitudinal Fig. 3 Resulting deformation for the nominal case of the initial design

5 Robust design of an automobile front bumper using design of experiments 1203 members respectively. For the barrier test condition, As can be seen in equation (4), the sensitivity of h i_max d is treated as 0. The thicknesses of the longitudinal to each thickness should be calculated to obtain the 0 members defined as design parameters should be variance of h. On the other hand, when the outer i_max determined by considering crashworthiness as well array is adopted to evaluate the statistics in lieu of as static and dynamic stiffness. In the applied equations (3) and (4), the orthogonal array is used vehicle, their thicknesses are fixed as 1.6 mm. to calculate the mean and variance of h. In this i_max The related points to measure d and d are research, each random variable is assumed to be represented as P and Q respectively in Fig. 1(g). The characterized by a Gaussian distribution, its tolerance responses h and h are subject to change because of Dt is equal to 0.1 t:, and Dt =6s. Thus, per j j j tj the uncertainty with respect to the thickness of each cent of the random variable exists between t: 3s j tj member. That is, t and p can be regarded as random and t: +3s [12]. j tj variables, and h and h become the functions of random variables. When the nominal value of each 3.3 Worst-case analysis thickness is used, the time displacement curve for When the distribution of each random variable is loading case 7 in the initial design is as shown in normal and its function can be approximated as Fig. 4. linear, then the distribution of the function can be 3.2 Methods for evaluating statistics regarded as a normal distribution. Then, the worst case of the maximum relative displacement can be Three common methods for evaluating statistics such represented as as the mean and variance of a response according to the variations in random variables are the first-order w =m +3s, i_max hi_max hi_max i=1, 2 (5) statistical approximation method using Taylor series Using equation (5), at the initial design point, expansion, the orthogonal array known as the outer w satisfies the requirement for the first relative 1_max array, and Monte Carlo simulations [11]. Each displacement h, but w exceeds the allowable _max method has its strong and weak points, and the value d of relative displacement h. 2 2 details are described in reference [11]. In the present research, the first-order statistical approximation method and the outer array are adopted to evaluate the statistics of the relative displacements. 4 ROBUST DESIGN USING DOE Using the first-order statistical approximation From Table 2, case 7 is the severest among the seven method, the mean m and the variance s2 are loading cases, and its second relative displacement hi_max hi_max represented as [12] is very close to the allowable value d. The remaining 2 loading cases, however, give relative displacements m (t, p)$[h (t, p)], i=1, 2 (3) hi_max i_max t=t:,p=p: that are very marginal with respect to the allowable values. Therefore, robust design is applied, considers2 (t, p)$ 5 (t, p) i_max hi_max j=1 Cqh s2, i=1, 2 qt j D tj ing only loading case 7. After the robust optimum is t=t:,p=p: determined, the responses for the other loading (4) cases should be investigated to determine whether or not the two displacements are still marginal. 4.1 Definition of characteristic function The design for a front bumper is formulated as follows Find t=[t 1, t 2, t 3 ] (6) to mimimize W (t) (7) subject to w i_max (t, p) d i, i=1, 2 (8) In equations (6) to (8), t and p are composed of discrete variables, and furthermore h or w is i i_max determined from non-linear FE analysis. Thus, it is difficult to apply any gradient-based optimization Fig. 4 Time displacement curve method in solving equations (6) to (8). In this

6 1204 Kwon-Hee Lee and Il-Kwon Bang research, the DOE strategy using the orthogonal Table 3 Levels of design array is adopted to overcome the current limitation. variables (mm) Conventional DOE applying the orthogonal array does not consider the constraints defined in Level t 1 t 2 t 3 equation (8). Thus, the characteristic function W(t, p) is defined to include the constraint feasibility as follows [13, 14] W(t, p)= W (t) W 0 +V(t, p) (9) which two design variables are assigned for the first two columns and the remaining design variable for the fourth column [15]. Under the rule, the L (34) 9 V(t, p)=b 2 max w (t, p) i=1 i_max (10) d i D orthogonal array is made as shown in Table 4. The L (34) orthogonal array produces nine characwhere W (3.713 kg) is the weight of the bumper teristic function values, evaluating the mean and 0 9 beams and stay at initial design. Robust design variance for each experiment. In the present research, implies robustness of the objective function and the two approximation methods are applied to obtain constraint functions. However, robustness of the the statistics used to calculate the characteristic weight function is much less important than that function defined by equation (9). One method is the of the constraint function in equations (7) to (8). first-order statistical approximation method, and the Therefore, robust design in this research means con- other is the outer array method. For nine experiments straint robustness, as emphasized by the scale factor, and initial design, the statistics, w (i=1, 2), and i_max which is set to 0.2. characteristic function values determined from two In equation (10), the penalty function is equal approximation methods as well as their weights can to zero when the constraints of equation (8) are be summarized as in Table 5. satisfied but has a positive value when any constraint In the case of the first-order statistical approxiis violated. The characteristic function defined in mation method, the sensitivities in equation (4) are equation (9) is adopted as the response for DOE. calculated by using the central difference method. The sensitivity with respect to each variable is 4.2 Robust design using an orthogonal array For each design variable, the number of levels is set Table 4 L (34) orthogonal array (mm) 9 to 3. The second level is set up as the initial thickness. Experiment t t Error (level) t The first and third levels are fixed by the lower and 3 upper ones around the initial thickness, respectively, considering the standard table used in a steel com pany. The levels of design variables for an orthogonal array are determined as shown in Table 3. Then, an appropriate orthogonal array is selected. For a problem with three design variables and three levels, the L (34) orthogonal array is recommended, in 9 Table 5 Summary of L 9 (34) orthogonal array and initial design of experiments [m, s, w (mm)] First-order statistical method Outer array method Weight Experiment (kg) m h1_max s h1_max m h2_max w 1_max w 2_max W m h1_max s h1_max m h2_max w 1_max w 2_max W Initial

7 Robust design of an automobile front bumper using design of experiments 1205 Table 6 Sensitivity information using the first-order statistical method h 1_max h 2_max Experiment qt 1 qt 2 qt 3 qt 4 qt 5 qt 1 qt 2 qt 3 qt 4 qt Initial represented in Table 6. The sensitivities of design become t and s2 respectively [11]. For example, the j tj parameters p, the components of which are t and t, outer array for the first experiment in Table 4 is 4 5 are very small. That is, the effect of longitudinal presented in Table 8. The outer array method needs member thickness on bumper stiffness is not 9 9 finite element analyses, but the first-order severe. Thus, for the outer array method, only design statistical approximation method needs 11 9 finite variables t, the components of which are t, t, and t, element analyses. From Table 5 it can be seen that 3 are considered in the calculation of the statistics. the characteristic function value of the first-order When the combinations of design variables are fixed, statistical approximation method is almost the same the levels of the outer array are defined as shown in as that of the outer array method. Table 7. The three levels for the jth design variable Then, with the L (34) orthogonal array and 9 are determined so that their mean and variance characteristic function values, the optimum levels are determined on the basis of the DOE strategy. Table 7 Levels of design variables for the This process is called analysis of means (ANOM), outer array (mm) and the results are represented in Table 9. From Table 9, the optimum levels are determined as Level t 1 t 2 t 3 [level 1 level 3 level 3]=[1.0 mm 1.4 mm 2.3 mm], 1 t 3/2s t 3/2s t 3/2s and they are shown in bold type. Its estimated 1 t t2 3 t3 2 t1 t t characteristic function is calculated as t + 3/2s 1 t1 t + 3/2s 2 t2 t + 3/2s 3 t3 Ŵ=m +m +m t1 t2 t3 2m: (11) Table 8 L 9 (34) outer array for the first experiment of Table 4 (mm) Experiment t 1 t 2 Error (level) t 3 h 1_max h 2_max Table 9 Analysis of means (ANOM) for the characteristic function First-order statistical method Outer array method Design variable Level 1 Level 2 Level 3 Level 1 Level 2 Level 3 t t t

8 1206 Kwon-Hee Lee and Il-Kwon Bang Table 10 True and approximated statistics for h 2_max (mm) Approximation method First order Outer array Monte Carlo statistical method method simulation Design m h2_max m h2_max m h2_max Initial Optimum where m,m,m, and t1 t2 t3 m: are the summations of 5 CONCLUSIONS the characteristic function values to the optimum levels of t, t, and t divided by 3 and the mean of This research leads to the followings conclusions: 3 nine characteristic function values respectively. The 1. A discrete and robust design for a bumper that estimators determined from the two approximation can be applied at the protodesign stage was methods have values of and respectproposed, accounting for the lightweight design ively. The combination of optimum levels coincides criterion. The DOE strategy was a reliable design with that of the third experiment of Table 4 by methodology for the structural design of a bumper. chance. Therefore, a confirmation experiment to 2. The stiffness requirements for bumper design verify its estimator is not required. were defined in terms of relative displacements under the regulations related to automobile 4.3 Stochastic analysis using Monte Carlo bumpers. However, distributions in the relative simulations displacements result from distributions in the For the initial and optimum designs, the statistics of thicknesses. Thus, the worst-case analysis was h can be calculated by Monte Carlo simulations. introduced to enhance the constraint feasibility, 2_max In general, in mechanical engineering, a sample size which led to robust design. Finally, robust design between 50 and 100 for Monte Carlo simulation is was performed by defining the characteristic sufficient [16]. In Table 10, the true statistics for the function to include the weight effect and the initial and optimum designs are calculated from 50 constraint feasibility. At the same time, discrete Monte Carlo samples, and they are compared with design was performed, considering the standard table used in a steel company and adopting the the values determined from the two approximation orthogonal array. methods. 3. By applying the suggested method to a front For the test of normality, the statistic built in bumper, the weight of the bumper was slightly Origin [17] is adopted. With a significance level of increased by 2.2 per cent, satisfying the imposed 0.05, the two distributions with 50 samples satisfy the constraint which cannot meet the second requirenormality criterion. Thus, the distributions can be ment in the initial design. In this study, the design regarded as Gaussian, which shows that the worst-case variables were composed of only the size variables, analysis is a reasonable approach. excluding the shape variables. In addition to the Finally, the responses for the other loading cases size variables, the shape variables controlling the are investigated by conventional FE analyses. The beam cross-section can be studied to obtain a results are shown in Table 11, and they are still bumper of lighter weight. Future work will examine marginal with respect to the allowable values. the shape variables. Table 11 Relative displacements at the optimum design ACKNOWLEDGEMENT (mm) Loading case h 1 h 2 This work was supported by the Second-Phase of Brain Korea 21 Project in REFERENCES cfr/title49/part581.html

9 Robust design of an automobile front bumper using design of experiments Shah, P. and Danne, A. Stochastic analysis of mvsa/regulations/mvsrg/210/mvsr215.html frontal crash model. NAFEMS Seminar, Wiesbaden, 3 Fenton, J. Handbook of vehicle design analysis, 1996, Germany, 7 8 May 2003, pp pp (SAE International). 17 Origin 7.5 (OriginLab Corporation), Kim, S. H., Kim, M. H., and Ha, S. K. Design and structural analysis of bumper for automobiles SAE International Congress and Exposition, Detroit, Michigan, 1998, paper Clausen, A. H., Hopperstad, O. S., and Langseth, M. Stretch bending of aluminum extrusions for car APPENDIX bumpers. J. Mater. Processing Technol., 2000, 102(1/3), Notation 6 wp29regs41-60.html d displacement of the pendulum in 0 7 Yim, H. J., Kim, M. S., Park, J., Heo, S. J., and the x direction Park, D. K. Shape optimization of bumper beam d displacement of the lumped-mass cross section for low speed crash SAE World 1 point P in the x direction Congress, Detroit, Michigan, April 2005, paper d displacement of the inner beam at y=0 inthexdirection 8 Peace, G. S. Taguchi methods: a hands-on approach, 1995 (Addison-Wesley, Massachusetts). h, h first and second relative 9Roy,R.K.Design of experiments using the Taguchi displacements approach, 2001 (John Wiley & Sons, Inc., New York). h, h maximum h and maximum h in 10 PAMCRASH solver notes manual (Version 2000), 1_max 2_max the time interval 2000 (ESI). p, p: design parameter vector and its 11 Phadke, M. S. Quality engineering using robust mean design, 1989 (Prentice-Hall, Englewood Cliffs, New Jersey). t, t: design variable vector and its 12 Lee, K. H. and Park, G. J. Robust optimization mean considering tolerances of design variables. Comput. t:, s2 mean and variance of the jth Struct., 2001, 79(1), j tj random variable 13 Lee, K. H., Shin, J. K., Song, S. I., Yoo, Y. M., and w worst case of h (i=1, 2) Park, G. J. Automotive door design using structural i_max i_max W weight of bumper beams and stay optimization and design of experiments. Proc. IMechE, Part D: J. Automobile Engineering, 2003, 217(10), d, d allowable displacements with 14 Lee, K. H., Joo, W. S., Song, S. I., Cha, I. R., and respect to h and h Park,G.J.Optimization of an automotive side door m, s2 mean and variance of h beam, considering static requirement. Proc. IMechE, hi_max hi_max i_max (i=1, 2) Part D: J. Automobile Engineering, 2004, 218(1), W, Ŵ characteristic function and its Lee, K. H. and Park, G. J. Robust optimization in estimator discrete design space for constrained problems. Am. V, b penalty function and scale factor Inst. Aeronaut. Astronaut. J., 2002, 40(4), to enhance the feasibility