1.Introduction
An automobile ball joint is the element that can be rotated in every degree of freedom for the steering, connecting suspension and steering systems. In existing research1,2, the quality of caulking process and the structural performance were predicted by performing numerical analysis.
The caulking process of the ball joint is the work of assembling stud, plug, socket and seat. Through this process, the ball joint is assembled as the plastic deformation generates at the top of the socket.
Structural responses of pullout strength, gap stiffness, operating torque, etc. are commonly considered in developing a ball joint. A car maker or part manufacturer has its own design requirements related to the structural responses. It is known that among the structural responses, pullout strength and gap stiffness are the most important performance in the design process. The pullout strength is the required force to pull the stud out from the ball joint assembly when applying the vertical load on the stud after fixing the bottom1,2. If the pullout strength is less than the allowable value, the ball joint is considered as an infeasible design. The gap stiffness is evaluated base on the value of displacement that generates when applying load on the stud after fixing the socket. If the gap stiffness is too high or too low, it degrades the performance of NVH.
Jang and Sin1,2 calculated the pullout strength and the gap stiffness using the commercial software called the DAFUL3,4. One of the research objectives was to inspect the caulking quality using threedimensional dynamic analysis. In addition, in Ref. [1], two shape design variables were defined and the optimum design was suggested by applying the metamodel based optimization technique. In Ref. [2], by applying the design of experiments, the optimum design considering the pullout strength and gap stiffness was suggested.
The deterministic numerical analysis in the existing research outputs a constant pullout strength and gap stiffness since it does not include any noise factor. However, the pullout strength and the gap stiffness have the distributions due to the variation on the noise factor, respectively. Thus, it is more realistic to suggest its distribution rather than the deterministic value in predicting the pullout strength and the gap stiffness. In this research, the noise factors are selected as the material properties of the seat and the tolerances of design variables. The material of the seat, one of the components of ball joint, is nylon. The material property of nylon fluctuates relatively heavily when the material test is conducted repeatedly. Thus, nylon’s Young’s modulus and yield strength are considered as the noise factors. In addition, the dimensions of the stud and the plug have a huge influence on the pullout strength and the gap stiffness. Thus, they are defined as the design variables, while their tolerances are set up as the noise factors. Then, the parameter design scheme proposed by the Taguchi method is applied for the robust design. The final design is recommended by considering the worst case of the structural responses.
The threedimensional flexible multibody analysis in Refs. [1] and [2] was performed using a commercial software, DAFUL3,4. The pullout strength and gap stiffness analysis was sequentially performed, following the caulking process. This sequential analysis has a strong advantage in that it can be analyzed by considering the residual stress. However, one analysis time for the initial design took 71 hours for a caulking analysis, 10 hours for a pullout strength calculation by using the 3GHz PC2. Furthermore, the pullout strength calculations could be carried out dozens or hundreds times to obtain a robust design. Thus, when using the threedimensional analysis, it is impossible to compute the robust design solution due to the long calculation time. In this research, the twodimensional analysis substitutes for the threedimensional analysis using a commercial software, Abaqus5.
2.PullOut Strength Analysis of the Ball Joint
2.1.Threedimensional analysis of the ball joint using DAFUL
The ball joint used in this study is the product being installed in the midsize car of A company. This ball joint consists of stud, plug, socket and seat, and the threedimensional analysis using the flexible multibody dynamics was already conducted in the existing research 2. This research is the followup study of Ref. [2]. The base design of the ball joint was completed by using the CATIA. Based on this, the threedimensional dynamic analysis was performed, and the finite element model of each component was shown as in Fig. 1. When analyzing the caulking process, the roller and the pusher were modeled as the rigid body1,2 .
The analysis of the caulking process can be summarized as follows. First, a temporarily assembled ball joint is set in the caulking machine. Then, the bottom of the socket is fixed to a jig, and the pusher goes down to fix the temporarily assembled ball joint. After that, the pusher stops, and two rolling rollers drop to the top of the socket. At this time, the plastic deformation generates at the top of the socket, leading to bending the top of the socket and attaching it to the plug. The contacts that should be considered in the analysis are represented in Fig. 2.
The caulking process can be evaluated qualitatively through the threedimensional analysis, and inspecting a plastically deformed shape of the socket. The pullout strength was sequentially performed, following the caulking analysis in the existing research1,2 . However, the threedimensional analysis induces the excessive computer calculation time. For the boundary condition in the finite element analysis, all the degree of freedoms that define the outer diameter of the socket are fixed. The pullout strength is determined by investigating the forcedisplacement curve. In the base design of ball joint, the pullout strength was calculated as 33kN and shown in Fig. 3. The caulking analysis using the commercial software DAFUL took 71 hours, and the pullout strength analysis took 10 hours. The flow stressstrain curve, the material and material property of each component, is included in the Ref. [2].
Stressstrain curve, the material and material property of each component, is included in the Ref. [2].
The contact boundary condition is given to the combination with the contacting part. The contact boundary conditions applied to the combination of parts are plug and seat, plug and stud, socket and seat, seat and stud, pusher and plug, roller and socket.
2.2.Twodimensional analysis of the ball joint using Abaqus
The caulking analysis using threedimensional flexible multibody dynamics is essential for evaluating the quality of the caulking. On the other hand, it is more efficient to adopt the twodimensional analysis rather than the threedimensional analysis for the pullout strength prediction. Especially, a number of analyses is required to consider the effects of noises in the robust design. If the pullout strength is obtained from the threedimensional analysis, it will be impossible to implement the robust design due to excessive computing time. Thus, in this research, the twodimensional analysis is adopted to perform the pullout strength analysis of the ball joint.
The CAD model for the twodimensional analysis is shown in Fig. 4. Because its geometry has the bilateral symmetry, the symmetric condition is imposed on the center line along the yaxis. The boundary condition, loading condition and contact condition are set up in the same way as the threedimensional analysis. For the pullout strength analysis using the twodimensional finite element, the commercial software, Abaqus5 respectively.
In addition, the gap stiffness analysis is performed using the twodimensional finite element model. The boundary condition for the gap stiffness is the same as that for the pullout strength analysis. The loading condition was set up as the axial load with the magnitude of F0 applied on the stud. The gap stiffness of the base design was calculated as 0.3 mm.
The difference between the two results is about 10%. In general, a number of analysis are required to include the influence of noise effect in the robust design. When the pullout strength analysis is performed by threedimensional analysis, it is sometimes impossible finish the analysis due to excessive computation time. For this reason, we decided that it would be possible to replace the threedimensional analysis result by the twodimensional analysis result.Fig. 5
3.Robust Design of the Ball Joint Considering the Noise Factors
The uncertainty of the manufacturing tolerance or the material property may affect the performance of the ball joint. However, the existing research1,2 neglected these uncertainties, and the performance of ball joint was predicted as the deterministic value. In this research, these uncertainties are considered to determine the robust design. The robust design of the ball joint considering the structural responses is suggested by applying the parameter design suggested by Dr. Genichi Taguchi.
3.1.Parameter design in the Taguchi method
The parameter design is applicable for both the product design and the process design. The main purpose of the parameter design is to minimize the distribution of performance that is generated by the uncontrollable design and, at the same time, to find the combination of design variables that makes the mean value of characteristics approach the target value6~9,1011.
Dr. Taguchi suggested the use of orthogonal array in an efficient way in order to reduce the number of experiments, and defined the SN ratio (signaltonoise ratio) as the index to evaluate the robustness. The SN ratio is derived from the loss function according to the kinds of characteristic. The SN ratio is expressed as the following equations; they correspond to the characteristics of the smallerthebetter type, largerthebetter type, and nominalthebest type, respectively6~9,1011.
for smallerthebetter type characteristic
for largerthebetter type characteristic
for nominalthebest type characteristic
where n_{s} is the number of experiment considering the noise in ith experiment, y_{ij} is the characteristic measured in jth experiment of ith experiment, y_{i} is the sample mean of the characteristic of ith experiment, and s_{i}^{2} is the sample variance of the characteristic of ith experiment.
The robust design is determined as the combination that makes the SN ratio, which is defined in Eqs. (1)~(3), be maximized. Because the SN ratio is derived from the expectation of the loss function, the effect of the average and the standard deviation is related and coupled in many design problems^{1012}. Thus, this research adopts the worstcase analysis considering the average and standard deviation of response.
3.2.Definition of design variables and noise factors
The design variables that are expected to be the largest influence on the pullout strength are defined in Fig. 6. The shape design variable A is the radius of the ball stud, and the shape design variable B is defined as the angle between xaxis and the socket’s slope. The manufacturing tolerances of design variables A and B are set to △A=1.0mm and △B=1.0°, respectively. They are included in the noise factors. The material of the seat is nylon. However, the nylon tends to have a large variation in its material properties when conducting the material experiment. Thus, the nylon’s yield strength and Young’s modulus are added in the noise factors. The deviations of Young’s modulus E and yield strength σy are assumed as △E=1446 MPa and △σy=20 MPa, respectively.
The design variable, called the control factor in DOE (design of experiments), is set to threelevel in the design range. It is assumed that the design variable and noise factor have normal distribution as shown in Fig. 7. In Fig. 7, x , s and △x represent the mean of the design variable, the standard deviation and the tolerance. According to the normal distribution, the probability of the design variable and the noise factor being between the LSL (lower specification limit) and the USL (upper specification limit) is 99.7%. The distance between the LSL and the USL is 6s, thus △x=6s. The levels of the design variable and the noise factor are represented in Table 1. The initial design is assigned to the second level, the one step lower value than the initial design to the first level while the one step larger value to the third value. The three levels of each noise factor are determined so their mean and variance become A, B, E or σ_{y} and s_{A}^{2},s_{B}^{2},s_{E}^{2} and s _{ σy}^{2}, respectively^{6,8,12}. Fig. 8
3.3.Conducting of the experiments
Because the number of design variables and the number of levels are two and three, respectively, the number of experiments for the inner array is set to 32=9, considering the full combination. On the other hand, the number of the noise factors is four, and the number of levels is three. Thus, if the full combination experiment is chosen as the outer array, the number of experiments in the outer array becomes 34=81 for one row of the inner array, which requires 9×81=729 experiments for nine rows of inner array. That demands total 729 times finite element analyses. To prevent excessive computing time, the L9(34)orthogonal array is adopted as the outer array. When using the orthogonal array as the outer array, the number of the finite element analysis decreases from 729 to 81. The relation between the inner array and the outer array is shown in Fig. 6.
Through 81 times finite element analyses, the means, the variances and the SN ratios of pullout strength and gap stiffness are calculated for every row of the inner array. The pullout strength is classified as the largerthebetter type response. Thus, the SN ratio for the pullout strength is calculated using Eq. (2). On the other hand, though the gap stiffness could be classified as the nominalthebest type response, in the given design range, it could be considered as the smallerthebetter type response. Therefore, the SN ratio for gap stiffness is calculated using Eq. (1). The SN ratios, means, variances and worstcase values of pullout strength and gap stiffness in the inner array are summarized in Table 11. The worstcase responses of pullout strength and gap stiffness in this research are represented as(4)(5)
where p , G ,s_{p} and s_{G} are the means and the standard deviations of the pullout strength and the gap stiffness, respectively.
The pullout strength and the gap stiffness have their distributions due to the distributions of the noise factors. It is assumed that the distributions of the pullout strength and the gap stiffness are the normal distribution. When the worstcase response of the structural performance of the ball joint does not violate the border defined as the allowable value for the structural performance, it means that 99.7% of the ball joint products meets the design requirement. From Table 11, it can be seen that the worstcase responses of the pullout strength have larger than its allowable value fo in No. 7, 8, and 9. Only the worstcase responses of No. 1, 2 and 3 do not satisfy the design requirement related to the gap stiffness. Thus, we can select an optimum design as No. 9 since its worst case of the pullout strength has the largest value and its worst case of the gap stiffness has lower than the allowable value δ_{0}.
If we utilize the SN ratio as the index to obtain the robust design, when considering the pullout strength only, No. 9 is the best, while No. 4 is the best when considering the gap stiffness only. The tradeoff decision between the two responses will be made to determine the final robust optimum levels. But, if we do tradeoff only using the SN ratio, we will get the solution worse than that from using the worstcase analysis. In this research, No. 9 in the inner array is selected as the final robust solution. That means 99.7% of the ball joint products meets the design requirement. On the other hand, No. 5 in the inner array is the initial design, and just 44.8% of that meets the design requirement for the pullout strength.
The table below came out via an inner array of Fig.6 with outer arrays. For example, in the first outer array, both of the design variables A and B are one level, and experiments are performed in consideration of 1 level noise factor. 1 level of the design variable is described in Table 1, and the 2 level and 3 level are the same. Table 2Table 3Table 4Table 5Table 6Table 7Table 8Table 9Table 10Table 11
4.Conclusions
The robust design strategy applicable for the development process of the automobile ball joint is suggested, and the conclusions are as followsTable 11

(1) The existing threedimensional dynamic analysis is substituted with the twodimensional finite element analysis for predicting the pullout strength. To investigate the quality of the caulking process, threedimensional analysis is required, but for the calculation of the pullout strength and the gap stiffness, the twodimensional analysis has sufficient confidence.

(2) In this study, nylon’s Young’s modulus and yield strength are considered as the noise factors. In addition, the tolerance of the diameter of the stud's ball and the tolerance of the angle of the socket’s slope are also considered as the noise factors. It can be seen that the distributions of the pullout strength and the gap stiffness due to the noise factors should not be neglected. The distributions of the structural responses could be predicted by applying the DOE and the Taguchi method.

(3) We investigated the robust solutions determined from the SN ratio and the worstcase analysis. The final robust solution is selected considering the worstcase analysis, and 99.7% of this ball joint design meets the design requirement. The probability of design success is 55% higher than that of the initial design.

(5) This study focuses on the numerical anlysis. For the future study of this research, it is required to compare the numerical results with the experimental results.