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The Response Surface Methodology (RSM) approach is adopted for data analysis for an experimental model in order to determine
the optimal component tolerances in an assembly. The response variable is the total cost, which consists of quality loss and
tolerance cost. RSM is a combination of mathematical and statistical techniques, which provides desi...
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... resultant variance, 2 s , is estimated as t 2 s /9 C 2 Pm with the assumption that U s is equal to T s . The term C Pm is known process capability index. The general worst-case scenario for the tolerance analysis of linear assemblies is where t si is the component tolerance of X si , and n s is the number of components, X si , in assembly dimension s [8]. Figure 1 shows an example of a shaft assembly which consists of components X1, X2, X3, X4, and X5. The associated compo- nent tolerances are t1, t2, t3, t4, and t5. The functionality tolerance limit for Y is 0.16 mm. The constant, K, can be estimated from the loss due to failure divided by the functional limits of the product [4]. In this example, K is 3.0 × 10 4 . There is only one assembly dimension in this example, so q is 1. The possible value for s is 1. Hence, n 1 is 5 in this example. The resultant variance, 2 , at assembly dimension Y is esti- mated as t 2 Y /9C 2 Pm , in which C Pm is assumed to be 1. The worst case of tolerance analysis for linear assemblies in this example is t Y 5 i=1 ti, where ti is the component tolerance of Xi. The process capability limits, t Li and t Ui , are given in Table 1. The term ti is one of three tolerance levels, t Li , (t Li + t Ui )/2, and t Ui for each component tolerance, which is selected to have various combinations according to the factorial experiment design given in Table 3 [2]. The tolerance cost for each level Table 2. Since there is only one assembly dimen- sion in this example, m is equal to 5. Then, the tolerance cost is 5 i=1 C M (ti). The response value, TC, is found by Eq. (7). All relevant information needed for the response analysis is shown in 151284.0000t4 2 + 142224.0000t5 2 . The optimal solutions are: t1* = 0.032894 mm, t2* = 0.032085 mm, t3* = 0.029233 mm, t4* = 0.026804 mm, and t5* = 0.026656 mm. The predicted response, TC, at optimal tolerance solutions = $946.320915. The variance of prediction at optimal tolerance solutions = 2.9222. This implies that the predicting system should provide a description of the prediction error as well as the prediction. Then, the prediction process should result in an estimation of the probability distribution of the variable, TC. This permits risk to be objectively incorporated into the decision-making process. The effect on the five components is significant at a 5% confidence level. Test statistic, F, from Table 4 will be used to rank the order of importance of the components. The F value is the mean square for a factor divided by the mean square for error. These values indicate that components X4 and X5 should be closely controlled. In case a design improvement is needed, X4 and X5 are given first priority. For the purpose of analysis, we often plot the contours of the response surface as shown in Fig. 2 to visualise its 3D shape. Each contour represents a specific response value for combinations of the levels of the factors. It may not be easy to solve the above systems manually; therefore, computer software programs, such as SAS or S+, can be applied to find the solutions without ...
Citations
... Generally, the type of relationship between the design evaluation score and the set of design component tolerances is too complex or is unknown physically. Jeang [16] adopted the Response Surface Methodology (RSM) for data analysis for an experimental model to determine the optimal component tolerances in an assembly. In the fields of rotor dynamics, Liu et al. [17] developed a modified time-dependent excitation model for describing different kinds of defects in the outer ring of angular contact ball bearings. ...
Tolerance is one of major sources of uncertainty and it significantly contributes to the variation of dynamic responses of mechanical structures/products. In this paper, a robust tolerance design method based on possibilistic concepts is presented in order to reduce dynamic response variation of a rotor system due to uncertainties of design variables and assembly process. The robust design model is constructed with the objective of minimizing both the variation of system performance and manufacturing cost. The full factorial numerical integration (FFNI) method instead of the direct Monte Carlo simulation (MCS) is used to calculate the variance (standard deviation) of stochastic responses. The approach is performed by four steps: (1) construct a parameterized model including design and assembly process parameters, (2) analyze the uncertainty propagation of design and assembly process parameters with the statistical tolerance analysis performed by the full factorial numerical integration (FFNI) method instead of the MCS, (3) determine initial space for tolerance design and set up multi-objective optimization functions, and (4) obtain the Pareto front using the evolutionary algorithm NSGA-II and the robust tolerance design results. A rotor of low-pressure turbine compressor with initial unbalanced discs assembly is taken as an example and the robust tolerance design to reduce vibration responses and their variations is undertaken to demonstrate the capability and effectiveness of the proposed approach. The results show that vibration responses of the rotor of low-pressure turbine compressor can be significantly decreased and the qualified rate of maximum vibration responses of the low-pressure turbine compressor rotor system can be increased from 53.29% to over 99.87% if the initial unbalance amplitudes and phase angles of the discs are considered and optimum matching of each disc assembly is determined. The effectiveness of the proposed approach is validated with the Monte Carlo simulation analysis. It indicates the practical potential in industrial applications.
... Therefore different mathematical models can be used, which are suitable for specific problems, depending on the application. The methods range from mathematically simpler methods, such as the response surface method [13], to complex methods, such as those used in artificial neural networks [14]. As the Kriging method is used in this article, a brief description will be given. ...
Scattering manufacturing process parameters affect the quality of manufactured products, because the product properties deviate from the nominal design. If the desired quality requirements are not met, individual part tolerances can be adjusted in such a way that the required reject rates are achieved. In this context, different methods can be used to determine the individual contributors and to quantify the influence of the individual part tolerances on the quality of the manufactured product. In all existing approaches, however, only geometric tolerances are considered but not the varying manufacturing process parameters that are responsible for the geometrical part deviations. To overcome this, in this article, a novel approach is presented that allows determining the influence of manufacturing process parameters on quality characteristics for systems in motion. The novel approach is also highlighted in a case study of an X-ray shutter.
... In the literature, the several cost-tolerance relationships have been reported [6]. Jeang [7] presented a method to allocate optimal tolerances of a mechanical assembly by minimizing the total cost by using the response surface methodology (RSM) approach on the experimental data. Choi et al. [8] presented an approach for the optimal tolerance design by minimizing a combined single objective optimization problem including the cost and the Taguchi's loss functions. ...
Interactive and integrated design and manufacturing can be a useful strategy for designers to reach the efficient design through the cognitive or physical interactions. Process tolerance design is a key tool in the integrated design and manufacturing to reach a product with high quality and low cost. Since the optimal tolerance allocation involves several aspects of the design, manufacturing and quality issues, it is always a time consuming and difficult procedure, especially for complex products. Therefore, to overcome these difficulties, a computer-aided approach for optimal tolerance design of manufacturing process is needed in the design stage. In this paper, a novel interactive framework is introduced for computer-aided multi-objective optimal process tolerance design established upon entropy-based decision making. According to the proposed method, the optimal process tolerances of components are allocated through a multi-objective optimization problem where the process capability function and the overall manufacturing cost should be simultaneously optimized. To model the proper objective functions, the new formulations of process capability and manufacturing cost functions are proposed based on the design and customer’s requirements on the virtual model in CAD software, and the experimental observations, respectively. The non-dominated sorting genetic algorithm II is used for solving the multi-criteria optimization. For automated decision making to find the best process tolerances from the optimal Pareto solutions without objective weighting, an improved entropy-based TOPSIS is used. Based on the obtained optimal process tolerances and specifications, the process planning procedure can be carried out. Finally, to illustrate the capability of the proposed method and to validate it, a windmill transmission assembly as a case study is considered and the computational results are compared and discussed.
... Zhang presented the simultaneous tolerance which works in concurrent engineering contexts. The method can determine an appropriate machining process without using functional tolerances, and optimal machining tolerances, but decrease manufacturing costs [5]. Weill R proposed the tolerance requirements from the design stage to the manufacturing stage of the method to solve the problem of technology in the design of tolerance, which presented research of CAT [7]. ...
Tolerance design plays a major role in the quality of the product life cycle. Its purpose not only can ensure the products quality. At the same time, it monitors the product manufacturing costs. The traditional the tolerance design devotes to the optimization target of minimizing the manufacturing cost. But due to the manufacturing process of a product, there is a nonlinear relation between tolerance and cost. It disregards the problem of quality loss which cannot reach the effect of the high precision and low cost. Depending on the relationship between the manufacturing costs and tolerance, the tolerance optimization design method under the multiple correlation characteristic products is put forward, deducing the features related to quality of the product within multiple function relation between the loss and tolerance. Concentrating on the loss to the quality problems in the manufacturing process, manufacturing cost and quality loss tolerance model is created. The purpose is to improve product quality and reduce costs. Example results show that the product of multiple correlation characteristics of manufacturing cost and quality loss of tolerance optimization design model has certain validity, and compared with the traditional this tolerance design having great superiority.
... Resolving a tolerance synthesis problem requires numerous analysis and simulation techniques, including: statistical tolerance modeling and analysis methods; UQ; optimization algorithms; and tolerance cost estimation approaches. A number of tolerance synthesis methods have been proposed with a range of different analysis techniques1234567891011121314. The major challenge in realizing effective tolerance synthesis is the impractically large computational cost required as the complexity of the mechanical assembly under analysis increases [1,151617. ...
Statistical tolerance analysis and synthesis in assemblies subject to loading are of significant importance to optimized manufacturing. Modeling the effects of loads on mechanical assemblies in tolerance analysis typically requires the use of numerical CAE simulations. The associated uncertainty quantification (UQ) methods used for estimating yield in tolerance analysis must subsequently accommodate implicit response functions, and techniques such as Monte Carlo (MC) sampling are typically applied due to their robustness; however, these methods are computationally expensive. A variety of UQ methods have been proposed with potentially higher efficiency than MC methods. These offer the potential to make tolerance analysis and synthesis of assemblies under loading practically feasible. This work reports on the practical application of polynomial chaos expansion (PCE) for UQ in tolerance analysis. A process integration and design optimization (PIDO) tool based, computer aided tolerancing (CAT) platform is developed for multi-objective, tolerance synthesis in assemblies subject to loading. The process integration, design of experiments (DOE), and statistical data analysis capabilities of PIDO tools are combined with highly efficient UQ methods for optimization of tolerances to maximize assembly yield while minimizing cost. A practical case study is presented which demonstrates that the application of PCE based UQ to tolerance analysis can significantly reduce computation time while accurately estimating yield of compliant assemblies subject to loading.
... Cheng and Maghsoodloo [2] established optimization model for tolerance assignment to minimize the total cost, which includes manufacturing cost and Taguchi's quadric quality loss. Jeang [3] applied the response surface methodology to determine the optimal component tolerances, where the response variable is the sum of manufacturing cost and quality loss. Feng et al. [4] combined the manufacturing cost and quality loss as the objective function and presented an integer programming approach to simultaneously select component tolerances and suppliers based on process capability indices. ...
Due to the effect of various factors, the product quality characteristics are sometimes non-normal distribution. In this paper, two most commonly used non-normal distributions, triangular distribution and trapezoidal distribution, are studied based on asymmetric quadratic quality loss. With asymmetric quality loss, it is necessary to optimize the process mean in order to reduce the expected quality losses. In order to optimize the process mean of the triangular distribution, the different distances between the target value and the process mean are considered, and two mathematical models are proposed to calculate the expected quality loss. Because the probability density function of the trapezoidal distribution is a three-segment function, in order to optimize the process mean of the trapezoidal distribution, three models are established. Solving the proposed models, analytical solutions for the optimal process mean are obtained, and equations for the minimum quality loss are established. Considering the sum of manufacturing cost and the minimum quality loss as the objective function, tolerance model is established to calculate the optimal tolerances. Therefore, the optimal process mean and optimal tolerances are obtained for triangular distribution and trapezoidal distribution. At last, an example is used to illustrate the validity of the established model.
... The relevant works can 40 be classified into the following categories. Jeang (1999a) developed an application for parameter determination via RSM simulation to make the link between computer-aided design (CAD) and CAM AQ3 sys-45 tems more useful and effective. Rahim and Shaibu (2000) derived a model for economic selection of optimal target value. ...
Conventionally, process mean determination is performed prior to process tolerance design, independent of product design, such as product specification. In addition, previous studies considered the production costs and quality losses as deterministic values in mean and tolerance decisions. This study foregoes the assumptions of deterministic value and independent determination. For further cost reduction and quality improvement, process mean, process tolerance and product specification are simultaneously determined as controllable factors under the assumption that production costs and quality losses are random variables with given probabilistic distributions. At first, the various levels of mean, tolerance and specification are combined in accordance with the Box–Behnken experimental matrix, and used as inputs for a Monte Carlo simulation to obtain the simulated outputs. Then, these outputs are transferred to the total cost, which includes quality loss, production cost and penalty cost. Mean, tolerance and specification are treated as controllable factors, while total cost is a response value of interest. The design problem is analysed statistically using response surface methodology (RSM) in order to find the response function, which in turn is used as an objective function and optimised through mathematical programming (MP). A bicurve lens design is employed to demonstrate the proposed approach.
... Hence, the optimal tolerance values can be obtained by balancing quality loss and manufacturing cost. [14][15][16][17][18] Conventionally, RPD is performed prior to TD for economic considerations. Both RPD and TD aim at reducing quality loss, although TD may lead to higher manufacturing cost. ...
Robust parameter design (RPD) and tolerance design (TD) are two important stages in design process for quality improvement. Simultaneous optimization of RPD and TD is well established on the basis of linear models with constant variance assumption. However, little attention has been paid to RPD and TD with non-constant variance of residuals or non-normal responses. In order to obtain further quality improvement and cost reduction, a hybrid approach for simultaneous optimization of RPD and TD with non-constant variance or non-normal responses is proposed from generalized linear models (GLMs). First, the mathematical relationship among the process mean, process variance and control factors, noise factors and tolerances is derived from a dual-response approach based on GLMs, and the quality loss function integrating with tolerance is developed. Second, the total cost model for RPD-TD concurrent optimization based on GLMs is proposed to determine the best control factors settings and the optimal tolerance values synchronously, which is solved by genetic algorithm in detail. Finally, the proposed approach is applied into an example of electronic circuit design with non-constant variance, and the results show that the proposed approach performs better on quality improvement and cost reduction. Copyright © 2012 John Wiley & Sons, Ltd.
... Jeang [4,5] emphasized that the functional performance of the product should be accounted for in the tolerance design problem, specifically by using the quality-loss function. Several tolerance design methods based on quality loss [4,[6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22] have been developed for the purpose of finding optimum tolerances of components and assemblies. ...
... Several tolerance design methods based on quality loss [4,[6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22] have been developed for the purpose of finding optimum tolerances of components and assemblies. Jeang [5] also proposed using classical RSM to determine the optimum assembly and component tolerances by minimizing a total cost (TC) function, which is the sum of manufacturing cost (MC) and quality-loss cost (QLC). Classical RSM methods for experimental design, modeling, and optimization are discussed by Box and Draper [23], Khuri and Cornell [24], and Myers et al. [25]. ...
... In this article, component-amount (CA) and mixtureamount (MA) approaches are proposed as more appropriate than the classical RSM approach proposed by Jeang [5] for solving the tolerance design problem when using the worstcase method and the assembly characteristics are linear combinations of the component characteristics. The CA, MA, and RSM approaches use different kinds of experimental designs and models to solve the tolerance design problem, as discussed subsequently in the article. ...
The tolerance design problem involves optimizing component and assembly tolerances to minimize the total cost (sum of manufacturing cost and quality loss). Previous literature recommended using traditional response surface methodology (RSM) designs, models, and optimization techniques to solve the tolerance design problem for the worst-case scenario in which the assembly characteristic is the sum of the component characteristics. In this article, component-amount (CA) and mixture-amount (MA) experiment approaches are proposed as more appropriate for solving this class of tolerance design problems. The CA and MA approaches are typically used for product formulation problems, but can also be applied to this type of tolerance design problem. The advantages of the CA and MA approaches over the RSM approach and over the standard, worst-case tolerance-design method are explained. Reasons for choosing between the CA and MA approaches are also discussed. The CA and MA approaches (experimental design, response modeling, and optimization) are illustrated using real examples.
... Aspects such as design for quality, quality improvement and cost reduction, asymmetric quality losses, charts for optimum quality and cost, minimum cost approach, cost of assemblies, development of cost tolerance models (Feng and Kusiak, 1994; Jeang, 1998 Jeang, , 1999 Wu et al., 2003; Diplaris and Sfantsikopoulos, 1999) have been explored in the quality area of tolerance synthesis. Experiments (DOE) approach was used in robust tolerance design, the cases of " nominal the best " , " smaller the better " , " larger the better " , and asymmetric loss function, were investigated ( Jeang, 1997) and allocation of tolerances of products with asymmetric quality loss was presented (Wu et al., 2003). ...
Purpose
– Due to technological and financial limitations, nominal dimension may not be able achievable during manufacturing process. Therefore, tolerance allocation is of significant importance for assembly. Conventional tolerance analysis methods are limited by the assumption of the part rigidity. Every mechanical assembly consists of at least one or more flexible parts which undergo significant deformation due to gravity, temperature change, etc. The deformation has to be considered during tolerance design of the mechanical assembly, in order to ensure that the product can function as intended under a wide range of operating conditions for the duration of its life. The purpose of this paper is to determine the deformation of components under inertia effect and temperature effect.
Design/methodology/approach
– In this paper, finite element analysis of the assembly is carried out to determine the deformation of the components under inertia effect and temperature effect. Then the deformations are suitably incorporated in the assembly functions generated from vector loop models. Finally, the tolerance design problem is optimized with an evolutionary technique.
Findings
– With the presented approach, the component tolerance values found are the most robust to with stand temperature variation during the product's application. Due to this, the tolerance requirements of the given assembly are relaxed to certain extent for critical components, resulting in reduced manufacturing cost and high product reliability. These benefits make it possible to create a high‐quality and cost‐effective tolerance design, commencing at the earliest stages of product development.
Originality/value
– With the approach presented in the paper, the component tolerance values found were the most robust to withstand temperature variation during the product's application. Due to this, the tolerance requirements of the given assembly are relaxed to a certain extent for critical components, resulting in reduced manufacturing cost and high product reliability. These benefits make it possible to create a high‐quality and cost‐effective tolerance design, commencing at the earliest stages of product development.
