Reliability modeling-based tolerance design and process parameter analysis considering performance degradation

Highlights

• A reliability model with temperature-humidity-mechanical stress covariate is built.

• A two-stage parameters estimation method is proposed based on NLS and MLE.

• A novel tolerance design is developed considering reliability constraint.

• An allowable range of quality fluctuation is derived with reliability constraint.

• A real case is given to show the implementation of the proposed methods.

Abstract

Tolerance design and process parameter analysis have been widely used for reducing the fluctuation of product quality characteristics (QCs). Most existing works focus on finding the best compromise between manufacturing cost and tolerance of QCs. However, the performance degradation of products will lead to QCs deviating from design specifications, which is ignored in current researches. Motivated by the stress relaxation process of helical springs, we propose a more reliable tolerance design and process parameter analysis considering performance degradation. An accelerated degradation model is first constructed to reflect the degradation process of helical springs and the influence of the initial free length. Subsequently, considering the given reliability constraint, a tolerance design method based on degradation performance is developed to guide the quality improvement of helical springs. Furthermore, a new reliability model embodied with the influence of parameter fluctuation is proposed, and an allowable fluctuation range for the deviation and variance of the initial free length is derived, based on which the process parameter can be optimized to improve the manufacturing process. Finally, a real case study of helical springs is conducted to illustrate the implementation and effectiveness of the proposed method.

Introduction

Tolerance design and process parameter analysis have been extensively studied in the literature, and they are two effective methods to reduce fluctuations in product quality characteristics (QCs). Tolerance design is to determine the allowable fluctuation range of QCs, so as to minimize the manufacturing cost and ensure the normal function of products [1], while process parameter analysis is performed to improve the manufacturing process so that QCs of products can be within the given tolerance [2]. Most existing works focus on finding the best compromise between manufacturing cost and tolerance of QCs, regardless of the degradation effect. For some products, their QCs will continue to degrade during the use period, result in that the QCs are unable to remain within the initial tolerances and ultimately affect the normal operation of products. Therefore, ignoring the degradation effect will lead to that the tolerance design and process parameter analysis cannot meet the high-reliability requirement. For example, the free length is the most important QC of helical springs that are commonly used in mechanical devices such as spring-loaded safety valves (SLSVs) [3], and the free length determines the spring force provided by the helical spring when it is compressed to a fixed height. Due to the stress relaxation, the free length of helical springs will continue to degrade during the use period until exceeding the design specification, which will fail the mechanical device. Motivated by this illustrative example, this paper intends to develop a more reliable tolerance design and process parameter analysis considering the degradation performance of product QCs.

Tolerance of products refers to the allowable fluctuation range of QCs in terms of economy and function. Generally, limiting the fluctuation range implies the improvement in product quality and the increase in manufacturing cost [4]. To date, the studies on the tolerance design of products have been widely discussed in the literature, Hallmann et al. [5] provided a review of the state of the arts in the tolerance design, where the manufacturing cost and quality cost are commonly considered as the main constraints. Recent developments in tolerance design heightened the necessity of considering reliability constraints. Wang et al. [6] pointed out the importance of reliability in the tolerance design of DC hybrid contactor, and they developed a reliability tolerance design based on EDA software. From then on, the tolerance design considering reliability constraint has been applied to other products and manufacturing processes, such as mechanical assemblies [7], switched-mode power supply [8], and CNC machine tools [9]. However, all the literature mentioned above only focused on the influence of tolerance on the design reliability, i.e., the ability of the tolerance of QCs to meet the design requirement. For many products like helical springs, as the use time increases, their QCs will degrade during the use period, and result in the performance deviating from design specifications. Therefore, a more significant constraint on the tolerance design is the operational reliability, which is defined as the ability of products to work without failure under the stated conditions for a specified period. In view of this, Zhai et al. [10] developed a tolerance design of electronic circuits considering performance degradation and carried out an optimum allocation based on the predicted performance under the given task condition and task time. Nevertheless, the effect of the initial value of QCs on the degradation process is ignored in their works. For helical springs, when they are compressed into a fixed height, different initial free lengths will cause different mechanical stresses on helical springs, and the corresponding degradation performance will also show differences. Confronted with that, based on the initial tolerance designed from functional and economic perspectives, and considering the operational reliability constraint, this paper intends to develop a more reliable tolerance design of products based on the performance degradation affected by the initial value of QCs.

In addition, process parameters refer to manufacturing process control parameters that have a significant impact on the product quality, such as the axial pressure, initial rotational speed, and moment of inertia of the flywheel in solid-state welding processes [11]. Process parameter analysis aims to optimize manufacturing process control parameters to minimize the variance in QCs and the difference between the real value and the target value of QCs (i.e., the deviation in QCs) [2]. Recently, many different optimization models have been developed for this goal in different fields, such as the energy model developed by Bi and Wang [12] for the optimization of energy consumption in the production process, the SS-SVR meta-model studied by Zhang and Zhou [13] for the parameter optimization of laser magnetic welding process, and the gray relational analysis used in Yu et al. [14] for the process parameter optimization during the polishing process of optical components. However, to our best knowledge, existing works mainly focused on minimizing the deviation or variance of QCs and rarely considered how small the deviation and variance of QCs should be so that the manufacturing process can be accepted (i.e., the products produced by the manufacturing process satisfy the given requirement). In view of this, considering the relationship between QCs and the operational reliability of products, this paper aims to derive an allowable fluctuation range for the deviation and variance of QCs from the perspective of operational reliability.

Reliability modeling is the core tool to evaluate the operational reliability of products, and relevant studies have been widely developed in recent years. Some recent reviews of degradation modeling are available at Ye and Xie [15], Zhang et al. [16], and Kang et al. [17]. In general, there are three typical methods for degradation modeling: degradation amount distribution model [18], stochastic model (including Wiener process [19], Gamma process [20], and inverse Gaussian process [21]), and general path model. Among those models, the general path model, developed by Lu and Meeker [22] firstly, has become the most widely used degradation model as it is flexible to model the unit-to-unit variation and measurement errors. Recent developments in general path model include the multivariate path model proposed by Si et al. [23] and Lu et al. [24] for the reliability analysis of systems with multiple characteristics, and the linear path model embedded with self-recovery mechanism considered by Liu et al. [25] and Kong and Yang [26]. Therefore, considering the high applicability of the general path model, and inspired by Yuan et al. [27], this paper uses the general path model to describe the degradation performance of products. Furthermore, for highly reliable products like helical springs, it is still a challenge to obtain sufficient degradation data in a short time. Then, the accelerated degradation test (ADT) was developed to solve this problem. ADT is carried out by exposing and testing the product under high stressed conditions. Limon et al. [28] provided a review of the state of the arts in the optimal design of ADT, where the constant stress accelerated degradation test (CSADT) is considered the most primary and effective method [29,30]. The key to CSADT is to find the accelerated stresses that are sensitive to the product degradation. As the primary failure mode of helical springs is stress relaxation, many studies have shown that temperature and humidity have a significant effect on the stress relaxation, see Luo et al. [31] and Li et al. [32] for examples. Therefore, in this paper, temperature and humidity are listed as the principal accelerated stresses.

Consequently, motivated by the stress relaxation process of helical springs, this paper intends to develop a more reliable tolerance design of products considering performance degradation, and further derives an allowable fluctuation range for the deviation and variance of QCs to guide the process parameter analysis and improve the manufacturing process of products. In addition, considering the difficulty in obtaining the degradation signals, an accelerated degradation model is proposed by embodying the temperature-humidity-mechanical stress covariate into the general path model, through which the influence of the initial value of QCs on the operational reliability of products can be extracted from the CSADT.

The major contributions of this work are summarized as follows:

• In order to make the tolerance of products meet the high-reliability requirement, the proposed tolerance design takes into accordant with the performance degradation of products and the influence of the initial value of QCs.

• The proposed process parameter analysis quantifies the influence of process fluctuation on the operational reliability of products, and derives an allowable fluctuation range of the process fluctuation, which is critical for the improvement of the manufacturing process.

• While giving the tolerance design and process parameter analysis of helical springs, the proposed method also provides a framework of the tolerance design and process parameter analysis for products with degradation performance.

The rest of this paper is organized as follows. The detailed construction of the accelerated degradation model with the effect of initial free length and the estimation method of unknown parameters are presented in Section 2. Then, based on the accelerated degradation model, the tolerance design and process parameter analysis considering the reliability constraint are introduced in Section 3. Moreover, In Section 4, a real CSADT is designed to generate degradation information of helical springs, through which the implementation and effectiveness of the proposed method are illustrated. Finally, some conclusions are given in Section 5.