Process machining allowance for reliability analysis of mechanical parts based on hidden quality loss

Abstract


Introduction
During machining, a part often has multiple stages of manufacturing involving different machining methods.
Variations in the machining process may cause defects on the part. Machining allowance means the designer must remove some material from the part's surface or part blank. The purpose of machining allowances is to remove defects left by the previous process and ensure quality consistency14. In the actual processing of parts, as the machining allowance of the part increases, it is necessary to control and monitor the machine tool processing1317183738. The production cost will also increase.
However, when reducing the machining allowance, the part will have certain machining defects, and the quality will not be satisfactory. These defects may be subject to random fracture in the subsequent process19. In order to assess the quality of a Reliability is a quality indicator of a product. It depends on the product design and production process, which are determined by the design and production level. This is called inherent reliability. The reliability of mechanical parts ensures economic efficiency, service life, and quality stability. To predict and evaluate the product reliability during service and improve the quality reliability, Chen  Previous authors have mainly studied the machining allowance analysis of machining efficiency, cost, part deformation, and fracture. They have also combined the process capability index and part quality. Some have used the quality loss function to relate the part tolerance to the final manufacturing cost. Then, they used reliability to analyze the product quality based on the quality loss function. However, few have established a link between the process allowance of a part and its reliability. This paper is organized as follows.
Section 2 discusses the factors influencing machining allowance and its model selection. Section 3 derives a reliability prediction model based on the AQF and QEF for process machining allowance-hidden quality loss when the quality characteristic values follow a normal distribution. The derivation also incorporates the process capability index. Section 4 presents a case study to analyze how the process machining allowance affects the product reliability. It also compares the range of two reliability model choices. Section 5 concludes the paper.

Analysis of machining allowance and its influencing factors
The machining allowance is the amount of material that needs to be removed from a part in machining. It is allocated to each machining process according to the process machining allowances for each process.
In machining processes, it is generally assumed that the initial dimensional value of the part or part blank is located at the center of the tolerance specification. As shown in Fig. 1, the machining allowance for a given process is defined as the difference in process dimensions between two consecutive processes15. In other words, the machining allowance of a process is the difference between the dimension of this process and the dimension of the previous process −1 .
Given that tolerances influence process dimensions, the maximum machining allowance max and the minimum machining allowance min are established. In practical applications, the minimum machining allowance represents the lower threshold that must be maintained to ensure satisfactory machining quality. In conjunction with the annotations depicted in Fig.1, the minimum machining allowance min is calculated as the difference between the minimum size of process min and the maximum size of the preceding process ( −1)max . The maximum size of the preceding process ( −1)max is determined by adding its dimensional tolerance ( −1) to its minimum size ( −1)min , as expressed in the following equation: The maximum machining allowance max is calculated as the difference between the maximum size of process max and the minimum size of the preceding process ( −1)min . The maximum size of process max is determined by adding its dimensional tolerance to its minimum size min , as expressed in the following equation: The machining allowance tolerance is defined as the difference between the maximum value of the machining allowance max and its minimum value min . By applying Equations (1) and (2), the equation for calculating the machining allowance tolerance can be derived as follows: As shown in Fig.1, the initial dimension is positioned at the midpoint of the tolerance specification. The equation for computing the total machining allowance is presented below: where is the machining allowance for process , 0 is the initial dimensional tolerance, is the dimensional tolerance for process , is the dimension for process , is the total number of machining processes, and is the work sequence number, = 1,2, ⋯ , .
The Equation (4) shows that the total machining allowance is determined by the minimum machining allowance min and dimensional tolerance of each process. presented as Equation (5) and Equation (6), respectively. = + + + 0.001 ( −1) + ( −1) Rotary surfaces serve as the primary focus of this study, although flat machining types can also be analyzed using this method. The errors , , and are typically caused by clamping and can be combined into a single error term to represent clamping error. While is subject to human and machine influence, for this analysis, it is assumed that these objective factors are negligible and set = 0 . Under this assumption, Equation (6) simplifies to the following expression: = 0.002 ( −1) + 2 ( −1) In the above equation, is the machining allowance for the

Reliability modeling based on hidden quality loss
Taguchi's view of quality31 suggests that even when a product falls within specification limits, variations in its quality characteristics can result in quality losses. In other words, even products that meet specifications can generate what is known as hidden quality loss. The hidden quality loss is used to establish a relationship between the two to examine the impact of process machining allowance on part reliability. When parts leaving the factory have tolerances wider than their design tolerances, manufacturers can rework them to bring them within specification limits. Conversely, when parts leaving the factory have tolerances narrower than their design tolerances, they must be scrapped and cannot be reworked into qualified products.
These two scenarios result in an asymmetric quality loss function with respect to the target value. The AQF and QEF are used to conduct the relevant analysis. The dimensional tolerance of a part is linked to the process machining allowance tolerance through the use of PCI. By establishing a connection between process machining allowance and hidden quality loss, PCI enables us to estimate the hidden quality loss of qualified parts.
Then, a reliability model is constructed using the hidden quality loss derived from these two functions. This allows us to relate process machining allowance to reliability.
It is assumed that the quality characteristics follow a normal distribution, denoted as ~( , 2 ), where is the mean value of , is its standard deviation. In actual manufacturing, is the value of process machining allowance, is the mean of process machining allowance value, is the standard deviation of process machining allowance value. The normal distribution density function is given by ( ), while the standard normal distribution density function and distribution function are represented by ( ) and Φ( ) , respectively. The relevant equations are provided below:

Derivation of formula based on the AQF
According to the Taguchi quality loss function, the AQF is as follows: where ( ) is the loss of product quality per unit under the AQF, is the value of product quality characteristic, is the target value of quality characteristics, is the lower limit of quality specification, is the upper limit of quality specification, 1 is the quality loss caused when below the lower limit of quality specification, 2 is quality loss when the upper limit of the quality specification is exceeded, 1 , 2 are the mass loss coefficients to the left and right of the target value, respectively.
Estimation of expected quality cost ( ( )) based on Taguchi method and probability theory: represents the average quality loss for a batch of products and includes quality losses from non-qualified products. To estimate the quality loss of a batch of qualified products, the quality loss from non-qualified products must be transferred to qualified products. Let 0 denote the quality loss of a single qualified product, and let q represent the passing rate for a batch of products. The relevant equation is given below: Then the relationship between 0 and is as follows: Substituting Equation (14) into (16), the equation for 0 is given as follows： During the machining of parts, an excessively large machining allowance can increase the labor required, reduce productivity, and increase costs. Conversely, an excessively small machining allowance may increase the number of processes required and may not effectively eliminate errors and surface defects from previous processes, potentially resulting in scrap. Suppose assuming that the target value for machining allowance is at the center of the specification range (i.e., = − 2 and = = Δ ) and that the process is unbiased (i.e., = ). In that case, the equation for PCI can be expressed as follows: Then the PCI equation can also be expressed by the following equation: where is the upper deviation of machining allowance target value , is the lower deviation machining allowance target value , is the value of machining allowance tolerance and Δ is half of the tolerance range.
Equation (19) (18) and (19), the following expressions are derived: Substitute Equations (20), (21), and (22) into (17), then the following equation is given: Substitute Equation (18) into (23), then the following equation is given: Let = 2 . From Equation (3), we can derive that = + ( −1) , which leads to = In the machining process, the final dimensions of the machined parts must meet the design requirements. This forms a dimensional chain where the sum of each process's dimensional tolerances equals the final part's dimensional tolerance. Therefore, there must be a relationship between the dimensional tolerance of the previous and the current process. It is assumed that the dimensional tolerance of this process is times that of the previous process, where 0 < ≤ 1 based on experience. This means that = ( −1) . Thus, the following equation is given: Substitute Equation (7) into (25), the equation is as follows: The is defined as the machining allowance impact factor.
From Equation (26), it is clear that when obtaining a large amount of machining allowance data, is the standard deviation of the data sample. In a machining process, ( −1) , ( −1) , ( −1) are the known objective design parameters and is the variable machining allowance. Then is a function of . Equation (24) can be deformed as follows: At this point, the hidden quality loss 0 based on AQF is related to the machining allowance impact factor , which is influenced by the machining allowance of the process .
It is assumed that the number of parts in a batch is , then the total hidden quality loss of the batch is 0 . By transferring the hidden quality loss of all parts to the parts that cause larger quality loss, the number of simulated failures in a batch of qualified parts can be obtained as follows： This model ignores that when reaches its maximum value, the number of failures is at its minimum. Manufacturers often opt for machine products with wider tolerances to reduce loss 2 through reworking and repairing. However, when products are shipped with tolerances narrower than the design tolerances, they are often scrapped, resulting in a significant amount of unrepaired loss 1 . To accurately predict the number of simulated failure products in a batch of qualified products, it is necessary to attribute the implicit quality loss of each product to the product responsible for causing the quality loss. By letting = 1 + 2 , Equation (28) can be rewritten as follows: where and are the weights of the two quality losses.
Referring to the inherent product reliability model proposed by Liu et al. 23, the following equation is given: where ( ) is the inherent product reliability, ( ) is the failure rate， ( ) is the number of product failures.
Substitute Equation (29) into (30), the equation is as follows: Substitute Equation (27) into (31), the equation is as follows: At this point, the inherent reliability of the part is related to the machining allowance impact factor , which is influenced by the machining allowance of the process .
As shown in Fig. 2, initially increases, and increases correspondingly. When reaches a specific value, is used as the denominator in the model and grows as a square. Besides, it also grows faster than the numerator, so the growth rate of become slow.

Derivation of formula based on the QEF
After machining is completed and the diameter of the shaft deviates from the set target value, the growth rate of mass loss According to Equation (13), the following equation can be obtained based on QEF: Expressing Equation (16) in terms of , the equation for 0 is given: Substitute Equations (20), (21), and (22) into (36), then the following equation is given: Substitute Equation (18) into (37), then the following equation is given: Let = 2 . Equation (38) can be deformed as follows: Similarly, the hidden quality loss 0 based on QEF is related to the machining allowance impact factor , which is influenced by the process's machining allowance .
Substituting Equation (39) into (31), the equation for under QEF is obtained as follows: Similarly, the inherent reliability of the part is related to the machining allowance impact factor , which is influenced by the machining allowance of the process .
From Fig. 3, it can be obtained that as the machining allowance influence factor increases, decreases, i.e., the more robust the machining is, the more reliable it is. Compared to Fig. 2, this relationship is more consistent with actual production reliability predictions.

Results and discussion
As shown in Fig. 4, the long pin parts in a transparent hightemperature resistant theory teaching mold are used for reliability analysis of the process machining allowances. Pin parts are made of 35 hot rolled round steel. Its specification is 8.8 × 119 , as shown in Fig. 5 and 6.
The dimensional tolerance is ±0.40 , and the process route to be used is "Rough machining -Precision machining", as shown in Fig. 8.
Six specimens were selected, as shown in Fig. 4. The steps are as follows. In the first step, the outer dimensions of the raw material are measured at 30 , 60 and 90 from the end face with vernier calipers, as shown in Fig. 7. In the second step, the measurement results are sorted by specimen serial number as shown in Table 1. As shown in Table 1, the maximum and minimum values of the outside diameter of the six specimens meet the requirements of the part specifications.    At the end of the second process, a pin part that meets the specifications with a dimensional tolerance of 2 = ±0.40 .
The next step is the calculation of the parameters related to the two machining processes. Subsequently, the impact of process machining allowance on product reliability is analyzed and compared with the two methods developed above.

Calculation of relevant parameters during rough machining：
It is assumed that this processing adopts the grade processing method. The basic idea is to make the feature accuracy and surface roughness grade of this process three levels smaller than that of the previous process.  Table 2.
Substitute the data from Table 2 into Equation (7), 1 = 1.5586 . From Equation (3), it is obtained that 1 = 0 + 1 = 0.8 + 0.8 = 1.6 . Since the target value is located at the center of the machining specification, the tolerance of 1 is Substituting the data in Table 2 into Equation (26), it becomes the following equation: From Equation (41), it can be obtained that to make meaningful, i.e., The minimum machining allowance should be satisfied to remove the defects caused by the previous process.
The above machining allowances meet the requirements. To facilitate the calculation and analysis, the value range of 1 is 1.56~3.14 .
According to Equation (41), the value of is calculated concerning 1 and . The resulting graph is shown in Fig. 9. As can be observed from Fig. 9, increases as 1 increases and decreases as increases. The relationship between these variables is further illustrated in Fig. 10.
As shown in Fig. 10, the variables mentioned above exhibit a monotonic relationship. As the machining allowance 1 increases, so does the machining allowance impact factor .
Conversely, as the standard deviation decreases (indicating better production robustness), also increases. According to Equation (42), the variation of product inherent reliability with 1 and is shown in Fig. 11.
Their relationship can be seen more clearly in Fig. 12. As shown in Fig. 12, the variables mentioned above exhibit a monotonic relationship. Specifically, as the machining allowance within the design range increases, so does the product reliability.
Conversely, product reliability also improves as the standard deviation decreases (indicating better production robustness).
As can be observed from the two-dimensional variation surface Fig. 11. The relationship between and 1 , .
in Fig. 11, if it is stipulated that product reliability must satisfy a value of 0.9, there exists an optimal choice for values of 1 and . It also can be used to analyze the effect of the variation of machining allowance on the inherent reliability of the product. Equation (43) illustrates the variation of product inherent reliability concerning 1 and , as shown in Fig. 13.
The relationship between these variables is further elucidated in Fig. 14. As can be observed from the twodimensional variation surface in Fig. 13, if product reliability is specified to meet a value of 0.9, the range of values for 1 and is reduced compared to that shown in Fig. 11. The reason for this range narrowing can be derived from Equation (31), which primarily depends on 0 . 0 can be obtained through the derivation of AQF and QEF, respectively. As proposed earlier in this article, QEF is more accurate and better reflects production reality than AQF. It allows for a more precise analysis of the impact of changes in process allowance on the inherent reliability of the part.  Table 3. Substituting the data in Table 3 into Equation (26) According to Equation (44), the value of is calculated concerning 2 and . The resulting graph is shown in Fig. 15.
As can be observed from Fig. 15, increases as 2 increases and decreases as increases. The relationship between these variables is further illustrated in Fig. 16.
A similar result as in Fig. 10 can be obtained in Fig. 16. As it is shown in Fig. 16, the variables mentioned above exhibit a monotonic relationship. As the machining allowance 2 increases, so does the machining allowance impact factor .
Conversely, as the standard deviation decreases (indicating better production robustness), also increases.  According to Equation (45), the variation of product inherent reliability with 2 and is shown in Fig. 17.
Their relationship can be seen more clearly in Fig. 18. A similar result to that shown in Fig. 12 can be obtained in Fig. 18.
Comparing Fig. 18 with Fig. 12, it can be obtained that the rate of change of reliability concerning the process allowance and its standard deviation remains approximately the same in the finishing stage, despite the reduction of the machining allowance. This also serves to demonstrate the stability of the model to some extent. Comparing Fig. 17 with Fig. 11, it can be obtained that if the reliability is selected as 0.9, the range corresponding to the finishing stage is reduced compared to the roughing stage. This is following objective reality. As mentioned above, in the finishing stage, the machining allowance must not only be sufficient to remove defects from the previous stage, but also to ensure that the dimensions of the final produced part meet the design requirements. Consequently, the area must be reduced. It can also be used to examine the effect of variations in machining allowance on inherent product reliability.  According to Equation (46), the variation of product inherent reliability with 2 and is shown in Fig. 19.
Their relationship can be seen more clearly in Fig. 20. A similar result to that shown in Fig. 13 can be obtained in Fig. 19. Fig. 19. The relationship between and 2 , . The above analysis shows the stability as well as the applicability of the proposed model.

Conclusions
During the actual machining process of the part, different process machining allowances will lead to the changes of product reliability. Process machining allowance-reliability prediction methods are established based on AQF and QEF, respectively. Some consequences can be obtained.
1. During actual machining, excessive or minimal machining allowances can lead to changes of part dimensions and quality loss. In order to evaluate it, the product quality loss view is introduced, and AQF and QEF are adopted for comparison.
2. In order to use machining allowance tolerance as a benchmark to assess the level of machining quality for a given part, PCI is introduced to establish a relationship between process machining allowance and machining quality level.
3. It is assumed that quality characteristic values follow a normal distribution. Using the AQF and QEF, numerical models is generated to estimate the average hidden quality loss due to machining quality levels.
4. Sampling parts from a batch can obtain a more accurate estimate of the total hidden quality loss. The number of failures within the batch can be simulated by transferring the total hidden quality loss to those parts causing it. This allows for the establishment of the hidden quality loss-reliability prediction models.
The stability and reliability of both models are verified through case analysis. Further comparison of two model shown that the model based on QEF is more accurate than the one based on AQF. This finding has significant implications for manufacturers seeking to improve part quality and reduce machining costs.