In precision manufacturing, CNC gear grinding machines play a pivotal role in producing high-accuracy gears, where gear profile grinding ensures dimensional integrity and surface quality. However, operational failures in these machines, such as grinding cracks or misalignments, directly impact processing efficiency and equipment longevity. Traditional Failure Mode and Effects Analysis (FMEA) is widely used for reliability assessment but suffers from subjective evaluations and equal weighting of risk factors, leading to inaccurate risk prioritization. To address these limitations, this study proposes an enhanced FMEA methodology that integrates triangular fuzzy numbers (TFN) and the Analytic Hierarchy Process (AHP). By fuzzifying expert assessments and deriving differentiated weights for severity (S), occurrence (O), and detection (D) factors, this approach provides a more objective and structured framework for hazard analysis in gear grinding operations.
The core innovation lies in combining TFN to handle linguistic uncertainties in expert judgments with AHP for systematic weight allocation. In gear grinding processes, factors like grinding cracks or thermal distortions in the spindle system can severely affect gear accuracy. The improved FMEA quantifies these failure modes through fuzzy matrices, defuzzifies them using weighted averages, and computes risk priority numbers (RPN) with optimized weights. This method not only refines risk rankings but also identifies critical components, such as the spindle system, which is prone to failures like overheating or positioning errors during gear profile grinding. By focusing on high-hazard subsystems, maintenance strategies can be prioritized to enhance machine reliability and prevent defects like grinding cracks in finished gears.
Methodology: Integrating Triangular Fuzzy Numbers and AHP
The traditional FMEA calculates RPN as the product of S, O, and D, expressed as: $$RPN_i = S_i \times O_i \times D_i$$ where \(S_i\), \(O_i\), and \(D_i\) represent the severity, occurrence, and detection ratings for the \(i\)-th failure mode. However, this assumes equal importance of factors and relies on crisp numerical inputs, which may not capture expert ambiguity. The improved model incorporates TFN to represent fuzzy evaluations and AHP to assign weights, ensuring a more realistic hazard assessment.
Triangular fuzzy numbers are defined by a triplet \((a, b, c)\), where \(a\), \(b\), and \(c\) denote the minimum, most probable, and maximum values of a fuzzy set. The membership function \(\mu(x)\) is given by: $$\mu(x) = \begin{cases} 0, & x < a \\ \frac{x – a}{b – a}, & a \leq x < b \\ \frac{c – x}{c – b}, & b \leq x < c \\ 0, & x \geq c \end{cases}$$ For instance, in gear grinding, evaluations for factors like “occurrence of grinding cracks” can be fuzzified using semantic ratings such as Low (L: 0,1,3) or High (H: 7,9,10). Experts provide fuzzy ratings for each failure mode, which are aggregated into a fuzzy decision matrix. Suppose there are \(m\) failure modes and \(k\) experts with weights \(\zeta_i\). The aggregated fuzzy rating for a factor is computed as: $$x_{ij} = \frac{\zeta_1 x_{ij}^1 + \zeta_2 x_{ij}^2 + \cdots + \zeta_k x_{ij}^k}{k}$$ where \(x_{ij}^k\) is the TFN from the \(k\)-th expert. Defuzzification converts these fuzzy numbers into crisp values using the weighted average method: $$x_{ij} = \frac{a + 4b + c}{6}$$ This step transforms vague assessments into quantifiable scores for S, O, and D.
The AHP component establishes differentiated weights for S, O, and D based on their relative importance in gear grinding contexts. For example, in failure modes involving gear profile grinding inaccuracies, severity might outweigh occurrence due to its impact on product quality. A pairwise comparison matrix \(A = [a_{ij}]\) is constructed using a 1–9 scale, where \(a_{ij}\) indicates the relative importance of factor \(i\) over \(j\). The eigenvector method solves for weights, and consistency is verified through the consistency ratio \(CR\): $$CR = \frac{CI}{RI}, \quad CI = \frac{\lambda_{\text{max}} – n}{n – 1}$$ where \(\lambda_{\text{max}}\) is the largest eigenvalue, \(n\) is the matrix size, and \(RI\) is the random index. If \(CR < 0.1\), the weights are valid. The final RPN for each failure mode is calculated as: $$RPN_i = S_i \cdot w_S + O_i \cdot w_O + D_i \cdot w_D$$ where \(w_S\), \(w_O\), and \(w_D\) are the AHP-derived weights. This additive model replaces the multiplicative approach, reducing bias and improving sensitivity to high-risk scenarios like grinding cracks.
Case Study: Application to CNC Gear Grinding Machine
A CNC gear grinding machine, essential for high-precision gear manufacturing, comprises subsystems such as the spindle system, feed system, and CNC electrical system. Each subsystem is susceptible to failures that can induce grinding cracks or profile inaccuracies. For this analysis, the machine is divided into seven subsystems: spindle system, feed system, tool magazine system, cooling lubrication system, hydraulic system, CNC electrical system, and other components. Common failure modes include spindle overheating, bearing damage, and lubrication failures, which are critical in gear profile grinding operations.
To demonstrate the improved FMEA, the spindle system is analyzed as an example. Six failure modes are identified: component damage, spindle damage, overheating, excessive noise, positioning inaccuracy, and encoder faults. Five experts with varying expertise in gear grinding evaluated these modes using fuzzy semantic scales. Their weights were assigned based on experience, relevance, and qualifications, as summarized in Table 1.
| Expert | Qualification Score | Relevance Score | Experience Score | Weight (\(\zeta_i\)) |
|---|---|---|---|---|
| 1 | 5 | 5 | 5 | 0.3108 |
| 2 | 8 | 8 | 8 | 0.2703 |
| 3 | 2 | 2 | 2 | 0.1622 |
| 4 | 5 | 5 | 5 | 0.1351 |
| 5 | 8 | 8 | 8 | 0.1217 |
The experts provided fuzzy ratings for S, O, and D, which were aggregated into a fuzzy decision matrix. For instance, the “spindle damage” mode received fuzzy ratings of H for severity, L for occurrence, and M for detection. The aggregated TFN values were defuzzified using the weighted average formula. Table 2 presents the defuzzified scores for the spindle system’s failure modes.
| Failure Mode Code | Severity (S) | Occurrence (O) | Detection (D) |
|---|---|---|---|
| 0101 | 1.9597 | 7.7936 | 2.2074 |
| 0102 | 8.0413 | 1.9597 | 4.1085 |
| 0103 | 3.0474 | 5.2167 | 2.2571 |
| 0104 | 1.6623 | 5.8205 | 4.4059 |
| 0105 | 5.8655 | 6.1627 | 2.4798 |
| 0106 | 7.0050 | 4.1355 | 1.9101 |
Next, AHP was applied to determine weights for S, O, and D. For the “spindle damage” mode (0102), experts compared factors pairwise, resulting in a judgment matrix. The computed weights were \(w_S = 0.6110\), \(w_O = 0.1954\), and \(w_D = 0.1940\), with \(CR = 0.0158\), indicating consistency. Similarly, weights for other modes were derived, as shown in Table 3.
| Failure Mode Code | Severity Weight (\(w_S\)) | Occurrence Weight (\(w_O\)) | Detection Weight (\(w_D\)) |
|---|---|---|---|
| 0101 | 0.2665 | 0.4571 | 0.2764 |
| 0102 | 0.6110 | 0.1954 | 0.1940 |
| 0103 | 0.1705 | 0.5383 | 0.2912 |
| 0104 | 0.1628 | 0.5613 | 0.2759 |
| 0105 | 0.6013 | 0.1619 | 0.2368 |
| 0106 | 0.1937 | 0.2125 | 0.5938 |
The RPN for each mode was calculated by multiplying defuzzified scores with their respective weights and summing the products. For example, for mode 0102: $$RPN_{0102} = 8.0413 \times 0.6110 + 1.9597 \times 0.1954 + 4.1085 \times 0.1940 = 1.4996$$ Table 4 lists the RPN values for all spindle system modes, with “spindle positioning inaccuracy” (0105) having the highest hazard degree.
| Subsystem | Failure Mode Code | RPN |
|---|---|---|
| Spindle System | 0101 | 1.1352 |
| Spindle System | 0102 | 1.4996 |
| Spindle System | 0103 | 0.9590 |
| Spindle System | 0104 | 1.0747 |
| Spindle System | 0105 | 2.0664 |
| Spindle System | 0106 | 1.3524 |
This process was repeated for other subsystems, such as the feed system and CNC electrical system, where failures like bearing damage or voltage fluctuations could lead to grinding cracks in gear profile grinding. The subsystem hazard degrees, computed as the average RPN of their failure modes, are ranked in Figure 1. The spindle system emerges as the most critical, underscoring its impact on gear grinding accuracy and reliability.
Results and Validation
The improved FMEA identified the spindle system as the highest-hazard subsystem with an RPN of 1.3479, followed by the feed system (1.0221) and CNC electrical system (0.8402). Within the spindle system, failure modes like “spindle positioning inaccuracy” (RPN: 2.0664) and “spindle damage” (RPN: 1.4996) ranked highest, highlighting their propensity to cause grinding cracks or profile errors during gear grinding. These modes often involve thermal deformation or encoder faults, which are critical in maintaining precision in gear profile grinding operations.
To validate the method, a comparison with traditional FMEA was conducted. Traditional RPN, computed as \(S \times O \times D\) with equal weights, yielded different rankings. For instance, “spindle positioning inaccuracy” had a traditional RPN of 120, while “bearing damage” in the feed system scored 210. However, the improved model assigned higher priority to spindle-related failures due to their severe impact on gear quality, consistent with real-world observations in gear grinding facilities. The additive RPN calculation also resolved issues like tied rankings, providing a more discriminative hazard order.
The integration of TFN and AHP effectively addressed subjectivity and weight imbalance. In gear grinding, where factors like grinding cracks depend on multiple parameters, the fuzzy approach captured expert uncertainty, while AHP prioritized severity for modes affecting gear integrity. This synergy enhances decision-making for maintenance planning, focusing resources on high-risk components to prevent failures that could compromise gear profile grinding outcomes.
Conclusion
The proposed FMEA improvement, combining triangular fuzzy numbers and AHP, offers a robust framework for hazard analysis in CNC gear grinding machines. By fuzzifying expert evaluations and deriving context-sensitive weights, it mitigates the limitations of traditional FMEA, providing accurate risk prioritization. The case study confirms that the spindle system is the most critical subsystem, with failure modes like positioning inaccuracy and spindle damage posing significant risks of grinding cracks and profile inaccuracies. This method enables targeted maintenance strategies, improving reliability and efficiency in gear grinding operations. Future work could extend this approach to dynamic risk assessment or integrate it with real-time monitoring systems for proactive failure prevention in gear profile grinding applications.