Abstract
This study introduces a comprehensive framework that integrates reliability assurance and quality control in the Laser Powder Bed Fusion (L-PBF) gear manufacturing process via the Reliability-Quality-Reliability (RQR) chain model. The framework highlights the reciprocal link among manufacturing system dependability, process quality, and product reliability, hence enhancing gear performance and operating efficiency. The examination of L-PBF gears indicated that internal flaws, including absence of fusion, balling, keyhole porosity, and residual stress, significantly compromise structural integrity, while exterior issues such as surface wear, hardness variation, and geometric deviation exacerbate reliability deterioration. Experimental reliability graphs shown a significant decrease from about unity to below 0.1 during extended operation when both fault types were analyzed. The assessment of essential quality attributes verified the general stability of the process, with slight variations in wear resistance and geometric precision. Process parameters, including laser power, scan speed, and layer thickness, were recognized as critical factors affecting microstructural integrity, fatigue life, and wear reliability, while post-processing treatments like HIP markedly enhanced performance. The suggested RQR-based performance matrix offers a systematic framework for iterative process optimization and system-wide improvements, facilitating defect reduction, enhanced fatigue and wear reliability, and sustainable, high-quality manufacturing of L-PBF-manufactured gears. The framework is demonstrated using independent peer-reviewed datasets and should be interpreted as a methodological and benchmark validation rather than experimental post-process validation. Future work will focus on independent experimental verification of the proposed framework.
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Appendix A
Appendix A
A1: Production reliability degradation modelling framework based on RQR chain
Step 1: Collecting Key Quality Characteristics (KQCs), comprising Key Product Characteristics (KPCs) and Key Control Characteristics (KCCs), are critical for analysing production reliability. Variations in KCCs during manufacturing transfer to KPCs in workpieces, causing defects and reliability degradation. Big data on KQCs, including degradation parameters, quality inspection results, and defect counts, must be analysed to understand and model the degradation mechanism using the proposed RQR chain.
Step 2: Quantify process quality using co-effect relationships. Based on the identified KQCs from Step 1, product quality is evaluated through the co-effect relationship within the RQR chain. The process model as proposed by Chen and Jin (2005), is utilized to describe these KQCs.
The state of the product dimension, \({X}_{k}(t)\), evolves over time, influenced by a constant variable (\({\tau}_{lk}\)). degradation state vector \(\left({V}_{k}(t)\right)\), and noise vector \(\left({z}_{k}(t)\right)\). Effects are characterized by \({a}_{k}^{T}\) and \({\beta}_{k}^{T}\). while interactions are described by matrix \({\Gamma}_{k}\). Deviation from the standard dimension \({\gamma}_{k}\) is:
Degradation accumulates across stations, modelled as:
where \(\omega (t)={\omega}_{0}+{\omega}_{1}{e}^{\xi t}\) describes wear rate. The initial degradation state \(V\left({t}_{0}\right)\sim N\left({\mu}_{0},{\sigma}_{0}\right)\). and \({\epsilon}_{k}\sim N\left({\mu}_{c},{\sigma}_{k}\right)\). The mean state for \({V}_{k}(t)=0\) is:
Substituting into the deviation equation and Workpiece quality is defined based on tolerance \((k)\):
where \(T(k)\) represents the one-sided tolerance of KQCs and \(f(X(t))\) is a non-linear function of \(X(t)\), with \(X(t)=\left({X}_{1}(t),{X}_{2}(t),\dots ,{X}_{n}(t)\right)\).
Step 3: The model for reliability of exterior defects \(R\left({E}_{f}\right)\) accounts for cumulative variations considering the accumulation of variations, the model of reliability of exterior defects II \(R\left({E}_{f}\right)\) is expressed as follows:
For reliability modelling, a Taylor expansion linearizes \((x)\) :
and reliability is expressed as:
where \({\mu}_{x}=\left({\mu}_{{x}_{1}},{\mu}_{{x}_{2}},\dots ,{\mu}_{{x}_{n}}\right)\) is the average point of \(f(x)\), while the mean and variance of \(f(x)\) are \({\mu}_{f}=f\left({\mu}_{{x}_{1}},{\mu}_{{x}_{2}},\dots ,{\mu}_{{x}_{k}}\right)\) and \({\sigma}_{f}^{2}={\sum}_{i=1}^{n} {a}_{i}^{2}{\sigma}_{i}^{2}+{\sum}_{i=1}^{n} {\sum}_{\begin{array}{c}j\ne 1\\ j\ne 1\end{array}}^{n} {a}_{i}{a}_{j}\text{C}\text{o}\text{v}\left({x}_{i},{x}_{j}\right)\), respectively.
Step 4: For interior defects, the number of defects in station \(k\) is \({N}_{k}\), with \({N}_{k}^{l}\) as interior defects. Reliability \({R}_{k}\left({I}_{f}\right)\) is expressed as:
where \({\theta}_{k}(t)\) is the probability of defect failure by time \(t\). Considering detection \((\tau )\) and defect density \(\left({\lambda}_{k}\right)\), the probability of \({N}_{k}^{l}=m\) is:
leading to:
Assuming defect generation is i.i.d. and follows a Poisson distribution:
The probability for the number of interior defects to be \(m\) is expressed as follows:
For \({\theta}_{k}(t)\), the probability for interior defects to cause a failure at time \(t\) usually follows a Weibull distribution. Therefore, the following equation is obtained:
where \(a\) and \(b\) are the scale and shape parameters in the Weibull distribution, respectively. \({\prod}_{k=1}^{n} {R}_{k}\left({I}_{f}\right)\) can then be obtained as follows:
A.2: Sample calculation for exterior defects and interior defects for L-PBF gear
Step 1: Calculate Process Quality for external defects \((f(X(t)))\) From Table 3 explained in appendix A1 from steps 1 to step 5 using Eqs. A1.1A1.7. The data are used for analysis are from literature review and certain assumption.
where:
\(\Delta {X}_{k}\left(t\mid {V}_{k}(t)\right)={X}_{actual }-{X}_{design}\)
\(T(k)\) is the tolerance.
KQC 1: Lack of fusion porosity
Design: \(4.79\pm 0.39.T(k)=0.39\)
\(\Delta {X}_{k}\left(t{V}_{k}(t)\right)=4.974-4.79=0.184\)
\(f(X(t))=T(k)-\Delta {X}_{k}\left(t\mid {V}_{k}(t)\right)=0.39-0.184=0.206\)
Step 2: Calculate \({\mu}_{f}\) (Mean Deviation) The mean deviation \({\mu}_{f}\) and \({\sigma}_{f}\) (Standard Deviation) Assuming \(\text{C}\text{o}\text{v}\left({x}_{i},{x}_{j}\right)=0\), is given by:
where:
\({\sigma}_{i}\) is the standard deviation for each KQC based on the tolerances (\(T(k)\)).
\({a}_{i}=1\) (equal weights for all stations).
$$\begin{array}{c}{\sigma}_{f}=\sqrt{(0.39{)}^{2}+(40{)}^{2}+(100{)}^{2}+(0.5{)}^{2}}\\ {\sigma}_{f}=\sqrt{0.1521+1600+10000+0.25}\\ {\sigma}_{f}=0.351\end{array}$$
Step 3: Calculate \(\beta\) for standard normal CDF \(\Phi (x)\) gives the probability that a random variable \(Z\sim N(\text{0,1})\) is less than \(x\)
Step 4: By analyzing the defect data examine the influence of interior defects on production reliability is obtained as follows:
A3: Sample calculation for bending fatigue reliability (Section 6.2)
Assume:
Given strain \({\varepsilon}_{tot }=0.002\).
Material properties: \({\sigma}_{f}=450\text{M}\text{P}\text{a},{\varepsilon}_{f}=0.5,b=-0.1,c=-0.6\).
\(E=\text{200,000}\text{M}\text{P}\text{a}\)
Shape parameter \(\beta =1.5\).
Step 1: Predict \({N}_{f}\) From the strain-life equation:
Assume \({\sigma}_{m}=0.5{\sigma}_{f}\) (mean stress).
Solve for \({N}_{f}\).
Step 2: Reliability for \(N={10}^{6}\) Using the Weibull formula:
A4: Sample calculation for wear based reliability (Section 6.2):
Let’s assume the following example values:
Wear coefficient \(K=0.1\) (for DMLS 420 unpolished material).
Applied load \(F=0.25 \text{N}\),
Material hardness \(H=600\text{M}\text{P}\text{a}\),
Rotational speed \(RPM=400\),
Critical wear volume \({V}_{crit }=0.0005{ \text{m}\text{m}}^{3}\).
Characteristic life \(\eta =1000\) hours,
Shape parameter \(\beta =2\).
- 1.
Calculate wear volume at a specific time \(t\) :
$$\begin{array}{c}s(t)=\text{R}\text{P}\text{M}\times t=400\times t\\ V(t)=\frac{K\cdot F\cdot s(t)}{H}=\frac{0.1\cdot 0.25\cdot (400\cdot t)}{600}=\frac{10\cdot t}{600}=\frac{t}{60}\end{array}$$ - 2.
Find the failure time \({t}_{f}\) when \(V\left({t}_{f}\right)={V}_{crit:}\) :
$$\begin{array}{c}\frac{{t}_{f}}{60}=0.0005\\ {t}_{f}=0.0005\times 60=0.03 hours =1.8 minutes\end{array}$$ - 3.
Use the Weibull distribution to calculate failure probability at \({t}_{f}\) :
$$F\left({t}_{f}\right)=1 \text{e}\text{x}\text{p}\left(-{\left(\frac{{t}_{f}}{\eta }\right)}^{\beta }\right)$$$$\begin{array}{c}F\left({t}_{f}\right)=1-\text{e}\text{x}\text{p}\left(-{\left(\frac{1.8}{1000}\right)}^{2}\right)\\ F\left({t}_{f}\right)=1- \text{exp}\left({\left(-1.8\times {10}^{-3}\right)}^{2}\right)\end{array}$$ - 4.
Calculate reliability at \({t}_{f}\) :
$$R\left({t}_{f}\right)=1- F\left({t}_{f}\right)=1- 0=1$$
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Gupta, V.K., Chaturvedi, S.K. & Rai, R.N. Comprehensive reliability-quality integration framework for enhancing gear performance in laser powder bed fusion additive manufacturing process. Sci Rep (2026). https://doi.org/10.1038/s41598-026-52062-0
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DOI: https://doi.org/10.1038/s41598-026-52062-0
