Prior to processing experimental data, the convergence characteristics and noise robustness of the MISG algorithm were evaluated via numerical simulation. This evaluation establishes a controlled environment where true parameter values are known a priori, thereby enabling a direct quantitative assessment of the identification accuracy.

Synthetic vibration data were generated using the finite-element model described in Sect. 2. The system equations of motion were integrated using the fourth-order Runge–Kutta method (Ding et al., 2010; El-Mikkawy and Rahmo, 2003) with a time step dt=0.0001 s at a representative operating speed of 10 000 rpm. The reference radial stiffness was initialized at its nominal state k0=2.5×108 Nm-1, and the corresponding damping matrix was derived via the proportional relationship. To simulate realistic industrial measurement conditions, uncorrelated Gaussian white noise with a standard deviation was added to the displacement responses (He et al., 2025), matching the technical specifications of the eddy current sensors deployed in the physical experimental rig.

A Monte Carlo simulation with 1000 independent trials was conducted to evaluate the algorithm's performance. As illustrated in Fig. 1, the MISG algorithm with an innovation length of 3 exhibits superior convergence properties compared to the standard stochastic gradient (SG) method.

Quantitative analysis confirms that the MISG algorithm reduces convergence iterations from 606 to 75, yielding a 8.1× speedup and an 87.6 % reduction compared to the standard SG method. The MISG approach also decreases the final normalized root-mean-square error by 15 % with minimal steady-state tracking fluctuations. Despite an 18 % increase in per-iteration cost, the accelerated convergence reduces total computation time by 28 %. The results confirm that the algorithm is sufficiently robust in relation to measurement noise and computationally efficient for processing large-scale experimental datasets.

Following numerical verification, a systematic experimental campaign was conducted on a high-speed motorized spindle test rig to capture the true physical variability of the bearing parameters in Fig. 2.

The spindle speed spectrum from 1000 to 24 000 rpm was discretized into 50 uniformly distributed speed points. At each speed point, the spindle was operated under steady-state thermal conditions, and vibration data were acquired using high-precision eddy current sensors sampled at a frequency of 10 kHz. To explicitly quantify the parameter uncertainty, the identification process was repeated 20 times at each speed, resulting in a total dataset of 1000 identified parameter vectors.

The identification results reveal a distinct nonlinear degradation of bearing stiffness with increasing rotational speed, attributed to centrifugal softening and gyroscopic dilation effects. As summarized in Table 3, the radial stiffness of the front bearing decreases by approximately 25.2 % as the speed increases.

The full-range tracking trajectories and the statistical dispersion intervals are visually illuminated in Fig. 3.

The shaded regions in Fig. 3 represent the ±2σ uncertainty bounds derived from the experimental repetitions. Notably, the identified stiffness values exhibit minor yet distinct deviations of approximately 1.5 % to 3.0 % from the numerical baseline model. These discrepancies are attributed to installation-specific factors, such as the thermal expansion mismatch between the steel shaft and the aluminum housing and manufacturing tolerances in the bearing seats. This close yet imperfect agreement highlights the high fidelity of the proposed MISG identification process. It confirms the necessity of using experimentally identified parameters rather than idealized theoretical catalog values to achieve a high-fidelity robust design optimization.

To justify the use of multivariate normal (MVN) distributions in the uncertainty propagation model (Sect. 2), the identified parameters were subjected to rigorous statistical testing. While physical parameters such as stiffness are strictly positive, the coefficient of variation for all identified parameters was found to be low, ranging from approximately 6.0 % to 6.4 %. Consequently, the probability of sampling negative values from a fitted normal distribution is negligible, rendering the Gaussian assumption physically permissible. Normality was validated using the Shapiro–Wilk test, which is sensitive to deviations in the distribution tails. The test results for the main diagonal stiffness parameters are presented in Table 4.

Three of the four diagonal stiffness parameters yielded p values well above the 0.05 threshold, ranging from 0.420 to 0.829, confirming normality. The front-bearing radial stiffness exhibited a marginal departure with a p value of 0.025. However, given the large sample size of 1000, the Shapiro–Wilk test is highly sensitive to minor deviations in the distribution tails, and the practical impact on the Monte Carlo uncertainty propagation remains negligible. To further visualize the goodness of fit, quantile–quantile (Q–Q) plots were generated.

As shown in Fig. 4, the data points align closely with the theoretical dashed red line, with coefficients of determination R2>0.99 for all parameters. This linearity establishes a rigorous statistical foundation for the subsequent data-driven reliability-based robust design optimization (RBRDO) framework.

Furthermore, regarding the system's energy dissipation mechanisms, the operational damping is proportionally coupled with the identified radial stiffness rather than being modeled as an independent stochastic variable. To maintain physical consistency, the nominal damping ratios for the front and rear bearings are configured as 0.032 and 0.028, respectively. These values closely align with the baseline results from independent stationary impact modal tests (averaging 0.029) and fall strictly within the authoritative empirical range of 0.02–0.05 established in rotor dynamics literature (Cao and Altintas, 2004). Consequently, the system uncertainty is fundamentally governed by the four primary radial stiffness parameters, successfully eliminating parameter redundancy while preserving high modeling fidelity.

The bi-objective optimization framework was executed across three distinct target failure probabilities: Ptarget∈{1%,5%,10%}. For each discrete risk threshold, the NSGA-II algorithm generated a Pareto optimal front in the mean–standard deviation μy-σy space, quantifying the conflict between nominal vibration performance and robustness under parameter uncertainties.

A clear monotonic trend governs the topological evolution of the frontiers displayed in Fig. 5. Enforcing a more stringent target failure probability constraint severely restricts the available feasible design space, causing the Pareto fronts to shift toward regions characterized by an increased expected mean vibration but a suppressed statistical dispersion. Unlike traditional unconstrained or single-objective optimization scenarios where stricter constraints uniformly degrade the entire design point, the reliability-based robust design optimization formulation reveals a highly non-trivial physical mechanism. To guarantee a higher safety margin against identified bearing uncertainties, the optimization trajectory is compelled to sacrifice a portion of its deterministic performance to squeeze the distribution tails of the dynamic response. This phenomenon is quantitatively demonstrated by the transition from a 10 % to 1 % target failure probability, which yields a substantial 4.27 µm reduction in the response standard deviation at the cost of a 2.23 µm inflation in the expected peak vibration amplitude. This performance redistribution reflects the mechanical trade-off inherent in high-speed rotor assemblies, where ensuring reliability under uncertainty demands a targeted stiffening strategy.

To facilitate practical engineering decision-making and to isolate specific manufacturing configurations, a representative optimal compromise solution was selected from each Pareto front using the achievement scalarizing function (ASF) method configured with equal objective weight allocations w1=w2=0.5. This mathematical approach identifies the balanced design point that simultaneously minimizes nominal vibration magnitudes and suppresses operational sensitivity. Table 5 compiles the structural design vectors, statistical objectives, and deterministic state constraints evaluated for these three representative compromise configurations.

The results in Table 5 reveal a critical physical mechanism: the bearing preload and the span serve as the primary structural levers for system reliability enhancement. As the target failure probability constraint tightens from 10 % to 1 %, the front-bearing preload increases by 50 % from 2000 N to its absolute physical upper boundary of 3000 N. This aggressive loading directly maximizes the localized radial contact stiffness at the support nodes, narrowing the stochastic response dispersion caused by parametric bearing variations.

Concurrently, the optimization framework executes a major geometric modification by shortening the bearing support span from 749.9 to 323.2 mm. From a structural mechanics perspective, this substantial reduction in the support span alters the tilting lever arm of the shaft assembly, thereby maximizing the global flexural rigidity of the rotor-bearing system and driving the first natural frequency upward from 156.5 to 195.2 Hz. Crucially, all identified compromise solutions strictly satisfy the deterministic engineering guardrails, maintaining a minimum 20 % critical speed resonance margin and keeping structural stresses well below the material yield thresholds.

To guarantee the structural applicability and mathematical transparency of the established parameter identification loop, it is essential to delineate the boundaries of the multi-innovation stochastic gradient (MISG) method regarding error entanglement. In complex rotating machinery, algorithmic identification errors (epistemic uncertainty) and inherent structural variations (aleatory uncertainty) are inevitably coupled within the gathered experimental data. Under the linear-bearing assumption within small-displacement operating regimes, the convergence noise of the MISG algorithm is demonstrated to propagate linearly into the localized support matrices, exhibiting a strictly bounded variance that serves as a justified stochastic input layer for the subsequent RBRDO framework without misleading the optimization vector. However, for heavily loaded systems operating under massive imbalances or strong fluid–film nonlinearities where boundary parameters dynamically scale with displacement amplitudes, the current framework must be adapted into an amplitude-dependent iterative identification scheme. Acknowledging this limitation provides a clear operational boundary for the reliable stabilization and high-fidelity structural dynamic modeling of precision spindle systems.

While stricter reliability constraints generally reduce failure risk, they yield diminishing returns in vibration suppression performance. To explicitly quantify this trade-off, the marginal benefit ratio (MBR) is defined as the reduction in response standard deviation per percentage point reduction in the target failure probability, expressed as MBR=|Δμy||ΔPtarget|. Table 6 summarizes the metrics across the evaluated optimization transitions.

The data reveal a clear decline in optimization efficiency: the transition from 10 % to 5 % yields a high MBR of 0.55 µm%-1, which drops to 0.39 µm%-1 when tightening the constraint from 5 % to 1 %.

Crucially, capturing this minor variance reduction forces the bearing preload to its absolute physical limit of 3000 N, representing a 7.0 % increase over the 5 % design. This aggressive loading accelerates rolling-element wear and risks friction-induced thermal over-closure.

Balancing these trade-offs against ISO 20816-3:2022 Grade G2.5 standards and established spindle design criteria (ISO, 2022), the 5 % configuration provides the optimal engineering equilibrium. It compresses the verified failure probability to 0.35 %, suppresses the response standard deviation to 9.69 µm, and maintains a sustainable baseline for physical prototyping and validation.

To evaluate the proposed RBRDO framework, the optimized design is benchmarked against the unoptimized baseline, RDO, and RBDO frameworks under identical uncertainties. As illustrated in Fig. 6, the RDO approach minimizes the response standard deviation to 8.53 µm and the expected amplitude to 15.25 µm (by setting a 3000 N preload and a 0.3000 m span). However, lacking safety guardrails, its verified failure probability soars to an unacceptable 21.07 % (Fig. 6c). Conversely, the RBDO method restricts the failure probability to 0.30 % by adopting a conservative design (1850.00 N, 624.5 mm). Yet, without variance control, it yields an excessive standard deviation of 11.80 µm and an inflated amplitude of 16.93 µm, indicating high sensitivity to stochastic fluctuations.

By integrating reliability constraints with robust descriptors, the proposed RBRDO framework achieves an optimal compromise design (x=[2804.39N,536.8mm,62.5mm]T). It guarantees a low failure probability of 0.35 % while suppressing the standard deviation to 9.69 µm and maintaining a favorable expected amplitude of 16.44 µm, demonstrating superior joint-optimization capability.

Finally, it is critical to acknowledge that the accuracy of this data-driven probabilistic framework inherently relies on the availability of sufficient and unbiased experimental samples to construct precise probability density functions. When the available data volume is strictly constrained due to high testing costs or operational limits, alternative uncertainty quantification paradigms can be deployed to mitigate data dependency. For instance, non-probabilistic interval or bounded convex models, such as those established by Zhang et al. (Zhang et al., 2026) and Zhao et al. (Zhao et al., 2025), elegantly propagate parameter variations using only the hard physical boundaries of uncertain inputs instead of explicit probability shapes. Alternatively, if a probabilistic structure must be maintained under tight computational budgets, high-efficiency decoupled simulation protocols, such as the two-stage framework proposed by Khalid et al. (Khalid and Bansal, 2026), can be implemented to drastically reduce the required number of expensive function evaluations. Integrating these non-probabilistic interval descriptors or decoupled simulation frameworks into multi-axis high-speed spindle configurations represents a vital and promising direction for future methodological extensions.

To validate the proposed data-driven RBRDO framework, experimental testing was conducted on a high-speed motorized spindle. A high-precision sensor array monitored orthogonal displacement channels at the front- and rear-bearing boundaries to capture real-time lateral dynamics under multi-source uncertainties.

Prior to analyzing the broadband vibration amplitudes, the underlying modal evolution must be established. As illustrated in the integrated Campbell diagram (Fig. 7), the unoptimized baseline configuration features lower natural frequency trajectories, yielding three distinct critical speeds within the working range at 9765, 16 361, and 21 996 rpm. Upon implementing the optimized structural design vector, characterized by an optimal assembly preload of 2804.39 N and a shortened bearing support span of 536.8 mm, the static structural stiffness is substantially enhanced. Consequently, the natural frequency curves shift vertically upward, causing the first two critical speeds to migrate to 11 450 and 19 100 rpm, respectively. The third-order critical speed is successfully extrapolated to 25 800 rpm, safely clearing the 24 000 rpm maximum operational speed threshold.

The shift in critical speeds governs the overall frequency response. As shown in Fig. 8, the baseline spindle produces significant amplitude magnification and asymmetric localized resonance peaks with increasing rotational speed. These peaks display steep-left, shallow-right profiles induced by nonlinear bearing clearances and quadratic unbalance forces. The baseline design exceeds the safety threshold over multiple high-speed ranges, especially near the third-order critical speed.

In contrast, the optimized spindle maintains stable, constrained dynamic responses across the full speed spectrum. Its third-order critical speed falls outside the operating range, eliminating severe high-speed resonant amplification and confining vibrations to a mild sub-resonant region up to 24 000 rpm. The ±2σ uncertainty band verifies that the data-driven framework ensures full-speed operational safety and limits tool-tip vibration despite speed-dependent centrifugal softening and thermal expansion.

The validation metrics in Table 7 demonstrate high predictive fidelity, with relative modeling errors for both the expected mean vibration and response standard deviation remaining consistently well below 5 %. This close agreement confirms that the data-driven parameter identification accurately captured the actual physical variability of the spindle-bearing assembly.

To eliminate any potential experimental ambient bias and mathematically guarantee the optimization gains, a strict statistical verification campaign consisting of 20 independent physical replication runs was conducted at 24 000 rpm. The experimental verification metrics and statistical significance indicators across these independent trials are compiled in Table 7. A comprehensive analysis of the empirical datasets demonstrates that the optimized spindle assembly achieves an immense reliability gain, characterized by an extraordinary 158-fold reduction in the operational failure risk and a concurrent 16.84 % reduction in the expected mean peak vibration amplitude. Furthermore, the robust objective successfully suppresses the response standard deviation by 29.78 %, effectively halving the total response variance.

To highlight the distinct scientific contributions of the proposed data-driven RBRDO framework, Table 8 compares this study with existing robust optimization and reliability-based design methodologies in the contemporary literature.

Unlike traditional mean–variance RDO, which lacks explicit failure probability control, or standard RBDO methods that rely on theoretical FORM/SORM assumptions at a single reliability point, the proposed data-driven parametric framework offers superior robustness, flexibility, and empirical validity (Youn et al., 2005; Mohsine and El Hami, 2010).

The methodology achieves a 50.70 % experimental reduction in variance, surpassing the 22 % typically reported in simulation-based RDO (Gao et al., 2008). Furthermore, it enables systematic trade-off evaluations across three distinct reliability levels (1 %, 5 %, 10 %) rather than adhering to static boundaries. By establishing a rigorous statistical verification loop (p<0.001) from identified parameter distributions, this framework effectively bridges the gap between numerical modeling and physical reality, overcoming the inherent limitations of conventional interval analysis models restricted to unverified worst-case simulations (Zhang et al., 2026).