In modern mechanical systems, the high-speed gear shaft plays a critical role in transmitting power and motion, particularly in applications such as turbines, compressors, and gearboxes. The dynamic performance of the gear shaft directly influences the overall safety, efficiency, and reliability of the system. As operational speeds increase, the gear shaft is subjected to higher stresses, vibrations, and potential resonance issues, which can lead to failures if not properly addressed. Therefore, optimizing the design of the gear shaft to enhance its dynamic characteristics, such as critical speeds and vibration amplitudes, is essential. This study focuses on developing a multi-objective optimization framework for a high-speed gear shaft using the response surface method (RSM) combined with genetic algorithms. By integrating finite element analysis, parameter sensitivity studies, and reliability checks, we aim to achieve a gear shaft design that maximizes operational safety while minimizing critical speeds and associated vibrations.
The high-speed gear shaft considered in this work is part of a fan system, with a total length of 1084 mm and two impellers mounted at both ends. The operational speed is 10,000 rpm, and the gear shaft is made of carbon structural steel with material properties as summarized in Table 1. To facilitate analysis, the gear shaft model is simplified by treating impellers as concentrated masses and bearings as spring-damper elements, while neglecting minor features like fillets and grooves. The finite element model is constructed using SolidWorks and analyzed in ANSYS Workbench, with a hexahedral mesh of approximately 291,546 elements and 565,770 nodes, ensuring sufficient accuracy for dynamic simulations.
| Density (kg/m³) | Young’s Modulus (MPa) | Poisson’s Ratio | Yield Strength (MPa) |
|---|---|---|---|
| 7850 | 2×105 | 0.3 | 250 |
The dynamic behavior of the gear shaft is evaluated through modal analysis, considering both rotational prestress and gyroscopic effects. The natural frequencies and critical speeds are determined across a speed range of 0 to 25,000 rpm, with increments of 5,000 rpm. The governing equation for the rotor dynamics can be expressed as:
$$[M]\{\ddot{x}\} + [C]\{\dot{x}\} + [K]\{x\} = \{F\}$$
where [M] is the mass matrix, [C] is the damping matrix (including gyroscopic terms), [K] is the stiffness matrix, and {F} is the external force vector. The gyroscopic effect introduces splitting into forward (FW) and backward (BW) whirling modes, as shown in the Campbell diagram. For instance, at 10,000 rpm, the first critical speed (Ωc1) is 8,776.7 rpm, and the second critical speed (Ωc2) is 11,337 rpm, with corresponding maximum amplitudes of 13.302 mm and 14.643 mm, respectively. These values indicate that the initial design does not meet safety margins, as critical speeds fall within ±10–15% of the operational speed. Thus, optimization is necessary to shift critical speeds away from the operating range and reduce vibration amplitudes.
To optimize the gear shaft, a multi-objective optimization model is established, with design variables including impeller mass, impeller center of mass positions, bearing span, shaft diameter, bearing stiffness, and damping. The objectives are to minimize the first and second critical speeds (Ωc1 and Ωc2) while constraining their corresponding maximum amplitudes (A1 and A2). The mathematical formulation is:
$$\min \{ f_1(Z), f_2(Z) \}$$
$$\text{subject to: } g_1(Z) \leq 13.302, \quad g_2(Z) \leq 14.643$$
$$\text{where } Z = (z_1, z_2, \dots, z_n)^T$$
Here, f1(Z) and f2(Z) represent Ωc1 and Ωc2, respectively, and g1(Z) and g2(Z) are the amplitude constraints. The design variables and their ranges are listed in Table 2.
| Design Variable | Initial Value | Range |
|---|---|---|
| Shaft Radius (mm) | 37.5 | [33.75, 41.25] |
| Bearing Span (mm) | 235 | [211.5, 258.5] |
| Left Impeller Position (mm) | 110 | [99, 121] |
| Right Impeller Position (mm) | 970 | [873, 1067] |
| Impeller Mass (kg) | 10 | [9, 11] |
| Impeller Moment of Inertia (kg·mm²) | 60,000 | [54,000, 66,000] |
| Bearing Stiffness (N/mm) | 1×106 | [9×105, 1.1×106] |
| Bearing Damping (N·s/mm) | 200 | [180, 220] |
Parameter sensitivity analysis is conducted using Spearman correlation to identify the most influential variables on the critical speeds and amplitudes. The results, summarized in Table 3, show that the left impeller position, right impeller position, and impeller mass have significant effects on Ωc1 and A1, while the right impeller position dominantly influences Ωc2 and A2. This allows us to reduce the number of design variables to three for efficiency: left impeller position (X1), right impeller position (X2), and impeller mass (X3).
| Design Variable | Ωc1 | A1 | Ωc2 | A2 |
|---|---|---|---|---|
| Shaft Radius | 0.002 | 0.033 | 0.016 | -0.014 |
| Bearing Span | -0.035 | 0 | 0.043 | 0.154 |
| Left Impeller Position | 0.363 | 0.315 | 0.059 | 0.116 |
| Right Impeller Position | -0.663 | -0.685 | -0.973 | -0.670 |
| Impeller Mass | -0.415 | -0.420 | -0.132 | -0.161 |
| Impeller Moment of Inertia | 0.072 | 0.058 | -0.035 | -0.130 |
| Bearing Stiffness | 0.117 | 0.054 | -0.011 | -0.050 |
| Bearing Damping | -0.065 | -0.028 | -0.021 | -0.162 |
Using the optimal space-filling design of experiments (DOE), 25 sample points are generated to construct a Kriging-based response surface model. The accuracy of the model is verified through the coefficient of determination (R²), root mean square error (ERMS), and relative maximum absolute error (ERMA). The formulas for these metrics are:
$$R^2 = 1 – \frac{ESS}{TSS}$$
$$ESS = \sum_{i=1}^{n} (f_i(x) – \hat{f}_i(x))^2$$
$$TSS = \sum_{i=1}^{n} (f_i(x) – \bar{f}(x))^2$$
$$E_{RMS} = \sqrt{\frac{1}{n} \sum_{i=1}^{n} (f_i(x) – \hat{f}_i(x))^2}$$
$$E_{RMA} = \frac{\max |f_i(x) – \hat{f}_i(x)|}{\sqrt{\frac{1}{n} \sum_{i=1}^{n} (f_i(x) – \bar{f}(x))^2}}$$
where ESS is the residual sum of squares, TSS is the total sum of squares, n is the number of samples, fi(x) is the observed value, and \hat{f}_i(x) is the predicted value from the response surface. As shown in Table 4, all output parameters achieve R² = 1, ERMS ≈ 0, and ERMA = 0, confirming the high precision of the response surface model.
| Error Type | Ωc1 | Ωc2 | A1 | A2 |
|---|---|---|---|---|
| R² | 1 | 1 | 1 | 1 |
| ERMS | 5.84×10-6 | 1.93×10-5 | 1.4×10-8 | 6.4×10-8 |
| ERMA | 0 | 0 | 0 | 0 |
The multi-objective genetic algorithm (MOGA), a variant of NSGA-II, is applied to the response surface to find Pareto-optimal solutions. The algorithm parameters include an initial population of 10,000, 100 samples per iteration, a maximum Pareto percentage of 70%, and a convergence stability percentage of 2%. The Pareto front for Ωc1 and A1 shows that optimized Ωc1 ranges from 6,400 to 7,300 rpm, a reduction of 16.8–27.1% from the initial value, while A1 decreases to 7–10 mm, a 24.8–47.4% improvement. Similarly, for Ωc2 and A2, the optimized Ωc2 falls between 7,500 and 8,100 rpm (28.6–33.8% reduction), and A2 ranges from 10.8 to 12.8 mm (12.6–26.2% reduction). From the Pareto set, candidate point 1 is selected as the optimal design, with values: X1 = 100.3 mm, X2 = 1,066.6 mm, and X3 = 10.8 kg. The comparison in Table 5 highlights the improvements, where the intervals between critical speeds and the operational speed increase by 22.9% for Ωc1 and 10.8% for Ωc2, meeting safety requirements.
| Parameter | Initial Value | Candidate Point 1 | Candidate Point 2 | Candidate Point 3 |
|---|---|---|---|---|
| X1 (mm) | 110 | 100.27 | 102.38 | 101.06 |
| X2 (mm) | 970 | 1066.6 | 1066.7 | 1066.9 |
| X3 (kg) | 10 | 10.79 | 10.961 | 10.992 |
| Ωc1 (rpm) | 8776.7 | 6491.5 | 6488.2 | 6569.0 |
| Ωc2 (rpm) | 11337 | 7575.2 | 7602.7 | 7676.6 |
| A1 (mm) | 13.302 | 8.919 | 9.152 | 9.405 |
| A2 (mm) | 14.643 | 11.798 | 11.653 | 11.394 |
To ensure the robustness of the optimized gear shaft, a reliability analysis is performed using 10,000 Latin hypercube samples. Random variables include geometric dimensions, material properties, and external loads, assumed to follow normal distributions. The statistical characteristics are provided in Table 6. Sensitivity analysis of these random variables reveals that X1, X2, X3, and Young’s modulus significantly affect Ωc1 and A1, while X2 dominantly influences Ωc2 and A2. The cumulative distribution function (CDF) for the critical speeds indicates that the probability of Ωc1 and Ωc2 being below 8,500 rpm (i.e., within the safe margin of -15% operational speed) is 100% and 99.02%, respectively. This high reliability confirms the effectiveness of the optimization approach for the gear shaft.
| Parameter | Mean (μ) | Standard Deviation (σ) |
|---|---|---|
| Shaft Radius (mm) | 37.5 | 0.067 |
| Bearing Span (mm) | 235 | 0.333 |
| Left Impeller Position (mm) | 100.3 | 0.2 |
| Right Impeller Position (mm) | 1066.6 | 0.533 |
| Impeller Mass (kg) | 10.8 | 0.2 |
| Impeller Moment of Inertia (kg·mm²) | 60,000 | 1,200 |
| Bearing Stiffness (N/mm) | 1×106 | 2×104 |
| Bearing Damping (N·s/mm) | 200 | 4 |
| Young’s Modulus (MPa) | 2×105 | 6,000 |
| External Load (N) | 5,000 | 100 |
In conclusion, this study demonstrates a comprehensive framework for optimizing a high-speed gear shaft using response surface methodology and multi-objective genetic algorithms. By focusing on critical speeds and vibration amplitudes, we achieve a gear shaft design with enhanced dynamic performance and reliability. The optimized gear shaft shows significant improvements in safety margins, with critical speeds shifted away from the operational range and amplitudes reduced substantially. The reliability analysis further validates the design, ensuring a high probability of safe operation under uncertainties. This approach provides a valuable reference for optimizing gear shafts in high-speed applications, contributing to improved mechanical system safety and efficiency. Future work could explore additional factors such as thermal effects or nonlinear behaviors to further refine the gear shaft design.