Modeling and Analysis of Gyroscope Air-Floating Support Assembly Precision Under Uncertain Process Loads

Abstract

Air-floating supports are the core component of precision gyroscopes. The precision of their assembly directly determines the degree of gyroscope drift and plays a decisive role in the device’s output precision and service life. Existing air-floating support assembly precision analysis techniques have not yet considered interface deformation, resulting in poor theoretical accuracy and large deviations in actual performance. To address the issue of imprecise predictions in assembly precision for air-floating supports under process loads, this study performs a predictive analysis and impact assessment of assembly precision with uncertainties in the connection process. Firstly, based on bolt elastic interactions, a method for calculating interface assembly deformations is proposed and integrated into a predictive model under process loads. Secondly, numerical computations and assembly deformation tests are carried out to confirm the model’s accuracy. Finally, the distribution law of gyroscope drift is analyzed, considering uncertainty in the connection process. The results show that the maximum deviation of the assembly accuracy prediction model for air-floating supports is less than 5%. Drift in air-floating supports under normally distributed loads shows a skewed normal distribution pattern; the mean drift value decreased sequentially from about 0.03023°/h/g to 0.01995°/h/g, which was reduced by 34% with variation in the dispersion of the pretension force (from 15% to 5%). These findings provide novel insights to enhance assembly precision processes and optimize the fabrication of high-end, complex equipment.

1. Introduction

As modern industries accelerate their evolution toward high-end and intelligent systems, the demands for operational accuracy and reliability in advanced, complex equipment have become increasingly stringent [1]. Characterized by their high accuracy and stability, precision gyroscopes are critically important for ensuring the precision of navigation, attitude control, and motion control in aerospace and aviation systems [2,3]. As a core component of precision gyroscopes, the gyroscope air-floating support must exhibit exceptionally low drift errors to guarantee the long-term reliable operation of the gyroscope [4].

However, the process load applied to the air-floating support assembly is very likely to deform the support interface assembly, which affects the precision of the air-floating support assembly and increases the drift of the gyroscope. At the same time, the coupling effect of process load uncertainty in the on-site environment may degrade the performance of the gyroscope and can even lead to instrument jamming in serious cases [5]. Therefore, analytically investigating interface mechanical characteristics under assembly processes and accurately characterizing assembly precision under process loads is essential for guiding on-site assembly procedures. This approach represents a critical foundation for enhancing both the precision and stability of gyroscopes.

Even minute deviations during the assembly process of an air-floating support can compromise the uniformity of the gas film, leading to film degradation during service and directly affecting the gyroscope’s output accuracy and operational lifespan. Existing research on the precision of air-floating supports has predominantly focused on structural parameter design and manufacturing processes. For example, Zhang [6] investigated the effects of eccentricity, bearing radius clearance, and groove structure parameters on pressure distribution and load-carrying capacity. Yang [7] investigated the effects of three different groove structures (straight, helical, and herringbone) on bearing performance, showing that the helical groove structure had the worst effect. Qin [8,9] integrated the support structure and manufacturing techniques to compare gas interference torque in spherical air-floating gyroscopes using two throttling methods, namely, small orifices and throttling slits. Feng [10,11] developed a theoretical lubrication model for hemispherical dynamic pressure air-floating supports, compared the performance of these supports under various structural parameters and operating conditions, and analyzed the impact of nominal clearance and groove depth on bearing stability and unbalanced responses, providing a theoretical reference for bearing design and life evaluation. Regarding the impact of manufacturing errors on gyroscope precision, Liu [12,13] established mathematical models for float elliptical errors, triangular errors, throttling slit parallelism errors, and float radial eccentricity. They analyzed these effects on eddy current torque. Liang [14,15] investigated the relationships between gas film thickness, surface roughness, and slit width and load capacity, stiffness, and eddy current torque in air-floating-bearing gyroscopes. Li [16,17] examined the drift error induced by the nonlinear deformation of the gas film in dynamic pressure air-floating gyroscopes based on their structural characteristics and operating principles. Wang et al. [18] showed that double annular slot and nonuniform slot gas-bearing systems experience chaotic motions due to the nonlinear pressure distribution of gas film, unbalanced gas supply, and other factors under specific rotor mass and bearing number conditions. Meanwhile, the authors of [19] used various machine learning methods to efficiently predict chaotic behavior, significantly reducing computational costs and providing strong theoretical and practical guidance for designing gas-bearing systems in industrial applications. An et al. [20] found that the structural parameters of aerostatic bearings (aperture diameter, bearing clearance, groove length, and height) significantly affect the air film flow field and spindle motion through fluid–solid coupling. That study provides effective theoretical and methodological support for optimizing the structural design of aerostatic bearings and improving performance at high rotational speeds.

In summary, most research on air-floating supports has focused on the “design-processing” stage. The assembly stage is not discussed with respect to accuracy prediction, such that the theoretical results cannot be fully verified and applied to actual production. This limitation not only hinders enhancements in the overall performance of air-floating supports but also fails to provide comprehensive, scientifically grounded guidance for the gyroscope manufacturing process. Consequently, analyzing the assembly precision of gyroscope air-floating supports under process loads is a critical issue.

In analyzing the assembly precision of high-precision products, some scholars have established quantitative relationships between process loads and assembly precision by characterizing the coupling effects of various loads. Mu et al. [21,22] mapped small-displacement rotation vectors representing manufacturing errors and assembly deformations onto the tolerance domain, allowing for the establishment of an assembly precision analysis model that was validated for the assembly of high-pressure rotors in aeroengines. Sun et al. [23] extracted the differential morphological features of mating surfaces through surface morphology analysis and performed virtual assembly based on these characteristics. The calculated deviations—expressed via small-displacement rotation vectors—showed good agreement with the measured deviations. Other researchers have examined the precision of product assembly from the perspective of process loads. For example, Wang [24] developed a simulation model for the assembly of large reflector antennas based on the unit survival method, calculating the real-time assembly deformation of each panel and the resultant assembly error. Abasolo et al. [25,26] evaluated the elastic interactions between bolt groups in wind turbine towers using a four-parameter unit model and proposed an optimization method to achieve uniform bolt preloads. Zhu et al. [27,28,29] established a multi-bolt assembly model for flange structures based on annular beam elastic theory and optimized multi-step bolt tightening parameters, achieving favorable experimental results with large-scale pipeline connection structures. Wu et al. [30] studied the correlations between machine tool guideway straightness and bolt group tightening parameters, proposing an optimization method for bolt tightening sequences utilizing particle swarm and genetic algorithms. Zhang et al. [31] have derived an elastic interaction matrix from finite-element simulation data and subsequently established a model to predict product coaxiality. Cheng [32] introduced a novel method based on triangular facet models and the Laplace operator to calculate deformations in sheet metal parts during simulated assembly processes. Wang [33] combined finite element analysis with multi-level analysis to predict the deformation of horizontal stabilizer assemblies, providing a theoretical foundation for precisely analyzing small flexible components. Chang [34] developed a finite-element model for plate parts to analyze the influence of manufacturing tolerances and contact interference on assembly deformation. Wang et al. [35] investigated the elastic deformation of thin-walled structures generated during the clamping and positioning process. They established a local assembly deformation prediction model using the elastic beam model and determined its effect on the overall assembly error of the structure. Song et al. [36] calculated guide rail deformation under different tightening torques based on the pressure distribution of the bolts under tightening conditions.

In summary, during the actual assembly phase of precision equipment, process loads are affected by on-site conditions and subject to numerous uncertainties stemming from multiple sources. These include the technical proficiency of operators, tool precision, and other internal factors. These factors interact through cumulative, coupled, or amplified effects, ultimately profoundly impacting the final precision of the assembly. Therefore, considering the uncertainty of assembly process loads is crucial for optimizing on-site assembly conditions and enhancing product assembly quality.

In response to the above problems, this study (1) systematically explores the impact of process load uncertainty on the assembly accuracy of air-floating supports; (2) proposes an air-floating support interface assembly deformation prediction model; and (3) on this basis, for the first time, incorporates process load uncertainty into the air-floating support assembly accuracy analysis framework, revealing the influence of assembly process parameter fluctuations on drift in air-floating supports and clarifying the intrinsic relationship between spatial distribution nonuniformity in interface deformation and the accuracy of drift characteristics. The specific contents are as follows: First, based on the elastic interaction relationship of bolts, a method for calculating interface assembly deformation in air-floating bearings is proposed. Furthermore, we systematically establish a correlation between the process load in the assembly process and the assembly accuracy of air-floating bearings. Second, the interface assembly accuracy prediction model is verified through refined numerical calculations and assembly deformation tests. Finally, the influence of assembly process uncertainty on drift in gyroscope air-floating bearings is analyzed using actual preload test data, and the correlation between interface deformation nonuniformity and air-floating bearing drift is further explored, providing theoretical and methodological support for gyroscope air-floating bearing assembly process optimization and performance improvement.

2. Gyroscope Air-Floating Support Assembly Prediction Modeling

During the assembly process of gyroscope air-floating supports, the support interface is influenced by process loads, which can readily induce nonuniform interface deformations. This results in an eccentric error between the actual gas film and the ideal gas film centroid, thereby degrading assembly precision and leading to gyroscope drift errors. In this section, a method for calculating the interface assembly deformation of gyroscope air-floating supports is developed based on bolt interactions. A nonlinear fitting approach is employed to construct an assembly precision prediction model, and in conjunction with the gas lubrication equation, an association model between process loads and gyroscope air-floating support drift error is established. This provides a theoretical foundation for subsequent analyses of process uncertainties.

2.1. Interface Assembly Deformation Calculation Model

A gyroscope air-floating support comprises a rotor component and a stator component, interconnected by multiple bolts. As shown in Figure 1, the stator section consists of a motor shaft and an air-floating hemispherical bearing, while the rotor section comprises a rotor endcap and housing. The endcap of the support is mounted onto the housing using uniformly distributed screws, and the clearance between the hemispherical bearing and the rotor is maintained at the micron level to stabilize the dynamic pressure motor’s center of mass. During assembly, the bolt preload must be appropriately controlled to prevent deformation in the rotor end cap that could compromise instrument accuracy.

During the assembly process, bolts elongate under the bolt preload, and the connected components deform accordingly, as illustrated in Figure 2. The tightening process induces interface assembly deformation, affecting the uniformity of the gas film thickness. Moreover, bolts tightened later exert an elastic influence on those tightened earlier, thereby altering the magnitude of their bolt preload.

For analytical simplicity, the support structure is reduced to a schematic of a flange connection (Figure 3), consisting of upper and lower connecting components. In this configuration, the interface deformation of the connecting components is collectively influenced by six bolts.

The interface deformation of the connected parts under the process load comprises two parts. The first is the elastic deformation of the connected parts when the bolts are tightened; the second is the elastic deformation of the interaction between the bolts in other positions when they are tightened. However, the influence of non-adjacent bolts on the target bolt is smaller than that of the adjacent bolts, so only the influence of the adjacent bolts is considered here. For example, the elastic deformation at bolt No. 1 is not only influenced by its own action but also by adjacent bolts No. 3 and No. 6. The magnitude of this deformation can be expressed as follows:

$${\delta }_1={\delta }_{1,1}+{\delta }_{1,3}+{\delta }_{1,6}$$

(1)

After bolt No. 3 is tightened, based on the deformation coordination condition, the deformation of the connected component at bolt No.1 is equal to the elongation change in bolt No. 1 under the influence of bolt No. 3; that is,

$${\displaystyle \frac{\mathsf{\Delta }{F}_{1,3}}{K_{1,1}}}+{\displaystyle \frac{F_3}{K_{1,3}}}={\displaystyle \frac{\mathsf{\Delta }{F}_{1,3}}{K_B}}$$

(2)

where $\mathsf{\Delta }{F}_{1,3}$ represents the change in the bolt preload of bolt No. 1 induced by bolt No. 3.

Using the above equation, $\mathsf{\Delta }{F}_{1,3}$ can be determined. Consequently, the elastic deformation of bolt No. 1 caused by its elastic interaction with bolt No. 3 can be expressed as:

$${\delta }_{1,3}={\displaystyle \frac{\mathsf{\Delta }{F}_{1,3}}{K_{1,3}}}={\displaystyle \frac{F_3{K}_B{K}_{1,1}}{K_{1,1}{K}_{1,3}\cdot \left(K_B-K_{1,1}\right)}}={\displaystyle \frac{F_3{K}_B}{K_{1,3}\cdot \left(K_B-K_{1,1}\right)}}$$

(3)

Thus, the nodal elastic deformation at bolt No. 1 can be represented as:

$$\begin{array}{l}{\delta }_1={\delta }_{1,1}+{\delta }_{1,3}+{\delta }_{1,6}\\ \hspace{1em}=\left[{\displaystyle \frac{F_1}{K_{1,1}}}+{\displaystyle \frac{F_3{K}_B}{K_{1,3}\left(K_B-K_{1,1}\right)}}+{\displaystyle \frac{F_6{K}_B}{K_{1,6}\left(K_B-K_{1,1}\right)}}\right]+{\displaystyle \frac{F_3}{K_{1,2}}}+{\displaystyle \frac{F_6}{K_{1,6}}}\\ \hspace{1em}={\displaystyle \frac{F_1}{K_{1,1}}}+F_3\left[{\displaystyle \frac{1}{K_{1,3}}}+{\displaystyle \frac{K_B}{K_{1,3}\left(K_B-K_{1,1}\right)}}\right]+F_6\left[{\displaystyle \frac{1}{K_{1,6}}}+{\displaystyle \frac{K_B}{K_{1,6}\left(K_B-K_{1,1}\right)}}\right]\end{array}$$

(4)

In summary, the total elastic deformation at bolt No. 1 comprises the deformation of the connected component due to bolt No. 1’s own tightening and the elastic interaction deformations of bolts No. 3 and No. 6. This total deformation can be organized into three parts: the first is the deformation due to bolt No. 1’s tightening combined with the deformation caused by the change in its bolt preload when adjacent bolts No. 3 and No. 6 are tightened; the second and third are the tightening deformations of the connected component at bolt No. 1 induced by adjacent bolts No. 3 and No. 6.

According to Equation (4), the interface deformation of a gyroscope air-floating support can be expressed as:

$$\left[\begin{array}{c}{\delta }_1\\ {\delta }_2\\ {\delta }_3\\ {\delta }_4\\ {\delta }_5\\ {\delta }_6\end{array}\right]=\left[\begin{array}{cccccc}{\displaystyle \frac{1}{K_{1,1}}}& {\displaystyle \frac{1}{K_{1,2}}}+{\displaystyle \frac{K_B}{K_{1,2}\left(K_B-K_{1,1}\right)}}& 0& 0& 0& {\displaystyle \frac{1}{K_{1,6}}}+{\displaystyle \frac{K_B}{K_{1,6}\left(K_B-K_{1,1}\right)}}\\ {\displaystyle \frac{1}{K_{1,2}}}+{\displaystyle \frac{K_B}{K_{1,2}\left(K_B-K_{2,2}\right)}}& {\displaystyle \frac{1}{K_{2,2}}}& {\displaystyle \frac{1}{K_{2,3}}}+{\displaystyle \frac{K_B}{K_{2,3}\left(K_B-K_{2,2}\right)}}& 0& 0& 0\\ 0& {\displaystyle \frac{1}{K_{2,3}}}+{\displaystyle \frac{K_B}{K_{2,3}\left(K_B-K_{3,3}\right)}}& {\displaystyle \frac{1}{K_{3,3}}}& {\displaystyle \frac{1}{K_{3,4}}}+{\displaystyle \frac{K_B}{K_{3,4}\left(K_B-K_{3,3}\right)}}& 0& 0\\ 0& 0& {\displaystyle \frac{1}{K_{3,4}}}+{\displaystyle \frac{K_B}{K_{3,4}\left(K_B-K_{4,4}\right)}}& {\displaystyle \frac{1}{K_{4,4}}}& {\displaystyle \frac{1}{K_{4,5}}}+{\displaystyle \frac{K_B}{K_{4,5}\left(K_B-K_{4,4}\right)}}& 0\\ 0& 0& 0& {\displaystyle \frac{1}{K_{4,5}}}+{\displaystyle \frac{K_B}{K_{4,5}\left(K_B-K_{5,5}\right)}}& {\displaystyle \frac{1}{K_{5,5}}}& {\displaystyle \frac{1}{K_{5,6}}}+{\displaystyle \frac{K_B}{K_{5,6}\left(K_B-K_{5,5}\right)}}\\ {\displaystyle \frac{1}{K_{1,6}}}+{\displaystyle \frac{K_B}{K_{1,6}\left(K_B-K_{\mathrm{6,6}}\right)}}& 0& 0& 0& {\displaystyle \frac{1}{K_{5,6}}}+{\displaystyle \frac{K_B}{K_{5,6}\left(K_B-K_{\mathrm{6,6}}\right)}}& {\displaystyle \frac{1}{K_{\mathrm{6,6}}}}\end{array}\right]\cdot \left[\begin{array}{c}{F}_1\\ F_2\\ F_3\\ F_4\\ F_5\\ F_6\end{array}\right]$$

(5)

Assuming that the bolt preload of all six bolts is identical, Equation (5) can calculate the assembly interface deformation distribution of the support structure, as shown in Figure 4. The results indicate that even when identical bolt preloads are applied, the deformation magnitudes at each bolt location are not uniform. In practical assembly, however, it is difficult to achieve completely consistent bolt preloads across all six bolts, leading to significant nonuniformity in the interface deformation of air-floating support structures.

In summary, by establishing a computational model for the interface assembly deformation of air-floating supports under process loads, we can demonstrate that even under uniform bolt preload conditions, the interactions between bolts cause nonuniform deformation at the interface. In turn, this nonuniformity adversely affects the gas film of the air-floating support, ultimately reducing assembly precision.

2.2. Assembly Precision Prediction Model Under Process Loads

The rotor and stator of the air-floating are completely isolated by a gas film, allowing the rotor to operate at high speed and low friction. A simplified diagram of the air-floating is shown in Figure 5. The air-floating consists of a motor end cover and an air-floating with a spiral groove. The surface of the outer hemisphere is smooth, while the surface of the inner hemisphere is engraved with a spiral groove. The purpose of the air-floating is achieved by the air film between the inner and outer hemispheres; the uniformity of the air film directly affects the bearing’s performance. Under process loads during assembly, interface deformations affect the uniformity of the gas film thickness between the rotor and stator, resulting in an eccentricity error in the gas film that degrades the assembly precision of the air-floating support.

During assembly, the bolt preload applied by the bolts inevitably induces deformation at the support interface. This deformation causes eccentricities between the inner and outer hemispherical surfaces, leading to nonuniform variations in the gas film thickness and significantly impacting assembly precision. Under undeformed conditions, the centers of both hemispherical surfaces are located at (0, 0, 0) with a known radius R. When slight deformations occur due to assembly, the coordinates of points $P_i=(x(i),y(i),z(i))$ on the surface deviate from those of the ideal sphere but still approximately satisfy the spherical equation.

$${(x(i)-x_c)}^2+{(y(i)-y_c)}^2+{(z(i)-z_c)}^2=r^2$$

(6)

A nonlinear residual optimization objective function is designed, so that minimizing this function yields the best-fit sphere.

$$E=\sum _{i=1}^n{\left[\sqrt{{\left(x(i)-x_c\right)}^2+{\left(y(i)-y_c\right)}^2+{\left(z(i)-z_c\right)}^2}-r\right]}^2$$

(7)

By computing the vector difference between the center of the best-fit sphere and that of the ideal sphere, we can obtain the eccentricity components ${\epsilon }_x$ and ${\epsilon }_y$ along the gyroscope’s input and output axes, and ${\epsilon }_z$ along the self-rotation axis of the gyroscope motor. These three components represent the assembly precision of the air-floating support under process load conditions.

Furthermore, a correlation between the assembly precision of the gas film support and the resulting drift error is established. The governing equations of gas flow in lubrication are complex—velocity, density, pressure, and temperature are functions of both space and time. However, since air-floating supports operate under small-gap lubrication conditions (where the film thickness is much smaller than other dimensions), the Navier–Stokes equations can be simplified. To this end, the following assumptions are made:

(1)

The lubricating fluid is treated as an ideal fluid;

(2)

the thermal state of the lubricant is assumed to be isothermal;

(3)

the flow in the lubricant layer is considered to be laminar;

(4)

the viscosity in the gas film thickness direction is assumed to be constant.

Based on these assumptions, a gas lubrication characteristic model for hemispherical dynamic-pressure air-floating bearings is established, and the dimensionless lubrication equation for the air-floating support can be expressed as follows:

$$\sin \theta \cdot {\displaystyle \frac{\partial }{\partial \theta }}({\overline{h}}^3\sin \theta \cdot \overline{P}{\displaystyle \frac{\partial \overline{P}}{\partial \theta }})+{\displaystyle \frac{\partial }{\partial \varphi }}({\overline{h}}^3\overline{P}{\displaystyle \frac{\partial \overline{P}}{\partial \varphi }})=-\mathsf{\Lambda }{\displaystyle \frac{\partial (\overline{Ph})}{\partial \varphi }}+2\mathsf{\Lambda }\gamma \cdot {\displaystyle \frac{\partial (\overline{Ph})}{\partial \tau }}$$

(8)

where $\overline{h}$ represents the gas film thickness, $\overline{P}$ is the dimensionless gas pressure, $\mathsf{\Lambda }$ is the number of bearings, and $\gamma$ is the rotational speed coefficient.

Considering the eccentricity induced by the assembly deformation of the air-floating support, the gas film thickness in the lubrication equation is represented by

In the groove,

$$\overline{h}=1+H_g+{\epsilon }_x\sin \theta \cos \varphi +{\epsilon }_y\sin \theta \sin \varphi +{\epsilon }_z\cos \theta$$

(9)

Outside the groove,

$$\overline{h}=1+{\epsilon }_x\sin \theta \cos \varphi +{\epsilon }_y\sin \theta \sin \varphi +{\epsilon }_z\cos \theta$$

(10)

Eccentricity on the surface of the air-floating support results in a total tangential force along the circumferential direction, which can be expressed as:

$$F_{\tau }={\displaystyle \sum _{e\subset H}(p_{0}R{h}_0)\frac{\mathsf{\Delta }\overline{\mathrm{p}}}{4\mathsf{\Delta }\phi }{\displaystyle \underset{e}{\iint }\overline{h}}d\phi d\xi }$$

(11)

Furthermore, the assembly eccentricity brings about an air-float support interference moment calculated as:

$$M_0=\sum _{e\subset H}(p_0{R}^2{h}_0){\displaystyle \frac{\mathsf{\Delta }\overline{p}}{4\mathsf{\Delta }\phi }}{{\displaystyle \iint }}_e\overline{h}d\phi d\xi$$

(12)

Interference moments cause a gyroscope drift error which can be expressed as:

$${\omega }_{\mathrm{d}}={\displaystyle \frac{M_o}{H_r}}$$

(13)

where $H_r$ represents the momentum moment of the gyroscope motor’s air-floating support component.

According to the air-flotation support parameters shown in

Table 1. Calculation parameters of the air-floating support bearing capacity.

Air-Floating Support Parameters	Data
Radius, R/mm	25.0
Widths, b/mm	17.0
Air film thickness, c/μm	2.7
Groove depth, hg/μm	10
Number of grooves, Ng	10
Groove direction angle, βg/(°)	30
Effective angle of groove, θg/(°)	52
Groove width ratio, αg	0.5
Gas viscosities, μ/(Pa s)	1.79 × 10−5
Environmental pressure, Pa/Pa	1.013 × 10−5

, and using the above interference moment calculation model, the gas film pressure distributions—with and without assembly eccentricity—were computed as shown in Figure 6 and Figure 7.

Under ideal conditions with no eccentricity, the air film pressure distribution exhibits a high degree of symmetry and uniformity. The pressure maps show that the gas film pressure is regularly distributed in both the circumferential and axial directions, enabling the gas film to uniformly support the entire structure, thus preventing disturbances or deflections due to localized uneven forces. However, when eccentricity is present, the gas film pressure distribution changes markedly. The pressure distribution loses its symmetry, with the pressure peak in the central region notably shifting to one side and the pressure gradients in both the circumferential and axial directions increasing significantly. This nonuniform pressure distribution directly causes force imbalances within the system, which generate interference moments—in this case, the calculated interference moment is 3.784 × 10⁻⁶ Nm. For high-precision air-floating support systems, such an effect is far from negligible.

In summary, process loads inevitably induce assembly deformations at the air-floating support interface. A model for assembly precision based on interface deformation and established via a nonlinear fitting algorithm has been incorporated into calculating the air-floating support’s output drift. This has led to the development of a solution model for drift error under process loads, thereby providing a theoretical basis for analyzing their uncertain impacts.

3. Validation of Assembly Precision Prediction Model

In the assembly process of gyroscope air-floating supports, the bolt group’s process loads inevitably induce assembly deformations at the support interface. Taking the gyroscope air-floating support as an example, in this section, a detailed simulation model of the air-floating rotor component is developed and combined with assembly deformation tests to validate the previously proposed predictive model for assembly precision under process loads.

3.1. Numerical Calculations and Experiment Setups

Based on the air-floating support structure, a detailed rotor simulation model is established using the commercial finite element software ANSYS Workbench 2020R2, with material parameters listed in

Table 2. Parameters for numerical calculation of air-floating support.

Part	Material	Density $\mathsf{\rho } (\mathbf{g}/{\mathbf{c}\mathbf{m}}^3)$	Modulus of Elasticity E (GPa)	Poisson’s Ratio $\mathsf{\nu }$
Rotor Shell	65 Mn	7.82	211	0.288
Rotor Cover	65 Mn	7.82	211	0.288

. Bolt preloads are applied via beam elements to minimize the influence of bolt head dimensions on surface deformations. The shoulder fit is set as an interference friction contact, while the rotor interface is modeled as a frictional contact. The simulation process is divided into two stages: the shoulder pre-compression stage and the symmetric screw-tightening stage. In the first stage, pre-pressure is applied to the rotor cover, and the bottom of the rotor housing is fixed; in the second stage, while maintaining the fixed bottom constraint, the same tightening preload is sequentially applied to the screws in the following order: 1-4-2-5-3-6 (see Figure 8).

Simultaneously, assembly deformation tests of the air-floating rotor interface are conducted. The test specimen (Figure 9) comprises the upper connecting component (rotor cover) and the lower connecting component (rotor housing). A high-precision displacement measurement turntable captures the rotor’s interface assembly deformation. The test specimen is fabricated using 65 Mn steel, with a connecting component diameter of 100 mm, an upper component thickness of 5 mm, and a shoulder diameter of 72 mm. The surface roughness of the connecting and shoulder contact surfaces is Ra 1.6. In the tests, grade 8.8 full-thread M6 bolts with a length of 25 mm are used. As depicted in Figure 10, probe 1 measures the interface deformation due to bolt elastic interactions, while probe 2 measures the roundness of the rotor cover; the latter ensures the accurate clamping of the specimen, with retesting if necessary. A digital torque wrench (±2% accuracy) applies tightening torque to the uniformly distributed bolts via a torque control method, with a symmetric tightening sequence consistent with the simulation. A fixed tightening torque of 5 NM is applied to each bolt. At the same time, based on the experimentally determined torque–preload conversion coefficient, the actual preload force of a single bolt is calculated to be approximately 4250 N. The theoretical total clamping force of the six bolts under this torque condition is approximately 25,500 N. To eliminate the influence of manufacturing errors on the deformation measurement, no-load interface profile data are first acquired using a displacement probe; the deformation measured during bolt tightening is then corrected by subtracting the no-load profile, yielding the true interface assembly deformation on the bowl surface.

3.2. Comparison of Calculation Results

The assembly deformation data obtained from the simulation and experiments are extracted and compared with the theoretical calculations, as shown in Figure 11. In this figure, the blue line represents the numerically calculated interface deformation, the red line represents the theoretical model result, and the black line represents the experimental measurement. The theoretical model results agree well with the numerical and experimental data, with a maximum calculation error of less than 0.8 μm. The deformation curve indicates that under the influence of the external bolt preload, the rotor system exhibits nonuniform oscillatory interface deformations. Even with symmetrically applied bolt preload, the final interface deformation displays an asymmetric “double-peak” distribution, with maximum deformation occurring at bolts 2 and 5 (i.e., near the last-tightened bolt).

The assembly precision of the specimen is calculated based on the interface deformation measured by three methods, as summarized in the

Table 3. Comparison of assembly precision of air-floating support.

	${\mathit{\epsilon }}_{\mathit{x}}$	${\mathit{\epsilon }}_{\mathit{y}}$	${\mathit{\epsilon }}_{\mathit{z}}$
FEA	0.0084	0.0074	0.0286
Analytical	0.0078	0.0081	0.0273
Experiment	0.0088	0.0070	0.0289

below.

In summary, the comparison of numerical and experimental results demonstrates that the proposed interface assembly deformation calculation model achieves a maximum error of less than 0.8 μm. By fitting the assembly deformation results to derive the gas film eccentricity, the assembly precision of the air-floating support can be obtained. The assembly precision data from the three methods show that the maximum deviation between the theoretical model and the numerical and experimental results is less than 5%, verifying the accuracy of the theoretical model and establishing the relationship between process loads and assembly precision for air-floating supports.

4. Analysis of the Effect of Process Uncertainty on Drift in Air-Floating Supports

Prior studies have established a model for calculating the assembly precision of air-floating supports under process loads. However, the uncertainty of process loads induces fluctuations and deviations in actual assembly precision, resulting in drift errors in the support. To better quantify and control this effect, this section focuses on modeling uncertainty in the bolt preload associated with process loads, employing statistical and simulation methods to characterize the distribution of drift errors.

4.1. Modeling Preload Uncertainty

According to the classic torque–bolt preload relationship proposed by Motosh [37], the input tightening torque comprises three components: the friction torque at the bolt head end, $T_b$; the thread fastening torque, $T_p$; and the thread friction torque, $T_t$ (Equation (14)). Notably, the thread fastening torque, $T_p$, generates the bolt preload. When the input tightening torque, $T_{input}$, is constant, the dispersion in the bolt preload primarily arises from variations in the other two friction torque components, $T_b$ and $T_t$.

$$T_{input}=T_b+T_p+T_t$$

(14)

The relationship between the bolt preload and the thread fastening torque is expressed by Equation (15):

$$F=2T_p/({\displaystyle \frac{P}{\pi }}+{\mu }_p{D}_2\sec \beta )$$

(15)

where $D_2$ is the mean thread diameter, $P$ is the pitch, ${\mu }_p$ is the thread friction coefficient, and $\beta$ is the thread flank angle. This equation indicates a linear relationship between the thread torque and the bolt preload, allowing for its calculation based on the thread torque. In practice, however, factors such as component manufacturing processes, material properties, contact conditions, and assembly sequences cause significant fluctuations in the remaining friction torque, $T_b$ and $T_t$, under identical tightening torques, leading to considerable dispersion in the bolt preload.

To accurately characterize the distribution of bolt preloads in air-floating supports, single-bolt preload experiments were conducted, as illustrated in Figure 12.

By controlling the input torque to apply the bolt preload, the torque values and corresponding bolt preload were precisely recorded. To ensure the reliability of the statistical analysis, the experiment was repeated multiple times, yielding 100 sets of bolt preload data for a single bolt. A histogram of these data was constructed with an overlaid theoretical normal distribution curve, as shown in Figure 13. The red fitting curve—drawn using the calculated mean and standard deviation—matches the histogram closely, indicating that the data follow an approximately normal distribution. Accordingly, the single-bolt preload, F, is modeled as a normal distribution, $F\sim N(\mu ,{\sigma }^2)$, with a mean of $\mu$ = 4250 and a standard deviation of $\sigma$ = 19.250; thus, the preload for the air-floating support bolts can be adequately described by this normal distribution.

4.2. Analysis of Air-Floating Support Drift Under Process Uncertainty

Using the experimentally determined distribution parameters, uncertainty analysis samples were generated and incorporated into the assembly precision model developed for the air-floating supports, enabling an investigation into the effect of bolt preload uncertainty on gyroscope drift error. Statistical data were visualized using MATLABR2023b software. A scatter plot of the air-floating support drift error is presented in Figure 14, where the red solid line denotes the mean drift value and the red dashed lines indicate the drift value at ±3 sigma, and the blue circles is the drift value.

The drift error distribution histogram and the normal distribution curve fitting results are shown in Figure 15, with the blue bars showing the drift error data and the red curve showing the theoretical normal distribution curve. The drift error data (histogram) are mainly concentrated in the central area of the distribution, but there is a clear difference between the actual data and the theoretical normal distribution (red curve), especially in the area with large errors (the right tail of the histogram), which shows obvious skewness. This shows that the drift error does not strictly follow the normal distribution assumed in theory under actual assembly conditions. Further analysis of this phenomenon shows that, although the preload of a single bolt presents a relatively typical normal distribution, there is a complex elastic coupling effect on the assembly interface in the actual transmission process of the preload of multiple bolts (see the interface assembly deformation distribution curve shown in Figure 11). The resulting nonuniform deformation of the interface generates eccentricity in the air-floating assembly and ultimately affects the statistical characteristics of the drift error. Therefore, in the actual gyroscope structure design and formulation process, the preload size, spatial distribution, and coupling with assembly accuracy should be more deeply considered to better control the final drift error and improve the performance stability of the equipment.

Six symmetrically arranged bolts are employed to secure the endcap of the air-floating support structure. The bolt preload distribution of these six bolts is described by a covariance matrix, where the diagonal elements represent variance in the preload of each bolt. Smaller diagonal values indicate lower fluctuation and a more concentrated bolt preload distribution, while larger values indicate greater fluctuations and a more dispersed distribution.

Here, uncertainty analysis samples with three preload variance sizes are generated, with different sizes in the preload variance of each bolt to simulate different degrees of dispersion in the preload distribution, as shown in

Table 4. Preload variance parameter settings.

	Mean Value, $\mathit{\mu }$	Covariance Diagonal Factor, ${\mathit{\sigma }}_{\mathit{i}\mathit{i}}$	Characteristics of Preload Distribution
Dataset 1	4250	Smaller variance	5% preload dispersion
Dataset 2	4250	Medium-sized variance	10% preload dispersion
Dataset 3	4250	Greater variance	15% preload dispersion

. The case with the smallest variance (5%) is the actual measurement data obtained from the single-bolt preload test in Section 3 (the mean value is 4250 N, and the standard deviation is about 19.25 N). The specific settings are as follows.

These samples were incorporated into the air-floating support assembly precision model, and the resulting drift error distributions were statistically analyzed. The corresponding histograms, along with fitted normal distribution curves, are presented in the figures.

Figure 16 demonstrates that as the dispersion of the preload force gradually decreases, the mean value of the gyroscope drift shows a significant downward trend, from about 0.03023°/h/g under the 15% preload dispersion condition to about 0.01995°/h/g under the 5% dispersion condition, a decrease of 34%. At the same time, the distribution of the drift error changes from being relatively discrete to being more concentrated. This shows that reducing the volatility of the bolt preload force during assembly plays an important role in effectively reducing drift error. In the actual engineering assembly process, preload fluctuation should be reduced by improving the accuracy of the assembly tool and other measures, thereby improving the stability of the air-floating assembly, reducing drift error, and improving the overall performance and reliability of the equipment.

5. Discussion

Uniform deformation in the air-floating support interface ensures a stable gas film gap between the gyroscope’s rotor and stator, thereby maintaining the floating characteristics and dynamic balance of the air-floating. In contrast, significant nonuniform deformation at the interface degrades the gas film centroid, directly compromising the stability of the air-floating support. To further elucidate the relationship between process loads and assembly precision, the interface deformation uniformity is evaluated using characteristic parameters derived from the gray level co-occurrence matrix (GLCM).

The GLCM is a concept from digital image processing primarily used for texture analysis [38]. For an image with G gray levels, the core idea is to count the frequency at which pixel pairs with different gray levels, i and j, that co-occur with a given direction $\theta$ and step length $\delta$. This characterizes the spatial correlation of the image’s gray-level distribution. For an image quantized into L levels, the elements of the GLCM can be expressed in terms of joint probabilities, $P(i,j|\delta ,\theta )$.

$$\begin{array}{r}P(i,j|d,\theta )=\{(x,y)|f(x,y)=i,f(x+dx,y+dy)=j;\\ x,y=0,1,2,\dots N-1\}\end{array}$$

(16)

where $dx=d\cdot \cos \theta$ and $dy=d\cdot \sin \theta$. Depending on the specific application, appropriate $d$ and $\theta$ values are chosen to construct the spatial co-occurrence matrix.

To evaluate the interface assembly deformation, the deformation distribution can be treated as a grayscale image where each data point’s deformation corresponds to a gray level. The relationship between deformation values at different directions and distances is quantified by constructing the GLCM, thus revealing the texture characteristics of the deformation data. These characteristics describe the uniformity, symmetry, and local variations of the interface assembly deformation. In this study, the contrast (CON) parameter from the GLCM is used to evaluate deformation uniformity; a larger contrast value indicates greater local variation and a more nonuniform distribution.

$$CON={\displaystyle {\sum }_{\mathrm{i}}{\displaystyle {\sum }_j{(i-j)}^2{P}_{ij}}}$$

(17)

Based on the bolt preload uncertainty analysis model described earlier, bolt preload samples were generated and incorporated into the algorithm to compute the interface assembly deformation of the gyroscope air-floating support. This enabled the calculation of interface deformation data, the derivation of nonuniformity indicator parameters, and the computation of first-order drift error in the air-floating support. Pearson correlation analysis between the deformation nonuniformity indicator and the gyroscope’s static drift error (first-order drift) yielded a Pearson coefficient of 0.9394 (Figure 17), indicating a strong positive correlation.

Further linear regression analysis (Figure 18) confirms that the deformation nonuniformity indicator significantly predicts gyroscope drift, with a p-value well below 0.05 and a determination coefficient of R² = 0.882. This implies that the fluctuations in gyroscope drift can be largely explained by variations in interface deformation nonuniformity. These findings validate the critical impact of interface deformation nonuniformity on gyroscope drift and indicate that by influencing interface deformation nonuniformity, uncertainty in bolt preloads can substantially affect gyroscope accuracy. Consequently, interface deformation nonuniformity is an important process metric in guiding on-site assembly procedures for gyroscope air-floating supports.

6. Conclusions

Considering inaccurate predictions of assembly precision under process loads for gyroscope air-floating supports, this study proposed a method for calculating interface assembly deformation based on the elastic interactions among bolts and established an assembly precision solution model. The effects of process load uncertainty on gyroscope drift error were systematically analyzed. The main conclusions are as follows:

(1) A predictive model for the assembly precision of air-floating supports under process loads was established. During assembly, elastic interactions between bolts with uniform bolt preloads induce nonuniform deformation at the interface. Through employing a nonlinear fitting algorithm, the assembly precision was quantified, and the computational method was validated through numerical simulations and experimental tests, yielding a maximum error of less than 0.8 μm. This provides a robust theoretical basis for analyzing drift errors in air-floating supports.

(2) The influence of process loads on drift error in air-floating supports was demonstrated. An analysis of the drift error distribution under uncertain process loads revealed that, as the dispersion in bolt preloads decreases, the mean gyroscope drift reduces sequentially, from approximately 0.03023°/h/g to 0.01995°/h/g—a 34% reduction—with the distribution becoming more concentrated.

(3) A strong correlation between interface deformation nonuniformity and drift error was demonstrated. Utilizing the contrast parameter from the gray-level co-occurrence matrix to evaluate the nonuniformity of interface deformation, a high positive correlation (a Pearson correlation coefficient of 0.9394) was found between the nonuniformity indicator and the gyroscope drift error. This indicator is an effective process control metric for optimizing on-site assembly procedures.

This study provides theoretical guidance for the high-precision and high-stability assembly of gyroscopes. It also lays a methodological foundation for assembly process optimization and performance improvement through in-depth research on the influence of assembly process uncertainty on drift error in air-floating supports and the correlation between interface deformation nonuniformity and drift error. However, the present research was mainly based on the tightening process conditions of the bolt group and does not fully cover multi-source uncertainties such as different assembly methods, material types, and operating errors. In the future, it will be necessary to further deepen the research on the influence of multiple factors, build a more general prediction model, and consider bidirectional fluid–solid coupling research comparing the structure and the air film to improve the adaptability and robustness of complex on-site assembly processes.