Multi-Scale Modeling of Aerostatic Spindles Based on Shape Error Harmonic Analysis and Static Characteristic Evaluation

Abstract

Rotor machining errors strongly influence the air-film pressure distribution of aerostatic spindles and fundamentally limit performance enhancement. However, existing studies rarely provide a comprehensive statistical characterization based on measured manufacturing errors. To address this gap, a multi-scale modeling framework based on harmonic analysis of form errors is developed. Measured surface topography data from a batch of rotors are decomposed to establish a harmonic statistical model, which is then incorporated into a modified Reynolds equation together with macro-scale and micro-scale error components. The static performance of the aerostatic spindle is subsequently analyzed. Results show that low-order harmonics (1st–5th) dominate cylindricity errors, with amplitudes following a log-normal distribution. The statistical bounds are described by 3σ envelopes. When the eccentricity ε exceeds 0.3, barrel-shaped errors reduce the load capacity by more than 15%, whereas waist-drum-shaped errors exhibit a self-stabilizing tendency under small deviations. Performance degradation can be partially mitigated by adjusting the supply pressure and orifice diameter. This study addresses the research gap in understanding the impact of measured manufacturing errors on aerostatic spindle performance and provides a quantitative basis for tolerance allocation and performance optimization.

1. Introduction

Aerostatic spindles exhibit low friction, smooth rotation, high rotational accuracy, and minimal temperature rise at high speeds, and are extensively employed in ultra-precision machining [1,2,3]. Nevertheless, during manufacturing and assembly, rotor surfaces deviate from ideal cylindrical geometry due to process limitations and random system errors, resulting in shaft eccentricity and microscopic surface irregularities. Given that the designed air-film clearance of aerostatic spindles typically ranges from 10 to 30 μm, even small geometric deviations can significantly alter the air-film thickness distribution, induce a reconfiguration of the pressure field, and consequently affect the static characteristics, power consumption, and service life of the spindle system [4,5]. Therefore, precise characterization and assessment of the effects of geometric errors on aerostatic spindle performance constitute a critical challenge in the field of precision engineering.

Extensive research on the influence of rotor geometric errors has established a multi-level framework encompassing theoretical modeling, numerical simulation, and experimental validation. These studies collectively demonstrate that such errors are non-negligible. Pande and Somasundaram [6]’s study revealed that geometric form errors and surface roughness induced by manufacturing processes are key factors causing the performance of aerostatic radial bearings to deviate from the ideal state. Lin [7] performed a systematic theoretical analysis to examine the impact of surface roughness on the dynamic stiffness and damping properties of compensated hydrostatic thrust bearings, demonstrating that the roughness profile’s geometric pattern and height play a critical role in modulating the bearing’s dynamic response. Stout et al. [8] found that geometric deviations like parallelism, roundness, and tilt in aerostatic journal bearings directly reduce their load capacity, highlighting the critical impact of manufacturing variations. Yu et al. [9] used CFD simulations to study triangular surface textures on rotors, showing optimal improvements in load capacity and offering important reference for microstructure design. Cappa et al. [10] developed an ultra-precision aerostatic bearing system, and via analyzing the effects of manufacturing errors, bearing parameters and gas supply structures, as well as establishing a numerical gas waist-drum model, they verified that increasing the number of gas supply holes can improve gas film uniformity and thus reduce radial error motion to the nanoscale. Zhuang et al. [11] analyzed how surface waviness affects the static/dynamic performance and stability of orifice-type aerostatic thrust bearings, showing that radial waviness and higher rotational speeds exert a particularly strong influence, with findings validated via experimental tests. Zhang J et al. [12] compared vertical and horizontal tilts under purely static and hybrid lubrication conditions, indicating that vertical tilt mainly affects load capacity while horizontal tilt primarily shifts the stability boundary, revealing the sensitivity of system nonlinear response to error orientation. Wang et al. [13] proposed a numerical model based on linear perturbation theory to investigate the influence of journal rotation and bearing surface waviness on the dynamic performance of aerostatic journal bearings. Wang et al. [14] numerically investigated journal misalignment effects on the static and dynamic characteristics of aerostatic journal bearings, revealing their significant impact on static performance and dynamic stiffness yet negligible influence on damping. Chen et al. [15] systematically analyzed the effects of orifice compensation and intrinsic throttling on rotor–bearing stability, demonstrating that optimized orifice size can significantly suppress eccentricity-induced vortical instabilities, increasing the stability threshold by 15–20%; these studies established a comprehensive dynamic stability criterion framework based on the coupling of eccentricity, throttling, and stiffness. Sun et al. [16] investigated the influence of shaft shape errors on the dynamic characteristics of rotor-bearing systems, demonstrating that such errors increase the rotational speed at which fluid whip occurs. Chen et al. [17] introduced an innovative numerical framework for fluid–structure interaction in gas bearing–rotor systems, assessing how manufacturing imperfections including roundness, convexity, concavity and taper affect rotor dynamics, and showing that 1st- and 2nd-order roundness errors significantly amplify error motion, requiring stringent mitigation. Michal [18] examined how manufacturing and assembly errors affect the performance of multi-pad hydrostatic bearings in large-scale systems, developing compensation and prediction strategies and emphasizing the application of compliant support materials to mitigate such effects. Lee et al. [19] analyzed waviness errors in hydrostatic journal bearings, showing that load-carrying capacity varies with waviness amplitude and phase angle. Kim et al. [20] proposed a stochastic and contact-based approach to evaluate surface roughness effects in slider bearings, finding that roughness parameters exert a rapidly growing influence on friction and load capacity once asperity contact occurs. Ning [21] implemented the perturbation method to systematically examine the impact of five distinct manufacturing errors, encompassing taper error, ellipticity, and journal eccentricity, on the operational performance of air-lubricated gyroscopes, with results identifying taper error as the most influential factor. Mallya et al. [22] conducted a systematic theoretical investigation into the influence of rotor misalignment on the static performance of water-lubricated radial bearings with axial grooves, demonstrating that rotor tilt impairs the bearing’s load-carrying capacity and that horizontal and vertical misalignments exert distinct effects on the friction coefficient. Phalle [23] investigated the influence of journal misalignment on the static and dynamic performance of bearings under bushing wear conditions, demonstrating that bearings with higher offset coefficients exhibit superior overall performance.

Existing studies typically characterize geometric errors via single-form deterministic models, which fail to capture the statistical properties and spatial distributions of manufacturing-induced geometric deviations in practical production processes. As such, the current theoretical frameworks are inadequate to support tolerance optimization and the robust design of ultra-precision spindles. To mitigate this limitation, this study proposes a multi-scale spindle modeling framework based on the harmonic analysis of measured form errors. Surface topography data of rotors fabricated within the same production batch are decomposed into harmonic components, followed by statistical analyses to identify the dominant error orders and their probabilistic distributions. On this basis, a multi-scale geometric error model integrating macro-scale form deviations and micro-scale stochastic perturbations is established, incorporated into a modified Reynolds equation, and subsequently applied to perform performance solution. This work extends deterministic modeling approaches to a statistically characterized, manufacturing-informed predictive framework for the design of ultra-precision spindles.

Unlike bearings, rotors are moving components whose geometric errors periodically affect the air-film thickness with rotation, acting as the primary excitation source of nonlinear disturbances in the pressure field. Accordingly, the statistical characteristics of rotor errors and their influence on spindle static performance are investigated.

2. Extraction and Harmonic Analysis of Rotor Surface Topography Characteristics

2.1. Equipment and Methods for Feature Extraction

In this study, the Taylor Talyrond 585LT-500 ultra-high-precision roundness and cylindricity measuring instrument, manufactured by Taylor Hobson in Leicester, United Kingdom, was employed. With a resolution of 0.3 nm, it is capable of capturing geometric errors at the nanometer scale.

The relevant measurement parameters of the instrument are listed in

Table 1. Taylor Talyrond 585LT-500 measurement parameters.

Parameter	Value
Position resolution	0.02°
Positioning accuracy	±0.2°
Cylindricity measurement range	50~500 mm
Minimum positioning angle	0.1°
Radial error	±(0.02 + 0.0003 μm/mm) × R μm
Axial error	±(0.02 + 0.0003 μm/mm) × H μm

. The Taylor Talyrond 585LT-500 is suitable for high-precision characterization of spindle rotor surface topography and is equipped with a high-accuracy cylindricity algorithm, which enables both measurement reporting and the export of raw data.

As shown in Table 1, the roundness and cylindricity measuring instrument delivers high accuracy and resolution across its measurement range. Its roundness accuracy is ±(0.02 + 0.0003 μm/mm) × R μm and its cylindricity accuracy is ±(0.02 + 0.0003 μm/mm) × H μm, where R denotes the rotor radius and H denotes the axial measurement length. This performance ensures reliable and repeatable data acquisition, laying a solid foundation for subsequent analysis. Figure 1 illustrates the measurement procedure for the surface topography of the spindle labeled “S.” In Figure 1a, the overall measurement process is depicted: The rotor is mounted on the rotary table and rotated through one full revolution to capture the roundness profile of the section. Figure 1b presents the measured profile and roundness evaluation results for a specific section, where the red circular frame represents the reference circle, around which the rotor topography fluctuates; the inner “1100” indicates a runout of 1100 nm relative to the reference circle, while the outer “900” indicates a runout of 900 nm relative to the reference circle. The gray circular frame represents the reference circle, which serves as the ideal geometric benchmark for evaluating the rotor’s roundness error. The red arrow indicates the measured rotor surface profile, which fluctuates around the reference circle, clearly showing the magnitude and direction of the roundness error. The black arrows are evenly distributed along the circumference, dividing the circle into equal angular segments to facilitate the analysis of the error’s circumferential distribution. Figure 1c shows the complete surface topography of the “S” spindle; the left panel corresponds to the axial positions of the measured sections, while the right panel represents the radial runout of the least-squares circle centers of each section relative to the overall least-squares axis, with the four decimal places in the numerical results being software-fitted values that reflect the measurement precision of the instrument. The measurement results clearly demonstrate the complexity of the rotor surface topography. As a result, Fourier harmonic analysis—capable of decomposing complex profiles and revealing their composition features—is introduced to deconstruct the measured data.

The measurement results visually highlight the complexity of the rotor surface topography. Accordingly, Fourier harmonic analysis, which can decompose complex profiles and reveal their constituent features, is applied to the measured data for systematic characterization.

2.2. Harmonic Analysis of Rotor Surface Topography Characteristics

The core of harmonic analysis lies in decomposing complex, continuous roundness errors into a series of components with distinct frequencies and amplitudes. Widely used in precision machinery, it suits periodically varying geometric errors. When analyzing spindle rotor surface topography, the circumferential profile signal is a periodic function defined over the angular domain $\theta \in [0,2\pi ]$. Fourier series expansion decomposes the topography data into harmonic components with clear frequency characteristics [24,25,26].

In the cylindrical coordinate system, the radial error profile $\delta r(\theta ,z)$ of the rotor at axial position z is a periodic function, and it is expressed as Equation (1):

$$\left\{\begin{array}{l}\delta r(\theta ,z)={\displaystyle \frac{A_0(z)}{2}}+{\displaystyle \sum _{k=1}^{\infty }\left[A_k(z)\cos (k\theta )+B_k(z)\sin (k\theta )\right]}\\ A_k(z)={\displaystyle \frac{1}{\pi }}{\displaystyle {\int }_0^{2\pi }\delta r}(\theta ,z)\cos (k\theta )d\theta \\ B_k(z)={\displaystyle \frac{1}{\pi }}{\displaystyle {\int }_0^{2\pi }\delta r}(\theta ,z)\sin (k\theta )d\theta \end{array}\right.$$

(1)

where $k$ is the harmonic order, representing the number of fluctuation periods in the circumferential 2$\pi$ range; $A_k(z)$ and $B_k(z)$ are the Fourier coefficients of the k-th harmonic; $\theta$ denotes the circumferential angle; and $z$ denotes the axial coordinate.

Harmonic decomposition maps complex topography errors to frequency-domain components with clear physical meanings, thus identifying the dominant orders and composition rules of errors. This forms the basis for subsequent statistical error modeling and the prediction of spindle static performance. Systematic harmonic decomposition is performed on the measured profile. The process includes data acquisition and preprocessing, sampling rationality verification, discrete Fourier coefficient calculation, and harmonic energy proportion computation, ensuring the results are applicable in engineering [27].

The rotor roundness distribution characteristics are sampled at fixed angular intervals, yielding the discrete dataset $r\left\{k\right\}$ here, and $k$ represents topography data of different cross-sections. In this study, 18,000 sampling points are used per cross-section per revolution. The sampling density satisfies the Nyquist criterion, ensuring valid data acquisition.

During the measurement process, the measurement system may introduce low-frequency drift and random noise. To ensure the accuracy of harmonic analysis, preprocessing is required for the original topography error measurement data. According to [26], filtering should be applied to the raw data. The filtering effect is shown in Figure 2. Filtering acts as a high-frequency noise suppressor, yielding robust data for subsequent harmonic analysis.

Then, discrete Fourier transform is performed on the data. The complex coefficient $E[n]$ of the n-th harmonic is expressed as Equation (2):

$$E[n]={\displaystyle \sum _{k=0}^{N-1}r}[k]e^{-j2\pi nk/N}$$

(2)

where $r[k]$ is the preprocessed topography data, $N$ is the total number of data sampling points, $e^{-j2\pi nk/N}$ is the basis function of the DFT, and $E[n]$ contains amplitude and phase information.

Amplitude and phase of the harmonic are extracted from $E[n]$. The amplitude and phase of the n-th harmonic are expressed as Equation (3):

$$\left\{\begin{array}{l}{C}_n={\displaystyle \frac{2}{N}}|E[n]|\\ {\varphi }_n=\arg (E[n])\end{array}\right.$$

(3)

where $C_n$ is the error amplitude of the n-th harmonic, and ${\varphi }_n$ is the circumferential phase of the error.

To evaluate the relative importance of different harmonic orders in rotor geometric errors, a corresponding energy characterization index is defined as Equation (4). For the n-th harmonic, its energy is defined as follows:

$${\eta }_n={\displaystyle \frac{C_n}{{\displaystyle \sum _{k=1}^{N_{\mathrm{Truncate}}}{C}_k^2}}}$$

(4)

where $C_k^2$ is the square of the amplitude of the n-th harmonic, and $N_{\mathrm{Truncate}}$ is the total evaluation order.

This index can characterize the relative contribution of different harmonic orders to the overall geometric error.

The actual manufacturing errors of rotors exhibit both random discreteness and process consistency. Consequently, performance evaluation based on a single rotor or a single measurement may be highly sensitive to individual samples. Consequently, it is necessary to systematically analyze the harmonic components of rotor errors produced under the same process, as well as their statistical distribution characteristics, and to establish an error model with statistical significance. Firstly, harmonic analysis was conducted on the cross-sectional data of a single rotor to obtain the contribution distribution of errors along the axial direction, as shown in Figure 3a. Subsequently, the entire rotor was treated as a single sample for analysis, and the results from multiple sample groups are summarized in Figure 3b.

Figure 3a shows the harmonic contribution ratio of the shaft coded “S” across 20 cross-sections. The horizontal axis represents the numbers of different measured cross-sections, the vertical axis corresponds to harmonic orders, and the z-axis reflects the proportion of each harmonic component. Different colors represent different harmonic orders, visually distinguishing the energy distribution of each harmonic component. Low-frequency harmonics of orders 1–5 dominate all cross-sections, with their total proportion stably ranging from 70% to 85%, while the energy proportion of high-frequency components above order 8 is generally below 5%. This harmonic distribution indicates that the rotor errors have strong consistency in the axial direction, and their main topography features are determined by low-order harmonics. The machining process applied can be inferred from the axial distribution of harmonic contributions.

Figure 3b presents the statistical results of the overall harmonic contribution ratio of 20 shafts in the same batch. In the multi-sample comparison, low-order harmonics still maintain a dominant position with an average energy proportion exceeding 75%. Nevertheless, there exists a certain degree of discreteness in the relative proportion of harmonic amplitudes across different orders among various workpieces. These results demonstrate that, in addition to the systematic components determined by the process and equipment, machining errors inevitably superimpose random factors arising from tool wear, local vibration and material inhomogeneity.

The above rules reveal the source mechanism of rotor machining errors, providing a reference for harmonic truncation strategies and an analysis of key error components.

3. Establishment of Rotor Machining Error Model Based on Harmonic Analysis of Surface Topography

3.1. Rotor Topography Reconstruction Based on Harmonic Analysis

To assess the applicability of harmonic decomposition for characterizing rotor machining errors, the measured topography data are subjected to reconstruction and analysis. This enables verification of the contribution of harmonic components across different orders to the overall topography. Fourier coefficients of each order, derived from Equation (3), are superimposed sequentially by harmonic order up to the upper limit $N_{Truncate}$. This yields topography reconstruction results under varying frequency truncation conditions.

Harmonic reconstruction is performed on distinct measurement cross-sections. The rotor surface error $e(\theta )$, a periodic function composed of multiple harmonics, can be expressed as Equation (5):

$$e(\theta )={\displaystyle \sum _{n=1}^{N_{Truncate}}{C}_n}\cos (n\theta +{\varphi }_n)$$

(5)

To reconstruct the error using $N<N_{Truncate}$, the discrepancy between the reconstructed profile and the original profile is evaluated. Two metrics are employed: the Root Mean Square (RMS) error and the Peak-to-Valley (PV) error. The RMS reflects the overall magnitude of the error, while the PV measures the maximum local error deviation, as expressed in Equation (6).

$$\left\{\begin{array}{l}RM{S}_N=\sqrt{{\displaystyle \frac{1}{M}}{{\displaystyle \sum _{i=1}^M\left[e\left({\theta }_i\right)-{\widehat{e}}_N\left({\theta }_i\right)\right]}}^2}\\ {\mathrm{PV}}_N=\max |e({\theta }_i)-{\widehat{e}}_N({\theta }_i)|\end{array}\right.$$

(6)

where M denotes the cross-section label, $e\left({\theta }_i\right)$ represents the actual measured value, and ${\widehat{e}}_N({\theta }_i)$ denotes the reconstructed value.

Table 2. Harmonic reconstruction accuracy variation with order.

NTruncate	RMSN (nm)	PVN (nm)	NTruncate	RMSN (nm)	PVN (nm)
1	586.1	2450.8	10	235.84	1265.8
2	289.98	1483.6	15	194.49	927.14
3	289.98	1483.7	20	175.72	1069.3
5	266.35	1283.8	30	124.2	790.97

presents $RM{S}_N$ and $P{V}_N$ under different $N_{Truncate}$. Both $RM{S}_N$ and $P{V}_N$ show an overall decreasing trend and converge as the truncation order increases. When the order reaches 30, the improvement in $RM{S}_N$ is less than 2%, indicating that the energy contribution of harmonics at this stage is extremely small, and the reconstruction results have basically converged.

From the reconstruction process with different $N_{Truncate}$ values in Figure 4, harmonic reconstruction is performed for orders 10, 20, and 30 to analyze the effect of harmonic truncation order on reconstruction accuracy. The corresponding reconstruction proportions are approximately 80%, 93%, and 97%. The reconstruction results indicate that low-order harmonics can reconstruct most macroscopic geometric features, while high-order harmonics mainly supplement local details. Their contribution is concentrated at the micro-scale and has limited impact on the overall profile variation. By integrating the quantitative changes in error indicators and the morphological consistency of reconstruction curves, $N_{Truncate}=30$ is determined as the optimal fitting order for rotor surface error reconstruction.

Through quantitative verification via harmonic reconstruction, the machining error mathematical model established in this paper can accurately describe the spindle cylindricity machining error distribution with nanometer-scale precision.

The magnitude of errors is reflected by harmonics of different orders, while the harmonic order reveals their physical origin. This approach further elucidates the physical and process-related significance of manufacturing errors, providing guidance for process optimization. By correlating harmonics with typical sources of machining errors, the dominant factors and formation patterns of the errors can be identified through statistical analysis.

Using Equation (7), statistical analysis of the harmonic decomposition results from 10 rotor samples, provided we have the mean and standard deviation of the amplitude for each harmonic order.

$$\left\{\begin{array}{l}{\overline{C}}_n={\displaystyle \frac{1}{K}}{\displaystyle \sum _{k=1}^K{C}_{n,k}}\\ {\sigma }_n=\sqrt{{\displaystyle \frac{1}{K-1}}{\displaystyle \sum _{k=1}^K{(C_{n,k}-{\overline{C}}_n)}^2}}\end{array}\right.$$

(7)

where $K$ denotes the sample number, and $C_{n,k}$ represents the amplitude of the n-th harmonic order for the k-th sample.

Figure 5 presents the statistical distribution characteristics of the amplitudes of the first ten harmonics of the rotor’s circumferential errors. The dashed line in both subplots represents the overall trend of the corresponding indicator, showing the decreasing tendency of harmonic amplitude and its standard deviation with increasing order. From the statistical results of the mean and standard deviation, it is evident that the mean harmonic amplitude decreases with increasing order. The first-order harmonic exhibits the highest amplitude and the most pronounced fluctuation, corresponding to the eccentricity between the measurement reference and the rotor’s actual center, and represents the primary source of systematic error. The second-order harmonic has the next highest amplitude, reflecting typical ellipticity deviations. The third- and fourth-order harmonics correspond to triangular and quadrilateral shape errors, respectively, which are generally associated with periodic errors of machine tool components or repetitive positioning errors in the motion subsystem. The amplitudes of the fifth- to seventh-order harmonics decrease further, primarily arising from machine structural vibrations, periodic excitations in the drive system, or fluctuations in process stability during machining. Harmonics of the eighth order and above, along with their standard deviations, drop to very low levels. These high-order components are mainly attributed to the micro-texture of the machined surface, micro-scale tool vibrations, and measurement system noise, representing typical high-frequency stochastic processes that have limited influence on the overall rotor geometry.

Table 3. Statistics of harmonic amplitudes at each order and engineering associations.

Harmonic Order N	Average Amplitude (nm)	Standard Deviation (nm)	Engineering Relevance
1	925.14	63.936	Dominated by eccentricity error
2	491.54	54.949	Ellipticity
3	302.57	24.284	Triangular roundness
4	176.06	17.958	Quadrilateral roundness
5	137.57	17.626	Process system vibration
6	80.891	6.2839	Microscopic machining textures and high-frequency characteristics of machine tool systems
7	57.388	4.435
8	40.318	6.874
9	30.51	3.6627
10	19.06	1.5499

illustrates the statistical distribution of the amplitudes of the first ten harmonic components of the rotor circumferential error, together with their engineering implications. This relationship suggests that low-order harmonics dominate the macroscopic geometric deviations of the workpiece, whereas high-order harmonics predominantly represent micro-scale machining characteristics.

3.2. Boundary Characterization of Rotor Machining Errors Based on Harmonic Statistical Features

Harmonic analysis enables the characterization of error distribution and its physical association with the machining process. To further evaluate the relationship between the process and the statistical features of harmonics, the harmonic data of ten rotor samples were analyzed, focusing on the first ten harmonic orders. The amplitudes of the same-order harmonics across different rotors were statistically compiled to obtain the probability distributions of the first- to tenth-order harmonic amplitudes. As Figure 6 shows, the probability distributions of the first- to tenth-order harmonic amplitudes indicate that low-order harmonics exhibit significant dispersion among samples, whereas high-order harmonics are concentrated within a narrow range. The colored lines in the figure represent the corresponding harmonic orders, visually distinguishing the probability distribution characteristics of each order. The wide variation in low-order harmonic amplitudes reflects the influence of systematic process factors, such as non-uniform clamping and motion chain errors, while the concentrated distribution of high-order harmonics is primarily attributed to random surface texture or stochastic noise.

Harmonic amplitudes approximately conform to a log-normal distribution. This aligns with the formation mechanism of machining errors: superposition of multiple independent multiplicative random factors. Based on the $\mu \pm 3\sigma$ principle of the log-normal distribution, a statistical envelope of harmonic amplitudes is constructed. It characterizes the upper boundary of geometric errors in mass manufacturing, as expressed in Equation (8). In contrast to adopting the sample maximum as the limit input, the 3σ error envelope ensures coverage of 99.7% of manufacturing deviations. It also eliminates evaluation biases induced by individual abnormal samples. The 3σ error envelope is defined as the statistically derived limit working condition input. It serves to evaluate the robustness of spindle static performance against manufacturing errors.

$${\delta }_{\max }(\theta ,z)={\displaystyle \sum _{n=1}^N\left[\begin{array}{c}({\overline{A}}_n+3{\sigma }_{A_n})\cos (n\theta )+({\overline{B}}_n+3{\sigma }_{B_n})\sin (n\theta )\end{array}\right]}$$

(8)

where ${\overline{A}}_n$ and ${\overline{B}}_n$ are the means, and ${\sigma }_{A_n}$, ${\sigma }_{B_n}$ are the standard deviations. They are used to characterize the central tendency and dispersion degree of harmonic amplitudes in mass machining conditions.

The mean and standard deviation of harmonic coefficients summarized in

Table 4. Statistical summary of the mean and standard deviation of harmonic coefficients for each order of the shaft.

Harmonic Order	${\overline{\mathit{A}}}_{\mathit{n}}$	${\mathit{\sigma }}_{{\mathit{A}}_{\mathit{n}}}$	${\overline{\mathit{B}}}_{\mathit{n}}$	${\mathit{\sigma }}_{{\mathit{B}}_{\mathit{n}}}$	${\overline{\mathit{C}}}_{\mathit{n}}$	${\mathit{\sigma }}_{{\mathit{C}}_{\mathit{n}}}$
1	0.8045	0.0835	1.7692	0.0686	1.9452	0.0703
2	0.4499	0.0274	0.4103	0.0251	0.6094	0.0265
3	0.2519	0.0147	0.3945	0.0131	0.4311	0.0134
4	0.0103	0.0491	0.0111	0.0449	0.0573	0.0357
5	0.0171	0.0397	−0.0024	0.0480	0.0547	0.0330
6	−0.0020	0.0169	0.0027	0.0219	0.0248	0.0119
7	−0.006	0.0139	−0.002	0.0188	0.0201	0.0115
8	0.0023	0.0263	0.003	0.0195	0.0288	0.0151
9	−0.0029	0.0113	−0.0031	0.0134	0.0151	0.0094
10	0.0007	0.0080	−0.0091	0.0084	0.0104	0.0053

Unit: μm.

reveal the central tendency and dispersion of rotor error harmonics across different orders. The mean and standard deviation of low-order harmonics are generally larger than those of high-order harmonics, indicating that low-order harmonics play a more dominant role in shaping the rotor geometry. Analysis of the harmonic amplitudes across multiple samples further demonstrates the stochastic patterns of machining errors arising from the superposition of multiple contributing factors.

3.3. Construction of Rotor Machining Error Model with Random Characteristics

Macroscopic shape errors mainly stem from systematic process errors and have distinct geometric features. To simplify the analysis, conical, waist-drum-shaped, and barrel-shaped errors are focused on [28]. Figure 7 shows a 3D schematic, where color coding indicates the sign of radius deviation: blue represents negative values, while red represents positive values. They can be characterized by Equations (9)–(12).

Conical error refers to the linear variation in radius deviation along the axial direction.

$${\delta }_{\mathrm{taper}}(z)=k_{\mathrm{taper}}\cdot \left(z-{\displaystyle \frac{L}{2}}\right)$$

(9)

where $k_{\mathrm{taper}}$ is the taper coefficient, L is the rotor length, and $z$ is the axial position.

Waist-drum-shaped error means the radius deviation is positive at the rotor’s middle part, presenting a concave center.

$${\delta }_{waist}(z)=A_{waist}\cdot \left[1-{\left({\displaystyle \frac{2(z-L/2)}{L}}\right)}^2\right]$$

(10)

where $A_{waist}$ is the amplitude of the waist-drum-shaped error.

Barrel error refers to local protrusions at specific axial positions, which can be described by a Gaussian function:

$${\delta }_{\mathrm{bump}}(z)=\pm A_{\mathrm{bump}}\cdot \exp \left(-{\displaystyle \frac{{(z-z_0)}^2}{2{\sigma }_z^2}}\right)$$

(11)

where $A_{\mathrm{bump}}$ is the amplitude, $z_0$ is the central position of the protrusion, and ${\sigma }_z$ controls the width of the protrusion.

The total macroscopic shape error ${\delta }_{\mathrm{macro}}(z)$ is the linear superposition of the above items:

$${\delta }_{\mathrm{macro}}(z)={\delta }_{\mathrm{taper}}(z)+{\delta }_{\mathrm{waist}}(z)+{\delta }_{\mathrm{bump}}(z)$$

(12)

Micro-topography errors originate from random factors in the machining process, such as high-frequency vibration and tool wear. Their distribution characteristics can be obtained via harmonic analysis. To simulate the impact of random factors in batch machining and improve the model’s reliability and robustness, a statistically based 3σ error envelope model is used to define the circumferential random error ${\delta }_{\mathrm{micro}}(\theta ,z)$, as given by Equation (13).

$${\delta }_{\mathrm{micro}}(\theta ,z)={\displaystyle \sum _{n=1}^N\left[\begin{array}{l}(\frac{1}{M}{\displaystyle \sum _{m=1}^M{\overline{A}}_{n,m}(z)+3{\sigma }_{A_{n,m}}})\cos (n\theta )\\ +(\frac{1}{M}{\displaystyle \sum _{m=1}^M{\overline{B}}_{n,m}(z)+3{\sigma }_{B_{n,m}}})\sin (n\theta )\end{array}\right]}$$

(13)

where N is the harmonic order, M is the number of sample shafts manufactured with the same process, ${\overline{A}}_{n,m}(z)$, ${\overline{B}}_{n,m}(z)$ and ${\sigma }_{A_{n,m}}$, ${\sigma }_{B_{n,m}}$ are the axial means and standard deviations of the harmonic coefficients for the n-th harmonic order of the m-th sample shaft in the measured error data, respectively.

This model ensures that errors for 99.7% of manufactured rotors fall within this envelope.

By coupling the macroscopic shape error with the random micro-topography error, the comprehensive radial machining error $\Delta R(\theta ,z)$ of the rotor surface is obtained, as shown in Equation (14). Figure 8 illustrates the modeling process.

$$\Delta R(\theta ,z)={\delta }_{\mathrm{macro}}(z)+{\delta }_{\mathrm{micro}}(\theta ,z)$$

(14)

Accordingly, the revised expression for the gas film thickness has been finalized as Equation (15):

$$\begin{array}{l}h(\theta ,z)=h_m-\left[{\delta }_{\mathrm{macro}}(z)+{\displaystyle \sum _{n=1}^N\left[\begin{array}{l}(\frac{1}{M}{\displaystyle \sum _{m=1}^M{\overline{A}}_{n,m}(z)+3{\sigma }_{A_{n,m}}})\cos (n\theta )\\ +(\frac{1}{M}{\displaystyle \sum _{m=1}^M{\overline{B}}_{n,m}(z)+3{\sigma }_{B_{n,m}}})\sin (n\theta )\end{array}\right]}\right]\\ -e_x(z)\cos \theta -e_y(z)\sin \theta \end{array}$$

(15)

where $h_m$ denotes the average gas film clearance, while $e_x$, $e_{\mathrm{y}}$ are the eccentricity components. Subsequently, a rotor surface data repository was established, and an analysis was conducted on these data sources. It was found that the types of errors generated are closely related to the machining methods. Figure 9 illustrates the coupling results between different error types and the gas film. The colors represent the sign of the relative deviation: red indicates positive values, while blue indicates negative values.

This multi-scale model not only captures the dominant characteristics of macroscopic errors but also represents the stochastic nature of high-frequency surface textures. It serves as a unified input for subsequent analyses of the gas film pressure field and performance sensitivity, enabling the investigation of the impact of manufacturing errors on the static performance of the aerostatic spindle.

4. Calculation of Static Characteristics of Aerostatic Spindles Based on Rotor Machining Errors

4.1. Numerical Solution of the Reynolds Equation for Orifice-Controlled Gas Bearings

The research on gas lubrication is mainly carried out based on the mass conservation equation, motion equation, and energy conservation equation. According to the previous research of our team [29], the governing equation satisfied by the orifice-controlled gas-lubricated bearing can be obtained as Equation (16).

$$\left\{\begin{array}{l}{\displaystyle \frac{\partial }{\partial x}}\left(h^{3}p{\displaystyle \frac{\partial p}{\partial x}}\right)+{\displaystyle \frac{\partial }{\partial z}}\left(h^{3}p{\displaystyle \frac{\partial p}{\partial z}}\right)+12\eta {\displaystyle \frac{p_a}{{\rho }_a}}\rho \tilde{\nu }=12\eta {\displaystyle \frac{\partial (ph)}{\partial t}}+6\eta U{\displaystyle \frac{\partial }{\partial x}}\left(ph\right)\\ U=\omega R\end{array}\right.$$

(16)

where x is the circumferential direction, z is the axial direction, h is the gas film thickness, ω is the angular velocity of the shaft rotor, and R is the radius of the rotor.

After nondimensionalizing Equation (16), the gas film pressure p is solved using the finite difference method according to the computational flowchart shown in Figure 10. The procedure begins with the input of bearing geometry, operating conditions, and gas properties, followed by the initialization of the film thickness and pressure field. A nonlinear iterative scheme is then employed to obtain a converged pressure distribution, with physical consistency ensured through the verification of mass flow balance between the outlet and the throttle supply. Upon convergence, the main performance metrics, including load capacity, stiffness, and mass flow rate, are computed and output. The iterative procedure incorporates rigorous convergence controls, ensuring result accuracy and providing a reliable basis for analyzing how manufacturing errors affect spindle performance.

On the discrete grid, the Simpson’s rule is used to integrate the gas film pressure field over the bearing surface, yielding the load capacity components. The magnitude of the total load capacity can be calculated using Equation (17):

$$\overline{W}=\sqrt{{\overline{W}}_x^2+{\overline{W}}_y^2}$$

(17)

Based on the conservation of mass flow rate, the air consumption can also be calculated. By applying a small disturbance to the gas film, the static stiffness can be derived.

4.2. Simulation Parameter Setting and Model Validation

Based on the aforementioned theoretical model and numerical method, this study establishes the simulation parameters and verifies the reliability of the calculations by comparing with the literature data. The research object is a typical orifice-controlled radial gas bearing, and its key structural parameters and calculation conditions are presented in

Table 5. Calculation parameters.

Name	Value	Name	Value
D	100 mm	Ps	0.5 Mpa
L	130 mm	Num_O	2
h	20 μm	Num_Single_O	12
d0	0.2 mm	OL	0.5 mm

. Figure 11a illustrates the air-film pressure distribution at an eccentricity ratio of 0.05. The colors in the figure represent the relative pressure levels: red indicates higher relative pressure, while blue indicates lower relative pressure. It can be observed that the pressure on the eccentric side is relatively higher, which is consistent with the physical laws.

Using the same model simplification method, considering the same boundary conditions, the finite difference method is applied to solve the Reynolds equation. A comparison is made with the load capacity data from reference [30], and the results are shown in Figure 11b. The calculation results are generally consistent with the reference results, which verifies the rationality of the program in this paper.

5. Results and Discussion

5.1. Influence of the Coupling Between Machining Errors and Eccentricity Ratio on the Static Performance of the Spindle

Figure 12 illustrates the influence of eccentricity on the static performance of the aerostatic spindle under different coupled rotor errors. As shown in Figure 12a, the load capacity of the spindle increases approximately linearly with eccentricity for all types of manufacturing errors. The axial and circumferential distribution characteristics of different errors lead to significant variations in load capacity. This behavior results from the enhanced convergence effect of the gas film due to eccentricity, which gradually reduces the minimum film thickness, increases the local pressure gradient, and enlarges the effective integration area of the load-bearing pressure.

For waist-drum-shaped errors, within the eccentricity range of 0.3–0.4, the regions of minimum gap caused by the coupling of error and eccentricity spatially overlap, concentrating the pressure peak in the middle of the bearing and significantly enhancing it. Consequently, the load capacity increases by approximately 12–15% compared with that of an ideal rotor. As eccentricity further increases, the reduced film thickness in the overlapping region weakens the pressure peak, and the limited gas supply amplifies the sensitivity of the pressure distribution to errors, ultimately attenuating the growth trend of load capacity. Cylindrical (barrel-shaped) errors form a continuous low-gap region along the axial direction; at high eccentricities, the overall reduction in film thickness in this region tends to flatten or even split the pressure peak along the axis, reducing the effective load-bearing area and causing the growth of load capacity to slow or even decline. Conical errors, due to their axially diffused gas film profile, weaken the convergence effect induced by eccentricity, limiting the development of sufficient pressure gradients and constraining the enhancement of hydrodynamic effects.

Figure 12b shows that the gas consumption slightly decreases with increasing eccentricity for all error types. This is primarily due to the reduction in local film gaps caused by eccentricity, which restricts the flow cross-section and suppresses overall gas flow. Among the error types, waist-drum-shaped errors form a relatively concentrated pressure-support region at moderate eccentricities, yielding higher gas utilization efficiency and thus the lowest consumption. In contrast, conical errors, with axially diffused gaps, exhibit lower local flow resistance, allowing gas to escape more easily, resulting in relatively higher consumption.

Figure 12c indicates that spindle stiffness decreases overall with increasing eccentricity for all error types. This is attributed to the nonlinear saturation of the pressure response to perturbations in film thickness at higher eccentricities, leading to reduced equivalent stiffness. For waist-drum-shaped errors, when ε > 0.4, the local minimum film thickness becomes very small, significantly weakening the incremental response of the pressure peak to small displacements and exhibiting clear stiffness softening. For cylindrical errors, the relatively gentle axial distribution results in lower sensitivity of pressure to displacement perturbations, so stiffness variation remains relatively smooth.

Figure 12d shows that the ideal and waist-drum-shaped rotors exhibit the highest pressure peaks and flattest plateaus, sustaining a more stable high-pressure zone, whereas conical and barrel-shaped rotors have lower peaks and faster pressure decay. This confirms the self-stabilizing tendency of waist-drum rotors, whose pressure performance most closely matches that of the ideal rotor.

Analysis of the static performance indicates that, within the eccentricity range of 0.3–0.5, the coupling between manufacturing errors and eccentricity-induced gas film convergence is strongest, making spindle performance most sensitive to errors. In engineering practice, prolonged operation of the spindle in this range should be avoided, or rotor manufacturing errors should be strictly controlled to mitigate their adverse impact on static performance.

5.2. Influence of the Coupling Between Machining Errors and Rotational Speed on the Static Performance of the Spindle

Figure 13 presents the influence of coupling between machining error and rotational speed on spindle static performance. Figure 13a shows that the spindle-bearing capacity exhibits a typical S-shaped growth trend with increasing rotational speed, and gradually tends to saturation in the high-speed region. This reflects the transition of the spindle’s working mechanism from static pressure dominance to dynamic–static pressure coupling dominance. In the low-speed stage n < 10,000 r/min, the gas film pressure is mainly maintained by external air supply. The dynamic pressure term in the Reynolds equation contributes little. The coupling of various machining errors to the gas film thickness is effectively smoothed spatially by the static pressure effect, so the bearing capacity curves under different machining error morphologies basically coincide. With the increase in rotational speed, the shear-driven dynamic pressure effect gradually strengthens. The sensitivity of gas film pressure to gas film thickness changes increases significantly, which nonlinearly amplifies the local clearance disturbance caused by machining errors. Thus, in the medium-to-high-speed region n > 20,000 r/min, the pressure field distribution is significantly altered, leading to performance differentiation between different error morphologies.

For the waist-drum-shaped error, its geometric feature forms a local convergence zone in the axial middle. As the hydrodynamic effect strengthens, the shear-driven pressure increment in this zone increases significantly. This makes the pressure peak concentrated and stable, so the highest bearing capacity is achieved in the medium-to-high-speed region. Although the barrel-shaped error also forms a low-clearance region, its axial continuity leads to a more dispersed pressure peak distribution. The enhancement effect of hydrodynamic pressure is relatively limited. For the conical error, the gas film tends to diverge along the axial direction. This weakens the establishment of the shear-induced pressure gradient, so its bearing capacity is always lower than that of the previous two error morphologies.

Figure 13b shows that, under the conditions of barrel-shaped and conical errors, the gas consumption changes little with rotational speed, remaining around 9.0 L/min. This indicates that, under these error morphologies, the hydrodynamic-induced flow has limited influence on the overall supply flow rate. In contrast, the waist-drum-shaped error exhibits the lowest gas consumption of about 8.0 L/min in the low-speed region. As the rotational speed increases, the enhanced hydrodynamic effect in the local convergence zone promotes gas participation in bearing load. The gas consumption increases slightly and stabilizes at about 8.7 L/min, reflecting a relatively optimal balance between hydrodynamic enhancement and gas utilization efficiency.

Figure 13c shows that, under different error morphologies, the spindle stiffness gradually increases with rising rotational speed and tends to stabilize. This is because the increase in rotational speed enhances the response capacity of gas film pressure to small clearance disturbances, raising the pressure increment caused by unit displacement. However, in the high-speed region, limited by the minimum gas film thickness and air supply capacity, the pressure response to displacement disturbances gradually enters a saturation interval, slowing the growth trend of stiffness. Due to its stable local pressure peak structure, the waist-drum-shaped error achieves the highest stiffness level of about 60 N/μm. In contrast, the conical error, restricted by pressure gradient, has the lowest stiffness of about 40 N/μm.

Figure 13d demonstrates that the coupling of machining error profiles and rotational speed directly impacts fluid film pressure: Ideal and waist-drum-shaped rotors maintain the highest, most stable pressure peaks and plateaus at both ω = 0 and ω = 5000, while conical and barrel-shaped rotors show lower peaks and faster decay. From a local perspective, the waist-drum-shaped profile maintains a better pressure gradient, while the conical profile shows a poorer one. The inset confirms these plateau differences, validating the waist-drum rotor’s self-stabilizing pressure behavior, which closely matches the ideal rotor across speeds.

Comprehensive analysis shows that, in the rotational speed range of 20,000–40,000 r/min, the coupling effect between the hydrodynamic effect and geometric errors is most significant. Minor geometric disturbances can trigger obvious reconstruction of the pressure field, leading to high sensitivity of spindle performance to errors. This range is defined as the error-excited rotational speed window, which can serve as an important reference for spindle structure design, error control, and operating condition optimization.

Figure 14 reveals the influence of rotor geometric error amplitude on the static performance of aerostatic spindles, where the amplitude of Type I error is smaller than that of Type II. Figure 14a indicates that increasing the error amplitude consistently shifts the bearing capacity curve downward and significantly lowers its saturation level across different error morphologies. This indicates that the machining error amplitude not only affects the enhancement rate of the hydrodynamic effect, but also limits the maximum effective bearing capacity that can be established.

From the perspective of gas film thickness, the increase in error amplitude significantly amplifies the fluctuations of gas film thickness in the circumferential and axial directions. The originally continuous and smooth convergence zone is split into multiple discontinuous low-clearance regions locally. This discontinuous convergence characteristic weakens the ability to continuously establish the pressure gradient under shear drive, leading to a decrease in pressure peaks and a discrete spatial distribution, thus significantly reducing the overall bearing capacity. The amplitude of Type I error is smaller than that of Type II, so it can maintain a continuous convergence channel to a certain extent. In contrast, Type II error, with a larger amplitude, is more likely to destroy the continuous and gentle clearance distribution of the gas film, and its bearing capacity decreases by about 15–20% compared with Type I error. For the waist-drum-shaped error, under small amplitude conditions, it can form a relatively ideal local convergence zone in the axial middle, fully amplifying the hydrodynamic effect. Thus, it exhibits the optimal bearing performance under the same amplitude. However, this concave profile is highly sensitive to changes in error amplitude: As the amplitude increases, the local minimum gas film thickness decreases rapidly, causing the pressure peak region to shrink and become unstable, and the hydrodynamic gain advantage is lost quickly. This shows that the performance advantage of the waist-drum-shaped structure depends on high machining accuracy. Once the error amplitude exceeds a certain threshold, its geometric amplification effect will shift from beneficial to detrimental.

Figure 14b shows that the spindle static stiffness also increases with rising rotational speed and gradually stabilizes, but its sensitivity to error amplitude is significantly higher than that of bearing capacity. In the high-speed region (n > 40,000 r/min), the stiffness under Type II error decreases by about 25% compared with that under Type I error. The essence is that stiffness reflects the differential response of bearing capacity to gas film thickness disturbance. When the error amplitude increases, the non-uniformity of pressure distribution and the instability of local pressure peaks are significantly enhanced, causing the incremental response of pressure to small displacement changes to tend to saturation or even attenuation. Thus, the equivalent stiffness degrades noticeably.

In summary, the amplitude of machining errors affects pressure field stability by altering the continuity of the gas film. This directly influences the efficiency of hydrodynamic pressure establishment, and together with rotational speed, affects the stable operating boundary of the system. The ideal cylinder consistently exhibits optimal performance over the entire rotational speed range; the waist-drum-shaped structure can approach ideal performance under small error conditions, but its advantage is highly sensitive to error amplitude.

The above results quantitatively reveal the inherent relationship between geometric accuracy and spindle static performance: under low-speed working conditions, machining accuracy can be moderately relaxed on the premise of ensuring basic performance. For aerostatic spindles requiring high speed and high stiffness, rotor error amplitude must be strictly controlled to avoid bearing capacity attenuation and stiffness instability.

5.3. Influence of the Coupling Between Machining Errors and Supply Parameters on the Static Performance of the Spindle

5.3.1. Influence of the Coupling Between Machining Errors and Orifice Diameter on the Static Performance of the Spindle

Figure 15 reveals the influence of the coupling between machining error and orifice diameter on the static performance of the spindle. The results indicate that increasing the orifice diameter can systematically improve the spindle-bearing capacity, but it also significantly amplifies the system’s sensitivity to machining errors. There is a clear trade-off between the two in engineering design.

In terms of bearing capacity (Figure 15a), as the orifice diameter increases from 0.05 mm to 0.3 mm, the bearing capacity under all error morphologies shows an approximately linear growth. This is because the increase in orifice diameter reduces the supply impedance, increasing the mass flow rate of gas entering the gas film region per unit time, thus raising the average pressure level in the gas film. However, the reduction in supply constraints also makes the gas film pressure distribution more controlled by local geometric clearance changes, significantly increasing the sensitivity of the pressure field to rotor geometric errors. As a result, the bearing capacity gap between different error morphologies expands noticeably with increasing orifice diameter, indicating that, under high-supply-capacity conditions, the negative modulation effect of machining errors on pressure field distribution is further amplified.

Figure 15b shows that, as the orifice diameter increases, the gas consumption of all rotor types rises significantly. This trend stems from the weakened throttling effect, which makes gas more likely to escape through the bearing clearance, increasing the gas flow required per unit load. The ideal circular rotor exhibits high gas utilization efficiency under all orifice diameter conditions. In contrast, the waist-drum-shaped and conical rotors, due to local clearance mutations caused by geometric discontinuity, have more complex gas flow paths and increased additional flow losses. Under the maximum orifice diameter condition, their gas consumption is about 15% higher than that of the ideal rotor.

Figure 15c shows that orifice diameter and rotor profile together influence axial dimensionless pressure. Larger orifice diameters yield higher pressure peaks, while smaller orifices produce lower peaks and greater pressure differences across rotor profiles, confirming that larger orifices enhance fluid film pressure levels.

Further analysis of bearing efficiency reveals that the ideal circular rotor has an optimal efficiency balance point near an orifice diameter of 0.2 mm. At this diameter, the bearing capacity per unit gas consumption is about 18% and 32% higher than that of the waist-drum-shaped and conical rotors, respectively. This result indicates that a larger orifice diameter is not always beneficial. When the gas supply capacity exceeds a certain threshold, the gain in bearing capacity comes at the cost of a significant reduction in gas utilization efficiency.

Comprehensive analysis shows that the orifice diameter adjusts the constraint intensity of the gas supply system on the gas film pressure, directly affecting the amplification degree of geometric errors in the gas film pressure field. In the design of high-precision aerostatic spindles, the orifice diameter should be reasonably limited on the premise of meeting bearing requirements. This avoids excessive amplification of geometric error sensitivity, thereby achieving comprehensive optimization of bearing capacity, gas utilization efficiency, and error robustness.

5.3.2. Influence of the Coupling Between Machining Errors and Supply Pressure on the Static Performance of the Spindle

Figure 16 presents the influence of the coupling between machining error and supply pressure on the static performance of the spindle. The analysis reveals that increasing the supply pressure enhances the spindle-bearing capacity while also amplifying the system’s response to machining errors, indicating a clear coupling effect between the two.

In terms of bearing capacity (Figure 16a), as the supply pressure increases from 0.25 MPa to 0.6 MPa, the bearing capacity of all rotor types under the same error conditions shows an approximately linear growth. This trend mainly stems from the elevated supply pressure raising the overall pressure level in the gas film, enabling unit clearance changes to generate a larger pressure gradient response. However, the sensitivity of the pressure gradient to geometric clearance changes also increases, significantly amplifying the disturbance of local geometric errors on the pressure field distribution under high-pressure conditions. Thus, the bearing capacity difference between different error morphologies expands noticeably with increasing supply pressure, rising from about 200 N at 0.25 MPa to about 500 N at 0.6 MPa.

Figure 16b shows that gas consumption exhibits an approximately linear upward trend with supply pressure. As pressure increases, the driving force for gas leakage through the bearing clearance is significantly enhanced, leading to a higher mass flow rate per unit time under the same geometric conditions. The ideal cylindrical rotor, with uniform clearance distribution, can effectively suppress the formation of local high-leakage channels, maintaining the lowest gas consumption under all pressure conditions. In contrast, the waist-drum-shaped and conical rotors, due to the non-uniform distribution of geometric clearance along the axial and circumferential directions, are more prone to local leakage paths. At 0.6 MPa, their gas consumption is about 12% higher than that of the ideal cylinder.

Figure 16c demonstrates that, under identical error conditions, increasing the supply pressure from 0.3 MPa to 0.55 MPa elevates the peak dimensionless pressure and flattens the pressure plateau of the gas film. This indicates that higher supply pressures enhance the overall gas film pressure field, which in turn improves the static performance of the spindle.

The ideal cylindrical rotor achieves a relatively optimal balance between bearing capacity and gas consumption in the supply pressure range of 0.4–0.5 MPa. Within this range, the bearing capacity maintains a high growth rate while the enhanced leakage has not yet dominated system energy consumption, so its bearing capacity per unit gas consumption is about 15% and 25% higher than that of the waist-drum-shaped and conical rotors, respectively. When the supply pressure increases further, the gain in bearing capacity is gradually offset by the rapidly growing leakage loss, resulting in a decline in overall energy efficiency.

Comprehensive analysis shows that supply pressure amplifies the influence of gas film pressure gradient on gas film clearance changes, making the impact of machining errors on performance more significant under high-pressure conditions. Consequently, in high-precision aerostatic spindle systems, appropriately increasing the supply pressure can fully release the bearing potential under ideal geometric conditions, while, in working conditions with unavoidable geometric errors, the supply pressure level should be reasonably controlled to balance performance and energy consumption, thereby achieving coordinated optimization of the system’s comprehensive performance.

5.4. Quantitative Analysis of the Attenuation of Spindle Static Performance by the Ratio of Error Amplitude to Gas Film Thickness

Figure 17 systematically reveals the quantitative influence law of the ratio between error amplitude and gas film thickness on the attenuation of spindle static performance. Taking representative waist-drum-shaped and conical errors as research objects, the ratio between error amplitude and designed gas film clearance is calculated, and the spindle performance under different tolerance design conditions is analyzed. These findings suggest that an increase in this ratio significantly weakens the system’s bearing capacity and gas utilization performance.

When the ratio increases from 8% to 18.2%, the bearing capacity change rate of the barrel-shaped error (compared with the ideal rotor) rises from 8.7% to 22.1%, and the gas consumption increases from 5.7% to 12.9%. For the conical error, the response is more dramatic: Its bearing capacity change rate increases from 15.3% to 28.7%, and the gas consumption change rate rises from 7.9% to 15.9%. This indicates that the sensitivity of the conical error to gas film clearance changes is about 1.8 times that of the waist-drum-shaped error, resulting in stronger performance attenuation. This trend is readily observed from the pressure distribution.

An increase in the ratio significantly alters the spatial topological characteristics of the gas film clearance. Notably, under conditions of small error ratios, the performance change rate and the ratio show an approximately linear relationship. Within this range, the system can be regarded as a quasi-linear response region: Spindle performance can be predicted based on the ratio, or constraints can be proposed for spindle tolerance design according to performance and design requirements, which can effectively reduce economic costs.

Table 6. Effects of error shape and amplitude on performance parameters.

Error amplitude/air-film gap (%)		8	12.15	18.2
waist-drum shape error (%)	Load	8.7	13.9	22.1
	Air consumption	5.7	8.6	12.9
Conical error (%)	Load	15.3	21.5	28.7
	Air consumption	7.9	11.73	15.9

shows that, as the ratio between error amplitude and gas film clearance increases, the relative errors of the spindle system’s bearing capacity and gas consumption both show an upward trend. This indicates that, under the design condition of fixed gas film clearance, an increase in error amplitude may lead to reduced system stability or lower control accuracy. For the waist-drum-shaped error, when the ratio is small, its performance error can be quantitatively evaluated approximately using the ratio of performance gain (under ideal conditions) to error amplitude and gas film thickness, providing a reliable reference for the engineering evaluation of spindle performance under small error conditions.

The quantitative correlation model between error amplitude and performance attenuation constructed in this paper provides theoretical guidance for the performance prediction and tolerance design of aerostatic spindles. In high-precision spindle design, the amplitude of conical errors should be strictly controlled, while the waist-drum-shaped error has a relatively smaller impact on performance, so tolerance requirements can be moderately relaxed on the premise of ensuring system stability. This study provides differentiated tolerance design guidance for aerostatic spindles of different precision levels, achieving an optimized balance between manufacturing costs and performance requirements.

6. Conclusions

This study proposes a rotor machining error modeling and performance evaluation method based on harmonic statistical analysis and multi-scale integration. The main conclusions are as follows.

(1)

Fourier harmonic decomposition of measured rotor topography revealed that low-order harmonics (1st–5th) dominate cylindricity errors, and their amplitudes follow a log-normal distribution. A 3σ statistical envelope was established to characterize manufacturing variability, providing a probabilistic basis for tolerance analysis and design.

(2)

A multi-scale rotor error model incorporating macroscopic errors and microscopic stochastic surface errors was developed and incorporated into a modified Reynolds equation. The results show that geometric errors significantly alter gas film thickness and pressure distribution, leading to changes in load capacity and static stiffness.

(3)

The eccentricity range of ε = 0.3–0.5 was identified as an error-sensitive operating regime in which machining errors have the strongest influence on spindle performance. At high rotational speeds, low-frequency harmonic errors are amplified by hydrodynamic effects, forming an error-excited speed window. Supply pressure and orifice diameter were found to strongly couple with geometric errors, affecting both performance and sensitivity.

(4)

The ratio of error amplitude to gas film clearance was identified as a key parameter governing performance degradation. Conical errors have the most detrimental impact, whereas waist-drum-shaped errors exhibit a self-stabilizing tendency under small deviations. These findings provide quantitative guidance for tolerance allocation, spindle design optimization, and operating parameter selection.