A method for evaluating vibration performance of a hemispherical resonator based on comprehensive measurement of geometric error and unified parameterization

Abstract

The application discloses a kind of based on geometric error comprehensive measurement and unified parameterization hemispherical resonator vibration performance evaluation method, it is related to optical element processing field, to solve the lack of comprehensive evaluation method of hemispherical resonator vibration performance based on multiple source geometric error in prior art.The technical points include: measuring hemispherical resonator multi-source data, convert measurement data to unified coordinate system and unified circumferential reference;Further error separation is carried out to measurement data, obtain the circumferential distribution geometric error and position size geometric error of hemispherical resonator;Circumferential harmonic decomposition is carried out to circumferential distribution geometric error, obtain each error circumferential harmonic amplitude, circumferential harmonic phase and global circumferential harmonic complex amplitude, and position size geometric error is combined, and the geometric error parameter set of hemispherical resonator is constructed;Geometric structure is constructed to hemispherical resonator by geometric error parameter set, and the vibration performance of hemispherical resonator is carried out finite element analysis.

Description

Technical Field

[0001] This invention relates to the field of optical component processing technology, specifically to a method for evaluating the vibration performance of a hemispherical harmonic oscillator based on comprehensive measurement and unified parameterization of geometric errors. This method is applicable to the comprehensive measurement of errors, unified coordinate registration, error separation, circumferential harmonic decomposition, and vibration performance evaluation of hemispherical harmonic oscillators used in hemispherical resonator gyroscopes. Background Technology

[0002] The hemispherical resonator is the core sensing element of a hemispherical resonant gyroscope, and its vibration performance directly affects the accuracy, stability, and long-term reliability of the gyroscope. An ideal hemispherical resonator has good structural axisymmetry, and its pair of orthogonal operating modes should have nearly identical natural frequencies, a high quality factor, and good circumferential uniformity.

[0003] In actual manufacturing, hemispherical resonators typically undergo ultra-precision grinding, polishing, chemical treatment, cleaning, and precision adjustment. Influenced by factors such as tooling errors, clamping errors, material non-uniformity removal, measurement reference deviations, and machining path errors, hemispherical resonators inevitably possess various geometric errors. These include external surface shape errors, internal surface shape errors, shell thickness errors, end face shape errors, end face perpendicularity errors, concentricity errors of the inner and outer spherical surfaces, coaxiality errors of the support rod, and critical dimension errors.

[0004] Existing geometric testing methods for hemispherical harmonic oscillators often focus on a single surface or a single error index. For example, measuring only the outer surface profile error is insufficient to reflect the shell thickness variation determined by both the inner and outer surfaces; measuring only the end face morphology is insufficient to reflect the inclination of the end face normal relative to the support axis or rotation axis; measuring only roundness or coaxiality is insufficient to obtain the full-diameter surface shape distribution; and measuring only dimensional errors is insufficient to characterize the circumferential harmonic components in the surface shape error. Therefore, a single measurement method cannot comprehensively support the vibration performance evaluation of a hemispherical harmonic oscillator.

[0005] The technical problem to be solved by this invention is:

[0006] First, existing measurement methods mostly focus on a single surface or a single geometric index, making it difficult to comprehensively characterize the multi-source geometric errors of the hemispherical harmonic oscillator, including its outer surface, inner surface, shell thickness, end face, support rod, and key dimensions.

[0007] Second, data obtained from different measuring instruments have different measurement coordinate systems and circumferential phase references, and there is a lack of a unified method for coordinate registration and circumferential zero-degree reference establishment, which makes it difficult to fuse multi-source geometric data.

[0008] Third, existing measurement results are mostly in the form of point clouds, local surface shapes, cross-sectional roundness, or dimensional parameters, lacking a unified parametric expression for vibration performance evaluation;

[0009] Fourth, existing geometric error characterization methods are difficult to directly support subsequent analyses such as frequency splitting, quality factor, anchoring loss, thermoelastic damping, or ion beam tuning.

[0010] Fifth, there is a lack of methods to convert external surface shape errors, internal surface shape errors, shell thickness errors, end face shape errors, positional errors, and dimensional errors into a unified set of geometric error parameters, making it difficult to determine the impact of different error types on the vibration performance of the hemispherical harmonic oscillator.

[0011] Therefore, it is necessary to propose a method for evaluating the vibration performance of a hemispherical harmonic oscillator based on comprehensive measurement and unified parameterization of geometric errors. This method converts multi-source geometric measurement results into a unified set of geometric error parameters with clear mechanical meaning that can be used for vibration performance evaluation and adjustment calculations, thereby enabling the evaluation of the vibration performance of the hemispherical harmonic oscillator. Summary of the Invention

[0012] The technical problem to be solved by this invention is:

[0013] There is a lack of comprehensive evaluation methods for the vibration performance of hemispherical harmonic oscillators based on multi-source geometric errors in the existing technology.

[0014] The technical solution adopted by the present invention to solve the above-mentioned technical problems is as follows:

[0015] To address the aforementioned technical problems, this invention provides a method for evaluating the vibration performance of a hemispherical harmonic oscillator based on comprehensive measurement of geometric errors and unified parameterization, comprising the following steps:

[0016] S1. Establish a unified coordinate system and a unified circumferential reference for the hemispherical harmonic oscillator;

[0017] S2. Perform comprehensive measurements on the hemispherical harmonic oscillator to obtain multi-source measurement data of the hemispherical harmonic oscillator;

[0018] S3. Convert the multi-source measurement data of the hemispherical harmonic oscillator to a unified coordinate system and a unified circumferential reference.

[0019] S4. Based on the measurement data after coordinate registration, perform inner and outer surface registration, sphere center fitting, axis fitting and end face reference fitting, and further perform error separation to obtain circumferential distribution geometric errors including outer surface shape error, inner surface shape error, shell thickness error and end face shape error, as well as positional dimension geometric errors including end face perpendicularity error, inner and outer spherical concentricity error, support rod coaxiality error and key dimension error.

[0020] S5. Perform circumferential harmonic decomposition on circumferentially distributed geometric errors to obtain the circumferential harmonic amplitude, circumferential harmonic phase and global circumferential harmonic complex amplitude of each error, and combine with position and size geometric errors to construct a set of geometric error parameters for the hemispherical harmonic oscillator.

[0021] S6. The geometric structure of the hemispherical harmonic oscillator is constructed using the obtained set of geometric error parameters, and the vibration performance of the hemispherical harmonic oscillator is further analyzed by finite element method.

[0022] Further, the method for obtaining the hemispherical shell surface shape data and shell thickness data in step S2 is as follows: a spherical wavefront interferometer is used to measure multiple circumferential sub-apertures on the outer and inner surfaces of the hemispherical shell, and the hemispherical shell surface shape data and shell thickness data are obtained by global stitching; during the process, 5 circumferential sub-apertures are used for measurement, the circumferential angle interval between the centers of adjacent sub-apertures is 72°, the angle between the measuring optical axis and the support rod axis is 60°, and there is an overlap area of ​​not less than 25% of the effective area of ​​a single sub-aperture between adjacent sub-apertures.

[0023] Further, the method for obtaining the end face shape data in step S2 is as follows: a white light interferometer is used to measure the sub-aperture of the hemispherical resonator end face, and the end face shape data is obtained through global stitching; during measurement, the end face normal of the hemispherical resonator is adjusted to be consistent with the optical axis of the white light interferometer, and the axis of the support rod is made coaxial with the axis of the turntable; 72 end face sub-apertures are arranged in a circumferential array, the circumferential angle interval between adjacent sub-apertures is 5°, and the overlap rate between adjacent sub-apertures is greater than 25%. The full aperture shape reconstruction of the end face is achieved by optimizing the rigid body pose of the sub-apertures and minimizing the error of the overlapping area; and the end face sub-aperture measurement adopts the same circumferential zero-degree direction and positive angle direction as the hemispherical shell sub-aperture measurement.

[0024] Further, in step S3, the multi-source measurement data of the hemispherical harmonic oscillator is transformed to a unified coordinate system. Specifically, let the original measurement point of the q-th type of measurement data in its instrument coordinate system be... Then the measurement points after transformation to a unified coordinate system Represented as:

[0025]

[0026] In the formula, For measurement points after transformation to a unified coordinate system, Let q be the original measurement point in its instrument coordinate system. Let be the rotation matrix corresponding to the q-th type of measurement data. Let be the translation vector corresponding to the q-th type of measurement data.

[0027] Further, the error separation in step S4 specifically involves: determining the relative position vectors of the inner and outer sphere centers based on the fitted centers of the outer and inner spheres, and using these relative position vectors to characterize the concentricity error of the inner and outer spheres, wherein the relative position vectors of the inner and outer sphere centers are expressed as:

[0028]

[0029] The concentricity error of the inner and outer spherical surfaces is expressed as follows:

[0030]

[0031] In the formula, Let be the relative position vector of the inner and outer sphere centers. This is the concentricity error of the inner and outer spherical surfaces. Fit the center of the sphere to the outer sphere. Fit the center of the sphere to the inner sphere;

[0032] The perpendicularity error of the end face is determined by the angle between the normal vector of the end face fitting plane and the axial coordinate axis of the unified coordinate system, expressed as:

[0033]

[0034] In the formula, For end face perpendicularity error, The normal vector of the fitting plane for the end face. To unify the axial unit vector of the coordinate system;

[0035] The outer surface shape error is expressed as:

[0036]

[0037] In the formula, For the surface shape error, These are the positional parameters along the generatrix of the hemispherical harmonic oscillator. Circumferential angle, To measure the radius of the outer surface, The nominal radius of the outer surface;

[0038] The inner surface shape error is expressed as:

[0039]

[0040] In the formula, For the inner surface shape error, To measure the radius of the inner surface, The nominal radius of the inner surface;

[0041] The shell thickness error is obtained by registering the outer surface measurement data and the inner surface measurement data in a unified coordinate system, and is expressed as:

[0042]

[0043] In the formula, For shell thickness error, To measure the shell thickness, Nominal shell thickness;

[0044] The thickness of the measuring shell is calculated along the local normal of the hemispherical shell and is expressed as follows:

[0045]

[0046] In the formula, For the measurement points on the outer surface, For internal surface measurement points, This is the local normal vector of the hemispherical shell;

[0047] The end face shape error is expressed as:

[0048]

[0049] In the formula, For end face shape error, For the radial coordinates of the end face, For end face measurement points, As the end face reference point, The normal vector of the plane fitting the end face;

[0050] The critical structural dimension error is obtained by subtracting the actual structural dimension obtained from measurement or fitting from the corresponding nominal structural dimension, and is expressed as:

[0051]

[0052] Or it can be represented as a critical structural dimension error vector:

[0053]

[0054] In the formula, For the j-th critical structural dimension error, For the j-th term, the actual structural dimensions obtained by measurement or fitting are... Let J be the nominal structural dimension of the j-th item, and J be the number of critical structural dimension errors. This represents the error vector for key structural dimensions. This is the actual structural dimension vector. This is the nominal structural dimension vector.

[0055] Furthermore, the key structural dimensional errors include one or more of the following: outer spherical radius error, inner spherical radius error, measured support rod radius error, support rod length error, end face axial position error, spherical shell opening diameter error, and transition fillet radius error.

[0056] Furthermore, step S5 involves performing circumferential harmonic decomposition on the circumferentially distributed geometric errors, including: using position parameters along the generatrix direction of the hemispherical shell to decompose the outer surface shape error, inner surface shape error, and shell thickness error. and circumferential angle Represented as The end face shape error is expressed using the end face radial position parameter. and circumferential angle Represented as In the formula, It is a geometric error type;

[0057] For the external surface shape error, internal surface shape error, shell thickness error, and end face shape error, the circumferential angle is resampled and expressed as:

[0058]

[0059] In the formula, To unify the circumferential zero-degree direction, This represents the number of circumferential resampling points at a single busbar location.

[0060] The axisymmetric error components are first separated from the resampled geometric error function, and then the non-axisymmetric error components are extracted; among them, the axisymmetric error components of the outer surface shape error, inner surface shape error, and shell thickness error are... for:

[0061]

[0062] Its non-axisymmetric error components for:

[0063]

[0064] Axisymmetric error components of end face shape error for:

[0065]

[0066] Its non-axisymmetric error components for:

[0067]

[0068] The end face shape error is expressed as:

[0069]

[0070] In the formula, For the first The first type of geometric error Next-circular harmonic amplitude, For the first The first type of geometric error Sub-circular harmonic phase ;

[0071] For the external surface shape error, internal surface shape error, and shell thickness error, calculate their global circumferential harmonic complex amplitude values:

[0072]

[0073] In the formula, For the first Global first-order geometric error Second circumferential harmonic complex amplitude The weighting function is the direction of the busbar. and These represent the start and end positions of the bus directions participating in global harmonic synthesis. The imaginary unit;

[0074] For the end face shape error, calculate its global circumferential harmonic complex amplitude:

[0075]

[0076] In the formula, For the radial weighting function of the end face, and These represent the radial start and end positions of the end faces participating in global harmonic synthesis.

[0077] Furthermore, the vibration performance of the hemispherical harmonic oscillator described in S6 includes one or more of the following: frequency splitting, quality factor, anchoring loss, and thermoelastic damping.

[0078] Furthermore, S6 includes the following process:

[0079] S61. Construct the nominal geometric model of the hemispherical harmonic oscillator, and map the results of the geometric error parameter set to the nominal geometric model of the hemispherical harmonic oscillator respectively, thus constructing a finite element geometric model of the hemispherical harmonic oscillator containing geometric errors; for any characteristic surface S of the hemispherical harmonic oscillator, its finite element geometric points containing geometric errors are represented as:

[0080]

[0081] In the formula, S is the characteristic surface of the hemispherical harmonic oscillator, including: outer surface, inner surface, end face, transition fillet and outer circular surface of the support rod; (u,v) are the parametric coordinates of the characteristic surface; These are the finite element geometric points after mapping geometric errors; The nominal geometric point after taking into account critical dimension errors; The nominal structural size vector; This is the critical dimension error vector; This refers to the surface shape error or thickness error corresponding to the characteristic surface. The nominal normal vector of the characteristic surface; This is the position offset vector of the feature surface; For the small attitude deflection of the characteristic surface; This is the reference point corresponding to the attitude deflection;

[0082] S62. Perform modal analysis on the finite element geometric model of the hemispherical harmonic oscillator containing geometric errors, identify the second-order working modes of the hemispherical harmonic oscillator, and obtain the natural frequencies of the main modes respectively. and secondary mode natural frequencies And calculate the frequency decomposition of the primary and secondary working modes: ;

[0083] The fourth circumferential harmonic parameter in the circumferentially distributed geometric error is used for the calculation of the frequency splitting of the main and secondary working modes;

[0084] S63. In the finite element geometric model of the hemispherical harmonic oscillator containing geometric errors, establish a support structure and a perfect matching layer, and perform complex characteristic frequency analysis on the principal mode and the secondary mode respectively to obtain the complex angular frequency of the principal mode. and secondary mode complex angular frequencies And calculate the quality factor corresponding to the anchorage loss according to the following formula:

[0085]

[0086]

[0087] The parameters of the first to third circumferential harmonics, the perpendicularity error of the end face, the concentricity error of the inner and outer spherical surfaces, and the coaxiality error of the support rod are used to calculate the anchorage loss.

[0088] S64. Based on the quality factor corresponding to the anchorage loss of the principal mode. quality factor corresponding to submodal anchorage loss The modal variability of the quality factor corresponding to anchorage loss is calculated according to the following formula:

[0089]

[0090] S65. Based on the shell thickness error, establish a thermoelastic damping finite element model coupling the structural field and the temperature field, and calculate the modal energy storage for the main mode and the sub-mode respectively. and single-cycle heat dissipation energy And calculate the quality factor corresponding to thermoelastic damping. :

[0091] .

[0092] Compared with the prior art, the beneficial effects of the present invention are:

[0093] This invention acquires multi-source geometric information of a hemispherical harmonic oscillator, including its outer surface, inner surface, end face, support rod, inner and outer sphere centers, end face normal, and shell thickness. By registering geometric data from different measuring instruments using a unified coordinate system and a unified circumferential reference, the multi-source measurement data achieves a consistent spatial reference and circumferential phase reference. Furthermore, circumferential harmonic decomposition yields the amplitude, phase, and global complex amplitude of different orders of circumferential harmonics, providing quantifiable geometric error input for vibration performance evaluation. Finally, a unified set of geometric error parameters is constructed and directly applied to the finite element geometric reconstruction and vibration performance simulation of the hemispherical harmonic oscillator, improving the connectivity between geometric error measurement, error modeling, and vibration performance evaluation.

[0094] This invention is applicable to the manufacturing inspection, process evaluation, performance assessment, and error modeling before adjustment of hemispherical resonators for hemispherical resonator gyroscopes. It is also applicable to the comprehensive measurement and unified parameterization analysis of geometric errors of other axisymmetric thin-walled resonator structures, as well as error sensitivity analysis and ion beam adjustment calculations. It has good engineering applicability and promotion value. Attached Figure Description

[0095] Figure 1 This is a flowchart of the hemispherical harmonic oscillator vibration performance evaluation method based on comprehensive geometric error measurement and unified parameterization in an embodiment of the present invention;

[0096] Figure 2 This is a schematic diagram of the hemispherical harmonic oscillator structure, unified coordinate system, and unified circumferential reference in an embodiment of the present invention.

[0097] Figure 3 This is a schematic diagram of the multi-source geometric error measurement of the hemispherical harmonic oscillator in an embodiment of the present invention, wherein (a) is a schematic diagram of the aperture interferometry measurement of the spherical shell, (b) is a schematic diagram of the end face white light interferometry measurement, (c) is a schematic diagram of the roundness measurement, and (d) is a schematic diagram of the three-coordinate measurement.

[0098] Figure 4 This is a schematic diagram of unified conversion of multi-source measurement data, coordinate registration and error separation in an embodiment of the present invention, wherein (a) is a schematic diagram of multi-source measurement data, (b) is a schematic diagram of the establishment of a unified coordinate system and a unified circumferential reference, and (c) is a schematic diagram of error separation and generation of unified geometric error parameters;

[0099] Figure 5 The diagram below illustrates the unified parameterization of geometric errors, circumferential harmonic decomposition, and vibration performance evaluation in this embodiment of the invention. (a) is a schematic diagram of the unified geometric error parameter set, (b) is a schematic diagram of circumferential harmonic decomposition, (c) is a schematic diagram of the input for finite element geometric reconstruction with errors and vibration performance evaluation, and (d) is a schematic diagram of the vibration performance evaluation result. Detailed Implementation

[0100] To enable those skilled in the art to better understand the present invention, exemplary embodiments or examples of the present invention will be described below in conjunction with the accompanying drawings. Obviously, the described embodiments or examples are merely some, not all, of the embodiments or examples of the present invention. All other embodiments or examples obtained by those skilled in the art based on the embodiments or examples of the present invention without inventive effort should fall within the scope of protection of the present invention.

[0101] To make the above-mentioned objects, features and advantages of the present invention more apparent and understandable, specific embodiments of the present invention will be described in detail below with reference to the accompanying drawings.

[0102] like Figure 1 As shown, this embodiment provides a method for evaluating the vibration performance of a hemispherical harmonic oscillator based on comprehensive measurement of geometric errors and unified parameterization, including the following steps:

[0103] S1. Establish a unified coordinate system and a unified circumferential reference for the hemispherical harmonic oscillator.

[0104] like Figure 2 As shown, the hemispherical resonator 1 includes a hemispherical shell 2 and a support rod 3. The fitting axis of the support rod 3 or the rotation axis of the hemispherical resonator 1 is used as the axial coordinate axis of the unified coordinate system 4, and the meridional plane containing the center of the first sub-aperture in the sub-aperture interferometry of the spherical shell is used as the unified circumferential reference 5. Through the unified coordinate system 4 and the unified circumferential reference 5, the subsequent measurement data of the spherical shell, end face, support rod, and key dimensions have a consistent spatial reference and phase reference.

[0105] S2. Perform multi-source comprehensive measurements on the hemispherical harmonic oscillator to obtain multi-source measurement data of the hemispherical harmonic oscillator.

[0106] The multi-source measurement data of the hemispherical harmonic oscillator mentioned in step S2 includes: hemispherical shell surface shape data and shell thickness data, end face surface shape data; cross-sectional eccentricity data, cross-sectional azimuth data, concentricity data of inner and outer spheres and end face perpendicularity data; outer sphere radius data, inner sphere radius data, inner and outer sphere center position data, support rod radius data, support rod axis data, and key structural dimension data.

[0107] like Figure 3 As shown, a spherical wavefront interferometer is used to measure multiple circumferential sub-apertures on the outer and inner surfaces of the hemispherical shell, and the surface shape and thickness data of the hemispherical shell are obtained through global stitching. During measurement, the hemispherical resonator 1 is adjusted so that the axis of the support rod 3 is coaxial with the axis of the turntable, and the measuring optical axis is aligned with the axis of the support rod 3 through an angle adjustment mechanism. The included angle is adjusted using the XYZ adjustment mechanism to ensure that the focal point of the transmission sphere coincides with the center of the outer and inner spheres, respectively. Both the outer and inner surfaces are measured using five circumferential sub-apertures, with the circumferential angle interval between the centers of adjacent sub-apertures being [missing information]. The area between adjacent sub-apertures is not less than the effective area of ​​a single sub-aperture. The overlapping area. The corresponding sub-apertures on the outer and inner surfaces are sampled at the same circumferential angle position, and the meridional plane where the center of one of the sub-apertures is located is used as a unified circumferential reference, so as to ensure that the inner and outer surfaces have a consistent circumferential phase when calculating the shell thickness error.

[0108] A white-light interferometer was used to measure the sub-apertures on the end face of a hemispherical resonator, and the end face shape data was obtained through global stitching. During measurement, the hemispherical resonator 1 was adjusted so that the end face normal was aligned with the optical axis of the white-light interferometer, and the axis of the support rod 3 was coaxial with the axis of the turntable. Subsequently, 72 equally spaced sub-apertures were arranged on the end face, with an angular interval between adjacent sub-apertures. The overlap rate between adjacent sub-apertures is greater than Each end-face sub-aperture acquires local point cloud or height deviation data. A pose optimization target is constructed using the overlapping region of adjacent sub-apertures to solve for the rigid body pose of each sub-aperture. All end-face sub-aperture data are then fused onto a unified end-face reference plane to obtain the full-diameter end-face surface shape error. The center of the first end-face sub-aperture is aligned with the meridional plane containing the center of the first sub-aperture on the spherical shell, thus ensuring a consistent circumferential zero-degree reference between the end-face surface shape error and the spherical shell surface shape error.

[0109] A roundness meter was used to measure the roundness and eccentricity of the outer spherical surface, inner spherical surface, or related support sections at different heights of the hemispherical harmonic oscillator 1. The roundness error, eccentricity of the section center, and azimuth angle of the section center were obtained for each section. By comparing the positional changes of the section center at different heights, the concentricity error of the inner and outer spherical surfaces, the coaxiality error of the support rod, and the tilt of the end face normal relative to the reference axis were evaluated.

[0110] A coordinate measuring machine (CMM) was used to perform point cloud measurements on the outer spherical surface, inner spherical surface, end face, and support rod 3 of the hemispherical harmonic oscillator 1. The radius, center, and axis of the outer and inner spherical surfaces were obtained through spherical fitting; the axis and radius of the support rod 3 were obtained through cylindrical axis fitting; the end face position and normal were obtained through plane fitting; and key structural dimension data were obtained based on the difference between the measured or fitted actual structural dimensions and the corresponding nominal structural dimensions.

[0111] S3. Convert the geometric measurement data of the hemispherical harmonic oscillator to a unified coordinate system and a unified circumferential reference.

[0112] Let the original measurement point of the q-th type of measurement data in its measuring instrument coordinate system be... Then the measurement points after transformation to a unified coordinate system Represented as:

[0113]

[0114] In the formula, For measurement points after transformation to a unified coordinate system, Let q be the original measurement point in its instrument coordinate system. Let be the rotation matrix corresponding to the q-th type of measurement data. Let be the translation vector corresponding to the q-th type of measurement data.

[0115] S4. Based on the measurement data after coordinate registration, perform inner and outer surface registration, sphere center fitting, axis fitting, and end face datum fitting. Then, perform error separation to obtain circumferentially distributed geometric errors, including outer surface shape error, inner surface shape error, shell thickness error, and end face shape error, as well as geometric errors, including end face perpendicularity error, inner and outer spherical concentricity error, support rod coaxiality error, and critical dimension error.

[0116] Based on the measurement data transformed to a unified coordinate system and a unified circumferential datum in step S3, spherical fitting is performed on the outer surface measurement data to obtain the center and radius of the outer spherical fitting sphere; spherical fitting is performed on the inner surface measurement data to obtain the center and radius of the inner spherical fitting sphere; axis fitting is performed on the support rod measurement data to obtain the support rod fitting axis; and plane fitting is performed on the end face measurement data to obtain the end face fitting plane and end face normal.

[0117] The relative position vectors of the inner and outer sphere centers are determined based on the fitted centers of the outer and inner spheres, and these vectors characterize the concentricity error of the inner and outer spheres. The relative position vectors of the inner and outer sphere centers are expressed as follows:

[0118]

[0119] The concentricity error of the inner and outer spherical surfaces is expressed as follows:

[0120]

[0121] In the formula, Let be the relative position vector of the inner and outer sphere centers. This is the concentricity error of the inner and outer spherical surfaces. Fit the center of the sphere to the outer sphere. The center of the sphere is fitted to the inner sphere.

[0122] The perpendicularity error of the end face is determined by the angle between the normal vector of the end face fitting plane and the axial coordinate axis of the unified coordinate system, expressed as:

[0123]

[0124] In the formula, For end face perpendicularity error, The normal vector of the fitting plane for the end face. To unify the axial unit vector of the coordinate system;

[0125] The outer surface shape error is expressed as:

[0126]

[0127] In the formula, For the surface shape error, These are the positional parameters along the generatrix of the hemispherical harmonic oscillator. Circumferential angle, To measure the radius of the outer surface, The nominal radius of the outer surface;

[0128] The inner surface shape error is expressed as:

[0129]

[0130] In the formula, For the inner surface shape error, To measure the radius of the inner surface, The nominal radius of the inner surface;

[0131] The shell thickness error is obtained by registering the outer surface measurement data and the inner surface measurement data in a unified coordinate system, and is expressed as:

[0132]

[0133] In the formula, For shell thickness error, To measure the shell thickness, Nominal shell thickness;

[0134] The thickness of the measuring shell is calculated along the local normal of the hemispherical shell and is expressed as follows:

[0135]

[0136] In the formula, For the measurement points on the outer surface, For internal surface measurement points, This is the local normal vector of the hemispherical shell;

[0137] The end face shape error is expressed as:

[0138]

[0139] In the formula, For end face shape error, For the radial coordinates of the end face, For end face measurement points, As the end face reference point, This is the normal vector of the plane fitting the end face.

[0140] The critical structural dimension error is obtained by subtracting the actual structural dimension obtained from measurement or fitting from the corresponding nominal structural dimension, and is expressed as:

[0141]

[0142] Or it can be represented as a critical structural dimension error vector:

[0143]

[0144] In the formula, For the j-th critical structural dimension error, For the j-th term, the actual structural dimensions obtained by measurement or fitting are... Let J be the nominal structural dimension of the j-th item, and J be the number of critical structural dimension errors. This represents the error vector for key structural dimensions. This is the actual structural dimension vector. This is the nominal structural dimension vector.

[0145] The critical structural dimensional errors are obtained by subtracting the actual structural dimensions (obtained through measurement or fitting) from the corresponding nominal structural dimensions; these include: outer spherical radius error, inner spherical radius error, support rod length error, end face axial position error, spherical shell opening diameter error, and transition fillet radius error, which are respectively expressed as:

[0146]

[0147]

[0148]

[0149]

[0150] In the formula, For the error in the length of the support rod, This refers to the axial position error of the end face. This is the error in the diameter of the spherical shell opening. For the transition fillet radius error; superscript Indicates the actual structural dimensions obtained by measurement or fitting, superscript This indicates the corresponding nominal structural dimensions.

[0151] S5. Perform circumferential harmonic decomposition on the circumferentially distributed geometric errors to obtain the circumferential harmonic amplitude, circumferential harmonic phase and global circumferential harmonic complex amplitude of each error, and construct the geometric error parameter set of the hemispherical harmonic oscillator by combining the position and size geometric errors.

[0152] Step S5 involves performing circumferential harmonic decomposition on circumferentially distributed geometric errors, including: using position parameters along the generatrix of the hemispherical shell to decompose the outer surface shape error, inner surface shape error, and shell thickness error. and circumferential angle Represented as The end face shape error is expressed using the end face radial position parameter. and circumferential angle Represented as In the formula, It is a geometric error type;

[0153] For external surface shape errors, internal surface shape errors, shell thickness errors, and end face shape errors, the circumferential angles are resampled at equal angles along a uniform circumferential zero-degree direction and the same positive angular direction. For external surface shape errors, internal surface shape errors, and shell thickness errors, resampling is performed at each generatrix position; for end face shape errors, resampling is performed at each end face radial position. The resampled circumferential angles are expressed as follows:

[0154]

[0155] In the formula, To unify the circumferential zero-degree direction, This represents the number of circumferential resampling points at a single busbar location.

[0156] The axisymmetric error components are first separated from the resampled geometric error function, and then the non-axisymmetric error components are extracted; among them, the axisymmetric error components of the outer surface shape error, inner surface shape error, and shell thickness error are... for:

[0157]

[0158] Its non-axisymmetric error components for:

[0159]

[0160] Axisymmetric error components of end face shape error for:

[0161]

[0162] Its non-axisymmetric error components for:

[0163]

[0164] The end face shape error is expressed as:

[0165]

[0166] In the formula, For the first The first type of geometric error Next-circular harmonic amplitude, For the first The first type of geometric error Sub-circular harmonic phase ;

[0167] For the external surface shape error, internal surface shape error, and shell thickness error, calculate their global circumferential harmonic complex amplitude values:

[0168]

[0169] In the formula, For the first Global first-order geometric error Second circumferential harmonic complex amplitude The weighting function is the direction of the busbar. and These represent the start and end positions of the bus directions participating in global harmonic synthesis. The imaginary unit;

[0170] For the end face shape error, calculate its global circumferential harmonic complex amplitude:

[0171]

[0172] In the formula, For the radial weighting function of the end face, and These represent the radial start and end positions of the end faces participating in global harmonic synthesis.

[0173] Constructed set of geometric error parameters for hemispherical harmonic oscillators for:

[0174]

[0175] In the formula, For end face perpendicularity error, This is the concentricity error of the inner and outer spherical surfaces. To account for the coaxiality error of the support rod, For the first Key dimension error.

[0176] S6. The geometric structure of the hemispherical harmonic oscillator is constructed using the obtained set of geometric error parameters, and the vibration performance of the hemispherical harmonic oscillator is further analyzed by finite element method.

[0177] The vibration performance of the hemispherical harmonic oscillator includes one or more of the following: frequency fragmentation, quality factor, anchorage loss, and thermoelastic damping. The fourth circumferential harmonic parameter from the unified geometric error parameter set is used to evaluate the frequency fragmentation of the primary and secondary operating modes; the first to third circumferential harmonic parameters are used to evaluate the quality factor and its circumferential uniformity; the shell thickness error is used to evaluate the thermoelastic damping; and the end face perpendicularity error, support rod coaxiality error, and inner and outer spherical concentricity error are used to evaluate the support coupling loss or anchorage loss.

[0178] Specifically, the process includes the following steps:

[0179] S61. Using the unified geometric error parameter set as the geometric input for the finite element analysis of vibration performance, and under the assumption of small errors, mapping the unified geometric error parameter set to the nominal geometric model of the hemispherical harmonic oscillator according to dimensional error, surface shape error or thickness error, position error, and attitude error, respectively, to construct a finite element geometric model of the hemispherical harmonic oscillator containing geometric errors. For any characteristic surface S of the hemispherical harmonic oscillator, its finite element geometric points containing geometric errors are represented as follows:

[0180]

[0181] In the formula, S is the characteristic surface of the hemispherical harmonic oscillator, including the outer surface, inner surface, end face, transition fillet, and outer circular surface of the support rod; (u,v) are the parametric coordinates of the characteristic surface; These are the finite element geometric points after mapping geometric errors; The nominal geometric point after taking into account critical dimension errors; The nominal structural size vector; This is the critical dimension error vector; This refers to the surface shape error or thickness error corresponding to the characteristic surface. The nominal normal vector of the characteristic surface; This is the position offset vector of the feature surface; For the small attitude deflection of the characteristic surface; This is the reference point corresponding to the attitude deflection.

[0182] Among them, the surface shape error of the outer surface, the surface shape error of the inner surface, the shell thickness error, and the end face shape error are mapped to the finite element geometric model by the following formula:

[0183]

[0184] In the formula, The normal disturbance of a finite element geometric point caused by surface shape error or thickness error; This refers to the surface shape error or thickness error corresponding to the characteristic surface. Let be the nominal normal vector of the characteristic surface. This formula represents mapping the surface shape error or thickness error along the nominal normal vector of the characteristic surface to the finite element geometric model.

[0185] The concentricity error of the inner and outer spherical surfaces and the coaxiality error of the support rod are mapped to the finite element geometric model by the following formula:

[0186]

[0187] In the formula, The translational disturbance of a finite element geometric point caused by positional error; Let be the position offset vector of the characteristic surface. This formula represents the transformation of positional errors such as concentricity error of the inner and outer spheres and coaxiality error of the support rod into the overall position offset in the finite element geometric model.

[0188] The end face perpendicularity error and the support rod axis tilt error are mapped to the finite element geometric model using the following formula:

[0189]

[0190] In the formula, The rotational disturbance of the finite element geometric point caused by attitude error; For the small attitude deflection of the characteristic surface; This refers to the nominal geometric point without considering critical dimension errors. Here, represents the reference point corresponding to the attitude deflection; \times denotes the vector cross product. This formula represents the transformation of attitude-related errors such as end face perpendicularity error and support rod axis tilt error into a small rotational disturbance relative to the reference point.

[0191] Critical dimensional errors are mapped to the finite element geometric model using the following formula:

[0192]

[0193] In the formula, This is the actual structural dimension vector after taking into account critical dimension errors; The nominal structural size vector; This represents the critical dimensional error vector. This formula indicates that dimensional errors such as support rod length error, end face axial position error, spherical shell opening diameter error, and transition fillet radius error are incorporated into the finite element geometric model.

[0194] S62. Perform modal analysis on the finite element geometric model of the hemispherical harmonic oscillator containing geometric errors, identify the second-order working modes of the hemispherical harmonic oscillator, and obtain the natural frequencies of the main modes respectively. and secondary mode natural frequencies The frequency decomposition of the primary and secondary working modes is calculated according to the following formula:

[0195] Among them, the fourth circumferential harmonic parameter in the outer surface shape error, inner surface shape error, shell thickness error and end face shape error is used as the geometric error input for the frequency decomposition calculation of the main and secondary working modes;

[0196] S63. In the finite element geometric model of the hemispherical harmonic oscillator containing geometric errors, establish a support structure and a perfect matching layer, and perform complex characteristic frequency analysis on the principal mode and the secondary mode respectively to obtain the complex angular frequency of the principal mode. and secondary mode complex angular frequencies And calculate the quality factor corresponding to the anchorage loss according to the following formula:

[0197]

[0198]

[0199] Among them, the circumferential harmonic parameters from the first to the third order, the end face perpendicularity error, the concentricity error of the inner and outer spherical surfaces, and the coaxiality error of the support rod are used as the geometric error inputs for the anchorage loss calculation;

[0200] S64. Based on the quality factor corresponding to the anchorage loss of the principal mode. quality factor corresponding to submodal anchorage loss The modal variability of the quality factor corresponding to anchorage loss is calculated according to the following formula:

[0201]

[0202] S65. Map the shell thickness error to the finite element geometric model of the hemispherical harmonic oscillator, establish a thermoelastic damping finite element model coupling the structural field and the temperature field, and calculate the modal energy storage for the main mode and the sub-mode respectively. and single-cycle heat dissipation energy And calculate the quality factor corresponding to the thermoelastic damping according to the following formula:

[0203]

[0204] Among them, the shell thickness error or the shell thickness circumferential harmonic parameter is used as the geometric error input for the thermoelastic damping calculation;

[0205] The vibration performance evaluation results of the output hemispherical harmonic oscillator include the natural frequencies of the principal modes. Submodal natural frequencies , frequency decomposition of primary and secondary working modes Quality factor corresponding to principal mode anchorage loss Quality factor corresponding to submodal anchorage loss Modal variability of anchorage loss corresponding to quality factor Quality factor corresponding to thermoelastic damping .

[0206] The finite element geometric reconstruction model is constructed based on the surface shape error of the outer surface, the surface shape error of the inner surface, the shell thickness error, the surface shape error of the end face, the perpendicularity error of the end face, the concentricity error of the inner and outer spherical surfaces, the coaxiality error of the support rod, and the key structural dimension error. The nominal geometric model of the hemispherical harmonic oscillator is subjected to error mapping and mesh reconstruction, and the output is point cloud data, mesh data or finite element geometric reconstruction file containing geometric errors.

[0207] The modal frequency splitting evaluation model takes the amplitude of the 4th circumferential harmonic, the phase of the circumferential harmonic, or the complex amplitude of the global circumferential harmonic as input, performs working mode calculations on the finite element model containing geometric errors, and outputs the natural frequencies of the main and secondary working modes and their frequency splitting amounts.

[0208] The quality factor evaluation model takes the amplitude of the first to third circumferential harmonics, the phase of the circumferential harmonics, or the complex amplitude of the global circumferential harmonics as input, calculates the changes in modal energy distribution and circumferential non-uniformity caused by geometric errors, and outputs the quality factor, the decrease in quality factor, or the evaluation result of the circumferential uniformity of quality factor.

[0209] The thermoelastic damping evaluation model takes the shell thickness error function or shell thickness error distribution as input, calculates the thermoelastic damping characteristics of the hemispherical harmonic oscillator, and outputs the quality factor or thermoelastic damping loss evaluation result corresponding to the thermoelastic damping.

[0210] The anchorage loss assessment model takes the end face verticality error, support rod coaxiality error, and inner and outer spherical concentricity error as inputs to calculate the support coupling state and energy leakage characteristics, and outputs the anchorage loss or the quality factor evaluation result corresponding to the anchorage loss.

[0211] The tuning calculation model takes circumferential harmonic parameters, frequency splitting amount, quality factor evaluation results, and finite element geometric reconstruction file containing geometric errors as inputs, calculates the error components that need to be tuned, and outputs the tuning removal amount distribution, tuning region, or tuning calculation input file.

[0212] The output of the geometric error parameter set is at least one of the following: point cloud data, mesh data, busbar-circumferential angle parameter table, circumferential harmonic parameter table, finite element geometric reconstruction file, vibration performance evaluation model input file, vibration performance evaluation result file, or adjustment calculation input file.

[0213] This invention provides a system for evaluating the vibration performance of a hemispherical harmonic oscillator based on comprehensive measurement of geometric errors and unified parameterization, comprising:

[0214] The measurement data acquisition module is used to acquire interferometric measurement data of the spherical shell aperture, white light interferometric aperture measurement data of the end face, roundness measurement data, and coordinate measuring machine measurement data.

[0215] A unified registration module is used to convert the measurement data to a unified coordinate system and a unified circumferential reference.

[0216] The error separation module is used to extract one or more of the following: external surface shape error, internal surface shape error, shell thickness error, end face shape error, end face perpendicularity error, inner and outer spherical concentricity error, support rod coaxiality error, and critical dimension error.

[0217] The circumferential harmonic decomposition module is used to extract the circumferential harmonic amplitude, circumferential harmonic phase, and global circumferential harmonic complex amplitude.

[0218] The parameter set generation module is used to generate a unified geometric error parameter set for evaluating the vibration performance of hemispherical harmonic oscillators.

[0219] While the present invention has been disclosed above, its scope of protection is not limited to the embodiments described above. Those skilled in the art can make various changes, substitutions, and modifications to the present invention without departing from its concept and scope of protection, and all such changes, substitutions, and modifications should fall within the scope of protection of the present invention.

Claims

1. A method for evaluating the vibration performance of a hemispherical harmonic oscillator based on comprehensive measurement of geometric errors and unified parameterization, characterized in that: Includes the following steps: S1. Establish a unified coordinate system and a unified circumferential reference for the hemispherical harmonic oscillator; S2. Perform comprehensive measurements on the hemispherical harmonic oscillator to obtain multi-source measurement data of the hemispherical harmonic oscillator; S3. Convert the multi-source measurement data of the hemispherical harmonic oscillator to a unified coordinate system and a unified circumferential reference. S4. Based on the measurement data after coordinate registration, perform inner and outer surface registration, sphere center fitting, axis fitting and end face reference fitting, and further perform error separation to obtain circumferential distribution geometric errors including outer surface shape error, inner surface shape error, shell thickness error and end face shape error, as well as positional dimension geometric errors including end face perpendicularity error, inner and outer spherical concentricity error, support rod coaxiality error and key dimension error. S5. Perform circumferential harmonic decomposition on circumferentially distributed geometric errors to obtain the circumferential harmonic amplitude, circumferential harmonic phase and global circumferential harmonic complex amplitude of each error, and combine with position and size geometric errors to construct a set of geometric error parameters for the hemispherical harmonic oscillator. S6. The geometric structure of the hemispherical harmonic oscillator is constructed using the obtained set of geometric error parameters, and the vibration performance of the hemispherical harmonic oscillator is further analyzed by finite element method.

2. The method according to claim 1, characterized in that: The method for obtaining the hemispherical shell surface shape data and shell thickness data in step S2 is as follows: a spherical wavefront interferometer is used to measure multiple circumferential sub-apertures on the outer and inner surfaces of the hemispherical shell, and the hemispherical shell surface shape data and shell thickness data are obtained by global stitching; during the process, 5 circumferential sub-apertures are used for measurement, the circumferential angle interval between the centers of adjacent sub-apertures is 72°, the angle between the measuring optical axis and the axis of the support rod is 60°, and there is an overlap area of ​​not less than 25% of the effective area of ​​a single sub-aperture between adjacent sub-apertures.

3. The method according to claim 2, characterized in that: The method for obtaining the end face shape data in step S2 is as follows: a white light interferometer is used to measure the sub-aperture of the hemispherical resonator end face, and the end face shape data is obtained through global stitching; during measurement, the end face normal of the hemispherical resonator is adjusted to be consistent with the optical axis of the white light interferometer, and the axis of the support rod is made coaxial with the axis of the turntable; 72 end face sub-apertures are arranged in a circumferential array, the circumferential angle interval between adjacent sub-apertures is 5°, and the overlap rate between adjacent sub-apertures is greater than 25%. The full aperture shape reconstruction of the end face is achieved by optimizing the rigid body pose of the sub-apertures and minimizing the error of the overlapping area; and the end face sub-aperture measurement adopts the same circumferential zero-degree direction and positive angle direction as the hemispherical shell sub-aperture measurement.

4. The method according to claim 1, characterized in that: In step S3, the multi-source measurement data of the hemispherical harmonic oscillator is transformed to a unified coordinate system. Specifically, let the original measurement point of the q-th type of measurement data in its instrument coordinate system be... Then the measurement points after transformation to a unified coordinate system Represented as: In the formula, For measurement points after transformation to a unified coordinate system, Let q be the original measurement point in its instrument coordinate system. Let be the rotation matrix corresponding to the q-th type of measurement data. Let be the translation vector corresponding to the q-th type of measurement data.

5. The method according to claim 4, characterized in that: The error separation in step S4 specifically involves: determining the relative position vectors of the inner and outer sphere centers based on the fitted centers of the outer and inner spheres, and using these relative position vectors to characterize the concentricity error of the inner and outer spheres. The relative position vectors of the inner and outer sphere centers are expressed as follows: The concentricity error of the inner and outer spherical surfaces is expressed as follows: In the formula, Let be the relative position vector of the inner and outer sphere centers. This is the concentricity error of the inner and outer spherical surfaces. Fit the center of the sphere to the outer sphere. Fit the center of the sphere to the inner sphere; The perpendicularity error of the end face is determined by the angle between the normal vector of the end face fitting plane and the axial coordinate axis of the unified coordinate system, expressed as: In the formula, For end face perpendicularity error, The normal vector of the fitting plane for the end face. To unify the axial unit vector of the coordinate system; The outer surface shape error is expressed as: In the formula, For the surface shape error, These are the positional parameters along the generatrix of the hemispherical harmonic oscillator. Circumferential angle, To measure the radius of the outer surface, The nominal radius of the outer surface; The inner surface shape error is expressed as: In the formula, For the inner surface shape error, To measure the radius of the inner surface, The nominal radius of the inner surface; The shell thickness error is obtained by registering the outer surface measurement data and the inner surface measurement data in a unified coordinate system, and is expressed as: In the formula, For shell thickness error, To measure the shell thickness, Nominal shell thickness; The thickness of the measuring shell is calculated along the local normal of the hemispherical shell and is expressed as follows: In the formula, For the measurement points on the outer surface, For internal surface measurement points, This is the local normal vector of the hemispherical shell; The end face shape error is expressed as: In the formula, For end face shape error, For the radial coordinates of the end face, For end face measurement points, As the end face reference point, The normal vector of the plane fitting the end face; The critical structural dimension error is obtained by subtracting the actual structural dimension obtained from measurement or fitting from the corresponding nominal structural dimension, and is expressed as: Or it can be represented as a critical structural dimension error vector: In the formula, For the j-th critical structural dimension error, For the j-th term, the actual structural dimensions obtained by measurement or fitting are... Let J be the nominal structural dimension of the j-th item, and J be the number of critical structural dimension errors. This represents the error vector for key structural dimensions. This is the actual structural dimension vector. This is the nominal structural dimension vector.

6. The method according to claim 5, characterized in that: The critical structural dimensional errors include one or more of the following: outer spherical radius error, inner spherical radius error, support rod radius error (which has been measured), support rod length error, end face axial position error, spherical shell opening diameter error, and transition fillet radius error.

7. The method according to claim 6, characterized in that: Step S5 involves performing circumferential harmonic decomposition on circumferentially distributed geometric errors, including: using position parameters along the generatrix of the hemispherical shell to decompose the outer surface shape error, inner surface shape error, and shell thickness error. and circumferential angle Represented as The end face shape error is expressed using the end face radial position parameter. and circumferential angle Represented as In the formula, It is a geometric error type; For the external surface shape error, internal surface shape error, shell thickness error, and end face shape error, the circumferential angle is resampled and expressed as: In the formula, To unify the circumferential zero-degree direction, This represents the number of circumferential resampling points at a single busbar location. The axisymmetric error components are first separated from the resampled geometric error function, and then the non-axisymmetric error components are extracted; among them, the axisymmetric error components of the outer surface shape error, inner surface shape error, and shell thickness error are... for: Its non-axisymmetric error components for: Axisymmetric error components of end face shape error for: Its non-axisymmetric error components for: The end face shape error is expressed as: In the formula, For the first The first type of geometric error Next-circular harmonic amplitude, For the first The first type of geometric error Sub-circular harmonic phase ; For the external surface shape error, internal surface shape error, and shell thickness error, calculate their global circumferential harmonic complex amplitude values: In the formula, For the first Global first-order geometric error Second circumferential harmonic complex amplitude The weighting function is the direction of the busbar. and These represent the start and end positions of the bus directions participating in global harmonic synthesis. The imaginary unit; For the end face shape error, calculate its global circumferential harmonic complex amplitude: In the formula, For the radial weighting function of the end face, and These represent the radial start and end positions of the end faces participating in global harmonic synthesis.

8. The method according to claim 7, characterized in that: The vibration performance of the hemispherical harmonic oscillator described in S6 includes one or more of the following: frequency splitting, quality factor, anchoring loss, and thermoelastic damping.

9. The method according to claim 8, characterized in that: S6 includes the following process: S61. Construct the nominal geometric model of the hemispherical harmonic oscillator, and map the results of the geometric error parameter set to the nominal geometric model of the hemispherical harmonic oscillator respectively, thus constructing a finite element geometric model of the hemispherical harmonic oscillator containing geometric errors; for any characteristic surface S of the hemispherical harmonic oscillator, its finite element geometric points containing geometric errors are represented as: In the formula, S is the characteristic surface of the hemispherical harmonic oscillator, including: outer surface, inner surface, end face, transition fillet and outer circular surface of the support rod; (u,v) are the parametric coordinates of the characteristic surface; These are the finite element geometric points after mapping geometric errors; The nominal geometric point after taking into account critical dimension errors; The nominal structural size vector; This is the critical dimension error vector; This refers to the surface shape error or thickness error corresponding to the characteristic surface. The nominal normal vector of the characteristic surface; This is the position offset vector of the feature surface; For the small attitude deflection of the characteristic surface; This is the reference point corresponding to the attitude deflection; S62. Perform modal analysis on the finite element geometric model of the hemispherical harmonic oscillator containing geometric errors, identify the second-order working modes of the hemispherical harmonic oscillator, and obtain the natural frequencies of the main modes respectively. and secondary mode natural frequencies And calculate the frequency decomposition of the primary and secondary working modes: ; The fourth circumferential harmonic parameter in the circumferentially distributed geometric error is used for the calculation of the frequency splitting of the main and secondary working modes; S63. In the finite element geometric model of the hemispherical harmonic oscillator containing geometric errors, establish a support structure and a perfect matching layer, and perform complex characteristic frequency analysis on the principal mode and the secondary mode respectively to obtain the complex angular frequency of the principal mode. and secondary mode complex angular frequencies And calculate the quality factor corresponding to the anchorage loss according to the following formula: The parameters of the first to third circumferential harmonics, the perpendicularity error of the end face, the concentricity error of the inner and outer spherical surfaces, and the coaxiality error of the support rod are used to calculate the anchorage loss. S64. Based on the quality factor corresponding to the anchorage loss of the principal mode. quality factor corresponding to submodal anchorage loss The modal variability of the quality factor corresponding to anchorage loss is calculated according to the following formula: S65. Based on the shell thickness error, establish a thermoelastic damping finite element model coupling the structural field and the temperature field, and calculate the modal energy storage for the main mode and the sub-mode respectively. and single-cycle heat dissipation energy And calculate the quality factor corresponding to thermoelastic damping. : 。

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