Cradle turntable corner static stiffness model calculation and analysis method

Abstract

The invention provides a method for calculating and analyzing a corner static stiffness model of a cradle turntable, and relates to the technical field of precision machine tools. Firstly, the absolute linear displacement and the corner displacement of each key deformation position of a rotary table in the coordinate axis direction of a fixed coordinate system are assumed, and the absolute displacement difference of two parts in contact is the contact deformation of the parts; three-dimensional coordinate system transformation is adopted, force in a fixed coordinate system is transformed into a rotating coordinate system through a rotating matrix, and key parts such as a part combination surface and a rotating table bearing of the C shaft of the rotating table are analyzed under the rotating coordinate system; each roller of the bearing is sliced, and then a contact deformation model between the rollers and a raceway is established; the contact rigidity of the joint surface is calculated by adopting an MB parting principle aiming at analysis of the joint surface of the parts. Deformation among the workbench, the bearing, the swing frame, the mandrel, the bearing and the shell is comprehensively overlaid on the workbench surface, and the corner static rigidity of the cradle rotary table under the action of cutting force is constructed.

Description

Technical Field

[0001] This invention relates to the field of precision machine tool technology, and in particular to a method for calculating and analyzing the angular static stiffness model of a cradle turntable. Background Technology

[0002] The worktable of a precision CNC machine tool must withstand the combined effects of cutting force and gravity, which causes elastic deformation in key locations such as the rotary table bearings, the swing frame, and the joint surfaces where various components connect. The deformation at these locations will eventually be superimposed on the worktable, causing the position of the workpiece on the worktable in the coordinate system to change, directly affecting the machining accuracy.

[0003] YRT rotary table bearings are three-row roller bearings capable of withstanding combined loads. They are characterized by high precision, high reliability, and strong load-bearing capacity, and are widely used in rotating components of precision instruments such as machine tools and aerospace equipment. Under combined loads, the upper and lower rows of rollers bear axial loads, while the middle row bears radial loads. Using the Hertz contact principle, a contact deformation model between the rollers and raceways is established to determine the stress-deformation of the rotary table bearing. The contact surfaces between the various components of the rotary table are also key locations for elastic deformation. Given the geometric parameters, material properties, surface morphology, locking force, and spacing of the two contact surfaces, the normal and tangential static stiffness of the contact surfaces are determined. Then, based on the stress conditions, a stress-deformation model is established to calculate the displacements along each coordinate axis in the coordinate system.

[0004] Existing technologies suffer from the following shortcomings: a deformation transmission mechanism is lacking; the deformation transmission chain from the worktable to the bearing, swing frame, mandrel, bearing, and housing is not established, and the influence of coupled deformation between components is ignored; the tilting deviation of the rotating shaft after assembly and loading, and the resulting changes in the load direction and constraint reaction direction, are not considered, leading to a simplified treatment of the spatial force system equilibrium relationship; furthermore, due to the neglect of contact nonlinearity and multi-body coupled deformation, the evolution of angular displacement of the turntable during tilting and rotation cannot be accurately characterized. This results in inaccurate prediction of the cradle turntable's angular displacement and rotation accuracy, insufficient accuracy in calculating tilting error and torsional deformation, and difficulty in meeting the requirements for high-precision indexing and tilting machining. Summary of the Invention

[0005] To address the shortcomings of existing technologies, this invention provides a method for calculating and analyzing the angular static stiffness model of a cradle turntable; it solves the problems of unclear deformation transmission paths, inaccurate spatial force system coupling modeling, and inaccurate nonlinear contact models under complex loads.

[0006] To solve the above-mentioned technical problems, the technical solution adopted by the present invention is as follows:

[0007] A method for calculating and analyzing the angular static stiffness of a cradle turntable includes the following steps:

[0008] Step 1: Define the deformation location and fixed coordinate system;

[0009] The deformation locations specifically include: the mating surfaces of the contacting components, and the area between the rollers and raceways inside the turntable bearing;

[0010] The fixed coordinate system is specifically defined as follows: the intersection of the two rotation axes of the turntable is set as the origin of the fixed coordinate system, the rotation axis perpendicular to the worktable surface is set as the Z-axis, the rotation axis parallel to the worktable surface is set as the X-axis, and the axis perpendicular to the XOZ plane is set as the Y-axis.

[0011] Step 2: Calculate the linear and angular deformation at the deformation location, and transform the coordinate system for the deformation of the C-axis component;

[0012] Step 3: Solve for the nonlinear contact force and contact torque borne by the inner ring of the turntable bearing;

[0013] Step 3.1: Represent the deformation of each roller using the slicing method; divide each roller in each row into n equal slices, with each slice having a thickness of... l is the effective length of the roller;

[0014] Deformation of the upper rollers:

[0015] ;

[0016] Deformation of the lower rollers:

[0017] ;

[0018] Deformation of the middle row rollers:

[0019] ;

[0020] In the formula, the superscripts u, d, and m of the parameters indicate that the corresponding parameters belong to the upper row of rollers, the lower row of rollers, and the radial rollers, respectively; For axial deformation of the inner ring, For inner ring radial deformation; It is the pitch circle diameter of the bearing; It is the length of the bearing rollers; For the inner circle corner; For the first The position angle at each roller (Z is the number of rollers), at the point of maximum deformation. ; This is the initial axial clearance. This is the initial radial clearance; This represents the reduction in convexity at the j-th piece of the i-th roller.

[0021] Step 3.2: Calculate the normal load on each roller using the Hertz contact principle; the normal load at each slab roller in the upper row is:

[0022] ;

[0023] The normal load at each slice roller in the lower row is:

[0024] ;

[0025] The normal load at each slice roller in the middle row is:

[0026] ;

[0027] In the formula, Where is the effective diameter of the roller; k is the ratio of the effective diameter of the roller to the raceway diameter. The diameter of the raceway; Let be the thickness of each slice of the roller; then the normal load borne by the i-th roller in each row is:

[0028] ;

[0029] Step 3.3: Calculate the nonlinear contact force and contact torque of the inner ring of the YRT turntable bearing;

[0030] Specifically, the axial nonlinear contact force of the inner ring of the YRT turntable bearing is synthesized by the contact forces between each row of rollers and the raceway. Radial nonlinear contact force ; and the contact torque on the bearing coordinate axis. :

[0031] ;

[0032] In the formula, the superscripts u, d, and m of the parameters indicate that the corresponding parameters belong to the upper row of rollers, the lower row of rollers, and the radial rollers, respectively;

[0033] Step 4: Solve for the nonlinear contact force and contact torque between the mating surfaces of the turntable components;

[0034] Step 4.1: Calculate the parameters required for the MB fractal principle;

[0035] According to fractal theory, the cross-sectional area of ​​a microconvex body... The probability density function is:

[0036] ;

[0037] In the formula, The domain expansion factor of the cutoff area distribution is obtained through the transcendental equation:

[0038] ;

[0039] The maximum cross-sectional area of ​​the interfacial micro-convexity is calculated using the following formula:

[0040] ;

[0041] In the formula, erfc() represents the error complementarity function; This represents the actual contact area. Nominal contact area; The normal spacing between the mating surfaces is given; the initial spacing between the mating surfaces is given. If we define the motion of the two surfaces moving towards each other as positive, then... This represents the change in spacing. For integration variables; This represents the root mean square value of the surface profile of the mating surface.

[0042] Assuming the micro-protrusion shape of the surface profile is defined by the WM function, and assuming the surface roughness is statistically isotropic, the morphology of the contact surface of the micro-protrusion before deformation is defined as follows:

[0043] ;

[0044] In the formula, x is the contour displacement coordinate; L is the sampling length; G is the roughness amplitude; and D is the surface fractal dimension. For scale parameters; For fractal series index, Represents the order of the maximum fractal scale; Let be the random phase angle of the nth fractal component.

[0045] The critical cut-off area of ​​the micro-convexity is calculated by the following formula:

[0046] ;

[0047] In the formula, , H is the Poisson's ratio for the softer material, and H is the hardness of the softer material.

[0048] According to the fractal contact theory of the mating surface, the normal and tangential contact stiffness of a single micro-convex body on the mating surface are calculated as follows:

[0049] ;

[0050] In the formula , , , , These are the elastic modulus and Poisson's ratio of the two contacting materials, respectively. The equivalent shear modulus is given by where , The contact radius of a single micro-protrusion;

[0051] Step 4.2: When the parts have no angular deformation, calculate the contact stiffness of the plane according to the MB fractal principle;

[0052] From the formula The formula for normal contact stiffness is:

[0053] ;

[0054] Similarly, the formula for tangential contact stiffness is:

[0055] ;

[0056] In the formula, The maximum cross-sectional area of ​​the micro-convex body; The critical cut-off area of ​​the micro-convex body; , These are the normal contact stiffness and tangential contact stiffness of a single micro-protrusion on the mating surface, respectively.

[0057] The maximum cross-sectional area of ​​the micro-convex bodies at the two mating surfaces under normal preload can be obtained from the normal force formula. :

[0058] ;

[0059] Then calculate using the following formula:

[0060] ;

[0061] The conclusion is , The initial normal distance between the two mating surfaces under the action of normal preload force;

[0062] Under an applied load, the normal change in the distance between the two mating surfaces can be calculated as follows: Then the change in stiffness K with respect to the spacing is finally obtained. The formula:

[0063] ;

[0064] ;

[0065] In the formula, This represents the variation in the normal spacing between the mating surfaces; This represents the nominal contact area of ​​the planar mating surfaces; , For softer materials, Poisson's ratio is used.

[0066] Step 4.3: When the component has angular deformation, calculate the contact stiffness of the plane according to the MB fractal principle;

[0067] From the formula:

[0068] ;

[0069] The formula for normal contact stiffness is:

[0070] ;

[0071] Similarly, the formula for tangential contact stiffness is:

[0072] ;

[0073] Obtained from the formula for normal force :

[0074] ;

[0075] The maximum cross-sectional area of ​​the micro-protrusions on the two mating surfaces under normal preload is then calculated using the following formula:

[0076] ;

[0077] The conclusion is , which is the initial normal distance between the two mating surfaces under the action of normal preload.

[0078] The planar mating surface of the turntable is annular in shape, with an outer radius of R and an inner radius of r. Therefore, the contact area of ​​the planar mating surface is... The planar interface is divided into z parts along the perimeter and n parts along the radius. The area of ​​each infinitesimal interface element is... In the formula Let j be the j-th element along the radial direction, r be the inner radius, and R be the outer radius. Then, the deformation in the direction perpendicular to the force on each infinitesimal surface of the bonding plane is:

[0079] ;

[0080] This represents the linear deformation perpendicular to the plane under an applied load. Angular deformation of components ( These represent the angular deformations of the components around the X and Y axes in the coordinate system, respectively. This indicates the location of the area of ​​each micro-element.

[0081] Finally, the normal stiffness of each infinitesimal element of the planar interface is obtained. and tangential stiffness Regarding the variation in normal spacing The formula:

[0082] ;

[0083] ;

[0084] In the formula, The nominal contact area for each micro-element, This represents the nominal contact area of ​​the entire planar interface. , The first The maximum cross-sectional area and critical cross-sectional area of ​​a micro-convex body on a micro-element surface; For the first The variation in the normal spacing of each micro-element surface.

[0085] Step 4.4: Calculate the contact stiffness of the cylindrical surface based on the MB fractal principle;

[0086] The mating surface is divided into n infinitesimal mating surfaces at equal intervals along the circumference using a slicing method. The radial position angle of the k-th infinitesimal mating surface is... and contact area for:

[0087] ;

[0088] Where d is the diameter of the cylinder; l is the axial length of the joint surface.

[0089] Under an applied load, the normal change in the spacing between the k-th infinitesimal inter-element surfaces at the cylindrical surface joint is calculated as follows:

[0090] ;

[0091] in, and This indicates the deformation at the interface along the X and Y directions; Let be the radial position angle of the k-th infinitesimal interface.

[0092] Based on fractal contact theory, the initial normal spacing of the mating surfaces under preload is determined. The following equation can be used to obtain:

[0093] ;

[0094] in, The maximum cross-sectional area of ​​the micro-protrusion at the cylindrical mating surface under preload is calculated using the following formula:

[0095] ;

[0096] According to the fractal contact theory of the bonding surface, the first... The normal and tangential stiffness of the joint surfaces of each element are calculated as follows:

[0097] ;

[0098] In the formula, This represents the nominal contact area of ​​the cylindrical mating surfaces. Let be the nominal contact area of ​​the k-th infinitesimal interface; D and G are the fractal dimension and fractal roughness of the interface, respectively. ψ is the equivalent elastic modulus of the bonding surface; ψ is the domain expansion factor of the bonding surface. and Let be the maximum cross-sectional area and the critical cross-sectional area of ​​the k-th micro-element of the interfacial surface, respectively, calculated using the following formulas:

[0099] ;

[0100] In the formula, The initial normal spacing of the cylindrical mating surfaces is denoted as . Let be the change in the normal spacing of the k-th infinitesimal interface.

[0101] After obtaining the normal and tangential stiffness of all infinitesimal surfaces of the cylindrical segmented joint surface, the stiffness of all infinitesimal surfaces of the cylindrical joint surface at its axial midpoint in the three coordinate axes is calculated as follows:

[0102] ;

[0103] In the formula, , They are respectively the first of the mating surfaces The normal and tangential stiffness of the interface between individual elements;

[0104] Step 4.5: Calculate the nonlinear contact force and contact torque of a single turntable component based on the MB fractal principle.

[0105] Under the action of an external load, the normal and tangential contact forces of each infinitesimal surface of the planar mating surface are:

[0106] ;

[0107] In the formula, , Normal stiffness of each infinitesimal element of the planar interface and tangential stiffness ; , These represent the normal deformation and radial deformation of the planar interface, respectively.

[0108] Normal contact force F of the entire planar interface n Tangential contact force F t Contact torque about the coordinate axes of the mating surface coordinate system , for:

[0109] ;

[0110] In the formula, , These represent the normal and tangential contact forces of each infinitesimal surface of the planar interface.

[0111] For components whose contact surface is a combination of a plane and a cylinder, the nonlinear contact force along the three coordinate axes of the coordinate system is as follows:

[0112] ;

[0113] In the formula, , , Let be the stiffness of the cylindrical mating surface in the three directions of a fixed coordinate system; For the radial displacement Angle with the X-axis of the coordinate system:

[0114] ;

[0115] These represent the linear deformation of the component along the X, Y, and Z axes in the coordinate system.

[0116] The nonlinear contact torque of the part about the three coordinate axes in the coordinate system is:

[0117] ;

[0118] Step 5: Coordinate transformation of the applied load along the C-axis;

[0119] The turntable is considered to rotate only around the X-axis. The angle at which the turntable oscillates around the X-axis The position of the angle is determined by setting the intersection of the two rotation axes of the turntable as the origin of the rotation coordinate system. The rotation axis perpendicular to the worktable surface is defined as... The other axis of rotation is Axis, perpendicular to planar axis.

[0120] The rotation matrix for transforming from a fixed coordinate system to a rotating coordinate system about the X-axis is:

[0121] ;

[0122] The force generated by the tool in the fixed coordinate system is The workpiece's weight is In the rotated coordinate system:

[0123] ;

[0124] The forces in the three coordinate axes of the rotating coordinate system are obtained:

[0125] ;

[0126] These are the components of the external force along the three coordinate axes in the rotating coordinate system; These are the components of the workpiece's gravity along the three coordinate axes in the rotating coordinate system.

[0127] Step 6: Establish the overall equilibrium equation;

[0128] Change in the phase distance between the outer ring of the C-axis bearing and the planar infinitesimal interface of the pendulum frame The change in the normal distance between the outer ring of the C-axis bearing and the cylindrical micro-element mating surface of the pendulum frame. The outer ring of the C-axis bearing undergoes axial deformation relative to the swing frame. radial deformation C-axis bearing outer ring radial deformation With rotating coordinate system The included angle of the axis is Angular deformation of the bearing outer ring relative to the swing frame .in, These are the outer rings of the C-axis bearing in the rotating coordinate system. , , Straight line deformation in three directions; The position of each micro-element area of ​​the planar micro-element mating surface between the outer ring of the C-axis bearing and the swing frame. Let be the radial position angle of the mating surface between the outer ring of the C-axis bearing and the cylindrical micro-element of the swing frame. Rotate the coordinate system along... The bearing coordinate system is obtained by translating the axis downwards to the center of mass of the C-axis rotary table bearing. The three coordinate axes of the bearing coordinate system are as follows: , shaft and The axes are parallel. shaft and The axes are parallel. shaft and The axes coincide. Among them, Do not use the bearing coordinate system for the outer ring of the C-axis bearing. shaft and Angular deformation of the shaft.

[0129] Therefore, the outer ring of the C-axis bearing is in the rotating coordinate system , , The forces acting in the three directions are:

[0130] ;

[0131] These are the normal contact force and tangential contact force of the planar mating surface between the outer ring of the C-axis bearing and the swing frame, respectively. These represent the stiffness along the three axes of the rotating coordinate system of the cylindrical mating surface between the outer ring of the C-axis bearing and the swing frame.

[0132] C-axis bearing outer ring around bearing coordinate system , , The torques acting in the three directions are:

[0133] ;

[0134] These represent the normal contact forces of the planar mating surfaces between the outer ring of the C-axis bearing and the swing frame, relative to the bearing coordinate system axes. , The torque.

[0135] This refers to the axial deformation of the inner ring of the C-axis bearing. Radial deformation. The angular deformation of the bearing inner ring relative to the bearing outer ring. C-axis bearing inner ring radial deformation With rotating coordinate system The included angle of the axis is .in, These are the inner rings of the C-axis bearing in the rotating coordinate system. , , Linear deformation in three directions. Rotate the coordinate system along... The bearing coordinate system is obtained by translating the axis downwards to the center of mass of the C-axis rotary table bearing. The three coordinate axes of the bearing coordinate system are as follows: , shaft and The axes are parallel. shaft and The axes are parallel. shaft and The axes coincide. Among them, Do not refer to the bearing coordinate system for the inner ring of the C-axis bearing. shaft and Angular deformation of the shaft.

[0136] Therefore, the forces acting on the inner ring of the bearing along the three coordinate axes of the rotating coordinate system are:

[0137] ;

[0138] , These are the axial nonlinear contact force and radial nonlinear contact force of the inner ring of the C-axis YRT turntable bearing, which are the result of the contact forces between each row of rollers and raceways.

[0139] The torques of the bearing inner ring about the three coordinate axes in the translational coordinate system are:

[0140] ;

[0141] , The axial nonlinear contact forces between each row of rollers and raceway of the C-axis YRT rotary table bearing on the bearing coordinate system axes are respectively... , The torque.

[0142] Variation in the normal phase distance between the planar micro-element mating surface of the worktable and the bearing inner ring The change in the normal distance between the mating surfaces of the cylindrical micro-element of the worktable and the inner ring of the bearing. Axial deformation of the worktable relative to the bearing inner ring. radial deformation radial deformation of the worktable With rotating coordinate system The included angle of the axis is Angular deformation of the worktable relative to the inner ring of the bearing. .in, These are the worktable in the rotating coordinate system. , , Linear deformation in three directions; The position of each micro-element area of ​​the planar micro-element mating surface between the worktable and the bearing inner ring. θ represents the radial position angle of the cylindrical micro-element mating surface between the worktable and the bearing inner ring. The rotating coordinate system is positioned along... The worktable coordinate system is obtained by translating the axis downwards to the center of mass of the worktable. The three coordinate axes of the worktable coordinate system are as follows: , shaft and The axes are parallel. shaft and The axes are parallel. shaft and The axes coincide. Among them, Do not revolve the worktable around the worktable coordinate system. shaft and Angular deformation of the shaft.

[0143] Therefore, the forces acting on the worktable along the coordinate axes of the rotating coordinate system are:

[0144] ;

[0145] The normal and tangential contact forces at the planar mating surfaces between the worktable and the bearing inner ring; , , Stiffness in the three coordinate axes of the rotating coordinate system of the cylindrical mating surface between the split platform and the inner ring of the bearing.

[0146] The torque exerted on the worktable in the translation coordinate system about the coordinate axes is:

[0147] ;

[0148] The normal contact force between the worktable and the mating surface of the bearing inner ring plane is relative to the coordinate axes of the worktable coordinate system. , The torque.

[0149] The cutting force and gravity in the fixed coordinate system are transformed into the rotating coordinate system for force analysis, and balanced with the forces acting on the C-axis components:

[0150] ;

[0151] Torque balance is:

[0152] ;

[0153] Step 7: Calculate the linear and angular deformation of the turntable and calculate the static stiffness of the turntable in the three coordinate axes of the fixed coordinate system.

[0154] Based on the known applied load and the calculated angular deformation displacement of the turntable, the angular static stiffness of the turntable in the three coordinate axes of the worktable coordinate system can be calculated as follows:

[0155] ;

[0156] , respectively force For the worktable coordinate system , Torque of the coordinate axes.

[0157] The beneficial effects of adopting the above technical solution are as follows:

[0158] This invention provides a method for calculating and analyzing the angular static stiffness model of a cradle turntable. The comprehensive deformation modeling provided by this invention innovatively combines and superimposes the deformations of the mating surfaces between various components of the cradle turntable and key locations such as the turntable bearings: Step 3 uses a slicing method to more accurately determine the contact force between the rollers and raceways of the turntable bearings for nonlinear contact; Step 4 utilizes and improves the MB fractal principle to determine the contact stiffness between planes and between cylindrical surfaces, and thus the contact force for both contact types, for cases where a single component has both planar and cylindrical contacts with other components; Step 5 considers that the turntable rotates around the X-axis during operation, which complicates the stress and deformation analysis of the C-axis, so a coordinate system transformation method is used to convert the external load to a rotating coordinate system for calculation and analysis. Compared with traditional methods for calculating the angular static stiffness of cradle turntables, the breakthrough of this invention lies in: comprehensively considering the deformation transmission chain of the worktable-bearing-swing frame-spindle-bearing-housing; employing the Hertz slicing method to improve the calculation accuracy of the contact force between the roller raceways of the turntable bearings; and unifying the force balance between the fixed and rotating systems through a rotation matrix to solve the spatial attitude coupling problem. This method predicts the deformation of the turntable during machine tool operation, providing deformation parameters for subsequent compensation of positioning errors using control algorithms; and simultaneously calculates the angular static stiffness of the cradle turntable. Attached Figure Description

[0159] Figure 1 A schematic diagram showing the key deformation positions and numbers of the cradle turntable provided in an embodiment of the present invention;

[0160] Figure 2 This is a schematic diagram of the position of the rotary coordinate system and translation coordinate system of the turntable provided in an embodiment of the present invention;

[0161] Figure 3 This is a simplified structural diagram of the YRT turntable bearing provided in an embodiment of the present invention;

[0162] Figure 4 A schematic diagram of the area element division and position angle of the planar interface provided in this embodiment of the invention:

[0163] Figure 5 This is a schematic diagram showing the area micro-element division and position angle of the cylindrical mating surface provided in an embodiment of the present invention. Detailed Implementation

[0164] The specific implementation methods of this application will be further described in detail below with reference to the accompanying drawings and embodiments.

[0165] Example 1

[0166] A method for calculating and analyzing the angular static stiffness of a cradle turntable includes the following steps:

[0167] Step 1: Define the deformation location and fixed coordinate system;

[0168] The deformation locations specifically include: under the action of external loads, the worktable surface of the machine tool rotary table will experience linear and angular displacements. The displacement on the worktable surface is mainly due to the cumulative deformation of key locations inside the rotary table. The key locations where deformation occurs inside the rotary table are mainly at the mating surfaces of the contacting components (worktable and C-axis rotary table bearing, C-axis rotary table bearing and swing frame), and at the rollers and raceways inside the rotary table bearing (C-axis rotary table bearing).

[0169] The fixed coordinate system is specifically defined as follows: the intersection of the two rotation axes of the turntable is set as the origin of the fixed coordinate system, the rotation axis perpendicular to the worktable surface is set as the Z-axis, the rotation axis parallel to the worktable surface is set as the X-axis, and the axis perpendicular to the XOZ plane is set as the Y-axis.

[0170] In this embodiment, the fixed coordinate system of the turntable is as follows: Figure 1 As shown; when an external load is applied to the workpiece on the worktable, the force is transmitted to Figure 1 Positions ① to ④ cause linear and angular deformation of the parts at those locations. These deformations are ultimately superimposed on the worktable surface, resulting in errors.

[0171] Therefore, assuming the displacement from position ① to ③ along the three coordinate axes in the fixed coordinate system is... The displacements from ④ to ⑦ along the three coordinate axes in the fixed coordinate system are: ,and Figure 1 The displacement of the position corresponding to the marked position on the right is expressed as .

[0172] The displacement of the left-hand label position in the X, Y, and Z axes:

[0173] The outer ring of the bearing on the left side of axis A: ;

[0174] Inner ring of bearing on the left side of axis A: ;

[0175] Left spindle: ;

[0176] Left side shelf: .

[0177] The corresponding position on the right is located in the X, Y, and Z axes:

[0178] The outer ring of the bearing on the right side of axis A: ;

[0179] Inner ring of bearing on the right side of axis A: ;

[0180] Right side spindle: ;

[0181] Right side display shelf: ;

[0182] The C-axis label indicates linear displacement in the X, Y, and Z axes:

[0183] Bearing outer ring: ;

[0184] Inner ring of bearing: ;

[0185] Workbench: .

[0186] Angular displacement around the X, Y, and Z axes at the position marked on the C-axis:

[0187] Bearing outer ring: ;

[0188] Inner ring of bearing: ;

[0189] Workbench: ;

[0190] Step 2: Calculate the linear and angular deformation at the deformation location, and transform the coordinate system for the deformation of the C-axis component;

[0191] Step 3: Solve for the nonlinear contact force and contact torque borne by the inner ring of the turntable bearing;

[0192] Step 3.1: Use the slicing method to represent the deformation of each roller;

[0193] Since the contact between the rollers and raceways in YRT rotary table bearings is not an ideal linear contact, the slice method is used to calculate the contact force between the rollers and raceways, which yields a more accurate numerical solution. Each roller in each row is divided into n equal slices, and the thickness of each slice is... , where l is the effective length of the roller. For example... Figure 3 The simplified diagram of the YRT turntable bearing shown has three rows of rollers: upper, middle, and lower.

[0194] Step 3.2: Calculate the normal load on each roller using the Hertz contact principle.

[0195] Step 3.3: Calculate the nonlinear contact force and contact torque of the inner ring of the YRT turntable bearing;

[0196] Step 4: Solve for the nonlinear contact force and contact torque between the mating surfaces of the turntable components;

[0197] Step 4.1: Calculate the parameters required for the MB fractal principle;

[0198] The MB fractal model is an effective model for analyzing the micro-topographic features of contact surfaces. Assuming the shape of the micro-protrusions in the surface profile is defined by the WM function, and assuming the surface roughness is statistically isotropic, the contact surface morphology before deformation is defined as follows:

[0199] ;

[0200] In the formula, x is the contour displacement coordinate; L is the sampling length; G is the roughness amplitude; and D is the surface fractal dimension. For scale parameters; For fractal series index, Represents the order of the maximum fractal scale; is the random phase angle of the nth fractal component, usually taken as 1.5.

[0201] The mating surface is equivalent to an ideal plane and a rough plane, and the contact between the two surfaces is the interaction of micro-protrusions on the rough surface. When the cross-sectional area of ​​the micro-protrusions exceeds the critical cross-sectional area, the contact deformation between the micro-protrusions and the rigid plane is elastic deformation; when the cross-sectional area of ​​the micro-protrusions is less than the critical cross-sectional area, the contact deformation between the micro-protrusions and the rigid plane is plastic deformation.

[0202] Step 4.2: When the parts have no angular deformation, calculate the contact stiffness of the plane according to the MB fractal principle;

[0203] Step 4.3: When the component has angular deformation, calculate the contact stiffness of the plane according to the MB fractal principle;

[0204] like Figure 4 As shown, the planar mating surface of the turntable is annular in shape, with an outer radius of R and an inner radius of r. Therefore, the contact area of ​​the planar mating surface is... The planar interface is divided into z parts along the perimeter and n parts along the radius. The area of ​​each infinitesimal interface element is... In the formula Let j be the j-th element along the radial direction, r be the inner radius, and R be the outer radius. Then, the deformation in the direction perpendicular to the force on each infinitesimal surface of the bonding plane is:

[0205] ;

[0206] This represents the linear deformation perpendicular to the plane under an applied load. Angular deformation of components ( These represent the angular deformations of the components around the X and Y axes in the coordinate system, respectively. This indicates the location of the area of ​​each micro-element.

[0207] Finally, the normal stiffness of each infinitesimal element of the planar interface is obtained. and tangential stiffness Regarding the variation in normal spacing The formula:

[0208] ;

[0209] ;

[0210] In the formula, The nominal contact area for each micro-element, This represents the nominal contact area of ​​the entire planar interface. , The first The maximum cross-sectional area and critical cross-sectional area of ​​a micro-convex body on a micro-element surface; For the first The variation in the normal spacing of each micro-element surface.

[0211] Step 4.4: Calculate the contact stiffness of the cylindrical surface based on the MB fractal principle;

[0212] Step 4.5: Calculate the nonlinear contact force and contact torque of a single turntable component based on the MB fractal principle.

[0213] Step 5: Coordinate transformation of the applied load along the C-axis;

[0214] The force analysis of the C-axis in a fixed coordinate system is relatively complex. Therefore, a coordinate transformation is performed before analyzing the force on the C-axis rotary table bearing, as follows:

[0215] The turntable is considered to rotate only around the X-axis. The angle at which the turntable oscillates around the X-axis The position of the angle is determined by setting the intersection of the two rotation axes of the turntable as the origin of the rotation coordinate system. The rotation axis perpendicular to the worktable surface is defined as... The other axis of rotation is Axis, perpendicular to planar axis.

[0216] The rotation matrix for transforming from a fixed coordinate system to a rotating coordinate system about the X-axis is:

[0217] ;

[0218] The force generated by the tool in the fixed coordinate system is The workpiece's weight is In the rotated coordinate system:

[0219] ;

[0220] The forces in the three coordinate axes of the rotating coordinate system are obtained:

[0221] ;

[0222] These are the components of the external force along the three coordinate axes in the rotating coordinate system; These are the components of the workpiece's gravity along the three coordinate axes in the rotating coordinate system.

[0223] Step 6: Establish the overall equilibrium equation;

[0224] For C-axis components, the linear deformation in steps 3 and 4 above refers to the linear deformation in the three coordinate axes of the rotating coordinate system, and the angular deformation refers to the angular deformation in the three coordinate axes of the translational coordinate system. The calculated contact forces are also in the coordinate axes of the rotating coordinate system.

[0225] In this embodiment, as shown... Figure 1 As shown, the forces at position ③ are as follows:

[0226] In step 4; the change in the normal phase distance between the planar infinitesimal interface between the outer ring of the C-axis bearing and the pendulum frame. The change in the normal distance between the outer ring of the C-axis bearing and the cylindrical micro-element mating surface of the pendulum frame. The outer ring of the C-axis bearing undergoes axial deformation relative to the swing frame. radial deformation C-axis bearing outer ring radial deformation With rotating coordinate system The included angle of the axis is Angular deformation of the bearing outer ring relative to the swing frame .in, These are the outer rings of the C-axis bearing in the rotating coordinate system. , , Straight line deformation in three directions; The position of each micro-element area of ​​the planar micro-element mating surface between the outer ring of the C-axis bearing and the swing frame. Let be the radial position angle of the mating surface between the outer ring of the C-axis bearing and the cylindrical micro-element of the swing frame. Rotate the coordinate system along... The bearing coordinate system is obtained by translating the axis downwards to the center of mass of the C-axis rotary table bearing. The three coordinate axes of the bearing coordinate system are as follows: , shaft and The axes are parallel. shaft and The axes are parallel. shaft and The axes coincide. Among them, Do not use the bearing coordinate system for the outer ring of the C-axis bearing. shaft and Angular deformation of the shaft.

[0227] Therefore, the outer ring of the C-axis bearing is in the rotating coordinate system , , The forces acting in the three directions are:

[0228] ;

[0229] These are the normal contact force and tangential contact force of the planar mating surface between the outer ring of the C-axis bearing and the swing frame, respectively. These represent the stiffness along the three axes of the rotating coordinate system of the cylindrical mating surface between the outer ring of the C-axis bearing and the swing frame.

[0230] C-axis bearing outer ring around bearing coordinate system , , The torques acting in the three directions are:

[0231] ;

[0232] These represent the normal contact forces of the planar mating surfaces between the outer ring of the C-axis bearing and the swing frame, relative to the bearing coordinate system axes. , The torque.

[0233] In this embodiment, as shown... Figure 1 As shown, the forces at position ② are as follows:

[0234] In step 3 This refers to the axial deformation of the inner ring of the C-axis bearing. Radial deformation. The angular deformation of the bearing inner ring relative to the bearing outer ring. C-axis bearing inner ring radial deformation With rotating coordinate system The included angle of the axis is .in, These are the inner rings of the C-axis bearing in the rotating coordinate system. , , Linear deformation in three directions. Rotate the coordinate system along... The bearing coordinate system is obtained by translating the axis downwards to the center of mass of the C-axis rotary table bearing. The three coordinate axes of the bearing coordinate system are as follows: , shaft and The axes are parallel. shaft and The axes are parallel. shaft and The axes coincide. Among them, Do not refer to the bearing coordinate system for the inner ring of the C-axis bearing. shaft and Angular deformation of the shaft.

[0235] Therefore, the forces acting on the inner ring of the bearing along the three coordinate axes of the rotating coordinate system are:

[0236] ;

[0237] , These are the axial nonlinear contact force and radial nonlinear contact force of the inner ring of the C-axis YRT turntable bearing, which are the result of the contact forces between each row of rollers and raceways.

[0238] The torques of the bearing inner ring about the three coordinate axes in the translational coordinate system are:

[0239] ;

[0240] , The axial nonlinear contact forces between each row of rollers and raceway of the C-axis YRT rotary table bearing on the bearing coordinate system axes are respectively... , The torque.

[0241] In this embodiment, as shown... Figure 1 As shown, the forces at position ① are as follows:

[0242] The change in the normal phase distance between the planar micro-element mating surface of the worktable and the inner ring of the bearing in step 4. The change in the normal distance between the mating surfaces of the cylindrical micro-element of the worktable and the inner ring of the bearing. Axial deformation of the worktable relative to the bearing inner ring. radial deformation = radial deformation of the worktable With rotating coordinate system The included angle of the axis is Angular deformation of the worktable relative to the inner ring of the bearing. .in, These are the worktable in the rotating coordinate system. , , Linear deformation in three directions; The position of each micro-element area of ​​the planar micro-element mating surface between the worktable and the bearing inner ring. θ represents the radial position angle of the cylindrical micro-element mating surface between the worktable and the bearing inner ring. The rotating coordinate system is positioned along... The worktable coordinate system is obtained by translating the axis downwards to the center of mass of the worktable. The three coordinate axes of the worktable coordinate system are as follows: , shaft and The axes are parallel. shaft and The axes are parallel. shaft and The axes coincide. Among them, Do not revolve the worktable around the worktable coordinate system. shaft and Angular deformation of the shaft.

[0243] Therefore, the forces acting on the worktable along the coordinate axes of the rotating coordinate system are:

[0244] ;

[0245] The normal contact force and tangential contact force of the planar mating surface between the worktable and the bearing inner ring; , , Stiffness in the three coordinate axes of the rotating coordinate system of the cylindrical mating surface between the split platform and the inner ring of the bearing.

[0246] The torque exerted on the worktable in the translation coordinate system about the coordinate axes is:

[0247] ;

[0248] The normal contact force between the worktable and the mating surface of the bearing inner ring plane is relative to the coordinate axes of the worktable coordinate system. , The torque.

[0249] In step 5, the cutting force and gravity in the fixed coordinate system are transformed into the rotating coordinate system for force analysis, and balanced with the forces acting on the components at positions ①, ②, and ③ along the C-axis:

[0250] ;

[0251] Torque balance is:

[0252] ;

[0253] Step 7: Calculate the linear and angular deformation of the turntable and calculate the static stiffness of the turntable in the three coordinate axes of the fixed coordinate system.

[0254] The coordinate transformation in step 5 and the overall equilibrium equation established in step 6, under the condition of known cutting force and gravity, use the Newton-Raphson iterative method to calculate the angular displacement caused by the deformation of the turntable in the fixed coordinate system. Based on the known external load and the obtained angular deformation displacement of the turntable, the static stiffness of the turntable in the three coordinate axes of the worktable coordinate system can be calculated as follows:

[0255] ;

[0256] , respectively force For the worktable coordinate system , Torque of the coordinate axes.

[0257] Example 2:

[0258] In this embodiment, step 2 specifically includes the following steps:

[0259] Step 2: Deformation of C-axis components;

[0260] The force-deformation analysis of the C-axis in a fixed coordinate system is quite complex. Therefore, before performing the force analysis on the C-axis components, it is necessary to perform a coordinate transformation on the deformation of the C-axis components. In this example, the turntable can be considered to rotate only around the X-axis. The turntable is oscillating around the X-axis. The position of the angle is determined by setting the intersection of the two rotation axes of the turntable as the origin of the rotation coordinate system. The rotation axis perpendicular to the worktable surface is defined as... The other axis of rotation is Axis, perpendicular to planar axis.

[0261] The rotation matrix for transforming from a fixed coordinate system to a rotating coordinate system about the X-axis is:

[0262] ;

[0263] The straight line deformation of the outer ring in the fixed coordinate system X, Y, Z directions is as follows:

[0264] ;

[0265] Written in vector form:

[0266] ;

[0267] In the formula, , , These represent the linear displacements of the outer ring of the C-axis bearing in the X, Y, and Z directions of the fixed coordinate system.

[0268] The angular deformation of the outer ring of the C-axis bearing in the three directions of X, Y, and Z around a fixed coordinate system is as follows:

[0269] ;

[0270] Vector form:

[0271] ;

[0272] In the formula, These represent the angular displacements of the outer ring of the C-axis bearing around the fixed coordinate system's X, Y, and Z axes.

[0273] In the transformed coordinate system, the outer ring of the C-axis bearing is on the rotating coordinate axis. , , The deformation in three directions is as follows:

[0274] ;

[0275] In the formula, These are the outer rings of the C-axis bearing in the rotating coordinate system. , , Straight line deformation in three directions Do not use the C-axis bearing outer ring rotating around a coordinate system. , , Angular deformation of the three axes.

[0276] The linear deformation of the bearing inner ring in the X, Y, and Z directions of a fixed coordinate system is as follows:

[0277] ;

[0278] Written in vector form:

[0279] ;

[0280] In the formula, , , These represent the linear displacements of the inner ring of the C-axis bearing in the X, Y, and Z directions of the fixed coordinate system.

[0281] The angular deformation of the inner ring of the C-axis bearing in the X, Y, and Z directions of the fixed coordinate system is as follows:

[0282] ;

[0283] Vector form:

[0284] ;

[0285] In the formula, These represent the angular displacements of the inner ring of the C-axis bearing around the fixed coordinate system's X, Y, and Z axes.

[0286] The inner ring of the C-axis bearing after coordinate system transformation in the rotating coordinate system , , The deformations in the three directions are as follows:

[0287] ;

[0288] In the formula, These are the inner rings of the C-axis bearing in the rotating coordinate system. , , Straight line deformation in three directions Do not use the C-axis bearing inner ring rotating around a coordinate system. , , Angular deformation of the three axes.

[0289] The linear deformation of the worktable in the fixed coordinate axes X, Y, and Z directions is as follows:

[0290] ;

[0291] Written in vector form:

[0292] ;

[0293] In the formula, , , These represent the linear displacements of the worktable in the X, Y, and Z directions of the fixed coordinate system.

[0294] The angular deformation of the C-axis worktable in the fixed coordinate axes X, Y, and Z directions is as follows:

[0295] ;

[0296] Vector form:

[0297] ;

[0298] In the formula, These represent the angular displacements of the worktable around the X, Y, and Z axes of the fixed coordinate system.

[0299] The worktable after coordinate system transformation in the rotating coordinate system , , The deformation in three directions is as follows:

[0300] ;

[0301] In the formula, These are the worktable in the rotating coordinate system. , , Straight line deformation in three directions These are the coordinates of the worktable around the rotating coordinate system. , , Angular deformation of the three axes.

[0302] in , It is the torque of the external load on the C-axis.

[0303] Example 3

[0304] This embodiment innovatively combines the deformation of the mating surfaces between various components of the cradle turntable and the deformation of key locations such as the turntable bearings into a comprehensive superposition: Step 3 uses a slicing method to more accurately calculate the contact force between the rollers and raceways of the turntable bearings, addressing the nonlinear contact. Step 4 utilizes and improves the MB fractal principle to calculate the contact stiffness between planes and between cylinders, thus obtaining the contact force and contact torque for both contact types, considering that the turntable rotates around the X-axis during operation, which complicates the stress deformation analysis of the C-axis. Therefore, a coordinate system transformation method is used to convert the applied load to a rotating coordinate system for calculation and analysis. Compared with the traditional turntable deformation error calculation, the breakthrough of the algorithm in this embodiment is reflected in: comprehensively considering the deformation transmission chain of the worktable-bearing-swing frame-spindle-bearing-shell; using the Hertz slicing method to improve the calculation accuracy of the contact force between the rollers and raceways of the turntable bearings; and unifying the force balance between the fixed and rotating systems through a rotation matrix to solve the spatial attitude coupling problem. This method is applied to predict the deformation of the turntable of a machine tool during operation, providing deformation parameters for subsequent compensation of positioning errors using control algorithms; at the same time, the angular static stiffness of the cradle turntable is calculated.

[0305] The various embodiments in this application are described in a progressive manner. The same or similar parts between the various embodiments can be referred to each other. Each embodiment focuses on describing the differences from other embodiments.

[0306] The scope of protection of this application is not limited to the embodiments described above. Obviously, those skilled in the art can make various modifications and variations to this disclosure without departing from its scope and spirit. If such modifications and variations fall within the scope of the claims of this disclosure and their equivalents, then the intent of this disclosure also includes these modifications and variations. Figure 5 As shown.

Claims

1. A method for calculating and analyzing the angular static stiffness of a cradle turntable, characterized in that, Includes the following steps: Step 1: Define the deformation location and fixed coordinate system; Step 2: Calculate the linear and angular deformation at the deformation location, and transform the coordinate system for the deformation of the C-axis component; Step 3: Solve for the nonlinear contact force and contact torque borne by the inner ring of the turntable bearing; Step 4: Solve for the nonlinear contact force and contact torque between the mating surfaces of the turntable components; Step 5: Coordinate transformation of the applied load on the C-axis; Step 6: Establish the overall equilibrium equation; Step 7: Calculate the linear and angular deformation of the turntable and calculate the static stiffness of the turntable in the three coordinate axes of the fixed coordinate system.

2. The method for calculating and analyzing the angular static stiffness model of a cradle turntable according to claim 1, characterized in that, The deformation locations mentioned in step 1 specifically include: the mating surfaces of the contacting parts and the rollers and raceways inside the turntable bearing; The fixed coordinate system is specifically defined as follows: the intersection of the two rotation axes of the turntable is set as the origin of the fixed coordinate system, the rotation axis perpendicular to the worktable surface is set as the Z-axis, the rotation axis parallel to the worktable surface is set as the X-axis, and the axis perpendicular to the XOZ plane is set as the Y-axis.

3. The method for calculating and analyzing the angular static stiffness model of a cradle turntable according to claim 2, characterized in that, Step 3 specifically includes the following steps: Step 3.1: Represent the deformation of each roller using the slicing method; divide each roller in each row into n equal slices, with each slice having a thickness of... l is the effective length of the roller; Deformation of the upper rollers: ； Deformation of the lower rollers: ； Deformation of the middle row rollers: ； In the formula, the superscripts u, d, and m of the parameters indicate that the corresponding parameters belong to the upper row of rollers, the lower row of rollers, and the radial rollers, respectively; For axial deformation of the inner ring, For inner ring radial deformation; It is the pitch circle diameter of the bearing; It is the length of the bearing rollers; For the inner circle corner; For the first The position angle at each roller (Z is the number of rollers), at the point of maximum deformation. ; This is the initial axial clearance. This is the initial radial clearance; This represents the reduction in convexity at the j-th piece of the i-th roller. Step 3.2: Calculate the normal load on each roller using the Hertz contact principle; the normal load at each slab roller in the upper row is: ； The normal load at each slice roller in the lower row is: ； The normal load at each slice roller in the middle row is: ； In the formula, Where is the effective diameter of the roller; k is the ratio of the effective diameter of the roller to the raceway diameter. The diameter of the raceway; Let be the thickness of each slice of the roller; then the normal load borne by the i-th roller in each row is: ； Step 3.3: Calculate the nonlinear contact force and contact torque of the inner ring of the YRT turntable bearing; Specifically, the axial nonlinear contact force of the inner ring of the YRT turntable bearing is synthesized by the contact forces between each row of rollers and the raceway. Radial nonlinear contact force ; and the contact torque on the bearing coordinate axis. : ； In the formula, the superscripts u, d, and m of the parameters indicate that the corresponding parameters belong to the upper row of rollers, the lower row of rollers, and the radial rollers, respectively.

4. The method for calculating and analyzing the angular static stiffness model of a cradle turntable according to claim 3, characterized in that, Step 4 specifically includes the following steps: Step 4.1: Calculate the parameters required for the MB fractal principle; According to fractal theory, the cross-sectional area of ​​a microconvex body... The probability density function is: ； In the formula, The domain expansion factor of the cutoff area distribution is obtained through the transcendental equation: ； The maximum cross-sectional area of ​​the interfacial micro-convexity is calculated using the following formula: ； In the formula, erfc() represents the error complementarity function; This represents the actual contact area. Nominal contact area; The normal spacing between the mating surfaces is given; the initial spacing between the mating surfaces is given. If we define the motion of the two surfaces moving towards each other as positive, then... This represents the change in spacing. For integration variables; This represents the root mean square value of the surface profile of the mating surface; Assuming the micro-protrusion shape of the surface profile is defined by the WM function, and assuming the surface roughness is statistically isotropic, the morphology of the contact surface of the micro-protrusion before deformation is defined as follows: ； In the formula, x is the contour displacement coordinate; L is the sampling length; G is the roughness amplitude; and D is the surface fractal dimension. For scale parameters; For fractal series index, Represents the order of the maximum fractal scale; The random phase angle of the nth fractal component; The critical cut-off area of ​​the micro-convexity is calculated by the following formula: ； In the formula, , is the Poisson's ratio of the softer material, and H is the hardness of the softer material; According to the fractal contact theory of the mating surface, the normal and tangential contact stiffness of a single micro-convex body on the mating surface are calculated as follows: ； In the formula , , , , These are the elastic modulus and Poisson's ratio of the two contacting materials, respectively. The equivalent shear modulus is given by where , The contact radius of a single micro-protrusion; Step 4.2: When the parts have no angular deformation, calculate the contact stiffness of the plane according to the MB fractal principle; From the formula The formula for normal contact stiffness is: ； Similarly, the formula for tangential contact stiffness is: ； In the formula, The maximum cross-sectional area of ​​the micro-convex body; The critical cut-off area of ​​the micro-convex body; , These are the normal contact stiffness and tangential contact stiffness of a single micro-protrusion on the mating surface, respectively. The maximum cross-sectional area of ​​the micro-convex bodies at the two mating surfaces under normal preload can be obtained from the normal force formula. : ； Then calculate using the following formula: ； The conclusion is , The initial normal distance between the two mating surfaces under the action of normal preload force; Under an applied load, the normal change in the distance between the two mating surfaces can be calculated as follows: Then the change in stiffness K with respect to the spacing is finally obtained. The formula: ； ； In the formula, This represents the variation in the normal spacing between the mating surfaces; This represents the nominal contact area of ​​the planar mating surfaces; , For softer materials, Poisson's ratio is used. Step 4.3: When the component has angular deformation, calculate the contact stiffness of the plane according to the MB fractal principle; From the formula: ； The formula for normal contact stiffness is: ； Similarly, the formula for tangential contact stiffness is: ； Obtained from the formula for normal force : ； The maximum cross-sectional area of ​​the micro-protrusions on the two mating surfaces under normal preload is then calculated using the following formula: ； The conclusion is , is the initial normal distance between the two mating surfaces under the action of normal force preload; The planar mating surface of the turntable is annular in shape, with an outer radius of R and an inner radius of r. Therefore, the contact area of ​​the planar mating surface is... The planar interface is divided into z parts along the perimeter and n parts along the radius. The area of ​​each infinitesimal interface element is... In the formula Let j be the j-th element along the radial direction, r be the inner radius, and R be the outer radius; then the deformation in the perpendicular force direction of each infinitesimal element of the bonding plane is: ； This represents the linear deformation perpendicular to the plane under an applied load. Angular deformation of components ( These represent the angular deformations of the components around the X and Y axes in the coordinate system, respectively. Indicates the location of the area of ​​each infinitesimal element; Finally, the normal stiffness of each infinitesimal element of the planar interface is obtained. and tangential stiffness Regarding the variation in normal spacing The formula: ； ； In the formula, The nominal contact area for each micro-element, This represents the nominal contact area of ​​the entire planar interface. , The first The maximum cross-sectional area and critical cross-sectional area of ​​a micro-convex body on a micro-element surface; For the first The variation in the normal spacing of each infinitesimal surface element; Step 4.4: Calculate the contact stiffness of the cylindrical surface based on the MB fractal principle; The mating surface is divided into n infinitesimal mating surfaces at equal intervals along the circumference using a slicing method. The radial position angle of the k-th infinitesimal mating surface is... and contact area for: ； Where d is the diameter of the cylinder; l is the axial length of the joint surface; Under an applied load, the normal change in the spacing between the k-th infinitesimal inter-element surfaces at the cylindrical surface joint is calculated as follows: ； in, and This indicates the deformation at the interface along the X and Y directions; Let be the radial position angle of the k-th infinitesimal element interface; Based on fractal contact theory, the initial normal spacing of the mating surfaces under preload is determined. The following equation can be used to obtain: ； in, The maximum cross-sectional area of ​​the micro-protrusion at the cylindrical mating surface under preload is calculated using the following formula: ； According to the fractal contact theory of the bonding surface, the first... The normal and tangential stiffness of the joint surfaces of each element are calculated as follows: ； In the formula, This represents the nominal contact area of ​​the cylindrical mating surfaces. Let be the nominal contact area of ​​the k-th infinitesimal interface; D and G are the fractal dimension and fractal roughness of the interface, respectively. ψ is the equivalent elastic modulus of the bonding surface; ψ is the domain expansion factor of the bonding surface. and Let be the maximum cross-sectional area and the critical cross-sectional area of ​​the k-th micro-element of the interfacial surface, respectively, calculated using the following formulas: ； In the formula, The initial normal spacing of the cylindrical mating surfaces is denoted as . Let be the change in the normal spacing of the k-th infinitesimal interface. After obtaining the normal and tangential stiffness of all infinitesimal surfaces of the cylindrical segmented joint surface, the stiffness of all infinitesimal surfaces of the cylindrical joint surface at its axial midpoint in the three coordinate axes is calculated as follows: ； In the formula, , They are respectively the first of the mating surfaces The normal and tangential stiffness of the interface between individual elements; Step 4.5: Calculate the nonlinear contact force and contact torque on a single turntable component based on the MB fractal principle; Under the action of an external load, the normal and tangential contact forces of each infinitesimal surface of the planar mating surface are: ； In the formula, , Normal stiffness of each infinitesimal element of the planar interface and tangential stiffness ; , These represent the normal deformation and radial deformation of the planar interface, respectively. Normal contact force F of the entire planar interface n Tangential contact force F t Contact torque about the coordinate axes of the mating surface coordinate system , for: ； In the formula, , These represent the normal and tangential contact forces of each infinitesimal surface of the planar interface; For components whose contact surface is a combination of a plane and a cylinder, the nonlinear contact force along the three coordinate axes of the coordinate system is as follows: ； In the formula, , , Let be the stiffness of the cylindrical mating surface in the three directions of a fixed coordinate system; For the radial displacement Angle with the X-axis of the coordinate system: ； These represent the linear deformation of the components along the X, Y, and Z axes in the coordinate system. The nonlinear contact torque of the part about the three coordinate axes in the coordinate system is: 。 5. The method for calculating and analyzing the angular static stiffness model of a cradle turntable according to claim 4, characterized in that, Step 5 specifically involves: treating the turntable as rotating only around the X-axis. The angle at which the turntable oscillates around the X-axis The position of the angle is determined by setting the intersection of the two rotation axes of the turntable as the origin of the rotation coordinate system. The rotation axis perpendicular to the worktable surface is defined as... The other axis of rotation is Axis, perpendicular to planar axis; The rotation matrix for transforming from a fixed coordinate system to a rotating coordinate system about the X-axis is: ； The force generated by the tool in the fixed coordinate system is The workpiece's weight is In the rotated coordinate system: ； The forces in the three coordinate axes of the rotating coordinate system are obtained: ； These are the components of the external force along the three coordinate axes in the rotating coordinate system; These are the components of the workpiece's gravity along the three coordinate axes in the rotating coordinate system.

6. The method for calculating and analyzing the angular static stiffness model of a cradle turntable according to claim 5, characterized in that, Step 6 specifically involves: the change in the normal phase distance between the outer ring of the C-axis bearing and the planar micro-element mating surface of the swing frame. The change in the normal distance between the outer ring of the C-axis bearing and the cylindrical micro-element mating surface of the pendulum frame. ; Axial deformation of the outer ring of the C-axis bearing relative to the swing frame radial deformation C-axis bearing outer ring radial deformation With rotating coordinate system The included angle of the axis is Angular deformation of the bearing outer ring relative to the swing frame ;in, These are the outer rings of the C-axis bearing in the rotating coordinate system. , , Straight line deformation in three directions; The position of each micro-element area of ​​the planar micro-element mating surface between the outer ring of the C-axis bearing and the swing frame. Let be the radial position angle of the mating surface between the outer ring of the C-axis bearing and the cylindrical micro-element of the swing frame; rotate the coordinate system along... The bearing coordinate system is obtained by translating the axis downwards to the center of mass of the C-axis rotary table bearing. The three coordinate axes of the bearing coordinate system are as follows: , shaft and The axes are parallel. shaft and The axes are parallel. shaft and The axes coincide; among them, Do not use the bearing coordinate system for the outer ring of the C-axis bearing. shaft and Angular deformation of the shaft; Therefore, the outer ring of the C-axis bearing is in the rotating coordinate system , , The forces acting in the three directions are: ； These are the normal contact force and tangential contact force of the planar mating surface between the outer ring of the C-axis bearing and the swing frame, respectively. These represent the stiffness along the three axes of the rotating coordinate system of the cylindrical mating surface between the outer ring of the C-axis bearing and the swing frame. C-axis bearing outer ring around bearing coordinate system , , The torques acting in the three directions are: ； These represent the normal contact forces of the planar mating surfaces between the outer ring of the C-axis bearing and the swing frame, relative to the bearing coordinate system axes. , The torque; This refers to the axial deformation of the inner ring of the C-axis bearing. Radial deformation; angular deformation of the bearing inner ring relative to the bearing outer ring. C-axis bearing inner ring radial deformation With rotating coordinate system The included angle of the axis is ;in, These are the inner rings of the C-axis bearing in the rotating coordinate system. , , Straight line deformation in three directions; rotating the coordinate system along The bearing coordinate system is obtained by translating the axis downwards to the center of mass of the C-axis rotary table bearing. The three coordinate axes of the bearing coordinate system are as follows: , shaft and The axes are parallel. shaft and The axes are parallel. shaft and The axes coincide; among them, Do not refer to the bearing coordinate system for the inner ring of the C-axis bearing. shaft and Angular deformation of the shaft; Therefore, the forces acting on the inner ring of the bearing along the three coordinate axes of the rotating coordinate system are: ； , These are the axial nonlinear contact force and radial nonlinear contact force of the inner ring of the C-axis YRT turntable bearing, which are the result of the contact forces between each row of rollers and raceways. The torques of the bearing inner ring about the three coordinate axes in the translational coordinate system are: ； , The axial nonlinear contact forces between each row of rollers and raceway of the C-axis YRT rotary table bearing on the bearing coordinate system axes are respectively... , The torque; Variation in the normal phase distance between the planar micro-element mating surface of the worktable and the bearing inner ring The change in the normal distance between the mating surfaces of the cylindrical micro-element of the worktable and the inner ring of the bearing. ; Axial deformation of the worktable relative to the bearing inner ring radial deformation = radial deformation of the worktable With rotating coordinate system The included angle of the axis is Angular deformation of the worktable relative to the inner ring of the bearing ;in, These are the worktable in the rotating coordinate system. , , Linear deformation in three directions; The position of each micro-element area of ​​the planar micro-element mating surface between the worktable and the bearing inner ring. . is the radial position angle of the mating surface between the worktable and the cylindrical micro-element of the bearing inner ring; rotate the coordinate system along... The worktable coordinate system is obtained by translating the axis downwards to the center of mass of the worktable. The three coordinate axes of the worktable coordinate system are as follows: , shaft and The axes are parallel. shaft and The axes are parallel. shaft and The axes coincide; among them, Do not revolve the worktable around the worktable coordinate system. shaft and Angular deformation of the shaft; Therefore, the forces acting on the worktable along the coordinate axes of the rotating coordinate system are: ； The normal and tangential contact forces at the planar mating surfaces between the worktable and the bearing inner ring; , , Stiffness of the three coordinate axes of the rotating coordinate system of the cylindrical mating surface between the split platform and the inner ring of the bearing; The torque exerted on the worktable in the translation coordinate system about the coordinate axes is: ； The normal contact force between the worktable and the mating surface of the bearing inner ring plane is relative to the coordinate axes of the worktable coordinate system. , The torque; The cutting force and gravity in the fixed coordinate system are transformed into the rotating coordinate system for force analysis, and balanced with the forces acting on the C-axis components: ； Torque balance is: 。 7. The method for calculating and analyzing the angular static stiffness model of a cradle turntable according to claim 6, characterized in that, Step 7 specifically involves: based on the known applied load and the obtained angular deformation displacement of the turntable, the angular static stiffness of the turntable in the three coordinate axes of the worktable coordinate system can be calculated as follows: ； , respectively force For the worktable coordinate system , Torque of the coordinate axes.

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