An ultra-precision machining precision prediction method coupling motion precision and dynamic characteristics

Abstract

The application discloses a kind of coupling motion precision and dynamic characteristic ultra-precision machining precision prediction method, belongs to the technical field of ultra-precision machining and surface topography prediction.First, the spindle error transformation matrix model of ultra-precision machine tool is constructed, the error transformation matrix containing spindle geometric error information is established, and the parameters of the spindle dynamics equation are solved.Second, the spindle system dynamics equation coupled with real-time vibration response feedback is established, and the spindle real-time vibration response under steady-state operating condition is obtained to realize accurate reconstruction.Third, based on the spindle error transformation matrix and the spindle vibration response transformation matrix, an ultra-precision turning surface topography prediction model coupling geometric error and vibration response is constructed.Finally, the macro surface error and micro surface roughness of the machined surface are quantitatively characterized.The application can realize the quantitative prediction of the surface error and surface roughness of the machined surface, and provide a basis for ultra-precision machine tool machining precision analysis and process parameter optimization.

Description

Technical Field

[0001] This invention belongs to the field of ultra-precision machining and surface morphology prediction technology, and relates to a simulation prediction method for surface morphology of ultra-precision turning, and more particularly to an ultra-precision machining accuracy prediction method that couples motion accuracy and dynamic characteristics. Background Technology

[0002] Ultra-precision turning (SPDT) technology is widely used in the manufacturing of optical components, molds, and precision mechanical parts. The surface accuracy and surface roughness of the machined surfaces are key indicators for evaluating machining quality. In ultra-precision machining, the rotational accuracy of the air-bearing spindle of the ultra-precision machine tool is one of the core factors affecting machining quality. When the air-bearing spindle rotates at high speed, it is not only subject to geometric errors caused by manufacturing and assembly defects, but also to vibrations induced by cutting forces and unbalanced forces. Comprehensively considering the rotational geometric errors of the air-bearing spindle and the cutting vibration response during machining to predict the surface morphology of ultra-precision turned surfaces is an important way to ensure the quality of the machined surfaces.

[0003] Most existing surface topography prediction methods only consider the single-factor influence of the machine tool, which differs significantly from the actual surface topography of the machined workpiece. In ultra-precision freeform surface turning, the Archimedean spiral toolpath generation method with equal-angle discreteness is often used because it facilitates trajectory generation and interpolation, and also helps maintain relatively stable spatial sampling characteristics throughout the machining area. However, under this machining strategy, the actual cutting speed or commanded spindle / C-axis speed may vary significantly with the workpiece radius. For air-bearing spindles, their stiffness and damping are closely related to the rotational speed, so this speed variation will lead to changes in the dynamic characteristics of the spindle system during machining. At the same time, all measured geometric errors of the machine tool will also be transmitted through the kinematic link, forming direct spatial errors at the tool tip. Therefore, the final workpiece surface error depends not only on the direct kinematic transmission of geometric errors, but also on the combined influence of the coupling effect of geometric errors and rotational speed-related spindle dynamics. Therefore, it is necessary to establish a coupled model that simultaneously considers the kinematic accuracy transmission and spindle dynamics to achieve the prediction of workpiece surface shape errors and surface roughness.

[0004] Currently, research on surface morphology prediction for ultra-precision turning can be mainly divided into two categories: surface morphology prediction considering only the geometric errors of ultra-precision machine tools and surface roughness prediction considering only cutting dynamics. Chinese invention patent CN112387995B reconstructs the machined surface by dividing it into meshes and calculating the geometric position of the tool contact point, predicting the surface morphology based on simulations of ideal geometric kinematics. Chinese invention patent CN109753632A collects cutting force and vibration response data, and uses a radial basis function neural network to establish a nonlinear mapping model between signal characteristics and surface roughness.

[0005] Based on the above analysis, existing technologies typically treat the geometric error and vibration response of the air-bearing spindle as independent factors, lacking a comprehensive prediction model that couples the two. This makes it difficult to accurately reconstruct the surface morphology and fails to provide a reliable theoretical basis for optimizing the process parameters of ultra-precision machine tools. Summary of the Invention

[0006] The purpose of this invention is to address the problem in existing technologies where the prediction accuracy of workpiece surface morphology in ultra-precision machine tools is insufficient due to considering only the effects of geometric errors or vibration alone. This invention provides a method for predicting ultra-precision machining accuracy by coupling motion accuracy and dynamic characteristics. By establishing a geometric accuracy transfer model for an air-bearing spindle, this invention solves for the spindle dynamic equation parameters related to machining conditions and spindle geometric errors, thereby obtaining the steady-state real-time vibration response of the spindle. Based on this, and using the principle of spatiotemporal superposition and the cutting layer envelope mechanism, a surface morphology prediction model is established, thus achieving ultra-precision turning surface morphology prediction by coupling accuracy transfer and dynamic characteristic transfer.

[0007] To achieve the above objectives, the technical solution adopted by the present invention is as follows: A method for predicting the accuracy of ultra-precision machining by coupling motion accuracy and dynamic characteristics includes the following steps: The first step, based on multibody system theory, is to construct a spindle error transformation matrix model for ultra-precision machine tools, characterizing the geometric error mapping relationship of the ultra-precision machine tools, and establishing an error transformation matrix that includes spindle geometric error information. Specifically: A spindle error transformation matrix model based on multibody system theory is constructed. Based on multibody system kinematics theory, the machine tool's topology is defined, including the bed coordinate system. O B -X B Y B Z B Principal coordinate system O S -X S Y S Z S Tool coordinate system O T -X T Y T Z T and workpiece coordinate system O W -X W Y W ZW Two adjacent components in a machine tool that have direct assembly, connection, or motion constraints are defined as adjacent bodies. Based on the topological connections between these adjacent bodies, the machine tool's topology is constructed, and an ideal homogeneous transformation matrix between adjacent bodies is established. Simultaneously, the following definitions are made: This is the ideal homogeneous transformation matrix from the bed coordinate system to the principal axis coordinate system. Let be the ideal homogeneous transformation matrix from the bed coordinate system to the tool coordinate system.

[0008] Define the spindle error transformation matrix, which includes the spindle geometric error, as follows: The principal spindle error transformation matrix It consists of 6 geometric errors of the spindle, and its mathematical expression is as follows: (1) in, Represents the spindle error transformation matrix; The circumferential angle of rotation of the air-bearing main shaft in the main axis coordinate system; , , These are the air-bearing main shafts in the principal axis coordinate system. X,Y,Z Translational error in direction; , , The air-bearing main shaft rotates around the main shaft coordinate system. X,Y,Z Shaft tilt error.

[0009] The second step, based on the geometric error distribution characteristics of the air-bearing spindle obtained in the first step, considers the machining conditions and geometric errors, and solves for the parameters of the spindle dynamics equation. Specifically: the air film stiffness and damping characteristics related to the rotational speed are solved using the air film dynamics equation to obtain the stiffness and damping of the air-bearing spindle related to the rotational speed; simultaneously, the normal relative displacement between the tool tip and the workpiece and the cutting thickness characteristics caused by geometric errors are solved using the depth-of-cutting modulation mechanism to obtain the machining load related to the rotation angle. The specific steps are as follows: Step 2.1: Based on the compressible Reynolds equation in the gas lubrication theory, solve for the air film stiffness and damping of the main shaft at the rotational speed n, and obtain the stiffness and damping of the air-bearing main shaft related to the rotational speed. Step 2.1.1: To solve for the spindle air film stiffness and damping based on the compressible Reynolds equation in gas lubrication theory, a local thickness model of the spindle air film is first established. The air-bearing spindle consists of a front radial bearing, a rear radial bearing, and a thrust bearing. The rotor radius of the front radial bearing of the air-bearing spindle is... R The nominal radial clearance is The working length of the bearing is from the inlet end z 1. Export end z 2 determines that the axial length isL At a given spindle speed n At that time, the local thickness of the spindle air film is expressed as: (2) in, Indicates the local thickness of the air film on the main shaft; z is the circumferential angle of rotation of the air-bearing main shaft in the main axis coordinate system; z is the axial coordinate in the main axis coordinate system. n Main spindle speed; This is the nominal radial clearance of the bearing; , The reference sections at the rotor's center of mass are respectively at the rotational speed. n Below the lower edge of the principal axis coordinate system X, Y The equilibrium displacement in the direction.

[0010] Substituting equation (2) into the compressible Reynolds equation in gas lubrication theory, which is a mature gas film dynamics equation used in solving the gas film pressure distribution of air bearings, the gas film pressure distribution at different spindle speeds is obtained using the finite difference method; the spindle speed is obtained by integrating the bearing working area. n Lower film rotor in principal axis coordinate system X , Y The total reaction force in the direction is used to determine the spindle speed based on the force balance condition. n The equilibrium displacement below ( , ); at the determined equilibrium displacement ( , Based on the reference position, the rotor is subjected to time-varying pressure. t Small perturbation displacement and The incremental film reaction forces in the X and Y directions in the principal axis coordinate system are obtained. and Assuming that the air film force-displacement relationship in the neighborhood of the equilibrium displacement can be linearized, the incremental air film reaction force is expressed as a linear combination of stiffness and damping terms, as shown in formula (3): (3) in, and In the principal coordinate system X , Y Incremental film reaction force in the direction; and These are the rotor center along the lower edge of the main shaft coordinate system. X , Y Small perturbation displacement in the direction; and These are the rotor center along the lower edge of the main shaft coordinate system.X , Y The velocity of the disturbance in the direction; The front radial bearing stiffness matrix is ​​a 2×2 dimensional matrix that varies with rotational speed n. It is a 2×2 dimensional front radial bearing damping matrix that varies with rotational speed n.

[0011] For the rear radial bearing and thrust bearing of the spindle, the same analysis method as for the front radial bearing is used to obtain the stiffness matrix of the rear radial bearing. With damping matrix ; and thrust bearing axial stiffness matrix and the damping matrix is .

[0012] Step 2.1.2: In order to construct a dynamic model that can uniformly describe the constraint effect of the front radial bearing, rear radial bearing, and thrust bearing of the air-bearing spindle on the rotor, let the rotor reference section z at the rotor's center of mass be... m The axial distance to the front radial bearing section is Rotor reference section z m The axial distance to the rear radial bearing section is Based on the kinematics principle of rigid body small displacement, using the translational displacement and tilt angle of the rotor reference section as input, kinematic mapping relationships from the rotor to the local linear displacements at the front radial bearing, rear radial bearing, and thrust bearing are established respectively. A geometric transformation matrix for the front radial bearing capable of describing this kinematic mapping is then constructed. Geometric transformation matrix of rear radial bearing and the geometric transformation matrix of the thrust bearing The local stiffness and damping of the front and rear radial bearings and the thrust bearing are equivalently applied to the rotor reference section coordinate system, thereby obtaining the overall equivalent stiffness matrix and damping matrix of the spindle system in the full speed domain, as shown in formulas (4) and (5): (4) (5) Among them, among them, This represents the overall equivalent stiffness matrix of the spindle system; This represents the geometric transformation matrix that maps the translational displacement of the rotor reference section to the local radial translational displacement of the front radial bearing. This represents the local stiffness matrix of the front radial bearing; This represents the geometric transformation matrix that maps the translational displacement of the rotor reference section to the local radial translational displacement of the rear radial bearing; This represents the local stiffness matrix of the rear radial bearing; This represents the geometric transformation matrix that maps the translational displacement of the rotor reference section to the local axial translational displacement of the thrust bearing; This represents the local stiffness matrix of the thrust bearing; This represents the overall equivalent damping matrix of the spindle system; This represents the local damping matrix of the rear radial bearing; This represents the local damping matrix of the front radial bearing; This represents the local damping matrix of the thrust bearing. , and The specific form is as follows: (6) in, Rotor reference section at the rotor's center of mass position z m The axial distance to the front radial bearing section; Rotor reference section z m The axial distance to the rear radial bearing section.

[0013] Step 2.2: Based on the depth-of-cut modulation mechanism, the normal relative displacement between the tool tip and the workpiece and the cutting thickness characteristics caused by geometric errors are solved to obtain the machining load related to the rotation angle. Based on the ideal homogeneous transformation matrix and the spindle error transformation matrix obtained in the first step according to the machine tool kinematic chain topology, the definition of each coordinate system, and the ideal motion relationship, the ideal homogeneous transformation matrix from the bed coordinate system to the tool coordinate system will be used. Transformation matrix with principal axis error Matrix multiplication is performed to characterize the propagation relationship of geometric errors in the tool tip-workpiece normal direction, thus obtaining the relative normal displacement between the tool tip and workpiece caused by geometric errors. , represented as: (7) in, This represents the normal relative displacement between the tool tip and the workpiece caused by geometric errors; This represents the ideal homogeneous transformation matrix from the bed coordinate system to the tool coordinate system; This represents the spindle error transformation matrix.

[0014] Taking into account the feed motion during the cutting process, the depth of cut is given. and spindle rotation angle Actual chip thickness Represented as: (8) in, Indicates the angle The actual chip thickness below; For the back of the knife; This represents the normal relative displacement between the tool tip and the workpiece caused by geometric errors.

[0015] Given a feed rate, the machining load vector acting on the spindle system in the spindle coordinate system is expressed as: (9) in, Represents the machining load vector; , , This indicates that the coordinates are located at the lower edge of the principal axis coordinate system. X , Y , Z The cutting force component in the direction; , This indicates that the coordinates are rotated around the principal axis in the coordinate system. X shaft and Y The cutting torque of the shaft.

[0016] During machining, the cutting force components can be calculated using a specific cutting force model, along the lower axis of the spindle coordinate system. X Component of cutting force in the direction Represented as: (10) in, Indicates the lower edge of the principal axis coordinate system X The cutting force component in the direction; For the material along the principal axis coordinate system X Specific cutting force coefficient in the direction; f For feed rate; Indicates the angle The actual chip thickness.

[0017] During machining, the spindle coordinate system is rotated downwards. X shaft and Y The cutting torques of the shafts are expressed as follows: (11) (12) in, , These are respectively represented as the revolution around the principal axis coordinate system. X shaft and Y The cutting torque of the shaft; , , These are respectively represented as the lower edge of the principal axis coordinate system. X , Y , Z The cutting force component in the direction; , , This indicates the coordinates of the cutting point at the tool tip in the spindle reference coordinate system.

[0018] The third step involves establishing a spindle system dynamic equation coupled with real-time vibration response feedback based on the spindle film stiffness and damping related to rotation speed and the machining load related to rotation angle obtained in the second step. The real-time vibration response of the spindle under steady-state conditions is obtained through a time-domain iterative solution method, and the spindle vibration response vector in the steady-state phase is transformed into a spindle vibration response transformation matrix. Specifically: the spindle film stiffness and damping related to rotation speed and the machining load related to rotation angle obtained in the second step are processed using the spindle system dynamic equation coupled with real-time vibration response feedback; the spindle system dynamic equation is solved using a time-domain iterative solution method until the dynamic response satisfies the periodic steady-state convergence criterion, thus obtaining the real-time vibration response of the spindle under steady-state conditions; the real-time vibration response of the spindle under steady-state conditions is expressed by a homogeneous transformation matrix to obtain the spindle vibration response transformation matrix, thereby achieving accurate reconstruction of the system's dynamic characteristics during machining. The specific steps are as follows: Step 3.1: Based on the parameters obtained in the second step, establish the dynamic equations of the main shaft system.

[0019] Based on the toolpath and machining program design, the rotation speed is... n Spindle rotation angle feed rate f and the amount of back cut Mapped to time t Changing variables , , as well as .

[0020] Based on bearing-rotor dynamics theory, considering the effects of spindle speed, system stiffness, and damping, the dynamic equations of the spindle system are established as follows: (13) Where M is the system quality matrix; This represents the real-time vibration response vector of the spindle system. , , These are the rotor reference sections along the lower edge of the principal shaft coordinate system. X , Y , Z Translational displacement in the direction of motion. , The rotor rotates in the principal axis coordinate system. X shaft and Y The inclination angle of the axis; and These represent the overall equivalent damping matrix and the overall equivalent stiffness matrix of the spindle system, respectively. This represents the machining load vector.

[0021] Step 3.2: Based on the real-time vibration response of the spindle calculated using the spindle system dynamic equation established in Step 3.1, a parameter correction mechanism for vibration response feedback is established. The real-time vibration response of the spindle is fed back into the calculation process of the air film support characteristics and machining load, realizing time-varying correction of the air film support characteristics and machining load, thereby completing the dynamic update of the dynamic equation parameters.

[0022] Consider the real-time vibration response vector of the spindle system The local thickness of the spindle air film is dynamically corrected to obtain the corrected local thickness of the spindle air film, as shown in formula (14): (14) in, This indicates the local thickness of the spindle air film after correction; h 0 indicates the nominal radial clearance of the bearing; , The reference sections representing the rotor's center of mass position at time t along the lower axis of the principal axis coordinate system are respectively... X , Y The equilibrium displacement in the direction; , The reference sections representing the rotor's center of mass position at time t along the lower axis of the principal axis coordinate system are respectively... X , Y The vibration displacement component in the direction.

[0023] The local air film thickness of the spindle air film after dynamic correction Substituting into step 2.1, the equivalent stiffness and damping characteristics of the air-bearing spindle considering the real-time vibration response feedback of the spindle system are solved.

[0024] Consider the real-time vibration response vector of the spindle system The actual chip thickness is dynamically corrected to obtain the corrected actual chip thickness, as shown in formula (15): (15) in, This indicates the corrected actual chip thickness; The depth of cut at time t; This indicates the normal relative displacement between the tool tip and the workpiece; This represents the displacement of the air-bearing spindle vibration mapped onto the normal direction of the tool tip.

[0025] The actual chip thickness after dynamic correction Substituting into step 2.2, the machining load vector considering the real-time vibration response feedback of the spindle system is obtained.

[0026] Step 3.3: Based on the time-domain step integral algorithm, the dynamic equation of the main shaft system with coupled vibration response feedback is solved iteratively to obtain the steady-state real-time vibration displacement response vector of the main shaft that satisfies the periodic convergence criterion.

[0027] Set the initial displacement vector of the spindle system based on the machining conditions. and initial velocity vector The time span for solving the problem is defined as follows: T total Step size is This ensures that the sampling frequency is sufficient to cover the high-frequency vibration characteristics of the spindle.

[0028] Perform single-step dynamic response calculations and parameter feedback updates. At each time step... t k Within this process, the fourth-order Runge-Kutta method is used to solve the dynamic equations of the main shaft system in step 3.1. The instantaneous vibration response vector of the main shaft calculated at this moment is then used. The data is fed back in real time to step 3.2, synchronously updating the corrected local thickness of the spindle air film. and the corrected actual chip thickness The updated and corrected parameters of the main shaft dynamic equations are obtained.

[0029] Based on the synchronously updated and corrected parameters of the spindle dynamics equations, the overall equivalent stiffness matrix of the spindle system at the current moment is recalculated. Damping matrix and machining load vector Substitute the updated matrix and vector into the next time step. t k+1 The calculation of the main shaft system dynamic equations is performed, and time-domain iteration is continuously conducted. To determine whether the main shaft dynamic response has reached a periodic steady-state convergence state, the residuals of the displacement vectors in adjacent periods are calculated. : (16) in, It represents the residual of the displacement vector within adjacent periods; For time integration variables; T = 60 / n This is the rotation cycle of the air flotation main shaft.

[0030] When the residual When the convergence threshold is not less than the preset threshold, the spindle vibration response at the end of the current cycle is used as the initial state for the next cycle iteration, and the dynamic response calculation and parameter feedback update continue. When the residual... When the response time is less than a preset convergence threshold, the spindle dynamic response is considered to have reached a stable state. The spindle vibration response vector during the stable phase is then extracted. : (17) in, This represents the principal shaft vibration response vector during the steady-state phase. , , The instantaneous vibrational translational displacement of the principal axis reference section at time t in the principal axis coordinate system; , The instantaneous vibration tilt angles around the X and Y axes in the principal axis coordinate system.

[0031] In the above periodic steady-state convergence judgment, the convergence threshold is preset according to the accuracy requirements of the spindle vibration response calculation.

[0032] Step 3.4: Based on the spindle vibration response vector obtained in step 3.3 during the steady-state phase, the vibration response results obtained from solving the spindle system dynamic model are further transformed into a spindle vibration response transformation matrix. It is used to characterize the transmission effect of the spindle vibration response in spatial coordinate transformation.

[0033] Since the vibration magnitude in ultra-precision machining is a tiny displacement, we can ignore higher-order infinitesimal terms and consider the spindle vibration response vector during the steady-state phase. The translational displacement components and instantaneous vibration tilt angle are written into the homogeneous transformation matrix, and the principal shaft vibration response transformation matrix is ​​defined. for: (18) in, This represents the transformation matrix of the spindle vibration response; , , The instantaneous vibrational translational displacement of the principal axis reference section at time t in the principal axis coordinate system; , To rotate in the principal axis coordinate system X shaft and Y The instantaneous vibration tilt angle of the shaft.

[0034] The fourth step involves constructing a surface morphology prediction model for ultra-precision turning, based on the spindle error transformation matrix and spindle vibration response transformation matrix obtained in the first three steps. Specifically, the geometric error in the spatial domain and the vibration response in the time domain are coupled using the principle of spatiotemporal superposition to obtain the actual tool tip cutting point during the machining process. This actual tool tip cutting point is then processed in conjunction with tool geometric parameters and the cutting layer envelope mechanism to construct the surface morphology prediction model for ultra-precision turning, which couples geometric error and vibration response. The specific steps are as follows: Step 4.1: Based on the principle of spatiotemporal mapping and superposition, characterize the actual cutting point of the tool tip after coupling.

[0035] Based on the machine tool kinematic chain topology and spindle error transformation matrix determined in the first step. Combining the machining parameters determined in the second step with the spindle vibration response transformation matrix obtained in the third step Characterizes the actual cutting point of the tool tip after coupling. : (19) in, This represents the actual cutting point of the tool tip in the workpiece coordinate system; This represents the ideal homogeneous transformation matrix from the bed coordinate system to the principal axis coordinate system; Represents the spindle error transformation matrix; This represents the transformation matrix of the spindle vibration response; This represents the ideal homogeneous transformation matrix from the bed coordinate system to the tool coordinate system; The tool tip point in the tool coordinate system. The X-axis coordinate component of the tool tip in the tool coordinate system.

[0036] Define the actual tool tip cutting point The spatial position components in the workpiece coordinate system are respectively , , At the radius of the blade tip arc is At that time, the instantaneous cutting envelope surface generated by the tool tip on the workpiece surface in the workpiece coordinate system Represented as: (20) in, This represents the instantaneous cutting envelope generated by the tool tip on the workpiece surface in the workpiece coordinate system. , , These represent the actual cutting points of the tool tip. In the workpiece coordinate system X,Y,Z Spatial location component; The radius of the blade tip arc.

[0037] Step 4.2: Based on the cutting layer envelope mechanism, perform time-domain envelope processing on the instantaneous cutting envelope surface obtained in Step 4.1 to establish a surface morphology prediction model that couples geometric error and vibration response.

[0038] Arbitrary coordinate points on the workpiece surface Final residual height at the location The final residual height is determined by the minimum height of all instantaneous cutting envelope surfaces covering the coordinate point within the machining time range. The surface morphology prediction model is shown in formula (21): (twenty one) in, Indicates the final residual height; Indicates the instantaneous cutting envelope surface; T total This represents the total time span.

[0039] The fifth step, based on the surface morphology prediction model of coupled geometric error and vibration response obtained in the fourth step, quantitatively characterizes the macroscopic surface shape error and microscopic surface roughness of the processed surface. Specifically, digital signal processing theory is used to decouple the surface morphology by scale, obtaining the surface shape error component characterizing the macroscopic geometric accuracy and the surface roughness component characterizing the microscopic vibration response; the surface shape error component and the surface roughness component are quantitatively characterized to obtain the peak-valley values ​​of the surface shape error and the arithmetic mean height. The specific steps are as follows: In the fourth step, a surface morphology prediction model that couples geometric error and vibration response is obtained. According to the ISO 11562 standard cutoff wavelength λ c The final residual height was determined using a Gaussian low-pass filter. Convolution operations are performed to separate surface features with different frequency characteristics, including surface features and micro residual components. Extracting surface components: Removing the final residual height using a low-pass filter. High-frequency vibration and residual tool noise were analyzed to extract surface error components that characterize macroscopic geometric accuracy. .

[0040] Extracting microscopic residual components: from the final residual height Subtract the low-pass filtered surface shape component to extract the surface roughness component characterizing the micro-vibration response. : (twenty three) in, Represents the surface roughness component; Indicates the final residual height; This represents the surface shape error component.

[0041] Let the theoretical design surface be It represents any coordinate point on the workpiece surface. The theoretical design height at that point. The surface error components... With theoretical design surface A comparison is performed. The degree of deviation of the surface error component from the ideal surface is characterized by the peak-to-valley value (PV). (twenty four) Where PV represents the peak-to-valley value; Represents the surface shape error components; This represents the theoretically designed surface shape.

[0042] Surface roughness component Perform statistical calculations to obtain the arithmetic mean height. Within the defined sampling region A, the calculation formula is as follows: (25) in, The height is the arithmetic mean; A is the defined sampling area. This represents the surface roughness component.

[0043] By extracting the surface shape error component and the surface roughness component, the peak and valley values ​​of the surface shape error of the machined surface, as well as the arithmetic mean height, can be calculated, thereby realizing the prediction of the surface morphology of ultra-precision turning by coupling the transmission of precision and dynamic characteristics.

[0044] The beneficial effects of this invention are as follows: (1) Based on the multibody system theory, this invention constructs a spindle error transformation matrix to characterize the geometric error mapping relationship of ultra-precision machine tools, and establishes an error transformation matrix containing spindle geometric error information. This is beneficial to incorporate the geometric error of the air-bearing spindle into the subsequent normal relative displacement and surface morphology prediction process between the tool tip and the workpiece, and provides a basis for surface morphology prediction of coupling accuracy transfer and dynamic characteristic transfer.

[0045] (2) Based on the compressible Reynolds equation in the gas lubrication theory, this invention solves the stiffness and damping of the air-bearing spindle related to the rotational speed, and solves the machining load related to the rotational angle based on the cutting depth modulation mechanism. This enables the spindle dynamic equation parameters to simultaneously reflect the influence of machining conditions and geometric errors, thus improving the consistency of the spindle system dynamic equation parameter acquisition process.

[0046] (3) The present invention establishes the dynamic equation of the spindle system coupled with real-time vibration response feedback, and obtains the real-time vibration response of the spindle under steady-state working conditions through time-domain iterative solution method; at the same time, the real-time vibration response of the spindle is fed back to the calculation process of air film support characteristics and processing load, so as to realize the dynamic update of the dynamic equation parameters, which is beneficial to characterize the changes in the dynamic characteristics of the system during the processing.

[0047] (4) Based on the principle of spatiotemporal superposition and the cutting layer envelope mechanism, this invention constructs a prediction model for the surface morphology of ultra-precision turning that couples geometric error and vibration response. It can introduce the geometric error in the spatial domain and the vibration response in the time domain into the solution process of the actual cutting point and the final residual height, which is beneficial to reconstructing the macroscopic surface and microscopic surface of the machined surface.

[0048] (5) Based on digital signal processing theory, the present invention decouples the surface morphology by scale to obtain the surface error component and the surface roughness component, and quantitatively characterizes the peak and valley values ​​of the surface error and the arithmetic mean height, which can provide a basis for the evaluation of the surface morphology and the optimization of process parameters in ultra-precision turning.

[0049] In summary, this invention establishes a method for predicting the surface morphology of ultra-precision turning considering machining conditions by coupling the precision transmission of spindle geometric error, the transmission of air-bearing spindle dynamic characteristics, real-time vibration response feedback, and the cutting layer envelope mechanism. This method can achieve quantitative prediction of surface shape error and surface roughness of the machined surface, providing a basis for the analysis of machining accuracy and optimization of process parameters for ultra-precision machine tools. Attached Figure Description

[0050] Figure 1 A flowchart of a method for predicting the surface morphology of ultra-precision turning, considering machining conditions and the interactive coupling mechanism between precision transfer and dynamic characteristic transfer, is provided for embodiments of the present invention. Figure 2 This is a schematic diagram illustrating the spatiotemporal superposition principle of geometric error components and vibration error components provided in an embodiment of the present invention, wherein... Figure 2 (a) represents the surface components corresponding to the principal axis geometric errors predicted based on the multibody system. Figure 2 (b) represents the roughness component corresponding to the spindle vibration error predicted by dynamics. Figure 2 (c) shows the overall surface morphology after the two are superimposed; Figure 3 This is a simulation image of the three-dimensional morphology prediction of the ultra-precision turned surface provided in an embodiment of the present invention; Figure 4 A surface accuracy prediction diagram provided for an embodiment of the present invention; Figure 5 A surface roughness prediction diagram provided for an embodiment of the present invention. Detailed Implementation

[0051] The present invention will be further described below with reference to specific embodiments.

[0052] Taking a set of cutting conditions as an example, this paper describes a method for predicting the surface morphology of ultra-precision turning that considers both the machining conditions and the interactive coupling mechanism between accuracy transfer and dynamic characteristic transfer. A flowchart of an ultra-precision machining accuracy prediction method that couples motion accuracy and dynamic characteristics is shown below. Figure 1 As shown.

[0053] The first step, based on multibody system theory, is to construct a spindle error transformation matrix model for ultra-precision machine tools, characterizing the geometric error mapping relationship of the ultra-precision machine tools, and establishing an error transformation matrix that includes spindle geometric error information. Specifically: A spindle error transformation matrix model based on multibody system theory is constructed. Based on multibody system kinematics theory, the machine tool's topology is defined, including the bed coordinate system. O B -X B Y B Z B Principal coordinate system O S -X S Y S Z S Tool coordinate system O T -X T Y T Z T and workpiece coordinate system O W -X W Y W Z W Two adjacent components in a machine tool that have direct assembly, connection, or motion constraints are defined as adjacent bodies. Based on the topological connections between these adjacent bodies, the machine tool's topology is constructed, and an ideal homogeneous transformation matrix between adjacent bodies is established. Simultaneously, the following definitions are made: This is the ideal homogeneous transformation matrix from the bed coordinate system to the principal axis coordinate system. Let be the ideal homogeneous transformation matrix from the bed coordinate system to the tool coordinate system.

[0054] Define the spindle error transformation matrix, which includes the spindle geometric error, as follows: The principal spindle error transformation matrix It consists of 6 geometric errors of the spindle, and its mathematical expression is as follows: (1) in, Represents the spindle error transformation matrix; =0.15μm, =0.12μm, =0.10μm represents the air-bearing principal axis in the principal axis coordinate system. X , Y , Z Translational error in direction; =0.20arcsec =0.18arcsec =0.15arcsec represents the rotation of the air-bearing principal shaft in the principal shaft coordinate system. X , Y , Z Shaft tilt error.

[0055] The second step, based on the geometric error distribution characteristics of the air-bearing spindle obtained in the first step, considers the machining conditions and geometric errors, and solves for the parameters of the spindle dynamics equation. Specifically: the air film stiffness and damping characteristics related to the rotational speed are solved using the air film dynamics equation to obtain the stiffness and damping of the air-bearing spindle related to the rotational speed; simultaneously, the normal relative displacement between the tool tip and the workpiece and the cutting thickness characteristics caused by geometric errors are solved using the depth-of-cutting modulation mechanism to obtain the machining load related to the rotation angle. The specific steps are as follows: Step 2.1: Based on the compressible Reynolds equation in the gas lubrication theory, solve for the air film stiffness and damping of the main shaft at the rotational speed n, and obtain the stiffness and damping of the air-bearing main shaft related to the rotational speed. Step 2.1.1: To solve for the spindle air film stiffness and damping based on the compressible Reynolds equation in gas lubrication theory, a local thickness model of the spindle air film is first established. The air-bearing spindle consists of a front radial bearing, a rear radial bearing, and a thrust bearing. The rotor radius of the front radial bearing of the air-bearing spindle is... R =70mm, nominal radial clearance =12μm, bearing working length from inlet end z 1. Export end z 2 determines the axial length L =150mm. At a given spindle speed... n At 1000 rpm, the local thickness of the spindle air film is expressed as follows: (2) in, Indicates the local thickness of the air film on the main shaft; z is the circumferential angle of rotation of the air-bearing main shaft in the main axis coordinate system; z is the axial coordinate in the main axis coordinate system. n Main spindle speed; This is the nominal radial clearance of the bearing; , The reference sections at the rotor's center of mass are respectively at the rotational speed. n Below the lower edge of the principal axis coordinate system X, Y The equilibrium displacement in the direction.

[0056] Substituting equation (2) into the compressible Reynolds equation in gas lubrication theory, which is a mature gas film dynamics equation used in solving the gas film pressure distribution of air bearings, the gas film pressure distribution at different spindle speeds is obtained using the finite difference method; the spindle speed is obtained by integrating the bearing working area. n Lower film rotor in principal axis coordinate system X , Y The total reaction force in the direction is used to determine the spindle speed based on the force balance condition. n The equilibrium displacement below ( , ); at the determined equilibrium displacement ( , Based on the reference position, the rotor is subjected to time-varying pressure. t Small perturbation displacement and The incremental film reaction forces in the X and Y directions in the principal axis coordinate system are obtained. and Assuming that the air film force-displacement relationship in the neighborhood of the equilibrium displacement can be linearized, the incremental air film reaction force is expressed as a linear combination of stiffness and damping terms, as shown in formula (3): (3) in, and In the principal coordinate system X , Y Incremental film reaction force in the direction; and These are the rotor center along the lower edge of the main shaft coordinate system. X , Y Small perturbation displacement in the direction; and These are the rotor center along the lower edge of the main shaft coordinate system. X , Y The velocity of the disturbance in the direction; The front radial bearing stiffness matrix is ​​a 2×2 dimensional matrix that varies with rotational speed n. It is a 2×2 dimensional front radial bearing damping matrix that varies with rotational speed n.

[0057] For the rear radial bearing and thrust bearing of the spindle, the same analysis method as for the front radial bearing is used to obtain the stiffness matrix of the rear radial bearing. With damping matrix ; and thrust bearing axial stiffness matrix and the damping matrix is .

[0058] Step 2.1.2: In order to construct a dynamic model that can uniformly describe the constraint effect of the front radial bearing, rear radial bearing, and thrust bearing of the air-bearing spindle on the rotor, let the rotor reference section z at the rotor's center of mass be... m The axial distance to the front radial bearing section is Rotor reference section z m The axial distance to the rear radial bearing section is Based on the kinematics principle of rigid body small displacement, using the translational displacement and tilt angle of the rotor reference section as input, kinematic mapping relationships from the rotor to the local linear displacements at the front radial bearing, rear radial bearing, and thrust bearing are established respectively. A geometric transformation matrix for the front radial bearing capable of describing this kinematic mapping is then constructed. Geometric transformation matrix of rear radial bearing and the geometric transformation matrix of the thrust bearing The local stiffness and damping of the front and rear radial bearings and the thrust bearing are equivalently applied to the rotor reference section coordinate system, thereby obtaining the overall equivalent stiffness matrix and damping matrix of the spindle system in the full speed domain, as shown in formulas (4) and (5): (4) (5) Among them, among them, This represents the overall equivalent stiffness matrix of the spindle system; This represents the geometric transformation matrix that maps the translational displacement of the rotor reference section to the local radial translational displacement of the front radial bearing. This represents the local stiffness matrix of the front radial bearing; This represents the geometric transformation matrix that maps the translational displacement of the rotor reference section to the local radial translational displacement of the rear radial bearing; This represents the local stiffness matrix of the rear radial bearing; This represents the geometric transformation matrix that maps the translational displacement of the rotor reference section to the local axial translational displacement of the thrust bearing; This represents the local stiffness matrix of the thrust bearing; This represents the overall equivalent damping matrix of the spindle system; This represents the local damping matrix of the rear radial bearing; This represents the local damping matrix of the front radial bearing; This represents the local damping matrix of the thrust bearing. , and The specific form is as follows: (6) in, Rotor reference section at the rotor's center of mass position z m The axial distance to the front radial bearing section; Rotor reference section z m The axial distance to the rear radial bearing section.

[0059] Step 2.2: Based on the depth-of-cut modulation mechanism, the normal relative displacement between the tool tip and the workpiece and the cutting thickness characteristics caused by geometric errors are solved to obtain the machining load related to the rotation angle. Based on the ideal homogeneous transformation matrix and the spindle error transformation matrix obtained in the first step according to the machine tool kinematic chain topology, the definition of each coordinate system, and the ideal motion relationship, the ideal homogeneous transformation matrix from the bed coordinate system to the tool coordinate system will be used. Transformation matrix with principal axis error Matrix multiplication is performed to characterize the propagation relationship of geometric errors in the tool tip-workpiece normal direction, thus obtaining the relative normal displacement between the tool tip and workpiece caused by geometric errors. , represented as: (7) in, This represents the normal relative displacement between the tool tip and the workpiece caused by geometric errors; This represents the ideal homogeneous transformation matrix from the bed coordinate system to the tool coordinate system; This represents the spindle error transformation matrix.

[0060] Taking into account the feed motion during the cutting process, the depth of cut is given. =5μm and spindle rotation angle Actual chip thickness Represented as: (8) in, Indicates the angle The actual chip thickness below; For the back of the knife; This represents the normal relative displacement between the tool tip and the workpiece caused by geometric errors.

[0061] Given feed rate f =5mm / min, machining load vector acting on the spindle system in the spindle coordinate system Represented as: (9) in, Represents the machining load vector; , , This indicates that the coordinates are located at the lower edge of the principal axis coordinate system. X , Y , Z The cutting force component in the direction; , This indicates that the coordinates are rotated around the principal axis in the coordinate system. X shaft and Y The cutting torque of the shaft.

[0062] Assume the workpiece material is oxygen-free copper, and the material is along the lower axis of the principal coordinate system. X Specific cutting force coefficient in the direction =1500N / mm 2 In machining, the cutting force components can be calculated using a specific cutting force model, along the spindle coordinate system. X Component of cutting force in the direction Represented as: (10) in, Indicates the lower edge of the principal axis coordinate system X The cutting force component in the direction; For the material along the principal axis coordinate system X Specific cutting force coefficient in the direction; f For feed rate; Indicates the angle The actual chip thickness.

[0063] During machining, the spindle coordinate system is rotated downwards. X shaft and Y The cutting torques of the shafts are expressed as follows: (11) (12) in, , These are respectively represented as the revolution around the principal axis coordinate system. X shaft and Y The cutting torque of the shaft; , , These are respectively represented as the lower edge of the principal axis coordinate system. X , Y , Z The cutting force component in the direction; , , This indicates the coordinates of the cutting point at the tool tip in the spindle reference coordinate system.

[0064] The third step involves establishing a spindle system dynamic equation coupled with real-time vibration response feedback based on the spindle film stiffness and damping related to rotation speed and the machining load related to rotation angle obtained in the second step. The real-time vibration response of the spindle under steady-state conditions is obtained through a time-domain iterative solution method, and the spindle vibration response vector in the steady-state phase is transformed into a spindle vibration response transformation matrix. Specifically: the spindle film stiffness and damping related to rotation speed and the machining load related to rotation angle obtained in the second step are processed using the spindle system dynamic equation coupled with real-time vibration response feedback; the spindle system dynamic equation is solved using a time-domain iterative solution method until the dynamic response satisfies the periodic steady-state convergence criterion, thus obtaining the real-time vibration response of the spindle under steady-state conditions; the real-time vibration response of the spindle under steady-state conditions is expressed by a homogeneous transformation matrix to obtain the spindle vibration response transformation matrix, thereby achieving accurate reconstruction of the system's dynamic characteristics during machining. The specific steps are as follows: Step 3.1: Based on the parameters obtained in the second step, establish the dynamic equations of the main shaft system.

[0065] Based on the toolpath and machining program design, the rotation speed is... n Spindle rotation angle feed rate f and the amount of back cut Mapped to time t Changing variables , , as well as .

[0066] Based on bearing-rotor dynamics theory, considering the effects of spindle speed, system stiffness, and damping, the dynamic equations of the spindle system are established as follows: (13) Where M is the system quality matrix; This represents the real-time vibration response vector of the spindle system. , , These are the rotor reference sections along the lower edge of the principal shaft coordinate system. X , Y , Z Translational displacement in the direction of motion. , The rotor rotates in the principal axis coordinate system. X shaft and Y The inclination angle of the axis; and These represent the overall equivalent damping matrix and the overall equivalent stiffness matrix of the spindle system, respectively. This represents the machining load vector.

[0067] Step 3.2: Based on the real-time vibration response of the spindle calculated using the spindle system dynamic equation established in Step 3.1, a parameter correction mechanism for vibration response feedback is established. The real-time vibration response of the spindle is fed back into the calculation process of the air film support characteristics and machining load, realizing time-varying correction of the air film support characteristics and machining load, thereby completing the dynamic update of the dynamic equation parameters.

[0068] Consider the real-time vibration response vector of the spindle system The local thickness of the spindle air film is dynamically corrected to obtain the corrected local thickness of the spindle air film, as shown in formula (14): (14) in, This indicates the local thickness of the spindle air film after correction; h 0 indicates the nominal radial clearance of the bearing; , The reference sections representing the rotor's center of mass position at time t along the lower axis of the principal axis coordinate system are respectively... X , Y The equilibrium displacement in the direction; , The reference sections representing the rotor's center of mass position at time t along the lower axis of the principal axis coordinate system are respectively... X , Y The vibration displacement component in the direction.

[0069] The local air film thickness of the spindle air film after dynamic correction Substituting into step 2.1, the equivalent stiffness and damping characteristics of the air-bearing spindle considering the real-time vibration response feedback of the spindle system are solved.

[0070] Consider the real-time vibration response vector of the spindle system The actual chip thickness is dynamically corrected to obtain the corrected actual chip thickness, as shown in formula (15): (15) in, This indicates the corrected actual chip thickness; The depth of cut at time t; This indicates the normal relative displacement between the tool tip and the workpiece; This represents the displacement of the air-bearing spindle vibration mapped onto the normal direction of the tool tip.

[0071] The actual chip thickness after dynamic correction Substituting into step 2.2, the machining load vector considering the real-time vibration response feedback of the spindle system is obtained.

[0072] Step 3.3: Based on the time-domain step integral algorithm, the dynamic equation of the main shaft system with coupled vibration response feedback is solved iteratively to obtain the steady-state real-time vibration displacement response vector of the main shaft that satisfies the periodic convergence criterion.

[0073] Set the initial displacement vector of the spindle system based on the machining conditions. and initial velocity vector The time span for solving the problem is defined as follows: T total Step size is =10 -5 s, to ensure that the sampling frequency is sufficient to cover the high-frequency vibration characteristics of the spindle.

[0074] Perform single-step dynamic response calculations and parameter feedback updates. At each time step... t k Within this process, the fourth-order Runge-Kutta method is used to solve the dynamic equations of the main shaft system in step 3.1. The instantaneous vibration response vector of the main shaft calculated at this moment is then used. The data is fed back in real time to step 3.2, synchronously updating the corrected local thickness of the spindle air film. and the corrected actual chip thickness The updated and corrected parameters of the main shaft dynamic equations are obtained.

[0075] Based on the synchronously updated and corrected parameters of the spindle dynamics equations, the overall equivalent stiffness matrix of the spindle system at the current moment is recalculated. Damping matrix and machining load vector Substitute the updated matrix and vector into the next time step. t k+1 The calculation of the main shaft system dynamic equations is performed, and time-domain iteration is continuously conducted. To determine whether the main shaft dynamic response has reached a periodic steady-state convergence state, the residuals of the displacement vectors in adjacent periods are calculated. : (16) in, It represents the residual of the displacement vector within adjacent periods; For time integration variables; T = 60 / n This is the rotation cycle of the air flotation main shaft.

[0076] When the residual <10 -6 At this point, the spindle dynamic response is considered to have reached a steady state. The spindle vibration response vector for this steady-state phase is then extracted. : (17) in, This represents the principal shaft vibration response vector during the steady-state phase. , , The instantaneous vibrational translational displacement of the principal axis reference section at time t in the principal axis coordinate system; , The instantaneous vibration tilt angles around the X and Y axes in the principal axis coordinate system.

[0077] Step 3.4: Based on the spindle vibration response vector obtained in step 3.3 during the steady-state phase, the vibration response results obtained from solving the spindle system dynamic model are further transformed into a spindle vibration response transformation matrix. It is used to characterize the transmission effect of the spindle vibration response in spatial coordinate transformation.

[0078] Since the vibration magnitude in ultra-precision machining is a tiny displacement, we can ignore higher-order infinitesimal terms and consider the spindle vibration response vector during the steady-state phase. The translational displacement components and instantaneous vibration tilt angle are written into the homogeneous transformation matrix, and the principal shaft vibration response transformation matrix is ​​defined. for: (18) in, This represents the transformation matrix of the spindle vibration response; , , The instantaneous vibrational translational displacement of the principal axis reference section at time t in the principal axis coordinate system; , To rotate in the principal axis coordinate system X shaft and Y The instantaneous vibration tilt angle of the shaft.

[0079] The fourth step involves constructing a surface morphology prediction model for ultra-precision turning, based on the spindle error transformation matrix and spindle vibration response transformation matrix obtained in the first three steps. Specifically, the geometric error in the spatial domain and the vibration response in the time domain are coupled using the principle of spatiotemporal superposition to obtain the actual tool tip cutting point during the machining process. This actual tool tip cutting point is then processed in conjunction with tool geometric parameters and the cutting layer envelope mechanism to construct the surface morphology prediction model for ultra-precision turning, which couples geometric error and vibration response. The specific steps are as follows: Step 4.1: Based on the principle of spatiotemporal mapping and superposition, characterize the actual cutting point of the tool tip after coupling.

[0080] Based on the machine tool kinematic chain topology and spindle error transformation matrix determined in the first step. Combining the machining parameters determined in the second step with the spindle vibration response transformation matrix obtained in the third step Characterizes the actual cutting point of the tool tip after coupling. : (19) in, This represents the actual cutting point of the tool tip in the workpiece coordinate system; This represents the ideal homogeneous transformation matrix from the bed coordinate system to the principal axis coordinate system; Represents the spindle error transformation matrix; This represents the transformation matrix of the spindle vibration response; This represents the ideal homogeneous transformation matrix from the bed coordinate system to the tool coordinate system; The tool tip point in the tool coordinate system. The X-axis coordinate component of the tool tip in the tool coordinate system.

[0081] A schematic diagram of the coupling principle is shown below. Figure 2 As shown, Figure 2 The correspondence between the geometric error components and the vibration error components in the spatiotemporal domain is shown.

[0082] Define the actual tool tip cutting point The spatial position components in the workpiece coordinate system are respectively , , At the radius of the blade tip arc When the diameter is 1mm, the instantaneous cutting envelope surface generated by the tool tip on the workpiece surface in the workpiece coordinate system. Represented as: (20) in, This represents the instantaneous cutting envelope generated by the tool tip on the workpiece surface in the workpiece coordinate system. , , These represent the actual cutting points of the tool tip. In the workpiece coordinate system X,Y,Z Spatial location component; The radius of the blade tip arc.

[0083] Step 4.2: Based on the cutting layer envelope mechanism, perform time-domain envelope processing on the instantaneous cutting envelope surface obtained in Step 4.1 to establish a surface morphology prediction model that couples geometric error and vibration response.

[0084] Arbitrary coordinate points on the workpiece surface Final residual height at the location The final residual height is determined by the minimum height of all instantaneous cutting envelope surfaces covering the coordinate point within the machining time range. The surface morphology prediction model is shown in formula (21): (twenty one) in, Indicates the final residual height; Indicates the instantaneous cutting envelope surface; T total This represents the total time span.

[0085] Simulation results are as follows Figure 3 As shown, Figure 3 The final residual height is shown. The three-dimensional morphology prediction results of the ultra-precision turned surface are used to characterize the spatial distribution of the machined surface after coupling geometric errors and vibration response.

[0086] The fifth step, based on the surface morphology prediction model of coupled geometric error and vibration response obtained in the fourth step, quantitatively characterizes the macroscopic surface shape error and microscopic surface roughness of the processed surface. Specifically, digital signal processing theory is used to decouple the surface morphology by scale, obtaining the surface shape error component characterizing the macroscopic geometric accuracy and the surface roughness component characterizing the microscopic vibration response; the surface shape error component and the surface roughness component are quantitatively characterized to obtain the peak-valley values ​​of the surface shape error and the arithmetic mean height. The specific steps are as follows: In the fourth step, a surface morphology prediction model that couples geometric error and vibration response is obtained. Then, the standard cutoff wavelength was set according to ISO 11562. λ c =0.08mm, using a Gaussian low-pass filter to determine the final residual height The surface topography data is convolved to separate surface features with different frequency characteristics, including surface shape components and micro residual components. Extracting surface components: Removing the final residual height using a low-pass filter. High-frequency vibration and residual tool noise were analyzed to extract surface error components that characterize macroscopic geometric accuracy. .

[0087] Extracting microscopic residual components: from the final residual height Subtract the low-pass filtered surface shape component to extract the surface roughness component characterizing the micro-vibration response. : (twenty three) in, Represents the surface roughness component; Indicates the final residual height; This represents the surface shape error component.

[0088] Let the theoretical design surface be It represents any coordinate point on the workpiece surface. The theoretical design height at that point. The surface error components... With theoretical design surface A comparison is performed. The degree of deviation of the surface error component from the ideal surface is characterized by the peak-to-valley value (PV). (twenty four) Where PV represents the peak-to-valley value; Represents the surface shape error components; This represents the theoretically designed surface shape.

[0089] Surface accuracy prediction results are as follows Figure 4 As shown, Figure 4 The surface error components are shown. Compared to theoretical design surface Distribution of [something].

[0090] Surface roughness component Perform statistical calculations to obtain the arithmetic mean height. Within the defined sampling region A, the calculation formula is as follows: (25) in, The height is the arithmetic mean; A is the defined sampling area. This represents the surface roughness component.

[0091] Surface roughness prediction results are as follows Figure 5 As shown, Figure 5 The surface roughness components are shown. Distribution within sampling area A.

[0092] The ultra-precision turning surface morphology prediction method proposed in this patent, which takes into account the machining conditions and the interactive coupling mechanism between precision transmission and dynamic characteristic transmission, can comprehensively consider the geometric error and vibration information of the spindle system in the same prediction process, providing a methodological basis for the prediction of ultra-precision turning surface morphology.

[0093] The above embodiments are merely illustrative of the implementation methods of the present invention, but should not be construed as limiting the scope of the present invention. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of the present invention, and these modifications and improvements all fall within the protection scope of the present invention.

Claims

1. A method for predicting the accuracy of ultra-precision machining by coupling motion accuracy and dynamic characteristics, characterized in that, The method for predicting the surface morphology of ultra-precision turning includes the following steps: The first step is to construct a spindle error transformation matrix model for ultra-precision machine tools based on the kinematics theory of multibody systems, to characterize the geometric error mapping relationship of ultra-precision machine tools, and to establish an error transformation matrix that includes spindle geometric error information. The second step, based on the geometric error distribution characteristics obtained in the first step, takes into account the machining conditions and geometric errors, and solves the parameters of the spindle dynamics equation. Specifically, the air film stiffness and damping characteristics related to the rotational speed are solved using the air film dynamics equation to obtain the stiffness and damping of the air-bearing spindle related to the rotational speed; at the same time, the normal relative displacement between the tool tip and the workpiece and the cutting thickness characteristics caused by geometric errors are solved using the cutting depth modulation mechanism to obtain the machining load related to the rotation angle. The third step involves establishing the spindle system dynamic equations coupled with real-time vibration response feedback based on the spindle air film stiffness and damping related to rotation speed obtained in the second step, as well as the machining load related to rotation angle. The spindle system dynamic equations are then solved using a time-domain iterative solution method until the dynamic response satisfies the periodic steady-state convergence criterion, thus obtaining the real-time vibration response of the spindle under steady-state conditions. The real-time vibration response of the spindle under steady-state conditions is then expressed by a homogeneous transformation matrix to obtain the spindle vibration response transformation matrix, thereby achieving accurate reconstruction of the dynamic characteristics of the system during machining. The fourth step involves constructing a surface morphology prediction model for ultra-precision turning that couples geometric error and vibration response, based on the spindle error transformation matrix and the spindle vibration response transformation matrix. Specifically, the geometric error in the spatial domain and the vibration response in the time domain are coupled using the principle of spatiotemporal superposition to obtain the actual tool tip cutting point during the machining process. The actual tool tip cutting point is then processed in conjunction with the tool geometric parameters and the cutting layer envelope mechanism to construct the surface morphology prediction model for ultra-precision turning that couples geometric error and vibration response. The fifth step involves using the ultra-precision turning surface morphology prediction model obtained in the fourth step to quantitatively characterize the macroscopic surface shape error and microscopic surface roughness of the machined surface. Specifically, digital signal processing theory is used to decouple the surface morphology by scale, resulting in the surface shape error component characterizing the macroscopic geometric accuracy characteristics and the surface roughness component characterizing the microscopic vibration response. The surface shape error component and the surface roughness component are then quantitatively characterized to obtain the surface shape error peak and valley values, as well as the arithmetic mean height.

2. The method for predicting ultra-precision machining accuracy by coupling motion accuracy and dynamic characteristics according to claim 1, characterized in that, The first step specifically includes: Based on the kinematics theory of multibody systems, the topology of a machine tool is defined to include the bed coordinate system. O B -X B Y B Z B Principal coordinate system O S -X S Y S Z S Tool coordinate system O T -X T Y T Z T and workpiece coordinate system O W -X W Y W Z W Two adjacent components in a machine tool that have direct assembly, connection, or motion constraints are defined as adjacent bodies. Based on the topological connection relationships between the adjacent bodies, the topological structure of the machine tool is constructed, and an ideal homogeneous transformation matrix of features between adjacent bodies is established. Simultaneously, the following definitions are made: This is the ideal homogeneous transformation matrix from the bed coordinate system to the principal axis coordinate system. This is the ideal homogeneous transformation matrix from the bed coordinate system to the tool coordinate system; Define the spindle error transformation matrix, which includes the spindle geometric error, as follows: ; principal axis error transformation matrix It consists of 6 geometric errors of the spindle, expressed as follows: (1) in, Represents the spindle error transformation matrix; The circumferential angle of rotation of the air-bearing main shaft in the main axis coordinate system; , , The air-bearing main shaft is in the main shaft coordinate system. X,Y,Z Translational error in direction; , , The air-bearing main shaft rotates around the main shaft coordinate system. X,Y,Z Shaft tilt error.

3. The method for predicting ultra-precision machining accuracy by coupling motion accuracy and dynamic characteristics according to claim 2, characterized in that, The second step is specific to: Step 2.1: Based on the compressible Reynolds equation in gas lubrication theory, solve for the air film stiffness and damping of the main shaft at rotational speed n to obtain the stiffness and damping of the air-bearing main shaft related to the rotational speed. Step 2.1.1: To solve for the spindle air film stiffness and damping, a local thickness model of the spindle air film is first established; the air-bearing spindle consists of a front radial bearing, a rear radial bearing, and a thrust bearing; the rotor radius of the front radial bearing of the air-bearing spindle is... R The nominal radial clearance is The working length of the bearing is from the inlet end z 1. Export end z 2 determines that the axial length is L At a given spindle speed n At that time, the local thickness of the spindle air film is expressed as: （2） in, Indicates the local thickness of the air film on the main shaft; z is the circumferential angle of rotation of the air-bearing main shaft in the main axis coordinate system; z is the axial coordinate in the main axis coordinate system. n Main spindle speed; This is the nominal radial clearance of the bearing; , The reference sections at the rotor's center of mass are respectively at the rotational speed. n Below the lower edge of the principal axis coordinate system X, Y The equilibrium displacement in the direction; Substituting equation (2) into the compressible Reynolds equation in gas lubrication theory and integrating over the bearing working area, the spindle speed is obtained. n Lower film rotor in principal axis coordinate system X , Y The total reaction force in the direction is used to determine the spindle speed based on the force balance condition. n The equilibrium displacement below ( , ); at the determined equilibrium displacement ( , Based on the reference position, the rotor is subjected to time-varying pressure. t Small perturbation displacement and The incremental film reaction forces in the X and Y directions in the principal axis coordinate system are obtained. and Assuming that the air film force-displacement relationship in the neighborhood of the equilibrium displacement can be linearized, the incremental air film reaction force is expressed as a linear combination of stiffness and damping terms, as shown in formula (3): （3） in, and In the principal coordinate system X , Y Incremental film reaction force in the direction; and These are the rotor center along the lower edge of the main shaft coordinate system. X , Y Small perturbation displacement in the direction; and These are the rotor center along the lower edge of the main shaft coordinate system. X , Y The velocity of the disturbance in the direction; The front radial bearing stiffness matrix is ​​a 2×2 dimensional matrix that varies with rotational speed n. The front radial bearing damping matrix is ​​a 2×2 dimensional matrix that varies with rotational speed n. For the rear radial bearing and thrust bearing of the spindle, the same analysis method as for the front radial bearing is used to obtain the stiffness matrix of the rear radial bearing. With damping matrix ; and thrust bearing axial stiffness matrix and the damping matrix is ; Step 2.1.2: In order to construct a dynamic model that can uniformly describe the constraint effect of the front radial bearing, the rear radial bearing, and the thrust bearing of the air-bearing spindle on the rotor, let the rotor reference section z at the rotor's center of mass be... m The axial distance to the front radial bearing section is Rotor reference section z m The axial distance to the rear radial bearing section is Based on the kinematics principle of rigid body small displacement, using the translational displacement and tilt angle of the rotor reference section as input, kinematic mapping relationships from these displacements to the local linear displacements at the front radial bearing, rear radial bearing, and thrust bearing are established. A geometric transformation matrix for the front radial bearing that describes this kinematic mapping is then constructed. Geometric transformation matrix of rear radial bearing and the geometric transformation matrix of the thrust bearing The local stiffness and damping of the front and rear radial bearings and the thrust bearing are equivalently applied to the rotor reference section coordinate system to obtain the overall equivalent stiffness matrix and damping matrix of the spindle system in the full speed domain, as shown in formulas (4) and (5): （4） （5） Among them, among them, This represents the overall equivalent stiffness matrix of the spindle system; This represents the geometric transformation matrix that maps the translational displacement of the rotor reference section to the local radial translational displacement of the front radial bearing. This represents the local stiffness matrix of the front radial bearing; This represents the geometric transformation matrix that maps the translational displacement of the rotor reference section to the local radial translational displacement of the rear radial bearing; This represents the local stiffness matrix of the rear radial bearing; This represents the geometric transformation matrix that maps the translational displacement of the rotor reference section to the local axial translational displacement of the thrust bearing; This represents the local stiffness matrix of the thrust bearing; This represents the overall equivalent damping matrix of the spindle system; This represents the local damping matrix of the rear radial bearing; This represents the local damping matrix of the front radial bearing; This represents the local damping matrix of the thrust bearing; , and The specific form is as follows: （6） in, Rotor reference section at the rotor's center of mass position z m The axial distance to the front radial bearing section; Rotor reference section z m The axial distance to the rear radial bearing section; Step 2.2: Based on the depth-of-cut modulation mechanism, the normal relative displacement between the tool tip and the workpiece and the cutting thickness characteristics caused by geometric errors are solved to obtain the machining load related to the rotation angle. Based on the ideal homogeneous transformation matrix and the spindle error transformation matrix obtained in the first step according to the machine tool kinematic chain topology, the definition of each coordinate system, and the ideal motion relationship, the ideal homogeneous transformation matrix from the bed coordinate system to the tool coordinate system will be used. Transformation matrix with principal axis error Matrix multiplication is performed to characterize the propagation relationship of geometric errors in the tool tip-workpiece normal direction, thus obtaining the relative normal displacement between the tool tip and workpiece caused by geometric errors. , is represented as: （7） in, This represents the normal relative displacement between the tool tip and the workpiece caused by geometric errors; This represents the ideal homogeneous transformation matrix from the bed coordinate system to the tool coordinate system; Represents the spindle error transformation matrix; Taking into account the feed motion during the cutting process, the depth of cut is given. and spindle rotation angle Actual chip thickness Represented as: （8） in, Indicates the angle The actual chip thickness below; For the back of the knife; This represents the normal relative displacement between the tool tip and the workpiece caused by geometric errors; Given a feed rate, the machining load vector acting on the spindle system in the spindle coordinate system is expressed as: （9） in, Represents the machining load vector; , , This indicates that the coordinates are located at the lower edge of the principal axis coordinate system. X , Y , Z The cutting force component in the direction; , This indicates that the coordinates are rotated around the principal axis in the coordinate system. X shaft and Y The cutting torque of the shaft; During machining, the cutting force components can be calculated using a specific cutting force model, along the lower axis of the spindle coordinate system. X Component of cutting force in the direction Represented as: （10） in, Indicates the lower edge of the principal axis coordinate system X The cutting force component in the direction; For the material along the principal axis coordinate system X Specific cutting force coefficient in the direction; f For feed rate; Indicates the angle The actual chip thickness below; During machining, the spindle coordinate system is rotated downwards. X shaft and Y The cutting torques of the shafts are expressed as follows: （11） （12） in, , These are respectively represented as the revolution around the principal axis coordinate system. X shaft and Y The cutting torque of the shaft; , , These are respectively represented as the lower edge of the principal axis coordinate system. X , Y , Z The cutting force component in the direction; , , This indicates the coordinates of the cutting point at the tool tip in the spindle reference coordinate system.

4. The method for predicting ultra-precision machining accuracy by coupling motion accuracy and dynamic characteristics according to claim 3, characterized in that, The third step specifically includes: Step 3.1: Based on the parameters obtained in the second step, establish the dynamic equations of the main shaft system; Based on the toolpath and machining program design, the rotation speed is... n Spindle rotation angle feed rate f and the amount of back cut Mapped to time t Changing variables , , as well as ; Based on bearing-rotor dynamics theory, considering the effects of spindle speed, system stiffness, and damping, the dynamic equations of the spindle system are established as follows: （13） Where M is the system quality matrix; This represents the real-time vibration response vector of the spindle system; , , These are the rotor reference sections along the lower edge of the principal shaft coordinate system. X , Y , Z Translational displacement in the direction of motion. , The rotor rotates in the principal axis coordinate system. X shaft and Y The inclination angle of the axis; and These represent the overall equivalent damping matrix and the overall equivalent stiffness matrix of the spindle system, respectively. Represents the machining load vector; Step 3.2: Based on the real-time vibration response of the spindle system calculated by the dynamic equation of the spindle system established in Step 3.1, establish a parameter correction mechanism for vibration response feedback; feed the real-time vibration response of the spindle back to the calculation process of air film support characteristics and machining load to realize the time-varying correction of air film support characteristics and machining load, and thus complete the dynamic update of dynamic equation parameters. Consider the real-time vibration response vector of the spindle system The local thickness of the spindle air film is dynamically corrected to obtain the corrected local thickness of the spindle air film, as shown in formula (14): （14） in, This indicates the local thickness of the spindle air film after correction; h 0 indicates the nominal radial clearance of the bearing; , The reference sections representing the rotor's center of mass position at time t along the lower axis of the principal axis coordinate system are respectively represented. X , Y The equilibrium displacement in the direction; , The reference sections representing the rotor's center of mass position at time t along the lower axis of the principal axis coordinate system are respectively represented. X , Y The vibration displacement component in the direction; The local air film thickness of the spindle air film after dynamic correction Substituting into step 2.1, the equivalent stiffness and damping characteristics of the air-bearing spindle considering the real-time vibration response feedback of the spindle system are solved. Consider the real-time vibration response vector of the spindle system The actual chip thickness is dynamically corrected to obtain the corrected actual chip thickness, as shown in formula (15): （15） in, This indicates the corrected actual chip thickness; The depth of cut at time t; This indicates the normal relative displacement between the tool tip and the workpiece; This represents the displacement of the air-bearing spindle vibration mapped onto the tool tip normal. The actual chip thickness after dynamic correction Substituting into step 2.2, the machining load vector considering the real-time vibration response feedback of the spindle system is obtained; Step 3.3: Based on the time-domain step integral algorithm, the dynamic equation of the main shaft system with coupled vibration response feedback is solved iteratively to obtain the steady-state real-time vibration displacement response vector of the main shaft that satisfies the periodic convergence criterion. Set the initial displacement vector of the spindle system based on the machining conditions. and initial velocity vector The time span for solving the problem is defined as follows: T total Step size is This ensures that the sampling frequency is sufficient to cover the high-frequency vibration characteristics of the spindle; Perform single-step dynamic response calculations and parameter feedback updates; at each time step t k The fourth-order Runge-Kutta method is used to solve the dynamic equations of the main shaft system in step 3.1; the resulting instantaneous vibration response vector of the main shaft is then used. The data is fed back in real time to step 3.2, synchronously updating the corrected local thickness of the spindle air film. and the corrected actual chip thickness The updated and corrected parameters of the main shaft dynamics equations are obtained. Based on the synchronously updated and corrected parameters of the spindle dynamics equations, the overall equivalent stiffness matrix of the spindle system at the current moment is recalculated. Damping matrix and machining load vector Substitute the updated matrix and vector into the next time step. t k+1 In the calculation of the dynamic equations of the main shaft system, continuous time-domain iteration is performed; to determine whether the dynamic response of the main shaft has reached a periodic steady-state convergence state, the residuals of the displacement vectors in adjacent periods are calculated. : （16） in, It represents the residual of the displacement vector within adjacent periods; For time integration variables; T = 60 / n This is the rotation cycle of the air-float main shaft; When the residual When the response is not less than the preset convergence threshold, the spindle vibration response at the end of the current cycle is used as the initial state for the next cycle iteration, and the dynamic response calculation and parameter feedback update continue; when the residual... When the convergence threshold is less than the preset threshold, the spindle dynamic response is considered to have reached a stable state; the spindle vibration response vector during the stable phase is then extracted. : （17） in, This represents the principal shaft vibration response vector during the steady-state phase. , , The instantaneous vibrational translational displacement of the principal axis reference section at time t in the principal axis coordinate system; , The instantaneous vibration tilt angles around the X and Y axes in the principal axis coordinate system; Step 3.4: Based on the spindle vibration response vector obtained in step 3.3 during the steady-state phase, the vibration response results obtained from solving the spindle system dynamic model are further transformed into a spindle vibration response transformation matrix. , used to characterize the transmission effect of the spindle vibration response in spatial coordinate transformation; The principal shaft vibration response vector during the steady phase The translational displacement components and instantaneous vibration tilt angle are written into the homogeneous transformation matrix, and the principal shaft vibration response transformation matrix is ​​defined. for: （18） in, This represents the transformation matrix of the spindle vibration response; , , The instantaneous vibrational translational displacement of the principal axis reference section at time t in the principal axis coordinate system; , To rotate in the principal axis coordinate system X shaft and Y The instantaneous vibration tilt angle of the shaft.

5. The method for predicting ultra-precision machining accuracy by coupling motion accuracy and dynamic characteristics according to claim 4, characterized in that, In step 3.3, the convergence threshold is preset according to the accuracy requirements of the spindle vibration response calculation.

6. The method for predicting ultra-precision machining accuracy by coupling motion accuracy and dynamic characteristics according to claim 5, characterized in that, The fourth step is specifically as follows: Step 4.1: Based on the principle of spatiotemporal mapping and superposition, characterize the actual cutting point of the tool tip after coupling; Based on machine tool kinematic chain topology and spindle error transformation matrix Combining the machining parameters determined in the second step with the spindle vibration response transformation matrix obtained in the third step Characterizes the actual cutting point of the tool tip after coupling. : （19） in, This represents the actual cutting point of the tool tip in the workpiece coordinate system; This represents the ideal homogeneous transformation matrix from the bed coordinate system to the principal axis coordinate system; Represents the spindle error transformation matrix; This represents the transformation matrix of the spindle vibration response; This represents the ideal homogeneous transformation matrix from the bed coordinate system to the tool coordinate system; The tool tip point in the tool coordinate system. The X-axis coordinate components of the tool tip in the tool coordinate system; Define the actual tool tip cutting point The spatial position components in the workpiece coordinate system are respectively , , ; at the radius of the blade tip arc is At that time, the instantaneous cutting envelope surface generated by the tool tip on the workpiece surface in the workpiece coordinate system Represented as: （20） in, This represents the instantaneous cutting envelope generated by the tool tip on the workpiece surface in the workpiece coordinate system. , , These represent the actual cutting points of the tool tip. In the workpiece coordinate system X,Y,Z Spatial location component; The radius of the blade tip arc; Step 4.2: Based on the cutting layer envelope mechanism, perform time-domain envelope processing on the instantaneous cutting envelope surface obtained in Step 4.1 to establish a surface morphology prediction model that couples geometric error and vibration response. Arbitrary coordinate points on the workpiece surface Final residual height at the location The final residual height is determined by the minimum height of all instantaneous cutting envelope surfaces covering the coordinate point within the machining time range. The surface morphology prediction model is shown in formula (21): （21） in, Indicates the final residual height; Indicates the instantaneous cutting envelope surface; T total This represents the total time span.

7. The method for predicting ultra-precision machining accuracy by coupling motion accuracy and dynamic characteristics according to claim 6, characterized in that, The fifth step specifically involves: Set standard cutoff wavelength λ c The final residual height was determined using a Gaussian low-pass filter. Convolution operations are performed to separate surface features with different frequency characteristics, including surface features and micro residual components. Extracting surface components: Removing the final residual height using a low-pass filter. High-frequency vibration and residual tool noise were analyzed to extract surface error components that characterize macroscopic geometric accuracy. ; Extracting microscopic residual components: from the final residual height Subtract the low-pass filtered surface shape component to extract the surface roughness component characterizing the micro-vibration response. : （23） in, Represents the surface roughness component; Indicates the final residual height; Represents the surface error components; Let the theoretical design surface be It represents any coordinate point on the workpiece surface. The theoretical design height at the location; to include surface error components With theoretical design surface Perform a comparison; The deviation of the surface error components from the ideal surface is characterized by the peak-to-valley value PV: （24） Where PV represents the peak-to-valley value; Represents the surface error components; Represents the theoretical design surface type; Surface roughness component Perform statistical calculations to obtain the arithmetic mean height. Within the defined sampling region A, the calculation formula is as follows: （25） in, The height is the arithmetic mean; A is the defined sampling area. Represents the surface roughness component; By extracting the surface shape error component and the surface roughness component, the peak and valley values ​​of the surface shape error of the machined surface and the arithmetic mean height are calculated, ultimately realizing the prediction of the surface morphology of ultra-precision turning by coupling the transmission of precision and dynamic characteristics.

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