Method for calculating critical rotating speed of air static pressure main shaft

Abstract

The invention relates to the field of air floating spindles, in particular to a critical rotating speed calculation method of an air static pressure spindle, which comprises the following steps: S1, calculating dynamic rigidity and dynamic damping of an air static pressure spindle bearing by utilizing a finite difference method according to structural parameters of the air static pressure spindle bearing; s2, calculating the critical perturbation frequency of the aerostatic spindle according to the dynamic stiffness and dynamic damping of the aerostatic spindle bearing; s3, according to the dynamic rigidity and the dynamic damping of the aerostatic spindle bearing, the inherent frequency of the aerostatic spindle core is calculated through a finite element method; and S4, a curve corresponding to the inherent frequency of the shaft core of the air static pressure main shaft and a curve corresponding to the critical perturbation frequency of the air static pressure main shaft are drawn under the rotating speed-perturbation frequency coordinate system through a drawing method, and when the two curves intersect, the abscissa where the intersection point is located is the critical rotating speed of the air static pressure main shaft. The method has the advantages of being high in prediction precision, simple in implementation method, suitable for air static pressure main shafts of different structures and the like.

Description

Technical Field

[0001] The present invention belongs to the technical field of aerostatic spindles, and particularly relates to a method for calculating the critical speed of an aerostatic spindle. Background Art

[0002] At present, due to its advantages of no friction, small temperature rise, high rotational accuracy, etc., the aerostatic spindle has become the main choice for the high-speed spindle system of ultra-precision machine tools. As the core component of an ultra-precision machine tool, the aerostatic spindle directly affects the machining accuracy of the machine tool. For the manufacturing of PCB circuit boards, it is necessary for the tool spindle to rotate at an ultra-high speed (rotation speed greater than 200,000 r / min) to ensure that a micro tool (diameter less than 0.5 mm) for micro milling and micro grinding maintains a certain linear speed to cut the workpiece. Under the condition of ultra-high-speed rotation, the vibration of the aerostatic spindle has an obvious impact on the machining quality of the workpiece. When the aerostatic spindle accelerates to near the critical speed, the vibration level of the aerostatic spindle increases sharply, often causing the aerostatic spindle to become unstable and the air film to fail. Seriously, it will lead to severe wear of the bearing and cause the bearing to be scrapped. Therefore, when designing an aerostatic spindle, it is necessary to accurately calculate the critical speed of the spindle and quickly pass through this critical speed during acceleration rotation, so as to improve the stability of the ultra-high-speed aerostatic spindle.

[0003] The invention patent application with the Chinese patent publication number CN116818324A, publication date December 19, 2023, and patent name "An Aerostatic Spindle Performance Detection Device", and the invention patent application with the Chinese patent publication number CN117516853A, publication date February 6, 2024, and patent name "A Modal Test System for High-Speed Aerostatic Spindles Based on Laser Shock Excitation" both measure the vibration and mode of the spindle through proprietary instruments and lack the calculation and measurement of the critical speed of the aerostatic spindle; the invention patent application with the Chinese patent publication number CN116907776A, publication date October 20, 2023, and patent name "A Method for Analyzing the Vibration Characteristics of a Whole High-Speed Motorized Spindle" analyzes the critical speed of an angular contact bearing high-speed motorized spindle; currently, there is a lack of relevant technical research on the calculation method of the critical speed of an aerostatic spindle. Summary of the Invention

[0004] In view of this, the present invention aims to provide a method for calculating the critical speed of an aerostatic spindle to solve the technical problem that there is a lack of relevant research on the calculation method of the critical speed of an aerostatic spindle in the prior art.

[0005] To achieve the above object, the technical solution of the present invention is realized as follows: A method for calculating the critical speed of an aerostatic spindle includes the following steps: S1: Calculate the dynamic stiffness and dynamic damping of the aerostatic spindle bearing based on its structural parameters using the finite difference method; S2: Calculate the critical perturbation frequency of the aerostatic spindle based on the dynamic stiffness and dynamic damping of the aerostatic spindle bearing; S3: Establish a finite element model of the aerostatic spindle using the finite element method based on the dynamic stiffness and dynamic damping of the aerostatic spindle bearing, and calculate the natural frequency of the spindle core of the aerostatic spindle; S4: Using the graphical method, respectively plot the curve corresponding to the natural frequency of the spindle core of the aerostatic spindle and the curve corresponding to the critical perturbation frequency of the aerostatic spindle in the rotational speed - perturbation frequency coordinate system. When the two curves intersect, the abscissa of the intersection point is the critical speed of the aerostatic spindle.

[0006] Furthermore, in step S1, the structural parameters of the aerostatic spindle bearing include the length of the bearing L , the inner diameter of the bearing D , the unilateral air film thickness of the bearing h m , then the process of calculating the critical perturbation frequency of the aerostatic spindle is as follows: Establish a dimensionless unsteady Reynolds equation: ; Among them, is the eddy current ratio, X is the horizontal coordinate of the dimensionless bearing, Z is the axial coordinate of the dimensionless bearing, P is the dimensionless air film pressure, Λ is the dimensionless bearing number, H is the dimensionless air film thickness, and τ is the dimensionless time coefficient; Solve the dimensionless unsteady Reynolds equation using the linear perturbation method. The spindle core center of the aerostatic spindle vortices at a frequency of w s , and the dimensionless spindle core position is expressed as: ; Among them, is the perturbation amount of the spindle core in the X direction, is the perturbation amount of the spindle core in the Y direction. The X direction is the horizontal direction, the Y direction is the vertical direction, is the time perturbation factor; Let ; Among them, is the perturbation amount of the air film, , P0 is the initial air film pressure, X0 is the initial horizontal position of the spindle core, and Y0 is the initial vertical position of the spindle core; The dimensionless air film thickness H is expressed as: ; Among them, is the angle in the circumferential direction of the bearing; Substitute the dimensionless film thickness H into the dimensionless unsteady Reynolds equation and neglect the higher-order terms to obtain the dimensionless steady Reynolds equation and the dimensionless perturbation Reynolds equation; The dimensionless steady Reynolds equation is: ; where P α is the steady-state film pressure, and H0 is the steady-state film thickness; Decompose the dimensionless steady Reynolds equation by using the central difference scheme through the finite difference method to obtain the steady-state film pressure P α and the steady-state film thickness H0; The term in the dimensionless perturbation Reynolds equation is: ; where P X is the first-order perturbation film pressure field in the X direction, P XX is the second-order perturbation film pressure field in the X direction, H X is the first-order perturbation film thickness in the X direction, H XX is the second-order perturbation film thickness in the X direction, and R is the bearing radius; The term in the dimensionless perturbation Reynolds equation is: ; Substitute the steady-state film pressure P α and the steady-state film thickness H0 into the term and term in the dimensionless perturbation Reynolds equation, and decompose the term and term of the dimensionless perturbation Reynolds by using the central difference scheme through the finite difference method to obtain P X and P XX under the perturbation condition; The term in the dimensionless perturbation Reynolds equation is: ; The term in the dimensionless perturbation Reynolds equation is: ; Decompose the term and term of the dimensionless perturbation Reynolds equation by using the central difference scheme through the finite difference method to obtain the first-order perturbation pressure field P Y in the Y direction and the second-order perturbation pressure field P YY under the perturbation condition; The P α and PX , P XX , P Y and P YY Calculate the dynamic stiffness and dynamic damping of the aerostatic spindle bearing: ; ; ; ; Among them, K XX and K YY are the main-direction dynamic stiffness, K XY and K YX are the cross dynamic stiffness, D XX and D YY are the main-direction dynamic damping, D XY and D YX are the cross dynamic damping.

[0007] Furthermore, in step S2, the aerostatic spindle whirls around the equilibrium position, and the dimensionless motion control equation in the X-Y coordinate system is: ; Among them, M is the rotor mass, — is the dimensionless symbol, is the first derivative symbol, is the second derivative symbol; By solving the dimensionless motion control equation, the critical mass and the critical perturbation ratio of the aerostatic spindle are obtained: ; ; Then the critical perturbation frequency of the aerostatic spindle is

[0008] Furthermore, in step S3, first, use the finite element method to establish the finite element model of the aerostatic spindle; secondly, use the Solid186 element to mesh the spindle core of the aerostatic spindle and assign material parameters to the spindle core of the aerostatic spindle; then, use Combine214 to establish elements at the position of the aerostatic spindle bearing and assign the calculated dynamic stiffness and dynamic damping of the aerostatic spindle bearing to the element attributes; finally, establish the boundary conditions and use the finite element software to automatically calculate the natural frequency of the spindle core of the aerostatic spindle.

[0009] Compared with the prior art, the present invention can achieve the following beneficial effects: Based on the dynamic analysis of the air static pressure spindle core and the accurate calculation of the dynamic performance of the air static pressure spindle bearings, the critical speed of the air static pressure spindle is calculated in the present invention. Therefore, the present invention has the characteristics of high prediction accuracy, simple implementation method, and applicability to air static pressure spindles of different configurations, thus providing a design direction for the optimal design of air static pressure spindles. BRIEF DESCRIPTION OF THE DRAWINGS

[0010] The drawings forming a part of the present invention are used to provide a further understanding of the present invention. The schematic embodiments and descriptions thereof of the present invention are used to explain the present invention and do not constitute an improper limitation to the present invention. In the drawings: Figure 1 is a schematic flow chart of the method for calculating the critical speed of the air static pressure spindle according to the embodiment of the present invention; Figure 2 is a schematic diagram of the calculation area of the finite difference method according to the embodiment of the present invention; Figure 3 is a schematic diagram of the calculation of the critical speed of the air static pressure spindle according to the embodiment of the present invention. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0011] In order to make the objectives, technical solutions and advantages of the present invention more clearly understood, the present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and do not constitute a limitation to the present invention.

[0012] It should be noted that, without conflict, the embodiments in the present invention and the features in the embodiments can be combined with each other.

[0013] In the description of the present invention, it should be understood that the terms "center", "longitudinal", "lateral", "upper", "lower", "front", "rear", "left", "right", "vertical", "horizontal", "top", "bottom", "inner", "outer", etc. indicate the orientation or positional relationship based on the orientation or positional relationship shown in the drawings, and are only for the convenience of describing the present invention and simplifying the description, rather than indicating or implying that the device or element referred to must have a specific orientation, be constructed and operated in a specific orientation, and therefore cannot be understood as a limitation to the present invention. In addition, the terms "first", "second", etc. are only used for descriptive purposes and cannot be understood as indicating or implying relative importance or implicitly indicating the quantity of the indicated technical features. Thus, the features defined with "first", "second", etc. can explicitly or implicitly include one or more of such features. In the description of the present invention, unless otherwise specified, the meaning of "plural" is two or more.

[0014] In the description of the present invention, it should be noted that unless otherwise clearly specified and defined, the terms "installation", "connection", and "coupling" shall be understood in a broad sense. For example, it may be a fixed connection, a detachable connection, or an integral connection; it may be a mechanical connection or an electrical connection; it may be a direct connection or an indirect connection through an intermediate medium, and it may be the communication inside two components. For those of ordinary skill in the art, the specific meanings of the above terms in the present invention can be understood according to specific circumstances.

[0015] The following will refer to Figures 1 - 3 and will describe the present invention in detail in conjunction with embodiments.

[0016] As Figures 1 - 3 shown, the embodiment of the present invention provides a method for calculating the critical speed of an aerostatic spindle, including the following steps: S1: According to the structural parameters of the aerostatic spindle bearing, use the finite difference method to calculate the dynamic stiffness and dynamic damping of the aerostatic spindle bearing.

[0017] The structural parameters of the aerostatic spindle bearing include: the length of the bearing L , the inner diameter of the bearing D , the unilateral air film thickness of the bearing h m .

[0018] There are throttle holes provided on the bearing, and the throttle holes are distributed in a double-row symmetric form in the axial direction of the bearing, and the number of throttle holes in each row is n, and the n throttle holes are evenly distributed in the circumferential direction of the bearing.

[0019] The process of using the finite difference method (Finite Differenee Method, FDM) to calculate the dynamic stiffness and dynamic damping of the aerostatic spindle bearing is as follows: Establish a dimensionless unsteady Reynolds equation: ; Among them, is the eddy current ratio, X is the horizontal direction coordinate of the dimensionless bearing, Z is the axial coordinate of the dimensionless bearing, P is the dimensionless air film pressure, Λ is the dimensionless bearing number, H is the dimensionless air film thickness, and τ is the dimensionless time coefficient.

[0020] Use the linear perturbation method to solve the dimensionless unsteady Reynolds equation. The center of the spindle core of the aerostatic spindle whirls at a w s frequency at the equilibrium position, and the dimensionless spindle core position can be expressed as: ; Among them, is the perturbation amount of the spindle core in the X direction, is the perturbation amount of the shaft core in the Y direction, the X direction is the horizontal direction, and the Y direction is the vertical direction. is the time perturbation factor.

[0021] Let ; where is the perturbation amount of the air film. , P0 is the initial pressure of the air film, X0 is the initial horizontal position of the shaft core, and Y0 is the initial vertical position of the shaft core.

[0022] The dimensionless air film thickness H can be expressed as: ; where is the angle in the circumferential direction of the bearing.

[0023] Substituting the dimensionless air film thickness H into the dimensionless unsteady Reynolds equation and neglecting the high-order terms, the dimensionless steady Reynolds equation and the dimensionless perturbation Reynolds equation can be obtained.

[0024] The dimensionless steady Reynolds equation is: ; where P α is the steady air film pressure, and H0 is the steady air film thickness.

[0025] By using the central difference scheme to decompose the dimensionless steady Reynolds equation with the FDM method, Figure 2 shows that the air film is unfolded into a plane and meshed. In the axial direction, it is divided into nx parts, and in the circumferential direction, it is divided into nz parts to obtain the steady air film pressure P α and the steady air film thickness H0.

[0026] The term in the dimensionless perturbation Reynolds equation is: ; where P X is the first-order perturbation air film pressure field in the X direction, P XX is the second-order perturbation air film pressure field in the X direction, H X is the first-order perturbation air film thickness in the X direction, H XX is the second-order perturbation air film thickness in the X direction, and R is the radius of the bearing.

[0027] The term in the dimensionless perturbation Reynolds equation is: ; Substitute the steady air film pressure P α and the steady air film thickness H0 into the Item and item, by using the finite difference method (FDM) with the central difference scheme to decompose the dimensionless perturbation Reynolds item and item, to obtain P X and P XX ; The item in the dimensionless perturbation Reynolds equation is: ; The item in the dimensionless perturbation Reynolds equation is: ; By using the FDM with the central difference scheme to decompose the item and item of the dimensionless perturbation Reynolds equation, to obtain the first-order perturbation pressure field P Y in the Y direction and the second-order perturbation pressure field P YY .

[0028] The P α , P X , P XX , P Y and P YY calculated are used to calculate the dynamic stiffness and dynamic damping of the aerostatic spindle bearing: ; ; ; ; where, K XX and K YY are the main-direction dynamic stiffness, K XY and K YX are the cross dynamic stiffness; D XX and D YY are the main-direction dynamic damping, D XY and D YX are the cross dynamic damping.

[0029] S2: According to the dynamic stiffness and dynamic damping of the aerostatic spindle bearing, calculate the critical perturbation frequency of the aerostatic spindle .

[0030] By using the dynamic stiffness and dynamic damping of the aerostatic spindle bearing, analyze the stability of the aerostatic spindle. The dimensionless motion control equation of the aerostatic spindle orbiting around the equilibrium position in the X - Y coordinate system (X is the horizontal direction, Y is the vertical direction) is: .

[0031] where \(M\) is the rotor mass, and \(-\) is a dimensionless symbol, is the symbol of the first derivative, is the symbol of the second derivative.

[0032] By solving the dimensionless motion control equation, the critical mass and the critical perturbation ratio of the aerostatic spindle are obtained: ; ; Then the critical perturbation frequency of the aerostatic spindle is

[0033] S3: According to the dynamic stiffness and dynamic damping of the aerostatic spindle bearing, use the finite element method to establish a finite element model of the aerostatic spindle, and calculate the natural frequency of the axis core of the aerostatic spindle.

[0034] First, use the finite element method (Finite Element Method, FEM) to establish a finite element model of the aerostatic spindle.

[0035] Secondly, use the Solid186 element to mesh the axis core of the aerostatic spindle, and assign material parameters such as density and elastic modulus to the axis core of the aerostatic spindle; Then, use Combine214 to establish an element at the position of the aerostatic spindle bearing, and assign the dynamic stiffness and dynamic damping of the aerostatic spindle bearing calculated in step S1 to the element attributes; Finally, establish the atmospheric boundary conditions of the finite element model of the aerostatic spindle, and use the finite element software to automatically calculate the natural frequency of the axis core of the aerostatic spindle.

[0036] S4: Using the graphical method, draw the curves corresponding to the natural frequency of the axis core of the aerostatic spindle and the curves corresponding to the critical perturbation frequency of the aerostatic spindle in the rotational speed - perturbation frequency coordinate system. When the two curves intersect, the abscissa of the intersection point is the critical speed of the aerostatic spindle.

[0037] When the natural frequency of the axis core of the aerostatic spindle is equal to the critical perturbation frequency of the aerostatic spindle, the damping of the aerostatic spindle is zero, the vibration of the aerostatic spindle rises steeply, and instability occurs. The rotational speed when the natural frequency and the critical perturbation frequency are equal is the critical speed of the aerostatic spindle. Using the graphical method, draw the curvature of the natural frequency and the curvature of the critical perturbation frequency in the rotational speed - perturbation frequency coordinate system (the abscissa is the rotational speed of the aerostatic spindle, and the ordinate is the perturbation frequency of the aerostatic spindle). The abscissa of the intersection point of the two curves is the critical speed of the aerostatic spindle. Taking the third - order natural frequency of the axis core of the aerostatic spindle as an example, such asFigure 3 as shown

[0038] The present invention innovatively proposes to predict the critical speed of the air-bearing spindle by accurately calculating the dynamic stiffness and dynamic damping of the air-bearing. The present invention has the characteristics of high prediction accuracy, simple implementation method, and applicability to air-bearings of different configurations, realizing efficient and accurate calculation of the critical speed of the air-bearing spindle, thereby providing a design direction for the optimal design of the air-bearing spindle.

[0039] It should be understood that various forms of the processes shown above can be used, steps can be reordered, added, or deleted. For example, the steps recited in the disclosure of the present invention can be executed in parallel, sequentially, or in a different order, as long as the desired results of the technical solution disclosed in the present invention can be achieved, and no limitation is imposed herein.

[0040] The above specific embodiments do not constitute a limitation to the protection scope of the present invention. Those skilled in the art should understand that various modifications, combinations, sub-combinations, and substitutions can be made according to design requirements and other factors. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention shall be included within the protection scope of the present invention.

Claims

1. A method for calculating the critical speed of an aerostatic spindle, characterized in that, The steps are as follows: S1: According to the structural parameters of the aerostatic spindle bearing, use the finite difference method to calculate the dynamic stiffness and dynamic damping of the aerostatic spindle bearing; S2: Calculate the critical perturbation frequency of the aerostatic spindle according to the dynamic stiffness and dynamic damping of the aerostatic spindle bearing; S3: According to the dynamic stiffness and dynamic damping of the aerostatic spindle bearing, use the finite element method to establish a finite element model of the aerostatic spindle and calculate the natural frequency of the spindle core of the aerostatic spindle; S4: Use the graphical method to respectively plot the curve corresponding to the natural frequency of the spindle core of the aerostatic spindle and the curve corresponding to the critical perturbation frequency of the aerostatic spindle in the rotational speed-perturbation frequency coordinate system. When the two curves intersect, the abscissa of the intersection point is the critical speed of the aerostatic spindle.

2. The critical speed calculation method of the aerostatic spindle according to claim 1, characterized in that In step S1, the structural parameters of the aerostatic spindle bearing include the length of the bearing L , the inner diameter of the bearing D , the unilateral air film thickness of the bearing h m , then the process of calculating the critical perturbation frequency of the aerostatic spindle is as follows: Establish a dimensionless unsteady Reynolds equation: ； wherein, is the swirl ratio, X is the horizontal coordinate of the dimensionless bearing, Z is the axial coordinate of the dimensionless bearing, P is the dimensionless film pressure, Λ is the dimensionless bearing number, H is the dimensionless film thickness, and τ is the dimensionless time coefficient; The dimensionless unsteady Reynolds equation is solved by the linear perturbation method, and the center of the air static pressure spindle vortexes at the equilibrium position with the w s frequency. The dimensionless spindle position is expressed as: ； Among them, is the perturbation amount of the shaft center in the X direction, is the perturbation amount of the shaft center in the Y direction. The X direction is the horizontal direction, and the Y direction is the vertical direction. is the time perturbation factor; Let ; Among them, is the perturbation amount of the air film, , P0 is the initial pressure of the air film, X0 is the initial horizontal position of the shaft center, and Y0 is the initial vertical position of the shaft center; The dimensionless film thickness H is expressed as: ； Among them, is the angle in the circumferential direction of the bearing; Substitute the dimensionless film thickness H into the dimensionless unsteady Reynolds equation and neglect the higher-order terms to obtain the dimensionless steady Reynolds equation and the dimensionless perturbation Reynolds equation; The dimensionless steady Reynolds equation is: ； where P α is the steady-state gas film pressure, and H0 is the steady-state gas film thickness; The dimensionless steady Reynolds equation is decomposed by using the central difference scheme through the finite difference method to obtain the steady gas film pressure P α and the steady gas film thickness H0; The terms in the dimensionless perturbation Reynolds equation are: as follows: ； Among them, P X is the first-order perturbation air film pressure field in the X direction, P XX is the second-order perturbation air film pressure field in the X direction, H X is the first-order perturbation air film thickness in the X direction, H XX is the second-order perturbation air film thickness in the X direction, and R is the bearing radius; The terms in the dimensionless perturbation Reynolds equation are: ​ ； Substitute the steady-state gas film pressure P α and the steady-state gas film thickness H0 into the terms and in the dimensionless perturbation Reynolds equation. Decompose the terms and of the dimensionless perturbation Reynolds by using the central difference scheme through the finite difference method to obtain P X and P XX under the perturbation condition; The terms in the dimensionless perturbation Reynolds equation are: ​ ； The term in the dimensionless perturbation Reynolds equation is: ； Decompose the terms of the dimensionless perturbation Reynolds equation by using the central difference scheme through the finite difference method and terms to obtain the first-order perturbation pressure field P Y in the Y direction and the second-order perturbation pressure field P YY ; The calculated P α , P X , P XX , P Y and P YY Calculate the dynamic stiffness and dynamic damping of the aerostatic spindle bearing: ； ； ； ； Among them, K XX and K YY are the main-direction dynamic stiffness, K XY and K YX are the cross dynamic stiffness, D XX and D YY are the main-direction dynamic damping, D XY and D YX are the cross dynamic damping.

3. The critical speed calculation method of the aerostatic spindle according to claim 1, characterized in that In step S2, the aerostatic spindle whirls around the equilibrium position, and the dimensionless motion control equation in the X-Y coordinate system is: ； where M is the rotor mass, — is a dimensionless symbol, is the symbol of the first derivative, is the symbol of the second derivative; By solving the dimensionless motion control equations, the critical mass and the critical perturbation ratio of the aerostatic spindle are obtained and the critical perturbation ratio : ； ； The critical perturbation frequency of the aerostatic spindle .

4. The critical speed calculation method of the aerostatic spindle according to claim 1, characterized in that In step S3, first, use the finite element method to establish a finite element model of the aerostatic spindle; secondly, use the Solid186 element to mesh the spindle core of the aerostatic spindle and assign material parameters to the spindle core of the aerostatic spindle; then, use the Combine214 to establish an element at the position of the aerostatic spindle bearing and assign the calculated dynamic stiffness and dynamic damping of the aerostatic spindle bearing to the element attributes; finally, establish the boundary conditions and use the finite element software to automatically calculate the natural frequency of the spindle core of the aerostatic spindle.

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