A Method for Grinding Spherical Surfaces with a Cup-Shaped Circular Arc Grinding Wheel Based on a Cradle-Type Five-Axis Machine Tool and for Error Compensation

Abstract

This invention discloses a method for grinding spherical surfaces using a cup-shaped arc grinding wheel on a cradle-type five-axis machine tool, and a method for error compensation. The method includes: acquiring the required dimensions and process parameters for grinding the spherical surface; acquiring the motion trajectory of the workpiece grinding the spherical surface; using a cradle-type five-axis machine tool, grinding the end face of the workpiece into a spherical surface using a cup-shaped arc grinding wheel according to the motion trajectory of the workpiece grinding the spherical surface; after processing, detecting the surface shape accuracy of the spherical surface, measuring the surface shape error of the spherical surface, calculating the position error of the cup-shaped arc grinding wheel, compensating for and adjusting the processing position of the cup-shaped arc grinding wheel, recalculating the motion trajectory of the workpiece grinding the spherical surface, and performing further processing until the surface shape error of the spherical surface is within the allowable range, thus completing the processing. This method improves the efficiency of spherical surface processing, increases the accuracy of the spherical radius, and solves the problem of limited accuracy in tool wear control.

Description

Technical Field

[0001] This invention belongs to the field of precision machining technology, and relates to a method for grinding spherical surfaces with a cup-shaped arc grinding wheel based on a cradle-type five-axis machine tool and for error compensation. Background Technology

[0002] Optical components are an indispensable part of optical imaging systems, widely used in aerospace, defense, medicine, and daily life. Optical components typically require rough grinding, fine grinding, and polishing processes. Grinding primarily removes material and shapes the blank, affecting the workpiece's forming accuracy. Polishing mainly removes damaged layers and reduces surface roughness, but the removal amount is minimal and time-consuming. Therefore, improving grinding accuracy can significantly reduce the time required for the subsequent polishing process. Optical components are usually made of hard and brittle materials, and grinding is generally used to process these materials. However, ordinary machining methods are not very accurate and are prone to subsurface damage. Ultra-precision grinding is a machining method that uses superhard abrasive wheels in conjunction with high-performance machine tools, achieving higher machining accuracy than ordinary grinding. Cup-shaped arc grinding wheels have a special shape and are highly efficient for machining spherical surfaces, thus being widely used in spherical grinding. Furthermore, spherical grinding is also widely used in aspherical surface manufacturing processes due to its ability to quickly remove large amounts of material, and it can also compensate for tool setting errors in spherical grinding.

[0003] Currently, the precision requirements for optical components in fields such as aerospace and advanced instruments are becoming increasingly stringent, but current spherical grinding methods cannot achieve a high-precision and high-efficiency machining process. Existing spherical forming methods have low machining efficiency, and the accuracy of the spherical radius is limited due to factors such as tool setting and tool wear control. Summary of the Invention

[0004] To address the problems existing in the prior art, this invention provides a method for grinding spherical surfaces using a cup-shaped arc grinding wheel on a cradle-type five-axis machine tool, along with error compensation. This method improves the efficiency of spherical surface machining, enhances the accuracy of the spherical radius, and solves the problem of limited accuracy in tool wear control.

[0005] This invention is achieved through the following technical solution:

[0006] A method for grinding spherical surfaces with a cup-shaped arc grinding wheel and for error compensation includes the following steps:

[0007] S1, obtain the various dimensions and process parameters required for grinding the spherical surface, and obtain the motion trajectory of the workpiece grinding the spherical surface;

[0008] S2, based on a cradle-type five-axis machine tool, according to the motion trajectory of the workpiece grinding the spherical surface, the workpiece to be processed changes with the swing angle of the cradle axis of the cradle-type five-axis machine tool, and the end face of the workpiece to be processed is ground into a spherical workpiece by a cup-shaped arc grinding wheel; the processing types of the spherical workpiece include convex spherical surface and concave spherical surface.

[0009] S3. After the machining is completed, the surface accuracy of the spherical workpiece is checked, the surface error of the spherical workpiece is measured, and the position error of the cup-shaped arc grinding wheel is calculated.

[0010] S4. Based on the position compensation amount of the cup-shaped arc grinding wheel, compensate and adjust the machining position of the cup-shaped arc grinding wheel, recalculate the motion trajectory of the workpiece grinding the spherical surface, and re-machine the spherical workpiece until the surface shape error of the spherical workpiece is within the allowable range, and the machining is completed.

[0011] Preferably, in S1, the dimensions and process parameters required for grinding the spherical surface are obtained, specifically including:

[0012] Diameter of the workpiece to be processed and height The center thickness of the spherical workpiece to be machined and spherical radius The distance between the center of the cradle's swing and the bottom surface of the workpiece when the workpiece is mounted on a cradle-type five-axis machine tool. The distance between the cradle shaft swing center and the center of the spherical workpiece to be machined. The rate at which the radius of a spherical workpiece decreases during grinding. The median diameter of the cup-shaped arc grinding wheel and the small radius of the abrasive region .

[0013] Preferably, the motion trajectory of the workpiece grinding the spherical surface in S1 is as follows:

[0014] When machining a convex spherical surface, the formula for the motion trajectory of the workpiece grinding the spherical surface is:

[0015]

[0016]

[0017] ;

[0018] In the formula: The radius of the envelope sphere formed by the truncated circle of the cup-shaped arc grinding wheel end face rotating at the initial grinding position and the rotation of the workpiece to be processed; f The rate at which the radius of a spherical workpiece decreases during grinding; The median diameter of the cup-shaped arc grinding wheel, The radius of the small arc of the abrasive grain area of ​​the cup-shaped arc grinding wheel; Grinding time; This refers to the Y-axis position of a cradle-type five-axis machine tool. This refers to the Z-axis position of a cradle-type five-axis machine tool. This refers to the A-axis position of a cradle-type five-axis machine tool. The distance between the center of the cradle shaft's swing and the center of the spherical workpiece;

[0019] Wherein, the radius of the envelope sphere is for:

[0020] ;

[0021] In the formula: Let be the radius of the sphere of the spherical workpiece. The center thickness of the spherical workpiece The diameter of the workpiece to be processed;

[0022] When machining a concave spherical surface, the formula for the motion trajectory of the workpiece grinding the spherical surface is:

[0023]

[0024]

[0025]

[0026] In the formula: The height of the workpiece to be processed. The distance between the center of the cradle's swing and the bottom surface of the workpiece when the workpiece is mounted on a cradle-type five-axis machine tool; This refers to the Y-axis position of a cradle-type five-axis machine tool. This refers to the Z-axis position of a cradle-type five-axis machine tool. This refers to the A-axis position of a cradle-type five-axis machine tool. Let be the radius of the envelope sphere formed by the truncated circle of the cup-shaped arc grinding wheel end face rotating at the initial grinding position and the rotation of the workpiece to be processed;

[0027] Wherein, the radius of the envelope sphere is for:

[0028] ;

[0029] In the formula: Let be the radius of the sphere of the spherical workpiece. The center thickness of the spherical workpiece The height of the workpiece to be processed.

[0030] Preferably, the specific processing procedure for S2 is as follows:

[0031] Based on a cradle-type five-axis machine tool, the workpiece to be processed is mounted on a rotary table, and a cup-shaped arc grinding wheel is mounted on the spindle. According to the motion trajectory of the workpiece grinding the spherical surface, the workpiece to be processed swings with the cradle axis of the cradle-type five-axis machine tool and rotates around its own axis. The spindle drives the cup-shaped arc grinding wheel to rotate to form a truncated circular grinding area. The swing angle of the cradle axis changes continuously during the grinding process. Through the grinding area of ​​the grinding wheel, it contacts the workpiece to be processed, and grinds the end face of the workpiece into a spherical workpiece.

[0032] Preferably, during the machining process, at the initial grinding position, the truncated circle of the end face of the cup-shaped arc grinding wheel rotates and forms an enveloping spherical surface with the rotation of the workpiece, and the radius of the enveloping spherical surface... With speed f With grinding time t The material is uniformly reduced or increased, and at any time at any position on the envelope sphere, the material removal method is bounded by the envelope sphere and shrinks or expands towards the workpiece. In any radial direction, the thickness of the removed material is uniform.

[0033] Preferably, the positional error of the cup-shaped arc grinding wheel in S3 is calculated as follows:

[0034] When the spherical workpiece is a convex spherical surface, the positional error of the cup-shaped circular arc grinding wheel is calculated using the following formula:

[0035]

[0036]

[0037] When the spherical workpiece is a concave spherical surface, the positional error of the cup-shaped arc grinding wheel is calculated using the following formula:

[0038]

[0039]

[0040] In the formula: The distance by which the center of the arc of the cup-shaped arc grinding wheel deviates from its ideal position in the horizontal Y direction. The distance by which the center of the arc of the cup-shaped arc grinding wheel deviates from its ideal position in the vertical Z direction. Let be the radius of the spherical workpiece after grinding. Let be the radius of the circular protrusion in the central region of the spherical workpiece. L is the radius of the small arc of the abrasive grain region of the cup-shaped arc grinding wheel; L is the distance from the center of the cup mouth of the cup-shaped arc grinding wheel to the center of the envelope spherical surface.

[0041] Preferably, in S2, at the initial machining position, there is an angle between the axis of the cup-shaped arc grinding wheel and the axis of the workpiece to be machined. That is, the cradle axis swing angle The grinding area of ​​the cup-shaped arc grinding wheel makes point contact with the end face of the workpiece; at the end of the machining process, there is an angle between the axis of the cup-shaped arc grinding wheel and the axis of the workpiece. That is, the cradle axis swing angle is The grinding area of ​​the cup-shaped arc grinding wheel is in line contact with the spherical surface of the workpiece.

[0042] Preferably, when machining a convex spherical workpiece, the mean diameter of the cup-shaped arc grinding wheel should meet the following conditions:

[0043] ;

[0044] When machining a concave spherical workpiece, the mean diameter of the cup-shaped arc grinding wheel should meet the following conditions:

[0045] ;

[0046] In the formula: The center thickness of the spherical workpiece Let be the radius of the sphere of the spherical workpiece. The radius of the small arc in the abrasive region of the cup-shaped arc grinding wheel. The angle between the line connecting the edge of the spherical surface to the center of the sphere and the axis. This refers to the median diameter of the cup-shaped arc grinding wheel.

[0047] Preferably, when machining a convex spherical workpiece, the cup depth of the cup-shaped arc grinding wheel is... Should meet:

[0048] , ( )

[0049] , ( )

[0050] In the formula: Let be the radius of the sphere of the spherical workpiece. The radius of the small arc in the abrasive region of the cup-shaped arc grinding wheel. The angle between the line connecting the edge of the spherical surface to the center of the sphere and the axis. The median diameter of the cup-shaped arc grinding wheel, This refers to the swing angle of the cradle shaft after processing.

[0051] A cradle-type five-axis machine tool is used for machining based on the aforementioned cup-shaped circular arc grinding wheel for truncated circular grinding of spherical surfaces and error compensation method.

[0052] Compared with the prior art, the present invention has the following beneficial technical effects:

[0053] This invention proposes a method for truncated circular grinding of spherical surfaces using a cup-shaped arc grinding wheel on a cradle-type five-axis machine tool, along with error compensation. By controlling the motion trajectory of the workpiece and the cup-shaped arc grinding wheel during the machining process, an envelope trajectory is achieved to remove material from the workpiece. The amount of material removed is uniform in any radial direction. The envelope spherical surface gradually approaches the desired spherical surface with the same center, ensuring uniform force at each contact point during machining and uniform material removal from the workpiece. This improves machining consistency and stability, as well as accuracy and tool durability.

[0054] Furthermore, based on the established relationship model between machining error and grinding wheel tool setting or wear error, the radius of the circular protrusion in the central region of the spherical workpiece after machining is detected. and the actual radius of the sphere Calculating and compensating for grinding wheel position errors reduces the impact of errors and improves machining accuracy.

[0055] Furthermore, the method proposed in this invention can process spherical surfaces of various sizes using grinding wheels of various sizes. Attached Figure Description

[0056] Figure 1 This is a schematic diagram of the CNC machine tool grinding installation according to the present invention;

[0057] Figure 2 This is a slant view of the CNC machine tool grinding installation according to the present invention;

[0058] Figure 3 This is a schematic diagram showing the dimensions and installation dimensions of the convex spherical workpiece;

[0059] Figure 4 This is a schematic diagram of the initial position during machining of a convex spherical surface;

[0060] Figure 5 This is a schematic diagram showing the finished position of the convex spherical surface machining process;

[0061] Figure 6 This is a schematic diagram of the error of the convex sphere enclosing the spherical surface;

[0062] Figure 7 This is a schematic diagram of machining errors for convex spherical surfaces;

[0063] Figure 8 This is a schematic diagram showing the dimensions and installation dimensions of the concave spherical workpiece;

[0064] Figure 9 This is a schematic diagram of the initial position during the machining of a concave spherical surface;

[0065] Figure 10 This is a schematic diagram of the error of the concave sphere enclosing the spherical surface;

[0066] Figure 11 This is a schematic diagram of machining errors for a concave spherical surface;

[0067] Figure 12 This is a schematic diagram of the minimum cutting tool radius for a convex spherical surface;

[0068] Figure 13 This is a schematic diagram illustrating the depth requirements of a cup-shaped arc grinding wheel;

[0069] Figure 14 This is a schematic diagram of the minimum cutting tool radius for a concave spherical surface;

[0070] In the attached diagram, 1 is the Y-axis moving platform; 2 is the Z-axis moving platform; 3 is the spindle; 4 is the cradle; 5 is the turntable; 6 is the workpiece; 7 is the cup-shaped arc grinding wheel; and 8 is the X-axis moving platform. Detailed Implementation

[0071] The present invention will be further described in detail below with reference to specific embodiments. These descriptions are for explanation purposes only and are not intended to limit the scope of the invention.

[0072] To enable those skilled in the art to better understand the present invention, the technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort should fall within the scope of protection of the present invention.

[0073] This invention discloses a method for grinding spherical surfaces using a cup-shaped circular arc grinding wheel on a cradle-type five-axis machine tool, along with error compensation. In a single grinding operation, the workpiece oscillates in the YOZ plane while simultaneously rotating around its own axis. The cup-shaped circular arc grinding wheel moves in tandem with the spindle within the YOZ plane, and the spindle also drives the cup-shaped circular arc grinding wheel to rotate. These two rotational motions together form an envelope spherical surface, the radius of which is the radius of the envelope spherical surface. During the machining process, the radius of the envelope spherical surface increases with velocity. f The material removal process decreases uniformly over time, with the material removal direction consistently pointing towards the center of the sphere along the radius. The thickness of the removed material is uniform until, after a certain period, the end face of the original cylindrical blank becomes a spherical surface with the required radius. After grinding, the spherical surface error is checked. Based on the established relationship model between the spherical surface error and the positional error of the cup-shaped arc grinding wheel, the actual spherical workpiece radius is measured. The radius of the circular protrusion in the central region of the spherical workpiece Calculate position error and Then compensate, and process again using the compensated trajectory until the dimensions meet the requirements.

[0074] Specifically: such as Figures 1 to 2As shown, this invention employs a cradle-type five-axis machine tool. The CNC machine tool can achieve three-axis movement (XYZ), as well as oscillation around the X direction and rotation around the Z direction. Specifically, it includes a Y-axis moving platform, a Z-axis moving platform, a spindle, a cradle, a rotary table, and an X-axis moving platform. The cradle can oscillate around the X direction. A rotary table is mounted on the cradle and can rotate around the Z direction. Spherical grinding is achieved by using a cup-shaped arc grinding wheel along a linkage trajectory of three axes: two linear axes and one cradle axis. The uniformity and consistency of material removal are achieved by continuously changing the oscillation angle of the cradle axis during grinding. Grinding error compensation is achieved by detecting the radius of the ground spherical workpiece and the size of the circular protrusion in the center area of ​​the workpiece, calculating the positional error of the cup-shaped arc grinding wheel, and changing the linkage trajectory. In the machine tool's components, when a person faces the machine tool, the left-right movement is the X-axis, the back-and-forth movement is the Y-axis, and the up-and-down movement is the Z-axis. The cradle axis, also called the A-axis, is the rotation axis, and its axis is parallel to the X-axis.

[0075] Specifically, the following steps are included:

[0076] Step 1) Before grinding, obtain the required dimensions and process parameters for grinding the spherical surface, including: the diameter of the cylindrical blank to be processed. and height The center thickness of the spherical workpiece to be machined and spherical radius The distance between the center of the cradle's swing and the bottom surface of the cylindrical blank when it is mounted on the machine tool. The distance between the center of the cradle's swing and the center of the spherical workpiece to be machined. The rate at which the radius of a spherical workpiece decreases during grinding. The median diameter of the cup-shaped arc grinding wheel and the small radius of the abrasive region Based on the above parameters, the spindle motion trajectory of the oscillating spindle machine tool, i.e., the X-axis position, is derived. Z-axis position Position relative to B-axis Relative to time The relationship of positional changes.

[0077] For a convex spherical surface, the Y-axis position Z-axis position Position relative to A-axis Relative to time The positional changes are as follows:

[0078]

[0079]

[0080]

[0081] in, and The formula contains The term is for simplifying the formula. It is derived from the dimensions and process parameters obtained above, and is used to simplify the formula. The physical meaning is that the radius of the envelope sphere formed by the rotating truncated circle of the cup-shaped arc grinding wheel end face at the initial grinding position and the rotating cylindrical blank is... , Calculated by the following formula:

[0082]

[0083] For a concave sphere, the Y-axis position Z-axis position Position relative to A-axis Relative to time The positional changes are as follows:

[0084]

[0085]

[0086]

[0087] in, and The formula contains The term is for simplifying the formula. It is derived from the dimensions and process parameters obtained above, and is used to simplify the formula. The physical meaning is that the radius of the envelope sphere formed by the rotating truncated circle of the cup-shaped arc grinding wheel end face at the initial grinding position and the rotating cylindrical blank is... , Calculated by the following formula:

[0088]

[0089] Step 2) Mount the cylindrical blank to be processed on the rotary platform. During the grinding process, the cylindrical blank oscillates with the cradle shaft and rotates around its own axis. Mount the cup-shaped arc grinding wheel on the machine tool spindle, and its rotation forms a truncated circular grinding area. The linear axis and the cradle shaft move along the trajectory in Step 1. The oscillation angle of the cradle shaft changes continuously during the grinding process. The abrasive grain area of ​​the grinding wheel contacts the cylindrical blank, grinding the end face of the cylindrical blank into a spherical surface.

[0090] Step 3) After one machining operation, measure the surface accuracy of the spherical workpiece, including the spherical error, such as the radius of the ground spherical workpiece. The radius of the circular protrusion in the central region of the spherical workpiece Calculate the positional error of the cup-shaped arc grinding wheel. and Each compensation and The linkage trajectory is adjusted, the machining trajectory is recalculated, and machining continues until the surface shape error of the spherical workpiece is within the allowable range, at which point machining is complete. The aforementioned cup-shaped arc grinding wheel position error refers to the distance by which the center of the arc of the cup-shaped arc grinding wheel deviates from its ideal position in both the horizontal Y direction and the vertical Z direction. , ,cause and There are many reasons for this, including tool setting error, grinding wheel wear error, and grinding wheel manufacturing error.

[0091] For a convex spherical surface and Calculated by the following formula:

[0092]

[0093]

[0094] For a concave spherical surface and Calculated by the following formula:

[0095]

[0096]

[0097] Furthermore, a trajectory model and method for grinding spherical surfaces using a cradle-type five-axis machine tool and a cup-shaped circular arc grinding wheel were derived and calculated. In grinding using a cradle-type five-axis machine tool, the workpiece is mounted on a rotary table and rotates with the rotary table, while the grinding wheel is mounted on the spindle and translates with the spindle. The workpiece oscillates in the YOZ plane and simultaneously rotates around its own axis. The spindle drives the cup-shaped circular arc grinding wheel to rotate. Due to its rotation, the cup-shaped circular arc grinding wheel forms a truncated circular grinding area on its end face. The truncated circular grinding area, combined with the rotational motion of the workpiece, forms the grinding area that envelops the spherical surface. The cup-shaped circular arc grinding wheel moves in tandem with the workpiece in the YOZ plane to complete the forming of the matching spherical surface. The tandem trajectory causes the enveloping spherical surface to shrink or expand on the workpiece. During the translational motion of the cup-shaped circular arc grinding wheel, the workpiece material is continuously removed along the boundary of the enveloping spherical surface until the workpiece oscillates at a certain angle, and the end face of the original cylindrical workpiece becomes a spherical surface.

[0098] Furthermore, at the initial machining position, there is an angle between the axis of the cup-shaped arc grinding wheel and the axis of the cylindrical blank. That is, the cradle swing angle is The abrasive grains of the cup-shaped arc grinding wheel make point contact with the end face of the cylindrical blank; at the end of the machining process, the cylindrical blank has been ground into a spherical workpiece, and the axis of the cup-shaped arc grinding wheel and the axis of the spherical workpiece are at an angle. That is, the cradle swing angle is The abrasive grains of the cup-shaped arc grinding wheel are in line contact with the spherical surface of the workpiece.

[0099] Furthermore, in the cross-section through the YOZ plane, the center of the small arc of the cup-shaped arc grinding wheel abrasive grain region is always on the workpiece axis.

[0100] Furthermore, during the machining process, the radius of the enveloping sphere decreases or increases uniformly with time at a speed f. At any time and at any position on the enveloping sphere, the material removal method is to shrink or expand towards the workpiece with the enveloping sphere as the boundary. In any radial direction, the thickness of the removed material is uniform, so as to reduce the excessive local grinding force of the grinding wheel or workpiece, which may cause tool wear or workpiece damage.

[0101] Furthermore, based on the established relationship model between spherical error and cup-shaped arc grinding wheel position error, the actual spherical workpiece radius is measured. The radius of the circular protrusion in the central region of the spherical workpiece Calculate position error and And compensation.

[0102] Furthermore, the spherical surface formed by grinding can be a convex spherical surface or a concave spherical surface.

[0103] Furthermore, for the center thickness after grinding... The radius of the sphere is The angle between the line connecting the edge of the sphere to the center of the sphere and the axis is . For workpieces of this type, the permissible mean diameter of the cup-shaped arc grinding wheel should meet the following requirements. For convex spherical surfaces, a "+" sign is used, and for concave spherical surfaces, a "-" sign is used. When machining convex spherical surfaces, the allowable depth of the cup-shaped arc grinding wheel should meet the following requirements.

[0104] , ( )

[0105] , ( )

[0106] Furthermore, the grinding process uses a cradle-type five-axis machine tool, where the workpiece swings with the cradle axis, and the cup-shaped arc grinding wheel remains vertical and performs translational motion.

[0107] Example 1

[0108] Specifically, the steps include:

[0109] Step 1), as Figure 1 , Figure 2 As shown, the cylindrical workpiece to be machined is mounted on the rotary table, and the cup-shaped arc grinding wheel is mounted on the spindle. Tool setting is performed according to the grinding wheel mounting dimensions and workpiece mounting dimensions. Each axis is set to its initial position, the rotary table speed and spindle speed are set, and the motion parameters and grinding time of the XYZ axes and cradle axis are input. t Spherical processing is achieved through envelope trajectory. After time t, the cradle swings to angle. The cylindrical workpiece is machined into a spherical workpiece, thus completing the spherical machining.

[0110] Before grinding, obtain the necessary dimensions and process parameters for grinding the spherical surface, including: the diameter of the cylindrical blank to be processed. and height The center thickness of the spherical workpiece to be machined and spherical radius The distance between the center of the cradle's swing and the bottom surface of the cylindrical blank when it is mounted on the machine tool. The distance between the center of the cradle's swing and the center of the spherical workpiece to be machined. The rate at which the radius of a spherical workpiece decreases during grinding. The median diameter of the cup-shaped arc grinding wheel and the small radius of the abrasive region Based on the above parameters, derive the spindle motion trajectory of the cradle-type five-axis machine tool, i.e., the Y-axis position. Z-axis position Position relative to A-axis Relative to time The positional changes are as follows:

[0111] The cup-shaped arc grinding wheel has a median diameter of [missing information]. The radius of the arc of the ground part of its end face is .

[0112] The cylindrical workpiece to be processed, such as Figure 3 As shown, the diameter of the cylindrical workpiece to be processed is... The height is The required thickness of the convex spherical workpiece is The radius of the sphere is The angle between the line connecting the edge of the sphere to the center of the sphere and the axis is . After the blank is mounted on the turntable, the distance between the bottom surface of the blank and the center of the cradle's swing is... The distance between the center of the cradle's swing and the center of the ball is .in:

[0113] The cup-shaped arc grinding wheel has a median diameter of [missing information]. The radius of the arc of the ground part of its end face is .

[0114] The cylindrical workpiece to be processed is as follows: Figure 3 As shown, the diameter of the cylindrical workpiece to be processed is... The height is The required thickness of the convex spherical workpiece is The radius of the sphere is The angle between the line connecting the edge of the sphere to the center of the sphere and the axis is . After the blank is mounted on the turntable, the distance between the bottom surface of the blank and the center of the cradle's swing is... The distance between the center of the cradle's swing and the center of the ball is .in:

[0115]

[0116] The initial position is as follows Figure 4 As shown, within the YOZ plane, with the cradle's swing center as the origin, the coordinates of the center of the cup-shaped circular arc grinding wheel's arc at any given moment are: At any given moment, the radius of the envelope sphere formed by the rotation of the cup-shaped arc grinding wheel and the rotation of the blank is... R At the initial position, the radius of the envelope sphere is... It shares a common center with the sphere formed after processing, satisfying:

[0117]

[0118] The cradle swing angle is :

[0119]

[0120] The initial coordinates of the arc center of the cup-shaped circular arc grinding wheel contact section relative to the cradle's swing center are ( ):

[0121]

[0122]

[0123] like Figure 5 As shown, after processing, the radius of the envelope sphere is... The cradle swing angle is ,

[0124]

[0125] During the processing, the radius of the envelope is controlled to decrease uniformly over time, from Reduce to Time Then the rate at which the radius decreases is:

[0126]

[0127] The radius of the envelope sphere at any time t R for:

[0128]

[0129] At any given moment, the angle of the cradle's swing satisfy:

[0130]

[0131] Combining equations (4), (5), (8), and (9), the trajectory of the center of the arc of the cup-shaped circular arc grinding wheel contact section relative to the center of the cradle's swing can be obtained as follows:

[0132]

[0133]

[0134]

[0135] Step 2) After one machining operation, measure the surface accuracy of the spherical workpiece, including the spherical error, such as the radius of the ground spherical workpiece. The radius of the circular protrusion in the central region of the spherical workpiece Calculate the positional error of the cup-shaped arc grinding wheel. and Each compensation and The linkage trajectory is adjusted, the processing trajectory is recalculated, and the processing is repeated.

[0136] like Figure 6 As shown, when the cup-shaped arc grinding wheel has a positional error, in the YOZ plane, the center of the arc of the cup-shaped arc grinding wheel deviates from the ideal position in the horizontal Y direction and the vertical Z direction, respectively. , Deviations in both directions cause changes in the radius of the envelope sphere, ultimately affecting the actual radius of the machined sphere. and There are many reasons for this, including tool setting error, grinding wheel wear error, and grinding wheel manufacturing error.

[0137] Combination Figure 5 Ideally, after machining, the Z-axis distance between the center of the middle diameter of the cup-shaped arc grinding wheel end face and the center of the enveloping spherical surface is L:

[0138]

[0139] Combination Figure 6 The cup-shaped arc grinding wheel has errors. , At that time, after machining, the Z-axis distance between the center of the cup-shaped arc grinding wheel's median diameter and the center of the enveloping spherical surface becomes... The radius of the enclosing sphere becomes ,

[0140]

[0141] The angle between the axis of the cup-shaped circular arc grinding wheel and the line connecting the center of the cup-shaped circular arc grinding wheel's arc to the center of the enveloping spherical surface becomes... :

[0142]

[0143]

[0144] like Figure 7 As shown, the cup-shaped arc grinding wheel has errors. , At this time, a circular bulge will appear at the center of the sphere, and the radius of the circular bulge is... The height of the entire convex spherical workpiece is from Become .

[0145] according to Figure 7 As shown, the radius of the circular protrusion in the central region of the spherical workpiece is obtained. :

[0146]

[0147] Combining equations (15) and (17), we can obtain:

[0148]

[0149] Combining equations (16) and (17), we can obtain:

[0150]

[0151] (Combined with step 2), the actual spherical radius is measured after processing. The radius of the circular protrusion in the central region of the convex spherical workpiece The result is obtained through equation (18). , derive from equation (19) ,Will and The machining trajectory shown in formulas (10) and (11) is compensated until the surface shape error of the spherical workpiece is within the allowable range, and the machining is completed.

[0152] For concave spherical surfaces, combining steps 1) and 2), similarly, cylindrical workpieces to be processed, such as... Figure 9 As shown, the diameter of the cylindrical workpiece to be processed is... The height is The thickness of the spherical workpiece after processing is The radius of the sphere is After the blank is mounted on the turntable, the distance between the bottom surface of the blank and the center of the cradle's swing is... .

[0153] Initial position as Figure 9 As shown, the cradle swing angle The cup-shaped arc grinding wheel rotates on its end face, creating a circular grinding motion. This circular grinding motion, combined with the workpiece's rotation, forms an enveloping spherical surface with an initial radius of [missing information]. ,satisfy:

[0154]

[0155]

[0156] The initial coordinates of the center of the contact segment of the cup-shaped circular arc grinding wheel, relative to the cradle's oscillation center, are ( ):

[0157]

[0158]

[0159] After processing, the radius of the envelope sphere is The cradle swing angle is ,

[0160]

[0161] During the processing, the radius of the envelope sphere is controlled from... Increase uniformly over time to The radius increases at a rate of The trajectory of the center of the arc of the cup-shaped circular arc grinding wheel contact section relative to the center of the cradle's swing can be obtained as follows:

[0162]

[0163]

[0164]

[0165] like Figure 10 As shown, when the cup-shaped arc grinding wheel has a positional error, in the YOZ plane, the center of the arc of the cup-shaped arc grinding wheel deviates from the ideal position in the horizontal Y direction and the vertical Z direction, respectively. , Deviations in both directions cause changes in the radius of the envelope sphere, ultimately affecting the actual radius of the machined sphere. and There are many reasons for this, including tool setting error, grinding wheel wear error, and grinding wheel manufacturing error.

[0166] Cup-shaped arc grinding wheels have positional errors. , At that time, after machining, the Z-axis distance between the center of the middle diameter of the end face of the cup-shaped arc grinding wheel and the center of the enveloping spherical surface becomes... The radius of the enclosing sphere becomes ,

[0167]

[0168] The angle between the axis of the cup-shaped circular arc grinding wheel and the line connecting the center of the arc of the contact section of the cup-shaped circular arc grinding wheel to the center of the envelope spherical surface becomes... :

[0169]

[0170]

[0171] like Figure 11 As shown, the cup-shaped arc grinding wheel has a positional error. , After grinding, a circular protrusion will appear at the center of the spherical surface, with a radius of [missing information]. The center thickness of the concave spherical workpiece is determined by Become .

[0172] according to Figure 10 , Figure 11 As shown, the radius of the circular protrusion in the central region of the spherical workpiece is obtained. :

[0173]

[0174] Combining equations (29) and (31), we can obtain:

[0175]

[0176] Combining equations (30) and (31), we can obtain:

[0177]

[0178] (Combined with step 2), the actual spherical radius is measured after processing. The radius of the circular protrusion in the central region of the spherical workpiece The result is obtained through equation (32). The result is obtained through equation (33). ,Will and The machining trajectory shown in equations (25) and (26) is compensated until the surface shape error of the spherical workpiece is within the allowable range, at which point the machining is completed.

[0179] The proposed method enables the machining of spherical surfaces of various sizes using cup-shaped arc grinding wheels of various sizes, such as... Figure 12 As shown, when machining a convex spherical surface according to the method of the present invention, the smallest permissible diameter cup-shaped arc grinding wheel should satisfy the requirement that its end face grinding surface exactly contacts the edge of the spherical surface of the workpiece. It is known that... ,at this time:

[0180]

[0181] After machining the convex spherical surface according to the method of the present invention, the inner diameter of the cup-shaped arc grinding wheel with the largest permissible diameter is exactly equal to that of the spherical surface, and the cup-shaped arc grinding wheel can just accommodate half of the spherical surface. At this time:

[0182]

[0183] like Figure 13 As shown, for the selected radius is The cup-shaped arc grinding wheel should have a cup depth that matches its radius to accommodate the workpiece. Then the bottom of the cup will not touch the workpiece:

[0184]

[0185] like Then the bottom of the cup will not touch the spherical surface:

[0186]

[0187] Similarly, such as Figure 14 As shown, after machining a concave spherical surface according to the method of the present invention, the smallest permissible diameter cup-shaped arc grinding wheel should satisfy the requirement that its end face grinding surface exactly contacts the edge of the spherical surface of the workpiece. It is known that... ,at this time:

[0188]

[0189] After machining the concave spherical surface according to the method of the present invention, the outer diameter of the cup-shaped arc grinding wheel with the largest permissible diameter is exactly equal to that of the spherical surface. At this time:

[0190]

[0191] The above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention in any way. Those skilled in the art can readily implement the present invention based on the accompanying drawings and the above description. However, any modifications, alterations, or variations made by those skilled in the art without departing from the scope of the present invention, utilizing the disclosed technical content, are equivalent embodiments of the present invention. Furthermore, any modifications, alterations, or variations made to the above embodiments based on the essential technology of the present invention are still within the protection scope of the present invention.

Claims

1. A method for grinding spherical surfaces with a cup-shaped arc grinding wheel and for error compensation, characterized in that, Includes the following steps: S1, obtain the various dimensions and process parameters required for grinding the spherical surface, and obtain the motion trajectory of the workpiece grinding the spherical surface; S2, based on a cradle-type five-axis machine tool, according to the motion trajectory of the workpiece grinding the spherical surface, the workpiece to be processed changes with the swing angle of the cradle axis of the cradle-type five-axis machine tool, and the end face of the workpiece to be processed is ground into a spherical workpiece by a cup-shaped arc grinding wheel; the processing types of the spherical workpiece include convex spherical surface and concave spherical surface. S3. After the machining is completed, the surface accuracy of the spherical workpiece is checked, the surface error of the spherical workpiece is measured, and the position error of the cup-shaped arc grinding wheel is calculated. S4. Based on the position compensation amount of the cup-shaped arc grinding wheel, compensate and adjust the processing position of the cup-shaped arc grinding wheel, recalculate the motion trajectory of the workpiece grinding the spherical surface, and process the spherical workpiece again until the surface shape error of the spherical workpiece is within the allowable range, and the processing is completed. S1 obtains the various dimensions and process parameters required for grinding the spherical surface, specifically including: Diameter of the workpiece to be processed and height The center thickness of the spherical workpiece to be machined and spherical radius The distance between the center of the cradle's swing and the bottom surface of the workpiece when the workpiece is mounted on a cradle-type five-axis machine tool. The distance between the cradle shaft swing center and the center of the spherical workpiece to be machined. The rate at which the radius of a spherical workpiece decreases during grinding. The median diameter of the cup-shaped arc grinding wheel and the small radius of the abrasive region ; The motion trajectory of the workpiece grinding the spherical surface in S1 is as follows: When machining a convex spherical surface, the formula for the motion trajectory of the workpiece grinding the spherical surface is: ； In the formula: The radius of the envelope sphere formed by the truncated circle of the cup-shaped arc grinding wheel end face rotating at the initial grinding position and the rotation of the workpiece to be processed; f The rate at which the radius of a spherical workpiece decreases during grinding; The median diameter of the cup-shaped arc grinding wheel, The radius of the small arc of the abrasive grain area of ​​the cup-shaped arc grinding wheel; Grinding time; This refers to the Y-axis position of a cradle-type five-axis machine tool. This refers to the Z-axis position of a cradle-type five-axis machine tool. This refers to the A-axis position of a cradle-type five-axis machine tool. The distance between the center of the cradle shaft's swing and the center of the spherical workpiece; Wherein, the radius of the envelope sphere is for: ； In the formula: Let be the radius of the sphere of the spherical workpiece. The center thickness of the spherical workpiece The diameter of the workpiece to be processed; When machining a concave spherical surface, the formula for the motion trajectory of the workpiece grinding the spherical surface is: In the formula: The height of the workpiece to be processed. The distance between the center of the cradle's swing and the bottom surface of the workpiece when the workpiece is mounted on a cradle-type five-axis machine tool; This refers to the Y-axis position of a cradle-type five-axis machine tool. This refers to the Z-axis position of a cradle-type five-axis machine tool. This refers to the A-axis position of a cradle-type five-axis machine tool. Let be the radius of the envelope sphere formed by the truncated circle of the cup-shaped arc grinding wheel end face rotating at the initial grinding position and the rotation of the workpiece to be processed; Wherein, the radius of the envelope sphere is for: ； In the formula: Let be the radius of the sphere of the spherical workpiece. The center thickness of the spherical workpiece The height of the workpiece to be processed; The specific processing procedure for S2 is as follows: Based on a cradle-type five-axis machine tool, the workpiece to be processed is mounted on a rotary table, and a cup-shaped arc grinding wheel is mounted on the spindle. According to the motion trajectory of the workpiece grinding the spherical surface, the workpiece to be processed swings with the cradle axis of the cradle-type five-axis machine tool and rotates around its own axis. The spindle drives the cup-shaped arc grinding wheel to rotate to form a truncated circular grinding area. The swing angle of the cradle axis changes continuously during the grinding process. Through the grinding area of ​​the grinding wheel, it contacts the workpiece to be processed, and grinds the end face of the workpiece into a spherical workpiece. During the machining process, at the initial grinding position, the truncated circle of the end face of the cup-shaped arc grinding wheel rotates and forms an enveloping spherical surface with the rotation of the workpiece. The radius of the enveloping spherical surface... With speed f With grinding time t The material is uniformly reduced or increased, and at any time at any position on the envelope sphere, the material removal method is bounded by the envelope sphere and shrinks or expands towards the workpiece. In any radial direction, the thickness of the removed material is uniform. The positional error of the cup-shaped arc grinding wheel in S3 is calculated as follows: When the spherical workpiece is a convex spherical surface, the positional error of the cup-shaped circular arc grinding wheel is calculated using the following formula: When the spherical workpiece is a concave spherical surface, the positional error of the cup-shaped arc grinding wheel is calculated using the following formula: In the formula: The distance by which the center of the arc of the cup-shaped arc grinding wheel deviates from its ideal position in the horizontal Y direction. The distance by which the center of the arc of the cup-shaped arc grinding wheel deviates from its ideal position in the vertical Z direction. Let be the radius of the spherical workpiece after grinding. Let be the radius of the circular protrusion in the central region of the spherical workpiece. L is the radius of the small arc of the abrasive grain region of the cup-shaped arc grinding wheel; L is the distance from the center of the cup mouth of the cup-shaped arc grinding wheel to the center of the envelope spherical surface.

2. The method for grinding spherical surfaces with a cup-shaped arc grinding wheel and for error compensation according to claim 1, characterized in that, In S2, at the initial machining position, there is an angle between the axis of the cup-shaped arc grinding wheel and the axis of the workpiece to be machined. That is, the cradle axis swing angle The grinding area of ​​the cup-shaped arc grinding wheel makes point contact with the end face of the workpiece; at the end of the machining process, there is an angle between the axis of the cup-shaped arc grinding wheel and the axis of the workpiece. That is, the cradle axis swing angle is The grinding area of ​​the cup-shaped arc grinding wheel is in line contact with the spherical surface of the workpiece.

3. The method for grinding spherical surfaces with a cup-shaped arc grinding wheel and for error compensation according to claim 1, characterized in that, When machining a convex spherical workpiece, the mean diameter of the cup-shaped circular arc grinding wheel should meet the following conditions: ； When machining a concave spherical workpiece, the mean diameter of the cup-shaped arc grinding wheel should meet the following conditions: ； In the formula: The center thickness of the spherical workpiece Let be the radius of the sphere of the spherical workpiece. The radius of the small arc in the abrasive region of the cup-shaped arc grinding wheel. The angle between the line connecting the edge of the spherical surface to the center of the sphere and the axis. This refers to the median diameter of the cup-shaped arc grinding wheel.

4. The method for grinding spherical surfaces with a cup-shaped arc grinding wheel and for error compensation according to claim 1, characterized in that, When machining a convex spherical workpiece, the cup depth of the cup-shaped circular arc grinding wheel. Should meet: ，( ) ，( ) In the formula: Let be the radius of the sphere of the spherical workpiece. The radius of the small arc in the abrasive region of the cup-shaped arc grinding wheel. The angle between the line connecting the edge of the spherical surface to the center of the sphere and the axis. The median diameter of the cup-shaped arc grinding wheel, This refers to the swing angle of the cradle shaft after processing.

5. A cradle-type five-axis machine tool, characterized in that, The method for grinding spherical surfaces with a cup-shaped arc grinding wheel and for error compensation as described in any one of claims 1-4 is used for machining using the aforementioned cradle-type five-axis machine tool.

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