Magnetorheological removal function curvature compensation and time-varying regulation and control method based on fluid mechanics

Abstract

The invention discloses a magnetorheological removal function curvature compensation and time-varying regulation and control method based on fluid mechanics, and belongs to the technical field of optical polishing. According to the implementation method, through a method of iteratively solving pressure intensity and iteratively calculating shear stress, the accuracy of calculating the pressure intensity and the shear stress by a Reynolds equation is improved, and the modeling precision of a magneto-rheological removal function is improved; a correction model of the magneto-rheological removal function is constructed, accurate modeling of the magneto-rheological removal function on a plane element, a spherical element and an aspheric element is achieved, and the prediction precision of the removal function of magneto-rheological polishing is improved; according to the derivative of the aspherical polynomial equation, the curvature radius of a sampling point on the aspherical element is accurately calculated, a compensation method for the curvature effect of the magneto-rheological removal function is established, and the problem that the magneto-rheological polishing precision is reduced due to the curvature effect is solved; a control method for changing the immersion depth and then dynamically regulating and controlling the time-varying removal function is provided, and the regulation and control capacity and the machining precision of the magneto-rheological polishing technology in the deterministic removal process are remarkably improved.

Description

Technical Field

[0001] The present invention belongs to the technical field of optical polishing, and relates to a method for compensating the curvature and time-varying regulation of a magnetorheological removal function based on fluid mechanics. Background Technique

[0002] Due to their significant advantages in improving imaging quality, optimizing system structure, reducing light energy loss, etc., high-precision aspherical optical elements play a core role in the fields of optical instruments, space laser communication, and aerospace, and have become an important foundation for promoting the progress of related technologies. As one of the key technologies for processing aspherical optical elements, the magnetorheological polishing method has become a hot topic of research and application at home and abroad due to its high efficiency and high precision, showing broad development potential and application prospects.

[0003] However, due to the complex changes in the curvature characteristics of aspherical optical elements, the removal function exhibits spatio-temporal dynamics, that is, there is a curvature effect, which significantly reduces the processing accuracy of the magnetorheological polishing technology. The existing curvature compensation strategies mainly rely on the method of approximately fitting local regions of complex surfaces using the removal function closest to a sphere. However, this method has significant bottlenecks: on the one hand, the large number of fitting regions significantly increases the computational complexity of compensation; on the other hand, the algorithm error in the fitting process will lead to a reduction in compensation accuracy, thus limiting the effectiveness and practical application level of curvature compensation.

[0004] In addition, the current magnetorheological optical polishing technology is based on the principle of computer-controlled optical surface forming (CCOS), establishing a constant removal function under specific working conditions, solving the dwell time distribution in combination with the surface shape error distribution of the workpiece, and achieving high-precision deterministic processing with the help of numerical control technology. However, this technology has many limitations, such as the sudden change in dwell time causing processing discontinuity, the frequent start and stop of the machine tool during processing reducing the service life of the machine tool and increasing the requirements for equipment stability, and at the same time, the insufficient dynamic performance of the machine tool significantly restricts the improvement of processing efficiency and precision. Thus, it restricts the further development and application effectiveness enhancement of the magnetorheological optical polishing technology as a whole.

[0005] The Preston equation reveals that regulating the time-varying removal function can also achieve high-precision deterministic processing. The magnetorheological polishing technology forms a polishing tool with a specific hardness through magnetic particles in the polishing fluid, and controls the tool shape, hardness, and pressure by factors such as magnetic field strength, liquid viscosity, tool rotation speed, and immersion depth, thereby controlling the material removal amount. Among them, the immersion depth has a significant impact on the peak removal efficiency and volume removal efficiency of magnetorheological polishing, and the controllability and time-varying characteristics of the magnetorheological removal function can be achieved by regulating the immersion depth. However, the complex coupling effect between the above factors significantly increases the difficulty of theoretical modeling of the magnetorheological removal function and theoretical curvature effect compensation. Summary of the Invention

[0006] To solve the problems of insufficient curvature effect compensation accuracy and weakened machining accuracy caused by sudden changes in dwell time in the existing magnetorheological finishing method, the object of the present invention is to provide a curvature compensation and time-varying regulation method for the magnetorheological removal function based on hydrodynamics. Based on hydrodynamics, the Reynolds equation is used to establish a relationship model between the magnetic field strength, immersion depth, tool size, liquid viscosity, polishing wheel speed and the magnetorheological removal function, so as to accurately model the removal function on plane components, spherical components and aspherical components, eliminate the curvature effect of aspherical components, and then accurately regulate the time-varying magnetorheological removal function by controlling the immersion depth, improve the machining accuracy of optical components, and enhance the correction ability of the magnetorheological finishing method for the surface shape error of components.

[0007] The object of the present invention is achieved by the following technical solutions:

[0008] The curvature compensation and time-varying regulation method for the magnetorheological removal function based on hydrodynamics disclosed by the present invention improves the accuracy of calculating the pressure and shear stress of the Reynolds equation by iteratively solving the method of nested iteration of pressure to calculate the shear stress, and then improves the modeling accuracy of the magnetorheological removal function; a correction model of the magnetorheological removal function is proposed to accurately model the magnetorheological removal function on plane components, spherical components and aspherical components, improve the prediction accuracy of the removal function of magnetorheological finishing, and then improve the machining accuracy of optical components; according to the derivative of the aspherical polynomial equation, the radius of curvature at the sampling points on the aspherical component is accurately calculated, and a compensation method for the curvature effect of the magnetorheological removal function is established, effectively solving the problem of the decline in magnetorheological polishing accuracy caused by the curvature effect; a control method for changing the immersion depth to dynamically regulate the time-varying removal function is proposed, significantly enhancing the regulation ability and machining accuracy of the magnetorheological finishing technology in the deterministic removal process.

[0009] The curvature compensation and time-varying regulation method for the magnetorheological removal function based on hydrodynamics disclosed by the present invention includes the following steps;

[0010] Step 1: According to the set immersion depth H l and the ribbon thickness H d , divide the spacing H h between the workpiece and the polishing wheel into several equal subintervals to obtain the corresponding height h of each interval; and preset the over-relaxation factor, pressure convergence condition, and nucleation range convergence condition to determine the shear yield strength τ 0; Utilize the property that magnetorheological fluid has the characteristics of Bingham fluid, discretize the Reynolds equation corresponding to each height h based on the difference method; adopt the successive over-relaxation iteration method to solve the Reynolds equation corresponding to each height h to obtain the analytical solution of the pressure. If the preset convergence condition is not satisfied, combine the pressure distribution with the over-relaxation factor and substitute it back into the Reynolds equation for solution until the preset pressure convergence condition is satisfied to obtain the convergent pressure and shear stress; define the solid nucleation range between the workpiece and the polishing wheel in the polishing area according to the shear yield strength. When the shear stress is less than the shear yield strength, the magnetorheological fluid presents a solid state, otherwise it is in a liquid state; update the apparent viscosity distribution according to the solid nucleation range, re-solve the Reynolds equation based on the new apparent viscosity distribution, and improve the solution accuracy of the shear stress and pressure analytical solution through iterative solution until the preset nucleation range condition converges to obtain the pressure and shear stress in the stable state.

[0011] The immersion depth H described in step one l , ribbon thickness H d and the spacing H between the workpiece and the polishing wheel h satisfy the equation relationship as shown in Equation (1):

[0012] H d = H h + H l (1)

[0013] The shear yield strength τ described in step one 0 has the relationship with the magnetic field strength as shown in Equation (2):

[0014]

[0015] In the formula, represents the volume fraction of ferromagnetic particles in the magnetorheological fluid, B is the magnetic induction intensity, r is the radius of ferromagnetic particles, δ is the net distance between adjacent ferromagnetic particles, μ 0 is the vacuum permeability, χ is the magnetic susceptibility of ferromagnetic particles, and ψ is the shear stress coefficient.

[0016] The Reynolds equation described in step one is as shown in Equation (3):

[0017]

[0018] In the formula, represents the dimensionless interval height, P represents the pressure distribution in the polishing area, represents the dimensionless pressure distribution in the polishing area, Λ represents the aspect ratio of the polishing area, represents the dimensionless apparent viscosity, represents the dimensionless surface spacing between the polishing wheel and the workpiece, h 0represents the minimum distance between the polishing wheel and the workpiece surface, R represents the size of the polishing tool, and U represents the linear velocity of the polishing wheel surface.

[0019] Step 2: Change the magnetic field strength, immersion depth, polishing tool size, and polishing wheel rotation speed. According to the iterative solution in Step 1, obtain the pressure and shear stress. Through the analysis of the solution results, the following law is obtained: The effects of the magnetic field strength, immersion depth, tool size, and polishing wheel rotation speed on the magnitude and distribution dimension of the pressure and shear stress all show a monotonic law, which is defined as the monotonic law.

[0020] Step 3: According to Preston's classical removal model, the dominant factor for material removal during the polishing process is pressure. In the magnetorheological polishing process, the shear force is the core factor that dominates the material removal efficiency, and pressure is a necessary factor for the existence of the shear force. Based on the monotonic law analyzed in Step 2, modify the removal function model by increasing the influence factor of the shear stress on the material removal efficiency to obtain the modified removal function model. Obtain the removal function at different immersion depths through the spot experiment method, and use the data fitting method to determine the fitting parameters in the modified removal function model to obtain the fitted modified removal function model. Solve the removal function of the magnetorheological polishing method according to the fitted modified removal function model to improve the prediction accuracy of the magnetorheological removal function.

[0021] Preston's classical removal model is shown in Equation (6):

[0022]

[0023] In the formula, E is the Preston coefficient, P is the pressure distribution in the polishing area, and V is the relative polishing speed between the polishing head and the workpiece.

[0024] The modified removal function model is shown in Equation (7):

[0025]

[0026] In the formula, C is the material removal coefficient, α represents the influence factor of pressure on the removal function, and C and α are fitting parameters. Substitute the pressure and shear stress obtained in Step 1 into Equation (7) to obtain the magnetorheological removal function in the polishing area.

[0027] Step 4: For an aspherical element, according to the ideal aspherical polynomial equation, by solving the first-order partial derivative and second-order partial derivative of the ideal aspherical polynomial equation, calculate the curvature radius of each sampling point of the aspherical surface. According to the calculated curvature radius of the aspherical surface, map the surface spacing between the polishing wheel and the workpiece at each sampling point of the aspherical surface to the surface spacing between the polishing wheel and the workpiece of the spherical element corresponding to the curvature. According to the geometric principle, construct a distance relationship model between the polishing wheel and the spherical element. Combining the distance relationship model between the polishing wheel and the spherical element, based on the removal function obtained by solving in Step 2 for the aspherical element, compensate and eliminate the curvature effect of the magnetorheological removal function on the aspherical surface, improve the accuracy of the compensated removal function under the complex curvature distribution, and further improve the machining accuracy of the magnetorheological processing for the aspherical element.

[0028] The aspherical polynomial equation described in Step 4 is shown in Equation (8).

[0029]

[0030] where r 2 = x 2 + y 2 , O is the vertex curvature radius, K is the eccentricity constant, a, b, c represent the quadratic term correction coefficients, and d, e, f, g are the cubic term correction coefficients. Based on the surface formula, calculate the first-order derivative and second-order derivative of the aspherical surface in the x and y directions, as shown in Equation (9).

[0031]

[0032] According to the definitions of the first-order derivative and second-order derivative, determine the first fundamental form coefficients E, F, G of the Gaussian curvature and the mean curvature, and the second fundamental form coefficients L, M, N. Their expressions are respectively:

[0033]

[0034] According to the first fundamental form coefficients and the second fundamental form coefficients, respectively determine the Gaussian curvature K G and the mean curvature K H . Their expressions are:

[0035]

[0036] Calculate and determine the normal principal curvature at each sampling point through the Gaussian curvature and the mean curvature, and according to the relationship that the curvature radius is the reciprocal of the normal principal curvature, calculate the curvature radius corresponding to each sampling point.

[0037] According to the geometric principle, for the surface spacing h between the polishing wheel and the spherical element, the distance relationship model between the polishing wheel and the spherical element is shown in Equation (12):

[0038]

[0039] In the formula, h 0 represents the distance between the lowest point on the polishing wheel and the component, and R 1 is the radius of the polishing wheel, x represents the coordinate of the polishing area, and R 2 represents the radius of curvature of the spherical optical component to be processed.

[0040] Combined with the distance relationship model (12) between the polishing wheel and the spherical component, the removal function corresponding to the aspherical component is obtained by solving based on Step 2, compensating and eliminating the curvature effect of the magnetic rheological removal function on the aspherical surface, improving the accuracy of the removal function under complex curvature distributions, and further improving the machining accuracy of the magnetic rheological aspherical component.

[0041] It further includes Step 5: In the magnetic rheological polishing machine tool, according to the compensated magnetic rheological removal function obtained in Step 3, dynamically regulate the immersion depth to achieve real-time adjustment of the magnetic rheological removal function, endowing the compensated removal function with time-varying characteristics, and improving the flexibility and accuracy of machining the aspherical component according to the dynamically regulated removal function.

[0042] Beneficial effects:

[0043] 1. For the method for curvature compensation and time-varying regulation of the magnetic rheological removal function based on hydrodynamics disclosed in the present invention, the over-relaxation iteration method is used to replace the mathematical approximation method to solve the pressure distribution in the Reynolds equation, and the iteration method is further used to judge the solid-state nucleation range of the magnetic rheological fluid. Compared with the method of solving the Reynolds equation by mathematical approximation, the solving accuracy of the pressure and shear stress is improved, and thus the modeling accuracy of the magnetic rheological polishing removal function is improved.

[0044] 2. For the method for curvature compensation and time-varying regulation of the magnetic rheological removal function based on hydrodynamics disclosed in the present invention, the magnetic rheological polishing removal function based on hydrodynamics is modeled, and the influence factor of the shear stress on the material removal efficiency of the magnetic rheological polishing is added to the existing magnetic rheological removal function model, which is more in line with the basic principle that the shear force is the dominant factor for material removal in magnetic rheological polishing, improving the accuracy of the magnetic rheological polishing removal function model, and thus improving the machining accuracy of the magnetic rheological polishing method for optical components.

[0045] 3. For the method for curvature compensation and time-varying regulation of the magnetic rheological removal function based on hydrodynamics disclosed in the present invention, the numerical analysis method is used to replace the closest sphere fitting method to calculate the radius of curvature at each sampling point on the surface of the aspherical component, and then the aspherical surface is mapped into a series of spherical surfaces, and the model of the removal function on the spherical component is accurately established by using the Reynolds equation, eliminating the curvature effect of the aspherical component during magnetic rheological polishing, and improving the machining accuracy of the magnetic rheological polishing method for the aspherical component.

[0046] 4. The curvature compensation and time-varying regulation method of the magnetorheological removal function based on fluid mechanics disclosed by the present invention, according to the correction model of the magnetorheological removal function, changes the distribution of the removal function by controlling the immersion depth. Compared with the polishing method of variable dwell time, it reduces the frequent start and stop of the machine tool, reduces the constraint of the dynamic performance of the machine tool on the machining accuracy, and improves the polishing quality. Description of the Drawings

[0047] Figure 1 It is a flowchart of the curvature compensation and time-varying regulation method of the magnetorheological removal function based on fluid mechanics of the present invention.

[0048] Figure 2 It is the verification result of the influence law of magnetic field intensity, immersion depth, tool size, and polishing wheel speed on the pressure and shear stress in terms of magnitude and distribution dimension in the embodiment. Figure (a) shows the influence law of immersion depth on the pressure magnitude and shear stress magnitude, Figure (b) shows the influence law of immersion depth on the size of the polishing area, Figure (c) shows the influence law of magnetic field intensity on the pressure magnitude and shear stress magnitude, Figure (d) shows the influence law of magnetic field intensity on the size of the polishing area, Figure (e) shows the influence law of polishing speed on the pressure magnitude and shear stress magnitude, Figure (f) shows the influence law of polishing speed on the size of the polishing area, Figure (g) shows the influence law of tool radius on the pressure magnitude and shear stress magnitude, and Figure (h) shows the influence law of tool radius on the size of the polishing area.

[0049] Figure 3 It is the magnetorheological removal function at different immersion depths on the quartz planar element obtained by the spot method in the embodiment.

[0050] Figure 4 It is the removal function distribution at different immersion depths solved in the embodiment. The surface distance between the polishing wheel and the workpiece in Figure (a) is 0.2 mm, the surface distance between the polishing wheel and the workpiece in Figure (b) is 0.35 mm, the surface distance between the polishing wheel and the workpiece in Figure (c) is 0.5 mm, and the surface distance between the polishing wheel and the workpiece in Figure (d) is 0.65 mm. Detailed Embodiment

[0051] Next, the technical solutions in the embodiments of the present invention will be clearly and completely described in conjunction with the drawings in the embodiments of the present invention. The described embodiments are only a part of the embodiments of the present invention, rather than all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative efforts shall fall within the protection scope of the present invention.

[0052] Embodiment

[0053] In this embodiment, under the conditions that the radius of the polishing wheel is 100 mm, the rotational speed of the polishing wheel is 260 r / min, the magnetic field current is 10 A, and the flow rate is 120 L / H, the immersion depth is set to range from 0.05 mm to 0.49 mm with a change rate of 0.02 mm. Using diamond as the abrasive of the magnetorheological fluid, a spot experiment of the removal function is carried out on a quartz plane mirror with a diameter of 100 mm to further illustrate a magnetorheological removal function curvature compensation and time-varying regulation method based on fluid mechanics of the present invention.

[0054] As Figure 1 shown, the specific implementation steps of the magnetorheological removal function curvature compensation and time-varying regulation method based on fluid mechanics disclosed in this embodiment are as follows:

[0055] Step 1. Set the immersion depth to 0.1 mm and the ribbon thickness to 0.6 mm, and divide the spacing H h between the workpiece and the polishing wheel into 20 equal intervals to obtain the corresponding height h for each interval; and preset the over-relaxation factor to 1.6 and the convergence accuracy tol P of the pressure to 0.0001 and the convergence accuracy tol H of the nucleation range to 0.01. Determine the shear yield strength τ 0 of the magnetorheological fluid through calculation; utilize the property that the magnetorheological fluid has Bingham fluid, and discretize the Reynolds equation corresponding to each height h based on the difference method; use the over-relaxation iteration method to solve the Reynolds equation corresponding to each height h to obtain the analytical solution of the pressure. If the preset convergence condition is not satisfied, combine the pressure distribution with the over-relaxation factor and substitute it back into the Reynolds equation for solution until the preset pressure convergence condition is satisfied to obtain the convergent pressure and shear stress; define the solid-state nucleation range between the workpiece and the polishing wheel in the polishing area according to the shear yield strength. When the shear stress is less than the shear yield strength, the magnetorheological fluid presents a solid state, otherwise it is a liquid state; update the apparent viscosity distribution according to the solid-state nucleation range, and re-solve the Reynolds equation based on the new apparent viscosity distribution. Through iterative solution, improve the solution accuracy of the shear stress and the pressure analytical solution until the preset nucleation range condition converges to obtain the pressure and shear stress in the stable state.

[0056] The immersion depth H l , ribbon thickness H d and the spacing H h between the workpiece and the polishing wheel in Step 1

[0057] satisfy the equation relationship, as shown in Equation (1): d H h = H l + H

[0058] The shear yield strength τ 0The relationship with the magnetic field strength is shown in Equation (2):

[0059]

[0060] In the formula, represents the volume fraction of ferromagnetic particles in the magnetorheological fluid, B is the magnetic induction intensity, r is the radius of ferromagnetic particles, δ is the net distance between adjacent ferromagnetic particles, μ 0 is the vacuum permeability, χ is the magnetic susceptibility of ferromagnetic particles, and ψ is the shear stress coefficient.

[0061] The Reynolds equation described in Step 1 is shown in Equation (3):

[0062]

[0063] In the formula, represents the dimensionless interval height, P represents the pressure distribution in the polishing area, represents the dimensionless pressure distribution in the polishing area, Λ represents the aspect ratio of the polishing area, represents the dimensionless apparent viscosity, represents the dimensionless surface spacing between the polishing wheel and the workpiece, h 0 represents the minimum spacing between the polishing wheel and the workpiece surface, R represents the size of the polishing tool, and U represents the linear velocity of the polishing wheel surface.

[0064] Step 2. Taking the polishing wheel radius of 100 mm, the polishing wheel rotation speed of 300 r / min, the magnetic field current of 10 A, and the immersion depth of 0.1 mm as the reference values, set control experiments with the polishing wheel radius, polishing wheel rotation speed, magnetic field strength, and immersion depth as single variables respectively. Change the magnetic field strength, immersion depth, polishing tool size, and polishing wheel rotation speed, and obtain the pressure and shear stress by iterative solution according to Step 1. Among them, the value range of the polishing wheel radius is 90 mm to 160 mm, and the sampling interval is 10 mm; the value range of the rotation speed is 100 r / min to 700 r / mm, and the sampling interval is 100 r / min; the value range of the magnetic field strength is 120 mT to 260 mT, and the sampling interval is 20 mT; the value range of the immersion depth is 0.07 mm to 0.14 mm, and the sampling interval is 0.01 mm. The fitting influence law curve is as Figure 2 shown. Through the analysis of the solution results, the following laws are obtained: The influences of the magnetic field strength, immersion depth, tool size, and polishing wheel rotation speed on the pressure and shear stress in terms of magnitude and distribution dimension all show a monotonicity law, which is defined as the monotonicity law.

[0065] Step 3. According to the Preston classical removal model, the dominant factor for material removal during the polishing process is pressure. In the magnetorheological polishing process, the shear force is the core factor that dominates the material removal efficiency, and pressure is a necessary factor for the existence of the shear force. Based on the monotonicity law obtained from the analysis in Step 2, the removal function model is corrected by increasing the influence factor α of the shear stress on the material removal efficiency, resulting in a corrected removal function model. The removal function at different immersion depths is obtained through the spot experiment method. In this embodiment, the processing time is set to 8 seconds, the change rate of the immersion depth is 0.02 mm, and diamond is used as the abrasive on a quartz plane mirror with a diameter of 100 mm to collect the magnetorheological removal function at immersion depths from 0.05 mm to 0.49 mm. The ZYGO interferometer is used for measurement, and the results are as Figure 3 shown. According to the Reynolds equation, the pressure distribution and shear stress distribution are calculated under the conditions of the initially set magnetic field current of 10 A, the tool rotation speed of 260 r / min, the polishing wheel radius of 100 mm, the immersion depth ranging from 0.05 mm to 0.49 mm, and the change rate of 0.02 mm, and then substituted into Equation (7). The data fitting method is used to determine the fitting parameters C = 5.0161 -10 and α = 0.1191 in the corrected removal function model. Substitute the fitting parameters into Equation (7) to obtain the fitted corrected removal function model. Solve the removal function corresponding to the aspherical element according to the fitted corrected removal function model to improve the prediction accuracy of the magnetorheological removal function.

[0066] The Preston classical removal model described in Step 3 is shown in Equation (6):

[0067]

[0068] In the formula, E is the Preston coefficient, P is the pressure distribution in the polishing area, and V is the relative polishing speed between the polishing head and the workpiece.

[0069] The corrected removal function model described in Step 3 is shown in Equation (7):

[0070]

[0071] In the formula, C is the material removal coefficient, α represents the influence factor of pressure on the removal function, and C and α are fitting parameters. Substitute the pressure and shear stress obtained in Step 1 into Equation (7) to obtain the magnetorheological removal function in the polishing area.

[0072] Step 4. Analyze the aspherical element with the following parameters: K = 1.1828, vertex curvature radius O = 1000 mm, quadratic term coefficients a = -1.214 -9 and b = 4.7478 -15 and c = 7.6871-20 、 The coefficient of the cubic term d = -1.896 -23 、 e = 7.0477 -28 、 f = 0.0, g = 0.0, and the diameter is D = 60 mm. For the aspherical element, according to the ideal aspherical polynomial equation, by solving the first-order partial derivative and the second-order partial derivative of the ideal aspherical polynomial equation, the radius of curvature of each sampling point of the aspherical surface is calculated. According to the calculated radius of curvature of the aspherical surface, the surface spacing between the polishing wheel and the workpiece at each sampling point of the aspherical surface is mapped to the surface spacing between the polishing wheel and the workpiece of the spherical element corresponding to the curvature. According to the geometric principle, a distance relationship model between the polishing wheel and the spherical element is constructed. Combining the distance relationship model between the polishing wheel and the spherical element, based on the solution obtained in Step 2, the removal function corresponding to the aspherical element is obtained, compensating for and eliminating the curvature effect of the magnetorheological removal function on the aspherical surface, improving the accuracy of the compensated removal function under the complex curvature distribution, and further improving the machining accuracy of the magnetorheological processing of the aspherical element.

[0073] The aspherical polynomial equation described in Step 4 is shown in Equation (8).

[0074]

[0075] In the formula, r 2 = x 2 + y 2 , O is the vertex radius of curvature, K is the eccentricity constant, a, b, c represent the quadratic term correction coefficients, and d, e, f, g are the cubic term correction coefficients. Based on the surface formula, the first-order derivative and the second-order derivative of the aspherical surface in the x and y directions are calculated, as shown in Equation (9).

[0076]

[0077] According to the definitions of the first-order derivative and the second-order derivative, the first fundamental form coefficients E, F, G of the Gaussian curvature and the mean curvature and the second fundamental form coefficients L, M, N are defined, and their expressions are respectively:

[0078]

[0079] According to the first fundamental form coefficients and the second fundamental form coefficients, the Gaussian curvature K G and the mean curvature K H are respectively determined, and their expressions are:

[0080]

[0081] The normal principal curvature at each sampling point is calculated by the Gaussian curvature and the mean curvature, and according to the relationship that the radius of curvature is the reciprocal of the normal principal curvature, the radius of curvature corresponding to each sampling point is calculated.

[0082] According to geometric principles, for the surface spacing h between the polishing wheel and the spherical element, the distance relationship model between the polishing wheel and the spherical element is shown in Equation (12):

[0083]

[0084] In the formula, h 0 represents the distance between the lowest point on the polishing wheel and the element, R 1 is the radius of the polishing wheel, x represents the coordinate of the polishing area, and R 2 represents the radius of curvature of the spherical optical element to be processed.

[0085] Combined with the distance relationship model (12) between the polishing wheel and the spherical element, the removal function corresponding to the aspherical element is obtained by solving based on Step 2, compensating and eliminating the curvature effect of the magnetorheological removal function on the aspherical surface, improving the accuracy of the removal function under complex curvature distributions, and further improving the machining accuracy of the magnetorheological aspherical element.

[0086] Step 5. Analysis Figure 2 (a) and Figure 2 (b) show that the immersion depth has a significant impact on the coverage range and removal efficiency of the magnetorheological removal function. Based on the influence law of the immersion depth on the magnetorheological removal function, in the magnetorheological polishing machine tool, according to the compensated magnetorheological removal function obtained in Step 3, the immersion depth is dynamically adjusted to achieve real-time adjustment of the magnetorheological removal function and endow the compensated removal function with time-varying characteristics. In this embodiment, the removal functions are calculated respectively when the surface spacing between the polishing wheel and the workpiece is 0.20 mm, 0.35 mm, 0.50 mm, and 0.65 mm, and the changes in the volume removal rate and the polishing area are observed, as Figure 4 shown. The flexibility and accuracy of the machining of the aspherical element are improved according to the dynamically adjusted removal function.

[0087] The above specific description further details the purpose, technical solution, and beneficial effects of the invention. It should be understood that the above is only a specific embodiment of the present invention and is not used to limit the protection scope of the present invention. Any modifications, equivalent replacements, improvements, etc. made within the spirit and principle of the present invention shall be included in the protection scope of the present invention.

Claims

1. A magnetorheological removal function curvature compensation and time-varying control method based on fluid mechanics, characterized by: The method comprises the following steps: Step 1: According to the set immersion depth H l and ribbon thickness H d , divide the distance H between the workpiece and the polishing wheel h The magnetorheological fluid is divided into several equal intervals, and the height h corresponding to each interval is obtained; and the super-relaxation factor, pressure convergence condition, and nucleation range convergence condition are preset to determine the shear yield strength τ0 of the magnetorheological fluid; the magnetorheological fluid has the characteristics of Bingham fluid, and the Reynolds equation corresponding to each height h is discretized based on the difference method; the super-relaxation iteration method is used to solve the Reynolds equation corresponding to each height h to obtain the analytical solution of the pressure. If the preset convergence condition is not met, the pressure distribution is combined with the super-relaxation factor and re-substituted into the Reynolds equation for solution until the preset pressure convergence condition is met to obtain the converged pressure and shear stress; according to the shear yield strength, the solid nucleation range of the workpiece and the polishing wheel in the polishing area is defined. When the shear stress is less than the shear yield strength, the magnetorheological fluid is solid, otherwise it is liquid; according to the solid nucleation range, the apparent viscosity distribution is updated, and the Reynolds equation is re-solved based on the new apparent viscosity distribution. The solution accuracy of the shear stress and pressure analytical solution is improved through iterative solution until the preset nucleation range condition converges to obtain the pressure and shear stress in the stable state; Step 2: Change the magnetic field strength, immersion depth, polishing tool size, and polishing wheel speed, and iterate and solve the pressure and shear stress according to step 1. The following law is obtained through analysis of the solution results: the influence of magnetic field strength, immersion depth, tool size, and polishing wheel speed on pressure and shear stress in terms of size and distribution dimension all show a monotonic law, which is defined as a monotonic law; Step 3: According to Preston's classic removal model, the dominant factor for material removal during polishing is pressure; in the magnetorheological polishing process, shear force is the core factor that dominates the material removal efficiency, and pressure is a necessary factor for the existence of shear force; based on the monotonicity law obtained from the analysis in step 2, the removal function model is corrected by increasing the influencing factor of shear stress on material removal efficiency to obtain a removal function correction model, the removal function at different immersion depths is obtained by the spot experiment method, the fitting parameters in the removal function correction model are determined by the data fitting method, and the fitted removal function correction model is obtained. The removal function of the magnetorheological polishing method is solved according to the fitted removal function correction model, thereby improving the prediction accuracy of the magnetorheological removal function; Step 4: for the aspheric element, according to the ideal aspheric polynomial equation, by solving the first-order partial derivative and the second-order partial derivative of the ideal aspheric polynomial equation, the curvature radius of each sampling point of the aspheric surface is calculated, and according to the calculated curvature radius of the aspheric surface, the surface distance between the polishing wheel and the workpiece at each sampling point of the aspheric surface is mapped to the surface distance between the polishing wheel and the workpiece of the spherical element with the corresponding curvature; According to geometric principles, a distance relationship model between the polishing wheel and the spherical element is constructed; combined with the distance relationship model between the polishing wheel and the spherical element, the removal function corresponding to the aspherical element is solved based on step two to compensate and eliminate the curvature effect of the magnetorheological removal function on the aspherical surface, thereby improving the accuracy of the compensated removal function under complex curvature distribution, thereby improving the magnetorheological processing accuracy of aspherical elements.

2. The magnetorheological removal function curvature compensation and time-varying control method based on fluid mechanics according to claim 1, characterized in that: The method also includes step five, in a magnetorheological polishing machine, dynamically adjusting the immersion depth according to the compensated magnetorheological removal function obtained in step three, realizing real-time adjustment of the magnetorheological removal function, giving the compensated removal function a time-varying characteristic, and improving the flexibility and accuracy of processing non-spherical components according to the dynamically adjusted removal function.

3. The method for curvature compensation and time-varying control of magnetorheological removal function based on fluid mechanics according to claim 1 or 2, characterized in that: The immersion depth H described in step 1 l , Ribbon thickness H d And the distance H between the workpiece and the polishing wheel h Satisfy the equation relationship, as shown in formula (1): H d =H h +H l (1) The relationship between the shear yield strength τ0 and the magnetic field strength described in step 1 is shown in formula (2): In the formula, represents the volume fraction of ferromagnetic particles in the magnetorheological fluid, B is the magnetic induction intensity, r is the radius of the ferromagnetic particles, δ is the net distance between adjacent ferromagnetic particles, μ0 is the vacuum permeability, χ is the magnetic susceptibility of the ferromagnetic particles, and ψ is the shear stress coefficient; The Reynolds equation in step 1 is shown in formula (3): In the formula, represents the dimensionless interval height, the pressure distribution in the polishing area of ​​the P table, represents the dimensionless pressure distribution in the polishing area, Λ represents the aspect ratio of the polishing area, represents the dimensionless apparent viscosity, represents the dimensionless distance between the polishing wheel and the workpiece surface, h0 represents the minimum distance between the polishing wheel and the workpiece surface, R represents the polishing tool size, and U represents the linear speed of the polishing wheel surface.

4. The method for curvature compensation and time-varying control of magnetorheological removal function based on fluid mechanics according to claim 3, characterized in that: The Preston classic removal model in step 3 is shown in formula (6): Where E is the Preston coefficient, P is the pressure distribution in the polishing area, and V is the relative polishing speed between the polishing head and the workpiece; The removal function correction model is shown in formula (7): Where C is the material removal coefficient, α represents the influence factor of pressure on the removal function, and C and α are fitting parameters. Substitute the pressure and shear stress obtained in step 1 into formula (7) to obtain the magnetorheological removal function in the polishing area.

5. The method for curvature compensation and time-varying control of magnetorheological removal function based on fluid mechanics according to claim 4, characterized in that: The aspheric polynomial equation described in step 4 is shown in equation (8): In the formula, r 2 =x 2 +y 2 , O is the vertex curvature radius, K is the eccentricity constant, a, b, c represent the quadratic correction coefficients, d, e, f, g are the cubic correction coefficients; based on the surface formula, the first-order derivative and second-order derivative of the aspheric surface in the x and y directions are calculated, as shown in formula (9): The first basic form coefficients E, F, G and the second basic form coefficients L, M, N of Gaussian curvature and mean curvature are defined according to the first-order derivative and the second-order derivative, and their expressions are: According to the first basic form coefficient and the second basic form coefficient, the Gaussian curvature K is determined respectively G and the mean curvature K H , whose expression is: The normal principal curvature at each sampling point is determined by calculating the Gaussian curvature and the mean curvature, and the curvature radius corresponding to each sampling point is calculated based on the relationship that the curvature radius is the inverse of the normal principal curvature; According to the geometric principle, for the surface distance h between the polishing wheel and the spherical element, the distance relationship model between the polishing wheel and the spherical element is shown in formula (12): In the formula, h0 represents the distance between the lowest point on the polishing wheel and the component, R1 is the radius of the polishing wheel, x represents the coordinates of the polishing area, and R2 represents the radius of curvature of the spherical optical element to be processed; Combined with the distance relationship model between the polishing wheel and the spherical element (12), the removal function corresponding to the aspherical element is obtained based on the solution of step 2, which compensates and eliminates the curvature effect of the magnetorheological removal function of the aspherical surface, improves the accuracy of the removal function under complex curvature distribution, and thus improves the magnetorheological processing accuracy of the aspherical element.

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