An ultrasonic elliptical vibration turning surface residual height prediction method

Abstract

This invention belongs to the field of ultrasonic elliptical vibration-assisted ultra-precision machining, and discloses a method for predicting the residual height of surfaces machined by ultrasonic elliptical vibration. Based on the principle of time-reversal symmetry in simple harmonic motion, and according to the principle that ultrasonic simple harmonic motion has displacement symmetry and time-reversal symmetry about the equilibrium position within a complete cycle, a numerical method is established to solve for the residual height introduced by ultrasonic elliptical vibration turning in the cutting direction. This method can be applied to the calculation of the theoretical residual height in the cutting direction for ordinary plane cutting and inclined plane cutting. This invention provides a new approach to calculating the theoretical residual height of surfaces machined by ultrasonic elliptical vibration, and can assist in judging the rationality of the selection of process parameters for ultrasonic elliptical vibration-assisted ultra-precision machining in actual production.

Description

Technical Field

[0001] This invention belongs to the field of ultrasonic elliptical vibration-assisted ultra-precision cutting, and more specifically, relates to a method for predicting the residual height of the surface after ultrasonic elliptical vibration turning. Background Technology

[0002] Ultrasonic elliptical vibration turning technology is widely used in the ultra-precision machining of difficult-to-machine materials such as ferrous metals and brittle materials. Due to its tool-workpiece separation cutting characteristics, this technology effectively reduces cutting forces, promotes the entry of cutting fluid into the cutting zone, and further improves cooling and lubrication. Based on these significant advantages, ultrasonic elliptical vibration turning technology broadens the application range of conventional single-point diamond turning, and is particularly significant in suppressing severe chemical wear during diamond tool cutting of ferrous metals.

[0003] In conventional circular arc-cut single-point diamond turning, the feed rate, spindle speed, and nose radius determine the theoretical residual height of the machined surface in the feed direction (radial). In ultrasonic elliptical vibration-assisted single-point diamond turning, in addition to the feed direction, due to the influence of the tool's elliptical vibration and cutting speed, periodic machining textures are introduced in the cutting direction (circumferential direction) under the combined effect of ultrasonic vibration frequency, vibration amplitude, and cutting speed. This is referred to as the theoretical residual height in the cutting direction. Therefore, the final microstructure and surface roughness of the machined surface are the superposition of the theoretical residual heights in the feed direction and the cutting direction. However, in actual production, the surface roughness requirements for ultra-precision machining are extremely high, generally requiring a nanometer-level finish. The additional machining textures introduced in the cutting direction by ultrasonic elliptical vibration turning have a significant impact on the surface roughness of ultra-precision machined surfaces. If the influence of process parameter selection on the residual height in the cutting direction cannot be quickly determined, considerable time and cost are required for process parameter debugging and optimization in actual machining, which is detrimental to improving machining efficiency.

[0004] CN113042823A discloses a method for predicting residual height in linear cutting under multiple process parameters. The method includes three steps: Step 1: Inputting the tool parameters and ultrasonic elliptical vibration cutting parameters for workpiece machining to obtain vibration machining parameters, and judging the surface machining condition of the part based on the tool parameters and ultrasonic elliptical vibration cutting parameters; Step 2: Measuring the machined semi-finished or finished product, obtaining the measurement model through surface reconstruction, and analyzing the residual height Rth1 after considering the blunt radius machining in actual machining; Step 3: Obtaining the vibration parameter equation of the machining trajectory and the tool rake and flank face parameter equations based on the machining parameters, and then analyzing the residual height Rth2 without considering the blunt radius. This invention considers the influence of the blunt radius on the tool. When the tool used for machining has a circular rake face and a conical flank face, it explores the influence of composite cutting process parameters, ultrasonic vibration parameters, and tool geometry parameters on the residual height in ultrasonic-assisted vibration cutting technology.

[0005] CN113901388A discloses a method for predicting the residual height of cutting along a curve under variable parameters. The method involves obtaining the curve equation based on the machined surface; establishing a variable-parameter elliptical vibration equation to obtain the elliptical trajectory equation for ultrasonic elliptical vibration machining; combining the elliptical trajectory equation with the curve equation obtained in step one to determine the coordinates of the intersection point of the curve and the elliptical trajectory; calculating the elliptical vibration trajectory equation of the nth tool contact point during the cutting process; then calculating the intersection point of the elliptical vibration trajectories of two adjacent tool contact points; and finally calculating the residual height between adjacent elliptical vibration trajectories.

[0006] The method for predicting the residual height of ultrasonic elliptical vibration cutting surfaces mentioned in patent CN113042823A focuses on ordinary plane studies, only considering the influence of tool speed along the cutting direction on the residual height under single-axis movement of the machine tool, and does not solve the problem of calculating the residual height of the surface when cutting inclined surfaces under multi-axis linkage of the machine tool. The method for predicting the residual height of cutting along curves under variable parameter conditions proposed in CN113901388A does not consider the influence of tool cutting speed on the calculation result of the intersection point when determining the intersection point of the elliptical vibration trajectory and the curve. However, the tool cutting speed actually affects the shape of the continuous relative motion trajectory of the tool, thus affecting the intersection point of the tool vibration trajectory and the curve.

[0007] Therefore, a new theoretical residual height prediction method for ultrasonic elliptical vibration turning needs to be developed to solve the problem of calculating the surface residual height when cutting inclined surfaces using multi-axis linkage ultrasonic elliptical vibration. Summary of the Invention

[0008] The purpose of this invention is to overcome the shortcomings of the prior art and propose a method for numerically solving the residual height in the cutting direction of ultrasonic elliptical vibration turning. This method enables the preliminary judgment of the rationality of process parameter selection in ultrasonic elliptical vibration-assisted ultra-precision cutting with multi-axis linkage using ordinary circular arc diamond tools. It helps operators quickly determine the appropriate range of process parameters, thereby reducing the time cost of process parameter debugging and optimization and improving processing efficiency.

[0009] To achieve the above objectives, this invention provides a method for predicting the residual height of surfaces machined by ultrasonic elliptical vibration. Based on the principle that ultrasonic simple harmonic vibration exhibits motion displacement symmetry and time reversal symmetry about the equilibrium position within a complete cycle, this method establishes a numerical approach to calculate the residual height introduced in the cutting direction during ultrasonic elliptical vibration turning. This method is applicable to calculating the residual height of surfaces in ordinary plane cutting and inclined plane cutting under multi-axis linkage. The specific steps of this method are as follows:

[0010] Step 1: Obtain the corresponding surface equation z for the target surface to be processed. s = f(y);

[0011] Step 2: Based on the equation of the surface to be machined and combined with the tool vibration mode, establish the cutting trajectory expression of the tool relative to the workpiece in ultrasonic elliptical vibration cutting. for:

[0012]

[0013] Among them, A y A z Let f represent the single-sided amplitude of the tool along the y and z directions of the machine tool coordinate system during ultrasonic elliptical vibration-assisted cutting, respectively, and let v be the vibration frequency of the ultrasonic elliptical vibration. c Represents the cutting speed along the y-direction, and k is the proportionality coefficient between the machine tool's z-axis guideway traverse speed and the y-axis cutting speed. Its specific value depends on the angle α between the target surface and the positive y-axis. d This is to ensure the cutting trajectory An offset along the z-axis added at a certain distance above the surface of the target to be processed;

[0014] Step 3: Starting from t=0, based on the tool relative cutting trajectory expression... The cutting trajectory was solved using the Newton-Raphson iterative method during the first vibration cycle. From the point on the target surface z to the surface to be processed s = f(y) is the cutting time t1 corresponding to the shortest straight distance along the z-axis, and the corresponding shortest straight distance along the z-axis is denoted as z. d1 ;

[0015] Step 4: Set the tool cutting path Translate z downward along the z-axis d1 Obtain the tool cutting trajectory

[0016]

[0017] The cutting trajectory at time t1 at this moment The points on the curve represent the cutting trajectory. The point of tangency A0 with the target surface to be machined is denoted as the cutting time corresponding to point A0. Based on the periodic characteristics of ultrasonic elliptical vibration of the cutting tool, along the cutting trajectory The direction of time progression can be determined within the next adjacent cycle, with the second tangent point being point D0. The time corresponding to point D0 can be represented as:

[0018]

[0019] Where T is the period of ultrasonic elliptical vibration of the tool;

[0020] Step 5: Define the moment when the tool re-enters the material during the first vibration cycle as t. c Set its corresponding cutting trajectory Point C0 on;

[0021] Step 6: Simultaneously, let points A0 and D0 start moving from their initial positions, where point A0 is on the cutting trajectory. The point D0 moves gradually along the direction of time advancement with a change in Δt (t increases by Δt), and the point D0 is on the cutting trajectory. The upward movement proceeds gradually in the direction of time regression with a change in Δt (t decreases by Δt). Once the relative magnitudes of the y-axis coordinates of the two moving points reverse, it indicates that they have reached the point of convergence C0. The time corresponding to point C0 at this point can be represented as:

[0022]

[0023] Where n is the point D0 on the cutting trajectory with a change Δt. The number of steps required to move along the backward time direction to the coincidence point C0;

[0024] Step 7: Based on the time t corresponding to the coincidence point C0 obtained in Step 6. c and the tool cutting trajectory expression obtained in step four The position coordinates of point C0 in the YOZ plane can be obtained as (y'(t)). c ),z'(t c ));

[0025] Step 8: Calculate the distance from point C0 to the ideal target surface z using the formula for the distance from a point to a line. s The distance result of f(y) is the theoretical residual height Rz in the cutting direction. v .

[0026] Furthermore, the formula for calculating the distance from a point to a line is as follows:

[0027]

[0028] Here, k and b are called the direction coefficients of the straight line equation, and c is a constant term representing the spatial position offset.

[0029] As a further preferred embodiment of the present invention, the parameter k related to the type of surface to be processed in step two is k = 0 when the surface is an end plane; when the surface is an inclined plane, k depends on the angle α between the inclined plane to be processed and the horizontal plane in the clockwise direction, specifically k = tanα.

[0030] As a further preferred embodiment of the present invention, the time change Δt in step six is ​​preferably set to... Where T is the period of ultrasonic elliptical vibration of the tool, and m represents the number of time segments, preferably set to be greater than or equal to 10. 5 The natural number.

[0031] Advantages and beneficial effects of the present invention:

[0032] 1. The method of the present invention takes into account the influence of the combined cutting speed of the tool end generated by the multi-axis linkage of the machine tool on the relative motion trajectory of the tool's elliptical vibration, and is also applicable to the calculation of the residual height of the cutting surface on a normal horizontal plane, thus having a certain degree of universality.

[0033] 2. Based on the characteristics of motion displacement symmetry and time reversal symmetry of ultrasonic simple harmonic vibration, this method can conveniently calculate the residual height introduced in the cutting direction when ultrasonic elliptical vibration is used to turn planes and inclined planes. It can assist relevant operators in making a preliminary judgment on the rationality of process parameter selection in actual processing, and make up for the shortcomings of long time cycle and cumbersome process adjustment and optimization in traditional trial cutting steps. Attached Figure Description

[0034] Figure 1 A schematic diagram of ultrasonic elliptical vibration-assisted cutting;

[0035] Figure 2 This is a schematic diagram of the principle of ultrasonic elliptical vibration cutting an inclined plane.

[0036] ( Figure 2 In the middle: 1 represents the initial cutting trajectory. 2 represents the cutting trajectory after translation.

[0037] Figure 3 This is a magnified view of a local detail of an ultrasonic elliptical vibration cutting inclined plane.

[0038] Figure 4 This is a schematic diagram of the ultrasonic elliptical vibration cutting plane. Detailed Implementation

[0039] The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not limiting, and should not be construed as limiting the scope of protection of the present invention.

[0040] A schematic diagram of the principle of ultrasonic elliptical vibration-assisted cutting is shown below. Figure 1 As shown, this method applies ultrasonic simple harmonic vibrations to a diamond tool along both the cutting direction and the cutting normal, thereby enabling the tool to remove material through elliptical vibration. Because ultrasonic elliptical vibration cutting has the periodic cutting characteristic of tool-workpiece separation, it introduces periodic textures along the cutting direction, referred to as the residual height in the cutting direction.

[0041] Figure 2 This diagram illustrates the principle of ultrasonic elliptical vibration cutting of an inclined plane. Figure 2 The analysis was carried out over several cycles, resulting in a magnified schematic diagram of the local details of the residual height generated in the cutting direction, as shown in the figure. Figure 3 As shown. In Figure 3 When the angle between the inclined plane to be machined and the positive y-axis is 0° (i.e. Figure 3 With a circumferential tilt angle of α = 0°, the entire cutting process will become a process of ultrasonic elliptical vibration cutting the plane, such as... Figure 4 As shown. Since the process of ultrasonic elliptical vibration cutting the plane is included in the special case where the inclination angle α is 0°, the following will refer to... Figure 3 The schematic diagram shown illustrates, in a more concrete way, the method for obtaining the numerical solution of the theoretical residual height in the cutting direction of ultrasonic elliptical vibration cutting in this invention:

[0042] Step 1: Obtain the corresponding surface equation z for the target surface to be processed. s =f(y), i.e., attached Figure 3 The surface indicated by the black dashed line in the middle;

[0043] The second step is to establish the cutting trajectory expression of the tool relative to the workpiece in ultrasonic elliptical vibration cutting based on the equation of the surface to be machined and the form of tool vibration.

[0044]

[0045] In the formula, A y A zThese represent the single-sided amplitudes of the tool along the y and z directions of the machine tool coordinate system during ultrasonic elliptical vibration-assisted cutting, respectively; f is the vibration frequency of the ultrasonic elliptical vibration; and v... c The cutting speed along the y-axis represents the magnitude of the cutting speed. k is the proportionality coefficient between the machine tool's z-axis guideway traverse speed and the y-axis cutting speed. It depends on the angle α between the target surface and the positive y-axis. Therefore, when the machined surface is a flat surface, k = 0, and the cutting trajectory is as follows: Figure 2 As shown; when the machined surface is an inclined plane, k = tanα, and the cutting trajectory is as follows. Figure 1 As shown, z d This is to ensure the cutting trajectory An offset along the z-axis that is added at a certain distance above the surface to be processed;

[0046] Furthermore, the relative speed v of the tool along the cutting direction and the cutting normal. y (t), v z (t) can be derived from the cutting trajectory expression. Differentiation yields:

[0047]

[0048] The third step is to start from t=0, based on the expression of the tool's relative cutting trajectory. The tool trajectory is solved using the Newton-Raphson iterative method during the first vibration cycle. From the point on the target surface z to the surface to be processed s = f(y) is the cutting time t1 corresponding to the shortest straight-line distance along the z-axis, and the straight-line distance along the z-axis corresponding to this time is set as z. d1 .

[0049] Step 4, such as Figure 2 As shown, the tool cutting path Translate z downward along the z-axis d1 Obtain the tool cutting trajectory

[0050]

[0051] The cutting trajectory at time t1 at this moment The points on the curve represent the cutting trajectory. The point of tangency with the target surface to be processed. For example... Figure 3 As shown, this tangent point is marked as point A0, and the cutting time is also denoted as... Based on the periodic characteristics of ultrasonic vibration of the cutting tool, along the cutting trajectory The direction of time progression can be determined within the next adjacent cycle, with the second tangent point being point D0. The corresponding time can be represented as:

[0052]

[0053] Where T is the period of ultrasonic elliptical vibration of the tool;

[0054] Step 5: Define the moment when the tool re-enters the material during the first vibration cycle as t. c Set its corresponding cutting trajectory Point C0 on;

[0055] Step 6: Simultaneously, let points A0 and D0 start moving from their initial positions, where point A0 is on the cutting trajectory. The point D0 moves gradually along the direction of time advancement with a change in Δt (t increases by Δt), and the point D0 is on the cutting trajectory. The upward movement proceeds gradually in the direction of time regression with a change in Δt (t decreases by Δt). Once the relative magnitudes of the y-axis coordinates of the two moving points reverse, it indicates that they have reached the point of convergence C0. The time corresponding to point C0 at this point can be represented as:

[0056]

[0057] Where n is the point D0 on the cutting trajectory with a change Δt. The number of steps required to move along the backward time direction to the coincidence point C0, and the time change Δt are preferably set as follows: m represents the number of time segments, and it is preferable to set m to be greater than or equal to 10. 5 _n_ natural numbers;

[0058] More specifically, the following will derive and explain in detail why points A0 and D0 can simultaneously reach the point of overlap C0(B0) according to the movement pattern in step six:

[0059] from Figure 3 In this equation, the oscillation cycle from point A0 to point D0 along the cutting trajectory is one period. A0 and D0 are both points of tangency between the envelope of the elliptical trajectory and the ideal target surface. The initial position coordinates of points A0 and D0 in the Cartesian YOZ plane can be expressed as follows: From the cutting trajectory expression It can be known that:

[0060]

[0061] in, These respectively represent the cutting trajectory The cutting time corresponding to the initial positions of A0 and D0;

[0062] exist Figure 3 In the middle, from point B0(y'(t) b ),z'(t b The machine moves along the cutting trajectory to point B1(y'(t)b1 ),z'(t b1 ()) is also a vibration period, derived from the cutting trajectory expression. Similarly, we can conclude that:

[0063]

[0064] Among them, t b t b1 These respectively represent the cutting trajectory The times corresponding to B0 and B1;

[0065] like Figure 3 As shown, points B0 and C0 coincide in the cutting trajectory, therefore we can conclude that:

[0066]

[0067] Where t c Indicates the cutting trajectory The time corresponding to point C0;

[0068] Next, with Figure 3 Point D0 on the tool path is the starting point, and the time corresponding to the initial position of this point is... The cutting times corresponding to points C0 and B1 are used The time intervals Δt1 and Δt2 are expressed as follows:

[0069]

[0070] In the formula, Δt1 and Δt2 represent the cutting trajectory, respectively. The time interval required for movement from C0 to D0 and from D0 to B1;

[0071] The next step will be to The t represented by Δt1 and Δt2 c and t b1 Substitute the relation In, and combined with the cutting trajectory expression The following relation R1 can be obtained:

[0072]

[0073] Since the direction of the resultant velocity of the tool's relative motion at the cutting moment corresponding to point D0 is parallel to the surface of the target being machined, we can obtain:

[0074]

[0075] Based on this, the expression for the relative motion velocity v of the tool along the cutting direction and the cutting normal is combined. y (t) and v z(t) can be further used to obtain relation R2:

[0076]

[0077] By combining the relations R1 and R2, we can obtain the following equation R3:

[0078]

[0079] Analysis shows that in equation R3 Since it is not always 0, in order to satisfy the equality relationship in R3, the following additional condition must be satisfied:

[0080] Δt1-Δt2=0;

[0081] Therefore, the time Δt1 required to move from C0 to D0 is equal to the time Δt2 required to move from D0 to B1. Furthermore, based on the periodicity of ultrasonic vibration, the time required to move from A0 to B0 is equal to the time Δt2 required to move from D0 to B1. Therefore, the time required to move from C0 to D0 is equal to the time required to move from A0 to B0. Thus, following the movement pattern in step four, points A0 and D0 can simultaneously reach the point of trajectory overlap, C0.

[0082] Step 7: Based on the cutting time t corresponding to the coincidence point C0 obtained in step 6. C and the cutting trajectory expression obtained in step four The position coordinates of point C0 in the YOZ plane can be obtained as (y'(t)). c ),z'(t c ));

[0083] Step 8: Calculate the distance from point C0 to the target surface using the point-to-line distance formula. The result is the theoretical residual height Rz in the cutting direction. v More specifically, the target surface to be processed is set in... Figure 3 The YOZ coordinate plane shown can be represented by a linear equation as follows:

[0084] ky+bz+c=0

[0085] Where k and b are called the direction coefficients of the straight line equation, and c is a constant term representing the spatial position offset;

[0086] Furthermore, Rz v It can be calculated as follows:

[0087]

[0088] Application Example 1:

[0089] The method for obtaining the theoretical residual height in the cutting direction of ultrasonic elliptical vibration turning in this invention is applicable to both planar and inclined surface machining. In this embodiment, planar cutting is used as an example, employing a vibration frequency of f = 40kHz (corresponding to a vibration period T = 1 / 40000s) and a cutting direction amplitude of A. y =2μm, cutting normal amplitude is A z The ultrasonic vibration parameters are 1 μm, and the cutting linear velocity is selected as v. c =10m / min, using a circular arc diamond turning tool with a rake angle of 0° and a clearance angle of 15°. The specific implementation method for obtaining the theoretical residual height numerical solution in the cutting direction for plane cutting is as follows:

[0090] First, for example, the plane to be processed can be represented by the equation z = 0 in the YOZ coordinate plane.

[0091] Next, the cutting trajectory expression of the tool relative to the workpiece is obtained based on the ultrasonic vibration parameters and cutting process parameters.

[0092] When cutting a plane, k = 0 in the formula. Furthermore, z is set... d =2mm, ensuring that the lower half of the cutting trajectory envelope is at a certain distance from the surface to be machined.

[0093] Next, starting from t=0, the tool path is solved within the first vibration cycle. The cutting time t1 corresponding to the shortest straight-line distance from the point on the surface to be machined along the z-axis is obtained using the Newton-Raphson iteration method, which yields t1 = 1.25 × 10⁻⁶. -5 s, at this time the corresponding shortest straight distance along the z-axis is z d1 =1.999mm.

[0094] Next, the tool path is... Translate z downward along the z-axis d1 Obtain the tool path

[0095]

[0096] Then the cutting trajectory corresponding to time t1 The points on the curve represent the cutting trajectory. The point of tangency with the inclined surface to be processed.

[0097] Then, in the cutting trajectory At time t1 and t1+T, the corresponding points are point A0 and point D0 respectively. Let them start moving simultaneously. Among them, point A0 moves step by step along the direction of time advancement on the cutting trajectory with a change amount of Δt (t increases by Δt), and point D0 moves step by step along the direction of time regression on the cutting trajectory with a change amount of Δt (t increases by Δt). Take Then the y coordinate values corresponding to the two moving points A0 and D0 on the cutting trajectory can be dynamically expressed as y'(t1+nΔt) and y'(t1+T-nΔt) respectively.

[0098] At first, y'(t1)<y'(t1+T). Next, let n = 1, and calculate the corresponding y coordinate values at this time as y'(t1+Δt) and y'(t1+T-Δt) respectively. If y'(t1+Δt)<y'(t1+T-Δt), then let n = n+1, and repeat the above operation until the condition y'(t1+nΔt)≥y'(t1+T-nΔt) is satisfied. At this time, the cutting time corresponding to the coincidence point C0 can be expressed as t c =t1+T-nΔt, and the corresponding coordinates on the cutting trajectory are (y'(t c ),z'(t c )).

[0099] Finally, through iterative calculation, when n = 136659, y'(t1+nΔt)≥y'(t1+T-nΔt) is just satisfied. At this time, the cutting time t c =3.40835×10 -5 s, and the theoretical residual height along the cutting direction can be calculated by the distance formula from a point to a line as

[0100] Application Example 2

[0101] In this example, taking inclined plane cutting as an example, it is set that the included angle between the target inclined plane to be machined and the positive direction of the y axis in the YOZ coordinate plane is 20°, and the vibration frequency is f = 40kHz (period ), the amplitude in the cutting direction is A y [[ID=3۴]]=2μm, the amplitude in the cutting normal direction is A z =1μm for ultrasonic vibration parameters, the cutting linear velocity is selected as v c =10m / min, and a circular arc edge diamond turning tool with a front angle of 0° and a back angle of 15° is used. The specific implementation manner for obtaining the numerical solution of the theoretical residual height in the cutting direction for inclined plane cutting is as follows:

[0102] First, for example, it is set that the inclined plane to be machined can be represented by the equation in the YOZ plane.<০০০০৩৮৩>

[0103] Next, based on the ultrasonic vibration parameters, cutting process parameters, and the inclination angle of the inclined plane to be machined, the cutting trajectory expression of the tool relative to the workpiece is obtained.

[0104]

[0105] When cutting the set inclined plane, in the formula In addition, set z d =2mm, ensuring that the lower half of the cutting trajectory envelope is at a certain distance from the surface of the inclined plane to be machined.

[0106] Next, starting from t=0, the tool path is solved within the first vibration cycle. The cutting time t1 corresponding to the shortest straight-line distance from the point on the inclined plane to the surface to be machined along the z-axis is obtained using the Newton-Raphson iteration method, which yields t1 = 1.500364 × 10⁻⁶. -5 s, at this time the corresponding shortest straight distance along the z-axis is z d1 = 1.9987631mm.

[0107] Next, the toolpath is... Translate z downward along the z-axis d1 Obtain the tool path

[0108]

[0109] Then the cutting trajectory corresponding to time t1 The points on the curve represent the cutting trajectory. The point of tangency with the inclined surface to be processed.

[0110] Then, in the cutting trajectory Let A0 and D0 be the points corresponding to time t1 and t1+T, respectively. Let them start moving simultaneously. Point A0 moves gradually along the cutting trajectory in the direction of time advancement with a change of Δt (t increases by Δt), while point D0 moves gradually along the cutting trajectory in the direction of time reversal with a change of Δt (t increases by Δt). Then the two moving points A0 and D0 are on the cutting trajectory The corresponding y-coordinate values ​​can be expressed as follows: And y'(t1+t-nΔt).

[0111] Initially, y'(t1) < y'(t1 + T). Next, let n = 1 and calculate the corresponding y-coordinate values at this time as y'(t1 + Δt) and y'(t1 + T - Δt) respectively. If y'(t1 + Δt) < y'(t1 + T - Δt), then let n = n + 1 and repeat the above operation until the condition y'(t1 + nΔt) ≥ y'(t1 + T - nΔt) is satisfied. At this time, the cutting moment corresponding to the coincidence point C0 can be expressed as t c = t1 + T - nΔt, and the corresponding coordinates on the cutting trajectory are (y'(t c ), z'(t c )).

[0112] Finally, through iterative calculation, when n = 165435, it just satisfies y'(t1 + nΔt) ≥ y'(t1 + T - nΔt). At this time, the cutting moment t c = 3.586779×1×10 -5 s, and the theoretical remaining height along the cutting direction can be calculated by the distance formula from a point to a line as

[0113] The above are only the preferred embodiments of the present invention. It should be noted that for those of ordinary skill in the art, without departing from the inventive concept, several modifications and improvements can still be made, and all of these belong to the protection scope of the present invention.

Claims

1. A method for predicting the residual height of a surface machined by ultrasonic elliptical vibration, characterized in that, The specific steps of this method are as follows: Step 1: Obtain the corresponding surface equation z for the target surface to be processed. s = f(y); Step 2: Based on the equation of the surface to be machined and combined with the tool vibration mode, establish the cutting trajectory expression of the tool relative to the workpiece in ultrasonic elliptical vibration cutting. Among them, A y A z denoted as , respectively, the single-sided amplitude of the tool along the y and z directions of the machine tool coordinate system during ultrasonic elliptical vibration-assisted cutting; f is the vibration frequency during ultrasonic elliptical vibration cutting; v c The value of z represents the cutting speed along the y-direction, and k is the proportionality coefficient between the machine tool's z-guide travel speed and the y-direction cutting speed, which depends on the angle α between the target surface and the positive y-axis. d This is to ensure the cutting trajectory An offset along the z-axis that is added at a certain distance above the surface to be processed; Step 3: Starting from t=0, based on the tool relative cutting trajectory expression... Solving the cutting trajectory within the first tool vibration cycle From the point on the target surface z to the surface to be processed s = f(y) is the cutting time t1 corresponding to the shortest straight distance along the z-axis, and the corresponding shortest straight distance along the z-axis is denoted as z. d1 ; Step 4: Set the tool path Translate z downward along the z-axis d1 Obtain the tool path Define the cutting trajectory corresponding to time t1. The points on the curve represent the cutting trajectory. The point of tangency A0 with the inclined surface to be machined is denoted as the cutting time corresponding to point A0. Based on the periodic characteristics of ultrasonic elliptical vibration of the cutting tool, along the cutting trajectory The direction of time progression can be determined within the next adjacent cycle, with the second tangent point being point D0, and the corresponding time is represented as: Where T is the period of ultrasonic elliptical vibration of the tool; Step 5: Define the moment t when the tool re-enters the material during the first vibration cycle of the tool. c Set its corresponding cutting trajectory Point C0 on; Step 6: Simultaneously, let points A0 and D0 start moving from their initial positions, where point A0 is on the cutting trajectory. The point D0 moves gradually along the cutting trajectory with a change in time Δt (t increases by Δt). The upward movement proceeds gradually in the direction of time regression with a change in Δt (t decreases by Δt). Once the magnitude relationship of the y-axis coordinates of the two moving points reverses, it indicates that they have reached the point of overlap, C0. The time corresponding to point C0 can be represented as: Where n is the point D0 on the cutting trajectory with a change Δt. The number of steps required to move along the backward time direction to the coincidence point C0; Step 7: Based on the time t corresponding to the coincidence point C0 obtained in Step 6. c and the tool cutting trajectory expression obtained in step four The position coordinates of point C0 in the YOZ plane can be obtained as (y'(t)). c ),z'(t c )); Step 8: Calculate the distance from point C0 to the ideal target surface z using the formula for the distance from a point to a line. s The distance result of f(y) is the theoretical residual height Rz in the cutting direction. v .

2. The method for predicting the residual height of an ultrasonic elliptical vibration-machined surface according to claim 1, characterized in that, The y-axis and z-axis mentioned in step two correspond to the y-direction and z-direction of the coordinate system of a multi-axis ultra-precision machining tool.

3. The method for predicting the residual height of an ultrasonic elliptical vibration-machined surface according to claim 1, characterized in that, The cutting tool is a circular arc-shaped diamond turning tool with a zero-degree rake angle.

4. The method for predicting the residual height of an ultrasonic elliptical vibration-machined surface according to claim 1, characterized in that, When the cutting tool undergoes ultrasonic elliptical vibration, the starting point along the y-axis is the equilibrium position of simple harmonic motion, and the phase difference between the vibration along the y-axis and the vibration along the z-axis is... That is, the major axis and minor axis of the elliptical vibration mode of the tool are parallel to the y-axis and z-axis, respectively.

5. The method for predicting the residual height of an ultrasonic elliptical vibration-machined surface according to claim 1, characterized in that, The proportionality coefficient k between the machine tool's z-axis guide rail movement speed and the y-axis cutting speed mentioned in step 1 is k = 0 when the surface is an end plane; when the surface is an inclined plane, k depends on the angle α between the inclined plane to be machined and the positive y-axis direction, k = tanα.

6. The method for predicting the residual height of an ultrasonic elliptical vibration-machined surface according to claim 1, characterized in that, The time change Δt is set as Where T is the period of the ultrasonic elliptical vibration of the tool, m represents the number of time segments, and m is greater than or equal to 10. 5 The natural number.

7. The method for predicting the residual height of an ultrasonic elliptical vibration-machined surface according to claim 1, characterized in that, Step 3: Solving for the tool path From the point on the target surface z to the surface to be processed s The cutting time corresponding to the shortest straight-line distance along the z-axis is obtained by the Newton-Raphson iteration method.

8. The method for predicting the residual height of an ultrasonic elliptical vibration-machined surface according to claim 1, characterized in that, The formula for calculating the distance from the point to the line mentioned in step eight is as follows: Here, k and b are called the direction coefficients of the straight line equation, and c is a constant term representing the spatial position offset.

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