JP5239650B2 - Wear amount simulation device for rotating tools - Google Patents

Description

  The present invention relates to a simulation apparatus for simulating the amount of wear of a rotary tool in grinding with a rotary tool.

  The grinding process is performed, for example, by relatively moving the grindstone rotating at high speed and the rotating workpiece, and bringing the grindstone and the workpiece into contact with each other. In such a grinding process, there has been a problem that chattering occurs and the grinding accuracy of the workpiece is lowered. Therefore, the above-mentioned problem has been solved by changing the grinding conditions according to the experience and intuition of the operator. In other words, after carrying out the grinding process, check the state of the machined surface, and if you find that there is a grinding burn or chatter phenomenon, change the grinding process conditions, perform the grinding process again, and change the machined surface condition. I was checking. For this reason, the same operation is repeated, and a large amount of work and work time are required.

  Therefore, the chatter vibration control method described in Japanese Patent Application Laid-Open No. 8-174379 (Patent Document 1) discloses that a chatter phenomenon suppression operation is automatically performed. Patent Document 1 describes that vibration of a machine tool is detected and a machining condition is changed based on the detection result. In Patent Document 1, as one of the chatter phenomenon, a relative displacement generated between a tool and a workpiece forms a swell on the workpiece surface, which acts as a variation in the depth of cut after one rotation of the workpiece. It is described that there is a regenerative self-excited vibration that occurs.

  By the way, in addition to forming waviness on the surface of a workpiece, waviness may be formed on the surface of a tool, for example, a grindstone. In other words, this means that the amount of wear differs depending on each part of the surface of the grindstone. The waviness formed on the surface of the grindstone is considered to be one of the causes of relative displacement between the work and the grindstone when machining the work. Furthermore, the waviness shape of the grindstone surface may be transferred to the work surface by processing the work piece with the grindstone having the waviness shape on the grindstone surface. As a result, it is considered that a waviness shape is formed on the workpiece surface.

Japanese Patent Laid-Open No. 2002-224955 (Patent Document 2) discloses a machine tool control system that calculates the wear amount of a grinding wheel and determines whether to perform grinding wheel correction processing. In Patent Document 2, for example, the turning power of the grinding wheel is detected based on the electric current of the electric motor that drives the grinding wheel, the grinding removal amount of the workpiece is calculated based on this turning power, and the accumulated grinding removal amount is further calculated. From this, it is possible to estimate the wear amount of the grinding wheel. Therefore, it is determined whether the grinding process is continued or the grinding wheel correction processing such as dressing is performed on the grinding wheel from the abrasion amount of the grinding wheel calculated by this method.
JP-A-8-174379 JP 2002-224554 A

  However, the machine tool control system described in Patent Document 2 calculates the removal amount of the workpiece based on the motor current, calculates the wear amount of the grinding wheel based on the removal amount, and uses the accumulated value as the grinding wheel. The configuration is used to determine when to perform the correction process. That is, even if the grinding wheel is worn, the grinding wheel has always been assumed to be a perfect circle. Therefore, even if the technique of Patent Document 2 is used, the waviness formed on the surface of the tool, which is considered to be one of the causes of the chatter phenomenon, is not considered.

  In the method described in Patent Document 1, it is necessary to measure the vibration of the machine tool during grinding. In the method described in Patent Document 2, it is necessary to measure the current value of the electric motor that drives the grinding wheel during grinding. Therefore, in any method, grinding must actually be performed, which leads to an increase in working time and a decrease in production efficiency. Therefore, in the grinding process, it is necessary to simulate the wear amount of the rotary tool before performing the grinding process.

  The present invention has been made in view of such circumstances, and provides a wear amount simulation device for a rotary tool capable of simulating the waviness shape of the surface of the rotary tool, which is considered to be one of the causes of chatter phenomenon in grinding. With the goal.

  Hereinafter, each means suitable for solving the above-described problems will be described with additional effects and the like as necessary.

(Means 1) An apparatus for simulating the wear amount of a rotary tool according to means 1 comprises:
Rotation for simulating the amount of wear of the rotary tool in processing performed by rotating the rotary tool while bringing the peripheral edge of the rotary tool into contact with the workpiece and relatively moving the workpiece and the rotary tool. A tool wear amount simulation device,
A workpiece removal amount calculation unit that calculates the removal amount of the workpiece by each of a plurality of rotational phase points on the periphery of the rotary tool;
A tool wear amount calculation unit for calculating a wear amount at a peripheral portion of the rotary tool by calculating a wear amount at each of the rotation phase points based on a removal amount of the workpiece at each of the rotation phase points;
It is characterized by providing.

  According to the present invention, in the simulation of machining with a rotary tool, the shape of the rotary tool is identified by a plurality of rotational phase points on the periphery thereof. Further, the amount of wear at each rotational phase point is calculated by calculating the amount of removal of the workpiece by each of the rotational phase points. Conventionally, the rotating tool in the same simulation reflects the amount of wear assuming a perfect circle, and therefore wears uniformly over the entire circumference. In this case, the waviness shape of the rotary tool, that is, the uneven wear of the rotary tool could not be simulated. However, with the above-described configuration, it is possible to calculate the wear amount according to the rotation phase at the periphery of the rotary tool. Therefore, it becomes possible to simulate the more accurate wear amount of the rotary tool. Therefore, it is possible to confirm by simulation that the uneven wear of the rotary tool, which causes the chatter phenomenon or the like, may occur under a predetermined processing condition before performing the processing.

(Means 2) In the rotating tool wear amount simulation apparatus of the means 1,
The rotary tool wear amount simulation device is:
In a coordinate system in which the workpiece is stationary, a trajectory calculation unit that calculates a movement trajectory of each rotation phase point with respect to the workpiece,
The workpiece removal amount calculation unit may calculate the removal amount of the workpiece based on the shape of the workpiece and the movement trajectory.

  According to this means, when the rotary tool rotates and the workpiece and the rotary tool move relative to each other, the movement trajectory of the rotational phase point that qualifies the rotary tool is calculated in the coordinate system where the workpiece is stationary. . Furthermore, the removal amount of the workpiece for each rotational phase point is calculated based on the shape of the workpiece and the movement trajectory of each rotational phase point. The removal amount of the workpiece per unit time or per unit movement amount can be calculated from the movement locus and the shape of the stationary workpiece.

(Means 3) In the rotating tool wear amount simulation apparatus of the means 2,
The rotary tool wear amount simulation device is:
A workpiece shape storage unit for storing the shape of the workpiece;
Based on the movement trajectory calculated by the trajectory calculation unit, a workpiece shape changing unit that changes the shape of the workpiece stored in the workpiece shape storage unit;
With
The workpiece removal amount calculation unit may calculate the removal amount of the workpiece by each of the rotation phase points in the workpiece shape sequentially changed by the workpiece shape changing unit.

  According to this means, when the removal amount of the workpiece of the means 2 is calculated, the shape of the workpiece stored in the workpiece shape storage unit is changed based on the movement locus of the rotation phase point. Further, the shape of the workpiece after the removal is stored in the workpiece shape storage unit, and is reflected in the calculation of the removal amount with the movement locus of the next rotational phase point. Thereby, the shape of the workpiece is sequentially changed every time the removal amount at one rotational phase point is calculated, and a more accurate removal amount can be calculated.

(Means 4) In the rotating tool wear amount simulation apparatus of the means 3,
The apparatus for simulating the amount of wear of a rotating tool brings the peripheral edge of the rotating tool into contact with the peripheral surface of the workpiece while rotating the rotating tool and rotating the workpiece, and rotates the workpiece and the rotating tool. In processing performed by relatively moving the tool, a device for simulating the amount of wear of the rotary tool,
The workpiece shape storage unit stores a workpiece line segment connecting a plurality of points on the periphery of the workpiece and the rotation center of the workpiece as the shape of the workpiece,
The wear amount simulation device further includes a machining point calculation unit that calculates a machining point that is an intersection of the workpiece line segment and the movement locus,
The workpiece shape changing unit changes the shape of the workpiece by changing the workpiece line based on the machining point,
The workpiece removal amount calculation unit may calculate the removal amount of the workpiece based on the workpiece line segment before the shape change and the workpiece line segment after the shape change.

  According to this means, in a wear amount simulation in machining performed by rotationally driving the workpiece and the rotary tool, the shape of the workpiece is recognized by line segments respectively connecting a plurality of points on the periphery and the rotation center. Then, the removal amount of the workpiece is calculated by calculating a machining point that is an intersection of a plurality of workpiece line segments that recognize the shape of the workpiece and the movement locus of the rotation phase point. Thereby, the shape of the workpiece can be recognized by a plurality of line segments, and the removal amount and the shape of the workpiece after removal can be calculated from the machining points. In other words, since the calculation can be performed by calculating the intersection of the workpiece line segment and the movement trajectory, the calculation load can be greatly reduced, and the wear amount simulation of the rotary tool can be facilitated.

(Means 5) In the rotary tool wear amount simulation apparatus of the means 4,
The plurality of points on the periphery of the workpiece line segment may be division points on the periphery obtained by equiangularly dividing the workpiece.

  According to this means, since the workpiece line segments are arranged at equal angles, there is no deviation in the circumferential direction, and the removal amount for each rotational phase point can be calculated with high accuracy. As a result, the wear amount of the rotary tool can be simulated with high accuracy. Furthermore, the calculation load for processing point calculation can be reduced.

(Means 6) In the rotating tool wear amount simulation apparatus of the means 4 or 5,
The workpiece shape changing unit changes a shape by changing a peripheral side end point of the workpiece line segment to the machining point on the workpiece line segment,
The workpiece removal amount calculation unit
In the two adjacent workpiece line segments, the area of the region formed by each of the peripheral side end points before and after the shape change is a fragment area,
The piece areas calculated for the shape changes of all two adjacent workpiece line segments may be integrated, and the removal amount of the workpiece may be calculated based on the integrated area.

  According to this means, when a predetermined workpiece line segment and the movement locus of the rotational phase point intersect and have a machining point, a line segment connecting the machining point and the rotation center of the workpiece is defined as a new predetermined workpiece. A line segment. By changing this to all the workpiece line segments, the shape of the workpiece is changed. Further, in two adjacent workpiece line segments, a triangular or quadrilateral fragment area formed by the end point before the shape change and the end point after the shape change (processing point) is calculated. The fragment area is calculated for at least a workpiece line segment having a machining point, and the amount of workpiece removal is calculated based on the integrated area. Therefore, the removal amount that is removed by grinding when the moving locus made of a curve comes into contact with the workpiece shape can be calculated based on the integrated area of the fragment areas, so that the calculation load can be greatly reduced.

(Means 7) In the rotary tool wear amount simulation apparatus of means 4 or 5,
The workpiece shape changing unit changes a shape by changing a peripheral side end point of the workpiece line segment to the machining point on the workpiece line segment,
The workpiece removal amount calculation unit
The length of the line segment connecting the peripheral side end points before and after the shape change is a fragment line segment length,
It is preferable to integrate the fragment line lengths calculated for the shape changes of all the workpiece line segments and calculate the removal amount of the workpieces based on the integrated length.

  According to this means, when a predetermined workpiece line segment and the movement locus of the rotational phase point intersect and have a machining point, a line segment connecting the machining point and the rotation center of the workpiece is defined as a new predetermined workpiece. A line segment. By changing this to all the workpiece line segments, the shape of the workpiece is changed. Further, the length of the segment line connecting the end point before the shape change and the end point (the machining point) after the shape change is calculated. The segment line length is calculated for at least a workpiece line segment having a machining point, and the removal amount of the workpiece is calculated based on the integrated length. Accordingly, since the removal amount that is ground and removed by contact between the movement locus formed by the curve and the workpiece shape can be calculated based on the integrated length of the fragment line segment length, the calculation load can be greatly reduced.

(Means 8) In the rotating tool wear amount simulation device of the means 7,
The workpiece removal amount calculation unit
In the shape change based on the movement trajectory of a predetermined number of adjacent rotation phase points, the ratio of each integrated length to the total sum of the calculated integrated lengths is calculated,
In the two adjacent workpiece line segments, the area of the region formed by each of the peripheral side end points before and after the predetermined number of the shape change, the fragment area,
When the piece area calculated for the shape change of all two adjacent workpiece line segments is integrated, and the removal amount of the workpiece is calculated based on the integrated area and the ratio of the integrated length, respectively. Good.

  According to this means, the removal amount of the workpiece is calculated for the shape change based on the movement trajectory of the predetermined number of adjacent rotation phase points. That is, first, for each of the predetermined number of integrated lengths calculated in the shape change of the workpiece, the total of the integrated lengths and the ratio of each integrated length to the total are calculated. Next, in two adjacent workpiece line segments, the triangular or quadrilateral fragment area formed by the end point before the shape change and the end point (work point) after the shape change a predetermined number of times is calculated. The fragment area is calculated for at least a workpiece line segment having a machining point, and the removal amount of the workpiece at each rotational phase point is calculated based on the ratio between the integrated area and the integrated length. Thereby, since the number of calculation of the fragment area can be reduced, the calculation load can be greatly reduced. Furthermore, since the removal amount is calculated collectively from a predetermined number of shape changes, the occurrence of calculation errors can be reduced.

(Means 9) In the rotary tool wear amount simulation apparatus according to any one of the means 2 to 8,
The movement trajectory calculated by the trajectory calculation unit is:
In the coordinate system in which the workpiece is stationary, the curve may be a curve approximated by an arc by point groups at at least three different times of the rotational phase points with respect to the workpiece.

  The actual movement trajectory of the rotational phase point is an arbitrary continuous curve determined according to the relative movement between the rotary tool and the workpiece. However, when the movement locus of the continuous rotational phase points is viewed in a minute time, it is possible to approximate the circular arc. Therefore, according to the present means, a curved line approximated by an arc by a point group of rotational phase points at at least three different times is used as a moving trace of minute time. Thereby, the calculation load of the movement locus of the rotation phase point can be reduced, and the calculation load for calculating the machining point that is the intersection of the movement locus and the workpiece line segment can be reduced. In addition, the point group used for the circular arc approximation may be extracted from a range where the workpiece and the rotary tool are in contact with each other or in the vicinity of the contact range. Preferably, at least one of the point groups used for the circular arc approximation is extracted from a range where the workpiece and the rotary tool are in contact with each other. Of course, all the point groups used for the circular arc approximation may be extracted from a range where the workpiece and the rotary tool are in contact with each other.

(Means 10) In the wear amount simulation apparatus for the rotary tool of any one of means 1 to 9,
The rotary tool is a grinding wheel,
The plurality of rotation phase points on the periphery of the rotary tool may be points on the periphery of the grinding wheel obtained by equiangular division of the grinding wheel.

  According to this means, it is possible to simulate the amount of wear in grinding using a grinding wheel as a rotary tool.

(Means 11) In the wear amount simulation apparatus for the rotary tool of any one of means 1 to 9,
The rotary tool is an end mill having a plurality of blades,
The plurality of rotational phase points on the periphery of the rotary tool may be the end points of the blades of the end mill.

  According to this means, it is possible to simulate the amount of wear in cutting using an end mill as a rotary tool.

(Means 12) In the apparatus for simulating the amount of wear of a rotating tool according to any one of means 1 to 11,
The rotary tool wear amount simulation device is:
A tool shape storage unit for storing a peripheral shape of the rotary tool;
A tool shape changing unit that changes a peripheral shape of the rotary tool stored in the tool shape storage unit based on the wear amount calculated by the tool wear amount calculating unit;
It is good to have.

  According to this means, the wear amount for each rotational phase point is calculated based on the removal amount of the workpiece calculated for each rotational phase point. Then, the shape of the rotary tool is changed according to the wear amount. Thereby, if the movement locus of the rotation phase point is calculated and updated, the wear amount of the workpiece reflecting the tool wear amount can be calculated. Therefore, a more accurate wear amount simulation can be performed.

  DESCRIPTION OF EMBODIMENTS Hereinafter, embodiments of a rotary tool wear amount simulation apparatus according to the present invention will be described with reference to the drawings.

<First embodiment>
A wear amount simulation apparatus for a rotary tool according to a first embodiment will be described with reference to FIGS. Here, the rotary tool wear amount simulation apparatus of the present embodiment will be described by taking as an example the case of performing grinding with the grinding machine 1. Specifically, while rotating the grinding wheel 30 as a rotary tool and rotating the cylindrical workpiece 40, the peripheral portion of the grinding wheel 30 is brought into contact with the peripheral surface of the workpiece 40, and the workpiece 40 and The case where the machining performed by moving the grinding wheel 30 relatively in the X-axis direction will be described as an example.

  FIG. 1 is a schematic diagram showing a grinding machine 1 and a rotary tool wear amount simulation device 20 according to the first embodiment. FIG. 2 is a block diagram showing the rotary tool wear amount simulation apparatus 20. FIG. 3 is a schematic diagram when the grinding wheel 30 in the wear amount simulation is viewed from the rotation axis direction. FIG. 4 is a schematic diagram showing the relationship between the movement trajectory 35 of the rotational phase point 32 of the grinding wheel 30 and the peripheral surface shape of the workpiece 40. FIG. 5 is an enlarged view of a portion of FIG. 4 showing the removal region 45 of the workpiece 40.

  As shown in FIG. 1, the grinding machine 1 includes a grinding wheel base 10, a head stock 11, a control unit 12, and a grinding wheel 30. The grinding wheel 30 is formed in a disk shape by a large amount of abrasive grains, and is pivotally supported on the grinding wheel base 10 so as to be rotatable about an axis. The grinding wheel platform 10 rotates the grinding wheel 30 around the grinding wheel axis according to a command from the control unit 24. In addition, the grinding wheel base 10 is configured to move the grinding wheel 30 in the grinding wheel axis direction (Z-axis direction, front-rear direction in FIG. 1) and the direction orthogonal to the grinding wheel axis (X-axis direction, FIG. 1) according to a command from the control unit 24. To the left and right).

  The headstock 11 pivotally supports the workpiece 40 so as to be rotatable around the spindle, and rotates the workpiece 40 around the spindle in response to a command from the control unit 12. The control unit 12 instructs the grinding wheel platform 10 and the head stock 11 to control the relative positions of the grinding wheel 30 and the workpiece 40 and the rotational speeds of the grinding wheel 30 and the workpiece 40. When the workpiece 40 has a cam shape or an eccentric shape, the control unit 12 also controls the rotation angle of the workpiece 40.

  That is, the grinding machine 1 contacts the circumferential surface of the grinding wheel 30 that is rotationally driven by the grinding wheel base 10 with the circumferential surface of the workpiece 40 that is rotationally driven by the headstock 11, and grinds the circumferential surface of the workpiece 40. It is a device that performs processing. Although not shown, the grinding machine 1 is provided with a cooling device that cools the periphery of the grinding point with a coolant.

  Next, the wear amount simulation device 20 of the rotary tool will be described. In this embodiment, the rotary tool simulation device 20 is a separate device from the grinding machine 1 as shown in FIG. However, as indicated by a dotted line in FIG. 1, the control unit 12 may be communicably connected. Further, the rotary tool simulation device 20 may be incorporated in the control unit 12.

  As shown in FIG. 2, the rotary tool simulation device 20 includes a workpiece shape storage unit 21, a tool shape storage unit 22, a command value storage unit 23, a locus calculation unit 24, and a machining point calculation unit 25. A workpiece removal amount calculation unit 26, a tool wear amount calculation unit 27, a workpiece shape change unit 29, and a tool shape change unit 28.

  The workpiece shape storage unit 21 stores the circumferential shape of the columnar workpiece 40. Here, it is assumed that the shape of the workpiece 40 is the same as the shape of the workpiece 40 in the axial direction. That is, immediately before the start of simulation, the workpiece shape storage unit 21 stores the peripheral surface shape of the part to be ground of the workpiece 40. In addition, after starting the wear amount simulation, the workpiece shape storage unit 21 stores the circumferential shape of the workpiece 40 that is sequentially changed by the workpiece shape changing unit 29.

  Here, the form which memorize | stores the surrounding surface shape of the workpiece 40 in this embodiment is demonstrated. Moreover, in this embodiment, in order to simplify description, the peripheral surface shape of the site | part made into the grinding object of the workpiece 40 is made into circular shape. As shown in FIG. 4, the workpiece 40 has a circular peripheral edge as an initial state, and is divided into equiangular angles around the rotation axis of the workpiece 40. Line segments respectively connecting a plurality of division points on the periphery of the workpiece 40 and the rotation center 41 are workpiece line segments 42 (42a, 42b, 42c in the case of explaining as individual workpiece line segments,... Is used). That is, the dividing points on the periphery are the end points 43 of the workpiece line segments 42a, 42b, 42c,... (43a, 43b, 43c,... Are used when described as individual end points). As described above, the workpiece shape storage unit 21 stores the workpiece line segments 42a, 42b, 42c,... At each phase of the workpiece 40 as the peripheral shape of the workpiece 40. More specifically, the peripheral surface shape of the workpiece 40 is recognized by the line segment length of the workpiece line segments 42a, 42b, 42c,.

  The tool shape storage unit 22 stores the peripheral shape of the grinding wheel 30 that is a rotary tool. Here, it is assumed that the shape of the grinding wheel 30 is the same as the shape of the grinding wheel 30 in the axial direction. That is, immediately before the start of simulation, the tool shape storage unit 22 stores a circular shape as the peripheral shape of the grinding wheel 30. In addition, after starting the wear amount simulation, the tool shape storage unit 22 stores the circumferential shape of the grinding wheel 30 changed by the tool shape changing unit 28.

  Here, the form which memorize | stores the surrounding surface shape of the grinding wheel 30 in this embodiment is demonstrated. As shown in FIG. 3, the grinding wheel 30 has a circular periphery at the initial state, and rotational phase points 32 a, 32 b, 32 c,... Are equiangular about the rotation axis on the periphery of the grinding wheel 30. Place multiple. As described above, the tool shape storage unit 22 stores the rotational phase points 32a, 32b, 32c,... At each phase of the grinding wheel 30 as the circumferential shape of the grinding wheel 30.

  The command value storage unit 23 stores command values for the X-axis value and the C-axis value for the grinding machine 1 in grinding. The X-axis value is a value that commands the separation distance in the X-axis direction between the rotation center 41 of the workpiece 40 and the rotation center 31 of the grinding wheel 30 at a certain time. The C-axis value is a C-axis value (C = ωt) that is a value for instructing the rotation angle of the workpiece 40 at a certain time. The command value is calculated based on grinding processing conditions such as the workpiece rotation speed, the grindstone rotation speed, and the feed amount (feed speed) of the grindstone. Note that ω is the angular velocity of the workpiece 40.

  The trajectory calculation unit 24 considers a coordinate system in which the workpiece 40 is stationary during the grinding process in which the grinding wheel 30 and the workpiece 40 are rotated and the two are relatively moved in the X-axis direction. In this coordinate system, the movement locus of the rotational phase point 32 a of the grinding wheel 30 is actually a cycloid curve 33. However, the movement locus of the rotational phase point 32a used for the simulation does not use the cycloid curve 33 but uses the following arc curve 35a that is easy to calculate.

  A method for calculating the arc-shaped movement locus will be described. Here, the movement locus 35a of the rotation phase point 32a will be described, but the movement locus 35b, 35c,... Of the other rotation phase points 32b, 32c,.

  First, three points on the cycloid-shaped curve 33 are extracted, and a moving trajectory 35 is calculated by approximating an arc based on the extracted points 34a to 34c. That is, the extraction points 34a to 34c are positions of the rotational phase point 32a at three different times within a minute time.

  At this time, as shown in FIG. 4, the extraction point 34a is the position of the rotational phase point 32a at time t1, which is located slightly outside the range where the workpiece 40 and the grinding wheel 30 are in contact with each other. The extraction point 34c is the position of the rotational phase point 32a at time t2 located in a portion on the opposite side to the extraction point 34a in the slightly outside range where the workpiece 40 and the grinding wheel 30 contact. The extraction point 34b is the position of the rotation phase point 32a at a time (t1 + t2) / 2 between the extraction points 34a and 34c. That is, the extraction point 34b is extracted from the vicinity of the center of the range where the workpiece 40 and the grinding wheel 30 are in contact. The extraction point 34 b is extracted from the vicinity of the center of the region where the cycloid-shaped curve 33 is located inside the periphery of the workpiece 40. The trajectory calculation unit 24 calculates the movement trajectory 35 by the number of rotational phase points 32a, 32b, 32c,... Arranged on the periphery of the grinding wheel 30 as described above.

  The machining point calculation unit 25 calculates a part of the workpiece 40 to be ground from the circumferential shape of the workpiece 40 stored in the workpiece shape storage unit 21 and the movement locus 35 of the grinding wheel 30. Specifically, as shown in FIG. 4, the machining point calculation unit 25 calculates a machining point 44 that is an intersection of a plurality of workpiece line segments 42 and a moving locus 35 that approximates an arc. This processing point calculation unit 25 is processing point 44 for each movement locus 35 of each rotation phase point 32a, 32b, 32c,... (44a, 44b, 44c,. Is used).

  The workpiece removal amount calculation unit 26 calculates a removal amount that each rotational phase point 32a, 32b, 32c,... Grinds with a single contact with the workpiece 40. The term “single contact” here refers to a period from when a certain rotational phase point 32 a starts to contact the current workpiece 40 to when the contact with the workpiece 40 is canceled. As shown in FIG. 4, the removal amount is a volume obtained by multiplying the area of the removal region 45 surrounded by the circumferential shape of the workpiece 40 and the movement locus 35 by the axial length of the workpiece 40. Since the actual removal amount is an amount proportional to the area of the removal region 45, in this embodiment, the removal region 45 is used as the removal amount without considering the axial length of the workpiece 40.

  Furthermore, in order to reduce the calculation load for calculating the area of the removal region 45, as shown in FIG. 5, the removal amount is obtained by integrating a plurality of removal fragments 46a to 46f. The removed piece 46 is an area of a triangular or quadrangular area formed by the end point 43 and the machining point 44 in two adjacent workpiece line segments 42. For example, the removed piece 46b is a rectangular region formed by the end points 43b and 43c and the machining points 44b and 44c in the two adjacent workpiece line segments 42b and 42c. The other removed pieces 46a and 46c to 46f are similarly calculated based on the end points 43a to 43g and the processing points 44a to 44f.

  Then, the workpiece removal amount calculation unit 26 integrates the plurality of removal pieces 46a to 46f, and calculates the removal amount based on the integrated area. In the present embodiment, the triangular removal fragments 46a and 46f at both ends in the removal region 45 are targeted for integration, but the removal fragments 46a and 46f may be excluded from integration because they are very small regions. Alternatively, a processing end point 47 that is an intersection point of the movement locus 35 and the peripheral edge of the workpiece 40 may be obtained to calculate a more accurate triangular removed piece 46.

  The tool wear amount calculation unit 27 calculates the wear amount of the grinding wheel 30 based on the removal amount of the workpiece 40 at each rotational phase point 32a, 32b, 32c,... Calculated by the workpiece removal amount calculation unit 26. To do. That is, the amount of wear is calculated for each rotational phase point 32 in accordance with the removal amount of each rotational phase point 32 ground by one contact with the workpiece 40. Here, the relationship between the removal amount and the wear amount for each rotation phase point 32 is obtained in advance by experiments or analysis.

  The tool shape changing unit 28 changes the circumferential shape of the grinding wheel 30 stored in the tool shape storage unit 22 based on the wear amount calculated by the tool wear amount calculating unit 27. Specifically, each rotational phase point 32a, 32b, 32c,... Wears at the rotational phase point 32a, 32b, 32c,. Accordingly, the rotational phase point 32 is moved in the direction approaching the rotational center 31 of the grinding wheel 30. The circumferential shape of the grinding wheel 30 is changed by moving the rotational phase point 32 with respect to all the wear amounts calculated by the tool wear amount calculation unit 27.

  The workpiece shape changing unit 29 changes the circumferential shape of the workpiece 40 stored in the workpiece shape storage unit 21 based on the movement track 35 calculated by the track calculation unit 24. Specifically, the machining point 44 on the workpiece line segment 42 calculated by the machining point calculation unit 25 based on the movement locus 35 is changed to a new end point 43 of the workpiece line segment 42. The peripheral shape of the workpiece 40 is obtained by moving the end points 43b, 43c,... Of the workpiece line segments 42b, 42c,. To change.

  A wear amount simulation example using the rotary tool wear amount simulation apparatus 20 will be described with reference to FIGS. FIG. 6A is a diagram of grinding by the movement locus 35a of the rotation phase point 32a, and FIG. 6B is a diagram of grinding by the movement locus 35b of the rotation phase point 32b. ) Is a diagram of grinding by the movement locus 35c of the rotational phase point 32c. FIG. 7 is a view of the workpiece 40d after grinding by the movement locus 35a to the movement locus 35c. FIG. 8 is an enlarged view of the grinding wheel 30 in the wear amount simulation.

  First, an operator or the like inputs the peripheral shape of the grinding wheel 30, the peripheral shape of the workpiece 40, processing conditions, simulation conditions, and the like. The workpiece shape storage unit 21 is set and stored so that the workpiece line segments 42a, 42b, 42c,... Are arranged at equal angles based on the peripheral surface shape of the workpiece 40 and the simulation conditions. The tool shape storage unit 22 sets and stores rotational phase points 32a, 32b, 32c,... Based on the peripheral surface shape of the grinding wheel 30 and simulation conditions. The command value storage unit 23 calculates and stores a command value based on the machining condition and the simulation condition.

  Next, the operation of the plurality of rotational phase points 32a to 32c calculated by the tool shape storage unit 22 will be described in detail. The trajectory calculation unit 24 is based on the rotational phase point 32a on the periphery of the grinding wheel 30 and the command values of the X-axis value and the C-axis value stored in the command value storage unit 23, and the rotational phase points at three different times. The three points 32a are extracted. Based on this extracted point, a circular arc is approximated to calculate the movement locus 35a. The trajectory calculation unit 24 similarly calculates the movement trajectories 35b and 35c for the other rotational phase points 32b and 32c.

  As shown in FIG. 6 (a), the machining point calculation unit 25 is an intersection of a plurality of workpiece line segments 42a, 42b, 42c,... Stored in the workpiece shape storage unit 21 and a movement locus 35a. Each machining point 44a, 44b, 44c,... Is calculated. Here, information on each workpiece line segment 42a, 42b, 42c,... And each machining point 44a, 44b, 44c,... Relating to the movement locus 35a is temporarily stored in the workpiece removal amount calculation unit 26. The

  Next, the workpiece shape changing unit 29 receives information on the workpiece line segments 42a, 42b, 42c,... And the machining points 44a, 44b, 44c,. , The end points 43a, 43b, 43c, ... of the workpiece line segments 42a, 42b, 42c, ... are changed to machining points 44a, 44b, 44c, .... The peripheral shape of the workpiece 40 is changed by changing the end points 43 for all the workpiece line segments 42 for which the machining points 44 are calculated. The changed workpiece shape is stored in the workpiece shape storage unit 21.

  Again, as shown in FIG. 6 (b), the machining point calculation unit 25 is the intersection of the updated plurality of workpiece line segments 42a, 42b, 42c,... And the movement locus 35b of the rotational phase point 32b. A certain processing point 44a, 44b, 44c,... Is calculated. Here, as with the movement locus 35a of the rotational phase point 32a, the workpiece line segments 42a, 42b, 42c,... And the machining points 44a, 44b, 44c,. The information is temporarily stored in the workpiece removal amount calculation unit 26.

  The workpiece shape changing unit 29 is similar to the movement locus 35a of the rotational phase point 32a, and each workpiece line segment 42a, 42b, 42c,... And the machining points 44a, 44b, 44c,. ..., the end point 43 of the workpiece line segment 42 is changed to a machining point 44, and the peripheral shape of the workpiece 40 is changed. The changed workpiece shape is stored in the workpiece shape storage unit 21. Then, as illustrated in FIG. 6C, the machining point calculation unit 25 calculates a machining point 44 that is an intersection of the plurality of workpiece line segments 42 updated again and the movement locus 35c.

  As described above, when the machining point calculation unit 25 calculates a plurality of machining points 44 related to one movement locus 35, the peripheral shape of the workpiece 40 is sequentially changed each time. As a result, the previously changed peripheral surface shape of the workpiece 40 is reflected in the calculation of the machining point 44 related to the next movement trajectory 35. That is, after calculating the machining points 44 related to the movement trajectories 35a to 35c, the workpiece line segment 42 of the workpiece 40d is stored in the workpiece shape storage unit 21, as shown in FIG.

  And the workpiece removal amount calculation part 26 is based on the information of the workpiece line segments 42a, 42b, 42c,... And the machining points 44a, 44b, 44c,. Then, the area of the removal region 45a in the workpiece 40a is calculated. The area of the removal region 45 a is calculated by approximation based on the integrated area obtained by integrating the areas of the removal pieces 46 formed by the end points of the two adjacent workpiece line segments 42 and the machining point 44. By repeating this process for each of the temporarily stored movement trajectories 35a to 35c, the removal amounts by the rotational phase points 32a to 32c are calculated with the areas of the removal regions 45a to 45c in the workpieces 40a to 40c. .

  Next, the tool wear amount calculation unit 27 calculates the wear amount at each of the rotation phase points 32a to 32c based on the removal amount of the workpiece 40 with respect to each of the movement trajectories 35a to 35c. In this embodiment, the product of the removal amount and the constant is used as the tool wear amount. Then, as shown in FIG. 8, the tool shape changing unit 28 moves the rotation phase points 32 a to 32 c in the direction approaching the rotation center of the grinding wheel 30 by the amount of the tool wear amount. Based on the rotational phase points 32 a ′ to 32 c ′ after movement, the changed peripheral shape of the grinding wheel 30 is stored in the tool shape storage unit 22. In this way, the wear amount of the rotary tool (grinding wheel 30) is simulated.

  Further, when the position of the rotational phase point 32 is changed and the shape is changed as a rotary tool, the trajectory calculation unit 24 converts the changed rotational phase point 32 and the command value stored in the command value storage unit 23. Based on this, the movement trajectory 35 of each rotational phase point 32 is calculated again. As a result, one rotational phase point 32 on the periphery of the grinding wheel 30 is changed before coming into contact with the workpiece 40 again. Therefore, since the peripheral surface shape of the grinding wheel 30 changed by the tool shape changing unit 28 is reflected in the next work shape change and the tool wear amount, a more accurate wear amount of the rotary tool can be simulated.

  In such a rotary tool wear amount simulation device 20, the calculated wear amount has always been reflected in a true circular rotary tool, whereas uneven wear on the circumferential shape of the grinding wheel 30 can be simulated. it can. In addition, the grinding wheel 30 is certified at the rotational phase point 32, the movement trajectory 35 of the rotational phase point 32 is calculated, and the workpiece 40 is certified by the workpiece line segment 42, thereby calculating the machining point and the workpiece. The calculation load in calculating the removal amount is greatly reduced. Therefore, smooth wear amount simulation of the rotary tool can be performed.

<Modification of First Embodiment>
In the first embodiment described above, the end point 43 of the workpiece line segment 42 is a division point on the circumference that is divided equiangularly around the rotation axis of the workpiece 40. In the case of the cylindrical workpiece 40, it is most preferable to divide it into equiangular parts. In addition, the end point 43 of the workpiece line segment 42 may be a division point on a circumference obtained by dividing the workpiece line segment 42 at an irregular angle around the rotation axis of the workpiece 40. In this case, other configurations are the same as those in the first embodiment.

  In the first embodiment, since the workpiece 40 is a columnar and non-eccentric rotating shaft, the workpiece 40 is illustrated by being equiangularly divided around the rotating shaft. That is, when the cross-sectional shape of the workpiece 40 is a non-circular shape, for example, a cam shape, or when it is a circular but eccentric rotating shaft, the command value of the C-axis value is inconstant For example, a configuration in which the rotation axis is divided at an irregular angle may be employed.

  As described above, the following division methods can be considered. For example, the workpiece 40 may be divided around the rotation axis so that the areas of the regions formed by two adjacent workpiece line segments 42 and the periphery of the workpiece 40 are equal. Alternatively, for example, the workpiece 40 may be divided around the rotation axis so that the lengths of the peripheral pieces of the workpiece 40 divided by the workpiece line segment 42 are equal. Thereby, it is possible to calculate the removal amount of the workpiece adapted to various grinding processes, and it is possible to simulate a more accurate wear amount of the rotary tool. However, the non-circular workpiece 40 such as a cam shape is preferably equiangular from the viewpoint of calculation load.

<Second embodiment>
The configuration of the second embodiment will be described with reference to FIGS. The configuration of the rotary tool wear amount simulation apparatus of the present embodiment is mainly directed to shape change based on movement trajectories of a plurality of rotational phase points by the workpiece removal amount calculation unit 26 of the first embodiment. The difference is that the removal amount is calculated. Since other configurations are the same as those of the first embodiment, detailed description thereof is omitted. Only the differences will be described below.

  In the first embodiment, each removal region 45 in the workpiece 40 is calculated by accumulating a plurality of removal pieces 46a to 46f, and this is performed each time for one contact by each rotation phase point. In the present embodiment, in order to further reduce the calculation load of the removal amount calculation, the workpiece removal amount calculation unit 26 first calculates a fragment line segment length for a single contact of the rotational phase point. That is, fragment line segments 43b-44b, 43c-44c, respectively connecting the end points 43b to 43f before the shape change of the workpiece 40 and the processing points 44b to 44f which are the end points after the shape change in FIG.・ Calculate and integrate each line segment length. Here, for example, the removal amount by one contact of the rotational phase point may be obtained by multiplying the integrated length of the segment line segment length by a constant. By doing in this way, since the removal amount of the workpiece for each rotation phase point can be calculated without calculating the removal area of the workpiece, the calculation load can be reduced.

  Further, in the present embodiment, a case will be described in which the shape change processing of the workpiece three times accompanying the grinding of the workpiece by the three adjacent rotation phase points 32a to 32c in the unit time is an object. Here, “three times” corresponds to an example of the “predetermined number” in the present application. The “predetermined number” may be a constant value set according to simulation conditions or the like, or may be the number of rotational phase points related to grinding of a workpiece within a unit time as in the present embodiment. . In addition, from the viewpoint of calculation load, it is preferable to adjust and set the number so as not to exceed the number of rotational phase points arranged on the rotary tool.

  The workpiece removal amount calculation unit 26 holds an integrated length of each segment line segment length in the grinding of the workpieces 40a to 40c. Next, for each accumulated length, a ratio to the total sum of accumulated lengths for three times is calculated. Further, the removal area is calculated by accumulating a plurality of removed pieces for the workpiece 40d that has undergone the shape change process three times. The calculation of the area of the removal region is the same as that for calculating the shape change process of three times as one contact and calculating the area of the plurality of removal fragments in the first embodiment, and thus the description thereof is omitted. And the removal amount of a workpiece is each calculated by multiplying the area of the removal area | region by the shape change process of the calculated three times of workpieces by the ratio of each integration length of the rotation phase points 32a-32c. That is, the removal area of the workpiece by the shape changing process three times is distributed according to the ratio of the accumulated lengths of the respective rotation phase points.

  By calculating the removal amount of the workpiece at each rotational phase point in this way, a predetermined number of fragment areas are collected, so that the number of calculations can be reduced and the calculation load can be greatly reduced. In addition, by appropriately setting the predetermined number, an error that occurs when the removal amount is calculated for each shape change is eliminated, so that the generation of calculation errors can be reduced as a whole.

<Third embodiment>
The configuration of the third embodiment will be described with reference to FIG. The apparatus for simulating the amount of wear of a rotary tool according to this embodiment is a case where machining is performed at a machining center (not shown). Specifically, the end mill 130 is applied as a rotating tool. That is, while rotating the end mill 130 that is a rotary tool, the workpiece 40 is brought into contact with the peripheral portion of the end mill 130 and the workpiece 40 and the end mill 130 are relatively moved in an arbitrary axial direction. This is the case. Here, FIG. 9 is a schematic view of an end face of the end mill 130 of the third embodiment.

  Here, the configuration of the present embodiment is mainly different in that the rotary tool of the first embodiment is an end mill 130. In connection with this, the point which makes the grinding machine 1 in 1st embodiment the machining center which applies the end mill 130 differs from the structure of 1st embodiment. In the case of such a configuration, the tool shape storage unit 22 of the simulation apparatus 20 may use the rotational phase point 32 as the end point of each blade of the end mill, as shown in FIG. When the rotary tool is the grinding wheel 30, a large amount of abrasive grains are in contact with the workpiece 40 for grinding, whereas in the case of the end mill 130, the cutting edge of each blade is in contact with the workpiece 40 for grinding. Because it does.

  Further, the tool shape changing unit 28 of the first embodiment moves the rotational phase point 32 in a direction closer to the rotation center 31 of the grinding wheel 30 with respect to the tool wear amount calculated by the tool wear amount calculating unit 27. The peripheral shape was changed. However, as shown in FIG. 9, even when the bottom blade 136 is a linear end mill 130, the outer peripheral blade (not shown) is generally spiral, and the direction in which the blade edge wears is not necessarily close to the rotation center 31. It is not the direction to do. Furthermore, the bottom blade may have a tip pocket, or the bottom blade may be curved in the radial direction unlike a shell end mill, and the wear direction varies depending on the shape of the end mill.

  Therefore, when the rotary tool is the end mill 130, the wear direction obtained from the actual measurement based on the shapes of the bottom blade 136 and the outer peripheral blade of the end mill 130 may be input to the wear amount simulation apparatus in advance. Thereby, the tool shape changing unit 28 can change the peripheral shape of the end mill 130 by moving the rotational phase point 32 in accordance with the input wear direction. Therefore, even if the rotary tool is the end mill 130, it is possible to simulate the wear amount adapted to the end mill 130.

<Other embodiments>
In 1st-3rd embodiment, although the simulation apparatus 20 of the rotary tool of this invention was used as the apparatus different from the grinding machine 1, you may combine with another simulation apparatus. For example, a NC data drawing simulation device, a chatter phenomenon simulation device, and a grinding processing condition determination simulation device may be used. By cooperating with these simulation devices, it is possible to obtain the rotational tool and workpiece position deviation, and more accurately simulate the wear amount of the rotational tool. In addition, when uneven wear of the rotary tool causes the chatter phenomenon, it is possible to change the machining conditions and the like so that the chatter phenomenon does not occur before actual grinding.

1st embodiment: It is a schematic diagram which shows the abrasion amount simulation apparatus 20 of the grinding machine 1 and a rotary tool. It is a block diagram which shows the wear amount simulation apparatus 20 of a rotary tool. It is a schematic diagram of the grinding wheel 30 in the wear amount simulation. It is a schematic diagram of the grinding process by the movement track | truck 35 and the workpiece 40. FIG. FIG. 5 is an enlarged view of a part of FIG. 4 showing a removal region 45. (A) It is a figure of the grinding process by the movement locus | trajectory 35a of the rotation phase point 32a, (b) It is a figure of the grinding process by the movement locus | trajectory 35b of the rotation phase point 32b, (c) By the movement locus | trajectory 35c of the rotation phase point 32c. It is a figure of a grinding process. It is the figure of the workpiece 40d after the grinding process by the movement locus | trajectory 35a-the movement locus | trajectory 35c. It is an enlarged view of the grinding wheel 30 in wear amount simulation. 3rd embodiment: It is a schematic diagram of the end surface of the end mill 130. FIG.

Explanation of symbols

DESCRIPTION OF SYMBOLS 1: Grinding machine 10: Grinding wheel base, 11: Spindle head, 12: Control part 20: Wear amount simulation apparatus of a rotary tool, 21: Workpiece shape memory | storage part 22: Tool shape memory | storage part, 23: Command value memory | storage part, 24 : Locus calculation unit 25: machining point calculation unit, 26: workpiece removal amount calculation unit, 27: tool wear amount calculation unit, 28: tool shape change unit, 29: workpiece shape change unit, 30: grinding wheel, 130a, 130b, 130c: End mill, 31: Center of rotation 32, 32a, 32b, 32c, 32a ′, 32b ′, 32c ′: Rotation phase point 33: Curve, 34: Extraction point, 35, 35a, 35b, 35c: Trajectory 136: Bottom Blade 40, 40a, 40b, 40c, 40d: Work piece, 41: Center of rotation 42, 42a, 42b, 42c, 42d, 42e, 42f, 42g: Work piece line segment 43, 43a, 43b 43c, 43d, 43e, 43f, 43g: end points 44, 44b, 44c, 44d, 44e, 44f: machining points 45, 45a, 45b, 45c: removal regions, 46: removal fragments, 47: machining end points

Claims (12)

Rotation for simulating the amount of wear of the rotary tool in processing performed by rotating the rotary tool while bringing the peripheral edge of the rotary tool into contact with the workpiece and relatively moving the workpiece and the rotary tool. A tool wear amount simulation device,
A workpiece removal amount calculation unit that calculates the removal amount of the workpiece by each of a plurality of rotational phase points on the periphery of the rotary tool;
A tool wear amount calculation unit for calculating a wear amount at a peripheral portion of the rotary tool by calculating a wear amount at each of the rotation phase points based on a removal amount of the workpiece at each of the rotation phase points;
A wear amount simulation apparatus for a rotary tool, comprising: The rotary tool wear amount simulation device is:
In a coordinate system in which the workpiece is stationary, a trajectory calculation unit that calculates a movement trajectory of each rotation phase point with respect to the workpiece,
2. The rotary tool wear amount simulation apparatus according to claim 1, wherein the workpiece removal amount calculation unit calculates the removal amount of the workpiece based on the shape of the workpiece and the movement trajectory. The rotary tool wear amount simulation device is:
A workpiece shape storage unit for storing the shape of the workpiece;
Based on the movement trajectory calculated by the trajectory calculation unit, a workpiece shape changing unit that changes the shape of the workpiece stored in the workpiece shape storage unit;
With
The workpiece removal amount calculation unit calculates the removal amount of the workpiece by each of the rotational phase points in the shape of the workpiece sequentially changed by the workpiece shape changing unit. The wear amount simulation apparatus for a rotary tool according to 2. The apparatus for simulating the amount of wear of a rotating tool brings the peripheral edge of the rotating tool into contact with the peripheral surface of the workpiece while rotating the rotating tool and rotating the workpiece, and rotates the workpiece and the rotating tool. In processing performed by relatively moving the tool, a device for simulating the amount of wear of the rotary tool,
The workpiece shape storage unit stores a workpiece line segment connecting a plurality of points on the periphery of the workpiece and the rotation center of the workpiece as the shape of the workpiece,
The wear amount simulation device further includes a machining point calculation unit that calculates a machining point that is an intersection of the workpiece line segment and the movement locus,
The workpiece shape changing unit changes the shape of the workpiece by changing the workpiece line based on the machining point,
The workpiece removal amount calculation unit calculates the removal amount of the workpiece based on the workpiece line segment before the shape change and the workpiece line segment after the shape change. Item 4. The wear amount simulation device for a rotary tool according to Item 3.   The wear amount simulation of the rotary tool according to claim 4, wherein the plurality of points on the periphery of the workpiece line segment are division points on the periphery obtained by equiangularly dividing the workpiece. apparatus. The workpiece shape changing unit changes a shape by changing a peripheral side end point of the workpiece line segment to the machining point on the workpiece line segment,
The workpiece removal amount calculation unit
In the two adjacent workpiece line segments, the area of the region formed by each of the peripheral side end points before and after the shape change is a fragment area,
5. The piece area calculated for the shape change of all two adjacent workpiece line segments is integrated, and the removal amount of the workpiece is calculated based on the integrated area. The wear amount simulation apparatus for a rotary tool according to claim 5. The workpiece shape changing unit changes a shape by changing a peripheral side end point of the workpiece line segment to the machining point on the workpiece line segment,
The workpiece removal amount calculation unit
The length of the line segment connecting the peripheral side end points before and after the shape change is a fragment line segment length,
6. The piece removal length calculated for the shape change of all the workpiece line segments is integrated, and the removal amount of the workpiece is calculated based on the integration length. The wear amount simulation apparatus of the described rotary tool. The workpiece removal amount calculation unit
In the shape change based on the movement trajectory of a predetermined number of adjacent rotation phase points, the ratio of each integrated length to the total sum of the calculated integrated lengths is calculated,
In the two adjacent workpiece line segments, the area of the region formed by each of the peripheral side end points before and after the predetermined number of the shape change, the fragment area,
The piece areas calculated for the shape changes of all two adjacent workpiece line segments are integrated, and the removal amount of the workpiece is calculated based on the integrated area and the ratio of the integrated lengths. The wear amount simulation apparatus for a rotary tool according to claim 7. The movement trajectory calculated by the trajectory calculation unit is:
9. The coordinate system in which the workpiece is stationary is a curve approximated by a circular arc by point groups at at least three different times of the rotational phase points for the workpiece. The wear amount simulation apparatus of a rotary tool as described in the item. The rotary tool is a grinding wheel,
The plurality of rotation phase points on the periphery of the rotary tool are points on the periphery of the grinding wheel obtained by equiangular division of the grinding wheel. Wear amount simulation device for rotary tools. The rotary tool is an end mill having a plurality of blades,
The wear amount simulation device for a rotary tool according to any one of claims 1 to 9, wherein the plurality of rotation phase points on a peripheral edge of the rotary tool are end points of each blade of the end mill. The rotary tool wear amount simulation device is:
A tool shape storage unit for storing a peripheral shape of the rotary tool;
A tool shape changing unit that changes a peripheral shape of the rotary tool stored in the tool shape storage unit based on the wear amount calculated by the tool wear amount calculating unit;
The wear amount simulation apparatus for a rotary tool according to any one of claims 1 to 11, comprising:

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