Method Of Deriving Natural Frequency Of Cutting Tool, Method Of Creating Stability Limit Curve, And Apparatus For Deriving Natural Frequency Of Cutting Tool

Abstract

An apparatus includes a machining executing part causing a machine tool to execute an operation of, while changing a spindle rotation speed of the machine tool in a stepwise manner, machining a workpiece by a predetermined distance or for a predetermined period of time at each spindle rotation speed, a displacement detector detecting a position displacement of a tool during the machining, a cutting force detector detecting a cutting force applied to the tool, a frequency analysis part performing frequency analysis on displacement data and cutting force data for each rotation speed to calculate a displacement spectrum and a cutting force spectrum, and a natural frequency deriving part calculating a compliance spectrum for each spindle rotation speed by deriving the displacement spectrum by the cutting force spectrum, calculating an integrated compliance spectrum by superimposing the compliance spectra, and deriving, as a natural frequency, a frequency showing the largest compliance value.

Claims

1. A method of deriving a natural frequency of a cutting tool used in machining an object to be machined with a machine tool, comprising: an actual machining step of, while changing a rotational speed of a spindle of the machine tool in a stepwise manner, machining the object to be machined with the cutting tool by a predetermined distance or for a predetermined period of time at each rotational speed; a detecting step of, during the actual machining step, detecting a position displacement occurring on the cutting tool and detecting a cutting force applied to the cutting tool; an analyzing step of performing frequency analysis on displacement data and cutting force data obtained at each rotational speed of the spindle in the detecting step to obtain a displacement spectrum and a cutting force spectrum; and a deriving step of: calculating, based on the displacement spectrum and the cutting force spectrum for each rotational speed of the spindle obtained in the analyzing step, a compliance spectrum for each rotational speed by dividing the displacement spectrum by the cutting force spectrum, calculating an integrated compliance spectrum by superimposing the calculated compliance spectra, and deriving, as a natural frequency of the cutting tool, a frequency showing a largest compliance value from the calculated integrated compliance spectrum.

2. The method according to claim 1, in which: the actual machining step is configured to, while changing a rotational speed of the spindle in a stepwise manner, at each rotational speed, machine the object to be machined by a predetermined distance or for a predetermined period of time by relatively moving the cutting tool with respect to the object to be machined through independent operations of a first axis and a second axis as two feed axes perpendicular to the spindle and perpendicular to each other or through a combined operation of the first axis and the second axis so that feed components for directions of the first axis and the second axis are contained; the detecting step is configured to detect a position displacement occurring on the cutting tool for each of feed directions of the first axis and the second axis at each rotational speed and detect a cutting force applied to the cutting tool at each rotational speed; the analyzing step is configured to perform frequency analysis on displacement data and cutting force data obtained for each of the feed directions at each rotational speed to obtain a displacement spectrum and a cutting force spectrum; and the deriving step is configured to, for each of the feed directions, calculate a compliance spectrum for each rotational speed by dividing the displacement spectrum by the cutting force spectrum and calculate an integrated compliance spectrum by superimposing the calculated compliance spectra, detect a frequency showing a largest compliance value from each of the calculated integrated compliance spectra, and derive the two detected frequencies as natural frequencies of the cutting tool for the feed directions.

3. A stability limit curve creating method, comprising: the steps of claim 1; and a curve creating step of calculating a damping ratio and an equivalent mass of a machining system including at least the cutting tool and the object to be machined based on the integrated compliance spectrum calculated in the deriving step and the natural frequency of the cutting tool, and creating a stability limit curve concerning regenerative chatter of the cutting tool based on the calculated damping ratio and equivalent mass and the natural frequency.

4. A stability limit curve creating method, comprising: the steps of claim 1; the deriving step being configured to derive, as natural frequencies of the cutting tool, at least two frequencies showing a maximal compliance value in decreasing order of the compliance value based on the calculated integrated compliance spectrum; and a curve creating step of calculating a damping ratio and an equivalent mass of a machining system including at least the cutting tool and the object to be machined corresponding to each of the natural frequencies based on the integrated compliance spectrum calculated in the deriving step and the natural frequencies of the cutting tool, and creating a stability limit curve concerning regenerative chatter of the cutting tool corresponding to each of the natural frequencies based on the calculated damping ratios and equivalent masses and the natural frequencies.

5. A stability limit curve creating method, comprising: the steps of claim 2; and a curve creating step of calculating a damping ratio and an equivalent mass of a machining system including at least the cutting tool and the object to be machined for each of the feed directions based on the integrated compliance spectra calculated in the deriving step and the natural frequencies of the cutting tool, and creating a stability limit curve concerning regenerative chatter of the cutting tool for each of the feed directions based on the calculated damping ratios and equivalent masses and the natural frequencies.

6. A stability limit curve creating method, comprising: the steps of claim 2; the deriving step being configured to derive, as natural frequencies of the cutting tool, at least two frequencies showing a maximal compliance value in decreasing order of the compliance value for each of the feed directions based on the integrated compliance spectrum calculated for the feed direction; and a curve creating step of calculating a damping ratio and an equivalent mass of a machining system including at least the cutting tool and the object to be machined corresponding to each of the natural frequencies for each of the feed directions based on the integrated compliance spectrum for the feed direction calculated in the deriving step and the natural frequencies of the cutting tool for the feed direction, and creating a stability limit curve concerning regenerative chatter of the cutting tool corresponding to each of the natural frequencies for each of the feed directions based on the calculated damping ratios and equivalent masses and the natural frequencies.

7. A stability limit curve creating method, comprising: the steps of claim 2; and a curve creating step of calculating a damping ratio and an equivalent mass of a machining system including at least the cutting tool and the object to be machined for each of the feed directions based on the integrated compliance spectrum for the feed direction calculated in the deriving step and the natural frequency of the cutting tool for the feed direction, and creating a stability limit curve concerning regenerative chatter of the cutting tool for a predetermined feed direction based on the calculated damping ratios and equivalent masses for the feed directions and the natural frequencies for the feed directions.

8. An apparatus for deriving a natural frequency of a cutting tool used in machining an object to be machined with a machine tool, comprising: a machining executing part causing the machine tool to execute an operation of, while changing a rotational speed of a spindle of the machine tool in a stepwise manner, machining the object to be machined by a predetermined distance or for a predetermined period of time at each rotational speed; a displacement detector detecting a position displacement occurring on the cutting tool during the execution of machining by the machine tool; a cutting force detector detecting a cutting force applied to the cutting tool during the execution of machining by the machine tool; a frequency analysis part performing frequency analysis on displacement data and cutting force data obtained at each rotational speed of the spindle by the displacement detector and the cutting force detector to obtain a displacement spectrum and a cutting force spectrum; a natural frequency deriving part, calculating, based on the displacement spectrum and the cutting force spectrum for each rotational speed of the spindle obtained by the frequency analysis part, a compliance spectrum for each rotational speed by dividing the displacement spectrum by the cutting force spectrum, calculating an integrated compliance spectrum by superimposing the calculated compliance spectra, and deriving, as a natural frequency of the cutting tool, a frequency showing a largest compliance value from the calculated integrated compliance spectrum.

9. The apparatus according to claim 8, in which: the machining executing part is configured to, while changing a rotational speed of the spindle in a stepwise manner, at each rotational speed, machine the object to be machined by a predetermined distance or for a predetermined period of time by moving the cutting tool through independent operations of a first axis and a second axis as two feed axes perpendicular to the spindle and perpendicular to each other or through a combined operation of the first axis and the second axis so that feed components for directions of the first axis and the second axis are contained, the displacement detector is configured to detect a position displacement occurring on the cutting tool for each of feed directions of the first axis and the second axis at each rotational speed; the cutting force detector is configured to detect a cutting force applied to the cutting tool at each rotational speed; the frequency analysis part is configured to perform frequency analysis on displacement data and cutting force data obtained for each of the feed directions at each rotational speed to obtain a displacement spectrum and a cutting force spectrum; and the natural frequency deriving part is configured to, for each of the feed directions, calculate a compliance spectrum for each rotational speed by dividing the displacement spectrum by the cutting force spectrum and calculate an integrated compliance spectrum by superimposing the calculated compliance spectra, detect a frequency showing a largest compliance value from each of the calculated integrated compliance spectra, and derive a natural frequency of the cutting tool from the two detected frequencies.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0067] FIG. 1 is a perspective view of a machine tool according to an embodiment of the present disclosure;

[0068] FIG. 2 is a block diagram showing a schematic configuration of a natural frequency deriving apparatus according to the embodiment;

[0069] FIG. 3 is an illustration showing a machining manner executed in a detection machining executing part according to the embodiment;

[0070] FIG. 4 is a spectrum waveform diagram showing a Y-axis direction displacement spectrum;

[0071] FIG. 5 is a spectrum waveform diagram showing a Y-axis direction cutting force spectrum;

[0072] FIG. 6 is a spectrum waveform diagram showing a Y-axis direction displacement spectrum after filtering;

[0073] FIG. 7 is a spectrum waveform diagram showing a Y-axis direction cutting force spectrum after filtering;

[0074] FIG. 8 is a spectrum waveform diagram showing a Y-axis direction compliance spectrum;

[0075] FIG. 9 is a spectrum waveform diagram showing superimposed Y-axis direction compliance spectra;

[0076] FIG. 10 is a spectrum waveform diagram showing a Y-axis direction integrated compliance spectrum;

[0077] FIG. 11 is a spectrum waveform diagram showing an X-axis direction integrated compliance spectrum;

[0078] FIG. 12 is an illustration showing a cutting model with two degrees of freedom;

[0079] FIG. 13 is an illustration for explaining calculation of a damping ratio; and

[0080] FIG. 14 is a diagram showing a stability limit curve.

DETAILED DESCRIPTION

[0081] Hereinafter, a specific embodiment of the present disclosure will be described with reference to the drawings. FIG. 1 is a perspective view of a machine tool used in this embodiment and FIG. 2 is a block diagram showing a natural frequency deriving apparatus and other elements according to this embodiment.

Schematic Configuration of Machine Tool

[0082] First of all, a machine tool 20 is schematically described. This machine tool 20 includes a bed 21, a column 22 erected on the bed 21, a spindle head 23 provided on a front surface (machining area side surface) of the column 22 to be movable in a direction of the Z axis indicated by arrow, a spindle 24 held by the spindle head 23 to be rotatable about an axis thereof, a saddle 25 provided on the bed 21 below the spindle head 23 to be movable in a direction of the Y axis indicated by arrow, a table 26 disposed on the saddle 25 to be movable in a direction of the X axis indicated by arrow, an X-axis feed mechanism 29 for moving the table 26 in the X-axis (first axis) direction, a Y-axis feed mechanism 28 for moving the saddle 25 in the Y-axis (second axis) direction, a Z-axis feed mechanism 27 for moving the spindle head 23 in the Z-axis (third axis) direction, and a spindle motor (not shown) rotating the spindle 24. Note that the X axis, the Y axis, and the Z axis are feed axes that are perpendicular to each other.

[0083] Note that operations of the X-axis feed mechanism 29, Y-axis feed mechanism 28, Z-axis feed mechanism 27, spindle motor (not shown), and other components are controlled by a controller 10 shown in FIG. 2. Specifically, an NC program stored in the controller 10 is executed as appropriate and an operation controller 11 controls the X-axis feed mechanism 29, the Y-axis feed mechanism 28, the Z-axis feed mechanism 27, the spindle motor (not shown), and other components in accordance with control signals based on the NC program.

[0084] Thus, in the machine tool 20, the X-axis feed mechanism 29, the Y-axis feed mechanism 28, the Z-axis feed mechanism 27, the spindle motor (not shown), and other components are driven under control by the controller 10, and thereby the spindle 24 is rotated about the axis thereof and the spindle 24 and the table 26 are relatively moved in a three-dimensional space. Accordingly, when the X-axis feed mechanism 29, the Y-axis feed mechanism 28, the Z-axis feed mechanism 27, the spindle motor (not shown), and other components are driven by the controller 10 in accordance with an NC program stored in the controller 10, a workpiece W placed and fixed on the table 26 is machined as appropriate by a tool T attached to the spindle 24. Note that the tool T used in this embodiment is an end mill.

[0085] Further, a display device 12 having a display is connected to the controller 10, and data and the like in the controller 10 can be displayed on the display of the display device 12.

Natural Frequency Deriving Apparatus

[0086] Next, a natural frequency deriving apparatus 1 according to this embodiment is described. The natural frequency deriving apparatus 1 according to this embodiment is, as shown in FIGS. 1 and 2, composed of an accelerometer 5 stuck to an outer peripheral surface of a lower end portion of the spindle head 23, a force detector 6 fixed on the table 26, a detection machining executing part 2, a frequency analysis part 3, and a natural frequency deriving part 4; the detection machining executing part 2, the frequency analysis part 3, and the natural frequency deriving part 4 are incorporated in the controller 10.

[0087] The accelerometer 5 detects an acceleration of the lower end portion of the spindle head 23, in other words, an acceleration transmitted from a cutting tool T (hereinafter, simply referred to as tool T) attached to the spindle 24. When the workpiece W is cut by the tool T rotating, cutting resistance from the workpiece W causes vibration on the tool T. The accelerometer 5 detects the vibration that is transmitted to the spindle head 23 from the tool T through the spindle 24 (vibration caused by the tool T) and outputs a signal corresponding to the vibration. Note that the accelerometer 5 can output components for two directions: the X-axis direction and the Y-axis direction. Further, a displacement can be detected by second-order integral of an acceleration; therefore, the output signal from the accelerator 5 can be regarded as detection of a displacement of the tool T.

[0088] The force detector 6 has a force sensor 6a incorporated therein and is fixed on the table 26; the force sensor 6a detects an external force applied thereon and outputs a signal corresponding to the external force. The workpiece W is mounted on the force detector 6. Accordingly, when the workpiece W is cut by the tool T in this state, a cutting force applied to the workpiece W by the tool T, in other words, a cutting force applied to the tool T as a reaction force of the cutting force applied to the workpiece W is detected by the force sensor 6a, and a signal corresponding to the cutting force is output.

[0089] The detection machining executing part 2 is a processing unit that transmits a control signal to the operation controller 11 to cause the operation controller 11 to control the machine tool 20, thereby causing the machine tool 20 to execute a machining operation for deriving a natural frequency of the tool T. Specifically, the detection machining executing part 2 has an NC program stored therein for causing the machine tool 20 to perform the machining operation shown in FIG. 3, and transmits a control signal based on the NC program to the operation controller 11 so as to cause the machine tool 20 to operate.

[0090] In the machining operation shown in FIG. 3, for example, an end mill is used as the tool T, and the spindle 24 is rotated in the arrow direction at an initial rotational speed that is set as appropriate (for example, 3300 [min.sup.1]). Depth of cut is set to a value which does not cause chatter (for example, 1 [mm]) and width of cut Ae and feed amount (mm/edge) are set as appropriate. The tool T and the workpiece W are relatively moved in the X-axis direction so as to move them to the position indicated by P.sub.1 and then to the position indicated by P.sub.2, whereby the workpiece W is machined by down cut by the tool T.

[0091] In this process, the distance between P.sub.1 and P.sub.2 is equally divided into n sections x.sub.1 to x.sub.n and the rotational speed of the spindle 24 is increased every section in a sequential stepwise manner. For example, when the rotational speed is to be increased by 10 [min.sup.1] every section and the rotational speed in the section x.sub.1 is 3300 [min.sup.1], the rotational speed in the section x.sub.2 is set to 3310 [min.sup.1] and the rotational speed in the section x.sub.3 is set to 3320 [min.sup.1]. Thus, the rotational speed is increased in increments of 10 [min.sup.1] in a stepwise manner until the section x.sub.n is reached. Note that, if the feed speed is constant, the sections have the same machining time. Therefore, it is also possible to conceive that the rotational speed is to be increased at intervals of a predetermined machining time.

[0092] After the machining in which the tool T and the workpiece W are relatively moved in the X-axis direction in the above-described manner is finished, the tool T and the workpiece W are relatively moved in the Y-axis direction so as to move them to the position indicated by P.sub.3 and then to the position indicated by P.sub.4, whereby the workpiece W is machined by down cut by the tool T.

[0093] In this process, similarly to the above, the distance between P.sub.3 and P.sub.4 is equally divided into i sections y.sub.1 to y.sub.i and the rotational speed of the spindle 24 is increased every section in a sequential stepwise manner. For example, when the rotational speed is to be increased by 10 [min.sup.1] every section and the rotational speed in the section y.sub.1 is 3300 [min.sup.1], the rotational speed in the section y.sub.2 is set to 3310 [min.sup.1] and the rotational speed in the section y.sub.3 is set to 3320 [min.sup.1]. Thus, the rotational speed is increased in increments of 10 [min.sup.1] in a stepwise manner until the section y.sub.i is reached

[0094] The detection machining executing part 2 causes the machine tool 20 to execute the above-described machining operation.

[0095] While the above-described machining operation is executed by the machine tool 20 under control by the detection machining executing part 2, the frequency analysis part 3 receives signals output from the accelerometer 5 and the force sensor 6a and processes an acceleration signal and a force signal for each section (that is, for each rotational speed of the spindle 24; the same is applied below).

[0096] That is, the frequency analysis part 3 performs frequency analysis by FFT on a Y-axis direction vibration component of the acceleration signal for each of the sections x.sub.1 to x.sub.n, and then performs second-order integration to convert it into a displacement spectrum for each section. A Y-axis direction displacement spectrum for a certain section, which was obtained in the above-described manner, is shown in FIG. 4.

[0097] Further, the frequency analysis part 3 also performs frequency analysis by FFT on a Y-axis direction component of the force signal for each of the sections x.sub.1 to x.sub.n to calculate a cutting force spectrum for each section. A Y-axis direction cutting force spectrum for a certain section, which was obtained in the above-described manner, is shown in FIG. 5.

[0098] Similarly, the frequency analysis part 3 performs frequency analysis by FFT on an X-axis direction vibration component of the acceleration signal for each of the sections y.sub.1 to y.sub.i, and then performs second-order integration to convert it into a displacement spectrum for each section. Further, the frequency analysis part 3 also performs frequency analysis by FFT on an X-axis direction component of the force signal for each of the sections y.sub.1 to y.sub.i to calculate a cutting force spectrum for each section.

[0099] Note that the reason why a displacement spectrum and a cutting force spectrum for the Y-axis direction are obtained for each of the sections x.sub.1 to x.sub.n is that, in down cut in which the feed direction is the X-axis direction, the tool T is displaced with a large amount in the Y-axis direction and the cutting force is large in the Y-axis direction. Similarly, the reason why a displacement spectrum and a cutting force spectrum for the X-axis direction are obtained for each of the sections y.sub.1 to y.sub.i is that, in down cut in which the feed direction is the Y-axis direction, the tool T is displaced with a large amount in the X-axis direction and the cutting force is large in the X-axis direction.

[0100] In the above-described manner, the frequency analysis part 3 calculates a Y-axis direction displacement spectrum and a Y-axis direction cutting force spectrum for each of the sections x.sub.1 to x.sub.n and calculates an X-axis direction displacement spectrum and an X-axis direction cutting force spectrum for each of the sections y.sub.1 to y.sub.i.

[0101] The natural frequency deriving part 4 performs a processing for deriving a natural frequency of the tool T based on the displacement spectra and cutting force spectra obtained by the processing in the frequency analysis part 3.

[0102] Specifically, the natural frequency deriving part 4 first filters the Y-axis direction displacement spectrum and Y-axis direction cutting force spectrum for each of the sections x.sub.1 to x.sub.n, which were calculated by the frequency analysis part 3, and the X-axis direction displacement spectrum and X-axis direction cutting force spectrum for each of the sections y.sub.1 to y.sub.i, which were calculated by the frequency analysis part 3, to remove noise therefrom. It is known that a frequency showing a peak in the displacement spectrum and the cutting force spectrum is an integer multiple of a frequency when a cutting edge of the tool T is brought into contact with the workpiece W (this frequency is referred to as cutting edge passing frequency). Therefore, noise components can be removed by extracting only frequency components in a predetermined width corresponding to integer multiples of the cutting edge passing frequency by filtering. The resultant spectrum after removing noise components from the Y-axis direction displacement spectrum of FIG. 4 is shown in FIG. 6. The resultant spectrum after removing noise components from the Y-axis direction cutting force spectrum of FIG. 5 is shown in FIG. 7. Note that the cutting edge passing frequency can be calculated by the following equation:

cutting edge passing frequency [Hz]=(rotational speed of spindle 24 [min.sup.1]number of edges)/60 [sec]

[0103] Subsequently, based on the Y-axis direction displacement spectrum and Y-axis direction cutting force spectrum after noise removal for each of the sections x.sub.1 and x.sub.n and the X-axis direction displacement spectrum and X-axis direction cutting force spectrum after noise removal for each of the sections y.sub.1 to y.sub.i, the natural frequency deriving part 4 calculates, for each of the sections x.sub.1 to x.sub.n and each of the sections y.sub.1 to y.sub.i, a compliance spectrum that is obtained by dividing the displacement spectrum by the cutting force spectrum. Note that compliance is a ratio between the cutting force as input and the displacement of the tool T as output thereto, and is defined as an input-to-output transfer function. An example of the compliance spectrum obtained in the above-described manner is shown in FIG. 8. FIG. 8 shows a Y-axis direction compliance spectrum for a certain section of the sections x.sub.1 to x.sub.n.

[0104] Subsequently, the natural frequency deriving part 4 integrally superimposes the obtained Y-axis direction compliance spectra for the sections x.sub.1 to x.sub.n to calculate a Y-axis direction integrated compliance spectrum, and integrally superimposes the obtained X-axis direction compliance spectra for the sections y.sub.1 to y.sub.i to calculate an X-axis direction integrated compliance spectrum. FIG. 9 shows, as an example, Y-axis direction compliance spectra for a section in which the rotational speed of the spindle 24 is set to 3600 [min.sup.1], a section in which the rotational speed of the spindle 24 is set to 4000 [min.sup.1], and a section in which the rotational speed of the spindle 24 is set to 4300 [min.sup.1] being superimposed. Further, FIG. 10 shows a waveform (Y-axis direction integrated compliance spectrum) obtained by integrally superimposing the Y-axis direction compliance spectra for the sections x.sub.1to x.sub.n and tracing peaks thereof. Similarly, FIG. 11 shows a waveform (X-axis direction integrated compliance spectrum) obtained by integrally superimposing the X-axis direction compliance spectra for the sections y.sub.1 to y.sub.i and tracing peaks thereof.

[0105] Subsequently, based on the calculated Y-axis direction integrated compliance spectrum and X-axis direction integrated compliance spectrum, the natural frequency deriving part 4 analyzes each of the integrate compliance spectra to derive, as a natural frequency of the tool T, a frequency showing the largest compliance value. As described above, compliance expresses [displacement (=output)/cutting force (=input)]. Therefore, a frequency showing the largest compliance value, that is, a frequency with the largest output to the input can be designated as a natural frequency of the tool T.

[0106] Note that the X-axis direction and Y-axis direction displacement spectra and the X-axis direction and Y-axis direction cutting force spectra, which are calculated by the frequency analysis part 3, can be displayed on the display of the display device 12. Similarly, the X-axis direction and Y-axis direction displacement spectra after noise filtering, the X-axis direction and Y-axis direction cutting force spectra after noise filtering, the X-axis direction and Y-axis direction compliance spectra, and the X-axis direction and Y-axis direction integrated compliance spectra, which are calculated by the natural frequency deriving part 4, also can be displayed on the display of the display device 12.

[0107] In the natural frequency deriving apparatus 1 according to this embodiment having the above-described configuration, first, the detection machining executing part 2 causes the machine tool 20 to operate so as to cut a workpiece W using the tool T. In this process, the rotational speed of the spindle 24 is increased every section from the section x.sub.1 to the section x.sub.n in a sequential stepwise manner when the tool T and the workpiece W are moved in the X-axis direction, and the rotational speed of the spindle 24 is increased every section from the section y.sub.1 to the section y.sub.i in a sequential stepwise manner when the tool T and the workpiece W are moved in the Y-axis direction.

[0108] While machining is being performed in the above-described manner under control by the detection machining executing part 2, based on signals output from the accelerometer 5 and the force sensor 6a, the frequency analysis part 3 calculates a Y-axis direction displacement spectrum and a Y-axis direction cutting force spectrum for each of the sections x.sub.1 to x.sub.n and calculates an X-axis direction displacement spectrum and an X-axis direction cutting force spectrum for each of the sections y.sub.1 to y.sub.i.

[0109] Based on the Y-axis direction displacement spectrum and Y-axis direction cutting force spectrum for each of the sections x.sub.1 to x.sub.n and the X-axis direction displacement spectrum and X-axis direction cutting force spectrum for each of the sections y.sub.1 to y.sub.i, which were calculated by the frequency analysis part 3, the natural frequency deriving part 4 calculates, for each of the sections x.sub.1 to x.sub.n and each of the sections y.sub.1 to y.sub.i, a compliance spectrum that is obtained by dividing the displacement spectrum by the cutting force spectrum. Thereafter, the natural frequency deriving part 4 calculates a Y-axis direction integrated compliance spectrum by integrally superimposing the obtained Y-axis direction compliance spectra and calculates an X-axis direction integrated compliance spectrum by integrally superimposing the obtained X-axis direction compliance spectra. Based on the calculated Y-axis direction integrated compliance spectrum and X-axis direction integrated compliance spectrum, the natural frequency deriving part 4 analyzes each of them to derive, as a natural frequency of the tool T, a frequency showing the largest compliance value.

[0110] Thus, according to this natural frequency deriving apparatus 1, a workpiece W is machined using an actual tool T whose natural frequency needs to be derived, and a natural frequency of the tool T is derived based on a displacement of the tool T and a cutting force applied to the tool T that are detected during the machining. Therefore, a more accurate natural frequency that takes into account influence of the workpiece W the tool T receives in actual machining can be derived.

[0111] Further, an impact hammer, which is used in the conventional method, is not used. Therefore, in deriving a natural frequency of the tool T, the problem of artificial variation does not occur, technical skilled are not required for obtaining appropriate data, and the cumbersome operation of selecting a hammer tip is also not required.

[0112] Further, a natural frequency of the tool T is derived for each feed direction. Therefore, a natural frequency of the tool T which better conforms to actual machining situation can be derived.

Creation of Stability Limit Curve

[0113] Next, a manner of creating a stability limit curve using a natural frequency of the tool T derived in the above-described manner is described.

[0114] First, basic principles for creating a stability limit curve are explained. The model shown in FIG. 12 is a physical model with two degrees of freedom which is configured to, like the machine tool 20 shown in FIG. 1, relatively move a tool T and a workpiece W in two feed axis directions. Based on this model, conditions which cause regenerative chatter vibration are obtained using an analysis method devised by Y. Altintas.

[0115] In this model, the equations of motion for the tool T are represented by the following equations 1 and 2.

x+2.sub.x.sub.xx+.sub.x.sup.2x=F.sub.x/m.sub.x (Equation 1)

y+2.sub.y.sub.yy+.sub.y.sup.2y=F.sub.y/m.sub.y (Equation 2)

[0116] In the equations, .sub.x is a natural frequency [rad/sec] in the X-axis direction of the tool T, .sub.y is a natural frequency [rad/sec] in the Y-axis direction of the tool T, .sub.x is a damping ratio [%] in the X-axis direction, and .sub.y is a damping ratio [%] in the Y-axis direction. Further, m.sub.x is an equivalent mass [kg] in the X-axis direction, m.sub.y is an equivalent mass [kg] in the Y-axis direction, F.sub.x is a cutting force [N] applied to the tool T in the X-axis direction, and F.sub.y is a cutting force [N] applied to the tool T in the Y-axis direction. Furthermore, x and y each represent a second-order derivative with respect to time and x and Y each represent a first-order derivative with respect to time.

[0117] The cutting forces F.sub.x and F.sub.y can be calculated by the following equations 3 and 4, respectively:

F.sub.x=K.sub.ta.sub.ph()cos()K.sub.rK.sub.ta.sub.ph()sin(); (Equation 3)

and

F.sub.y=+K.sub.ta.sub.ph()sin()K.sub.rK.sub.ta.sub.ph()cos(). (Equation 4)

[0118] In the equations, h() [m.sup.2] is a thickness with which an cutting edge cuts the workpiece W, a.sub.p [mm] is a depth of cut, K.sub.t [N/m.sup.2] is a c cutting rigidity in a circumferential direction, and K.sub.r [%] is a specific cutting rigidity in a radial direction.

[0119] The cutting forces F.sub.x and F.sub.y change in accordance with an angle of rotation [rad] of the tool T; therefore, the cutting forces F.sub.x and F.sub.y can be respectively obtained by integrating the cutting forces F.sub.x and F.sub.y between an angle .sub.st at which cutting is started and an angle .sub.ex at which the cutting is ended and calculating the average thereof. Further, the angle .sub.st and the angle .sub.ex can be geometrically determined based on the diameter D [mm] of the tool T, the width of cut Ae [mm], the feed direction, and whether the cutting is upper cut or down cut.

[0120] The eigenvalue for the above equations 1 and 2 is represented by the following equation 5:

=(a.sub.1(a.sub.1.sup.24a.sub.0).sup.1/2)/2a.sub.0, (Equation 5)

[0121] where

[0122] a.sub.0=.sub.xx(i.sub.c).sub.yy(i.sub.c)(.sub.xx.sub.yy.sub.xy.sub.yx)

[0123] a.sub.1=.sub.xx.sub.xx(i.sub.c)+.sub.yy.sub.yy(i.sub.c)

[0124] .sub.xx(i.sub.c)=1/(m.sub.x(.sub.c.sup.2+2i.sub.x.sub.c.sub.x+.sub.x.sup.2))

[0125] .sub.yy(i.sub.c)=1/(m.sub.y(.sub.c.sup.2+2i.sub.y.sub.c.sub.y+.sub.y.sup.2))

[0126] .sub.xx=[(cos 2.sub.ex2K.sub.r.sub.ex+K.sub.r sin 2.sub.ex)(cos 2.sub.st2K.sub.r.sub.st+K.sub.r sin 2.sub.st)]/2

[0127] .sub.xy=[(sin 2.sub.ex2.sub.ex+K.sub.r cos 2.sub.ex)(sin 2.sub.st2.sub.st+K.sub.r cos 2.sub.st)]/2

[0128] .sub.yx=[(sin 2.sub.ex+2p.sub.ex+K.sub.r cos 2.sub.ex)(sin 2.sub.st+2.sub.st+K.sub.r cos 2.sub.st)]/2

[0129] .sub.yy=[(cos 2.sub.ex2K.sub.r.sub.exK.sub.r sin 2.sub.ex)(cos 2.sub.st2K.sub.r.sub.stK.sub.r sin 2.sub.st)]/2.

[0130] In the equations, .sub.c is a frequency of chatter vibration.

[0131] When the real part and the imaginary part of the eigenvalue are represented by .sub.R and .sub.I, respectively, a depth of cut a.sub.plim and a spindle rotation speed n.sub.lim at a stability limit are represented by the following equations 6 and 7, respectively:

a.sub.plim=2.sub.R(1+(.sub.I/.sub.R).sup.2)/(NK.sub.t); (Equation 6)

and

n.sub.lim=60.sub.c/(N(2k+2 tan.sup.1(.sub.I/.sub.R))). (Equation 7)

[0132] In the equations, N is the number of edges of the tool T and k is an integer.

[0133] Using the equations 6 and 7, a stability limit curve can be created by, while changing the values of .sub.c and k of the equations in an arbitrary manner, calculating the limit depth of cut a.sub.plim and the spindle rotation speed n.sub.lim each time.

[0134] By the way, in the above-described natural frequency deriving apparatus 1, the X-axis direction cutting force F.sub.x and the Y-axis direction cutting force F.sub.y can be detected by the force sensor 6a. Therefore, the cutting rigidity K.sub.t [N/m.sup.2] and the specific cutting rigidity K.sub.r [%] can be calculated using the equations 3 and 4.

[0135] Further, when the X-axis direction natural frequency and Y-axis direction natural frequency of the tool T are represented by .sub.x and .sub.y, the damping ratios .sub.x and .sub.y of the machining system are calculated by, for example, the following equations 8 and 9, respectively:

.sub.x=(.sub.1x.sub.2x)/2.sub.x; (Equation 8)

and

.sub.y=(.sub.1y.sub.2y/2.sub.y. (Equation 9)

[0136] Note that, as shown in FIG. 13, .sub.1x and .sub.2x are frequencies corresponding to G.sub.x/2.sup.1/2 on the spectrum waveform when the largest value of the X-axis direction integrated compliance spectrum is G.sub.x, and .sub.1y and .sub.2y are frequencies corresponding to G.sub.y/2.sup.1/2 on the spectrum waveform when the largest value of the Y-axis direction integrated compliance spectrum is G.sub.y.

[0137] Further, the equivalent masses m.sub.x and m.sub.y are calculated by the following equations 10 and 11, respectively:

m.sub.x=1/(2G.sub.x.sub.x.sub.x.sup.2); (Equation 10)

and

m.sub.y=1/(2G.sub.y.sub.y.sub.y.sup.2). (Equation 11)

[0138] Thus, the cutting rigidity K.sub.t and the specific cutting rigidity K.sub.r are calculated using the equations 3 and 4 based on the cutting forces F.sub.x and F.sub.y obtained by the natural frequency deriving apparatus 1, and the damping ratios .sub.x and .sub.y and the equivalent masses m.sub.x and m.sub.y are calculated using the equations 8, 9, 10, and 11 based on the natural frequencies .sub.x and .sub.y. Based on the obtained natural frequencies .sub.x and .sub.y, cutting rigidity K.sub.t, specific cutting rigidity K.sub.r, damping ratios .sub.x and .sub.y, and equivalent masses m.sub.x and m.sub.y, the real part .sub.R and imaginary part .sub.I of the eigenvalue are calculated using the equation 5. Thereafter, as described above, using the equations 6 and 7, the limit depth of cut a.sub.plim and the spindle rotation speed n.sub.lim are calculated each time while the values of .sub.c and k are changed in an arbitrary manner, whereby a stability limit curve can be created.

[0139] An example of the stability limit curve created in the above-described manner is shown in FIG. 14.

[0140] Thus, according to the thus configured stability limit curve creating method, a stability limit curve corresponding to a machine tool 20 having two feed axes: X and Y axes that are perpendicular to a spindle 24 and perpendicular to each other can be created. Further, since this stability limit curve creating method is configured so that, as described above, more accurate natural frequencies which conform to actual machining situation, such as influence of an object to be machined the tool receives in actual machining, are obtained and a stability limit curve is created based on such natural frequencies, an accurate stability limit curve which better conforms to actual machining situation can be created.

[0141] Thus, an embodiment of the present disclosure has been described; however, the present disclosure is not limited thereto and can be implemented in other manners.

[0142] For example, although a so-called machining center is exemplarily used as the machine tool 20 in the above embodiment, the present disclosure is not limited thereto, and examples of the machine tool to which the present disclosure can be applied include all machine tools capable of machining using a cutting tool which has the possibility of causing regenerative chatter in cutting, such as a lathe and the like.

[0143] Further, although an end mill with two degrees of freedom is exemplarily used as the cutting tool T in the above embodiment, the present disclosure is not limited thereto, and the cutting tool to which the present disclosure can be applied may be a cutting tool with one degree of freedom, such as a cutting-off tool or the like.

[0144] Further, although the cutting force applied to the tool T is detected by the force sensor 6a in the above embodiment, the present disclosure is not limited thereto and the cutting force may be calculated based on a value of a current supplied to the spindle motor.

[0145] Further, the natural frequency deriving part 4 of the natural frequency deriving apparatus 1 in the above embodiment may be configured to, in the step of deriving a natural frequency, derive, as natural frequencies of the cutting tool, at least two frequencies showing a maximal compliance value in decreasing order of the compliance value for each of the X-axis and Y-axis feed directions based on the integrated compliance spectrum for the feed direction. Further, in the creation of a stability limit curve, a configuration is possible in which the damping ratio and the equivalent mass are calculated corresponding to each of the natural frequencies of the cutting tool for each of the feed directions based on the integrated compliance spectrum for the feed direction and the natural frequencies for the feed direction, and a stability limit curve is created corresponding to each of the natural frequencies based on the obtained damping ratios and equivalent masses and the natural frequencies.

[0146] In such a configuration, it is possible to create a stability limit curve for each of a plurality of possible natural frequencies of the cutting tool, and setting actual machining conditions with reference to such stability limit curves achieves a more stable machining in which regenerative chatter is more unlikely to occur.

[0147] Further, in the above stability limit curve creating method, a configuration is possible in which the damping ratio and the equivalent mass are calculated for each of the feed directions based on the integrated compliance spectrum obtained for the feed direction and the natural frequency of the cutting tool for the feed direction, and a cutting force applied to the cutting tool, or a damping ratio and an equivalent mass, as well as a natural frequency of the cutting tool for a predetermined arbitrary feed direction are estimated based on the obtained damping ratios and equivalent masses for the feed directions and the natural frequencies, and a stability limit curve concerning regenerative chatter of the cutting tool for the predetermined arbitrary feed direction is created.