JP7535446B2 - Apparatus and method for evaluating grinding burns on metal parts - Google Patents

Description

The present invention relates to a grinding burn evaluation device and a grinding burn evaluation method for evaluating grinding burns that occur on the surface of ground metal parts.

A corrosion surface observation method (JIS G 0561:2011) is known for observing grinding burns that occur on ground metal parts by corroding the surface using nitric acid and hydrochloric acid. However, this corrosion surface observation method requires cutting and polishing processes for the metal part to be measured, which requires a lot of man-hours, and has drawbacks such as difficulty in accurately evaluating grinding burns due to uneven corrosion caused by the range of corrosion and the corrosion time.

There is also a known hardness measurement method (JIS Z 2244:2009, JIS Z 2245:2016) that measures hardness by measuring the size of the indentation that occurs when an indenter is pressed against the surface of a metal part. However, this hardness measurement method has the disadvantage that it is difficult for the indenter to capture grinding burns that occur on the outermost layer of the metal part, and the hardness of the parent layer is shown as the measured value.

In response to this, a grinding burn evaluation device and grinding burn evaluation method have been proposed that uses diffracted X-rays to determine that grinding burns have occurred if the half-width of the diffracted X-rays diffracted on the surface of a metal part is equal to or less than a preset threshold value when the X-rays are irradiated onto the surface of the metal part. Patent Document 1 discloses a grinding burn inspection method for metal parts. This method uses diffracted X-rays that are affected by grinding burns that occur on the surface layer of the metal part, making it possible to evaluate grinding burns on the surface layer of the metal part.

JP 2004-271355 A

However, according to the method for evaluating grinding burns on metal parts in Patent Document 1, the measurement angle (angle Ψ) between the irradiated X-rays and the diffracted X-rays is one, so sufficient accuracy cannot be obtained to evaluate grinding burns that occur on the extreme surface layer of polycrystalline bodies such as metals that have multiple lattice planes. In addition, grinding burns on metal parts are uniformly judged when the half-width of the diffracted X-rays is below a certain value, i.e., 6° or less, but since the half-width differs depending on the lattice plane, this is inappropriate as an evaluation criterion. Furthermore, depending on the strength of the effect of the grinding process heat, the surface structure of the metal part may become a black layer that is softer than the normal tissue layer, or a white layer that is harder than the normal tissue layer, so that grinding burns may not always be evaluated accurately.

The present invention has been made against the background of the above circumstances, and its purpose is to provide a grinding burn evaluation device and a grinding burn evaluation method for metal parts that can more accurately evaluate grinding burns that occur on the surface of ground metal parts.

The gist of a first invention is that (a) a grinding burn evaluation device for metal parts that evaluates grinding burns occurring on a surface of a ground metal part, the device including: (b) an X-ray irradiation control unit that changes the angle Ψ between the incident optical axis of the X-ray incident on the metal part and a normal to the surface of the metal part, and irradiates the surface of the metal part with incident X-rays; (c) a diffracted X-ray detection control unit that detects an intensity profile of diffracted X-rays diffracted on the surface of the metal part for each of the angles Ψ; (d) an average half-width calculation unit that calculates an average half-width value obtained from the intensity profile of the diffracted X-rays for each of the angles Ψ detected by the diffracted X-ray detection control unit; and (e) a grinding burn evaluation unit that evaluates grinding burn of the metal part based on a comparison between the average half-width value and a grinding burn judgment threshold value that is preset to have a normal judgment region having a half-width lower than that of a white layer that is a re-hardened layer affected by the grinding heat and a half-width higher than that of a black layer that is a softened layer affected by the grinding heat.

The gist of the second invention is that, in the first invention, the grinding burn judgment threshold value is set to have the normal judgment region that is equal to or lower than an upper limit value for removing reject products whose half-value width shows a higher value than the normal judgment region due to the presence of the white layer , and is equal to or higher than a lower limit value for removing reject products whose half-value width shows a lower value than the normal judgment region due to the presence of the black layer.

The gist of the third invention is that in the first or second invention, the X-ray irradiation control unit irradiates the surface of the metal part with incident X-rays by varying the angle Ψ to multiple angles within a range of 0° to 45°.

The gist of the fourth invention is that in any one of the first to third inventions, the grinding burn judgment threshold is a value set based on the average half-width calculated from the intensity profile of the diffracted X-rays in the same manner as the X-ray irradiation control unit, by varying the angle Ψ and irradiating the surface of the metal part that has been evaluated as having normal grinding burn, and the average half-width calculated from the intensity profile of the diffracted X-rays in the same manner as the average half-width calculation unit.

The gist of the fifth invention is that in any one of the first to third inventions, the grinding burn judgment threshold is a value set based on the average half-width calculated from the intensity profile of the diffracted X-rays by electrolytically polishing the ground surface of the metal part to a depth that is not affected by grinding heat, irradiating the electrolytically polished surface of the metal part with incident X-rays while changing the angle Ψ, detecting the diffracted X-rays diffracted on the surface of the metal part in the same manner as the diffracted X-ray detection control unit, and calculating the average half-width from the intensity profile of the diffracted X-rays in the same manner as the average half-width calculation unit.

The gist of the sixth invention is that (a) a grinding burn evaluation method for evaluating grinding burns occurring on the surface of a ground metal part, the method including: (b) an X-ray irradiation step of irradiating the surface of the metal part with incident X-rays while changing the angle Ψ between the incident optical axis of the X-rays incident on the metal part and the normal to the surface of the metal part; (c) a diffracted X-ray detection step of detecting an intensity profile of diffracted X-rays diffracted on the surface of the metal part for each of the angles Ψ; (d) an average half-width calculation step of calculating an average half-width value obtained from the intensity profiles of the diffracted X-rays for each of the angles Ψ detected in the diffracted X-ray detection step; and (e) a grinding burn evaluation step of evaluating the grinding burn of the metal part based on a comparison between the average half-width value and a grinding burn judgment threshold value that is preset to have a normal judgment region having a half-width lower than that of a white layer that is a re-hardened layer affected by the heat of grinding and a half-width higher than that of a black layer that is a softened layer affected by the heat of grinding.

The gist of the seventh invention is that, in the sixth invention, the grinding burn judgment threshold value is set to have a normal judgment region that is equal to or lower than an upper limit value for removing reject products whose half-value width shows a higher value than the normal judgment region due to the presence of the white layer , and is equal to or higher than a lower limit value for removing reject products whose half-value width shows a lower value than the normal judgment region due to the presence of the black layer.

The gist of the eighth invention is that in the sixth or seventh invention, the X-ray irradiation step irradiates the surface of the metal part with incident X-rays by varying the angle Ψ to multiple angles within the range of 0° to 45°.

The gist of the ninth invention is that in any one of the sixth to eighth inventions, the grinding burn judgment threshold is a value set based on the average half-width calculated from the intensity profile of the diffracted X-rays in the same manner as in the X-ray irradiation process, by irradiating the surface of the metal part that has been evaluated as having normal grinding burn with incident X-rays while changing the angle Ψ, and by detecting the diffracted X-rays on the surface of the metal part in the same manner as in the diffracted X-ray detection process, in the same manner as in the half-width average calculation process.

The gist of the tenth invention is that in any one of the sixth to eighth inventions, the grinding burn judgment threshold is a value set based on the average half-width calculated from the intensity profile of the diffracted X-rays in the same manner as in the diffraction X-ray detection process, by electrolytically polishing the ground surface of the metal part to a depth that is not affected by grinding heat, irradiating the electrolytically polished surface of the metal part with incident X-rays while changing the angle Ψ in the same manner as in the diffraction X-ray detection process, and calculating the average half-width value from the intensity profile of the diffracted X-rays in the same manner as in the half-width average calculation process.

According to the grinding burn evaluation device for metal parts of the first invention and the grinding burn evaluation method for metal parts of the sixth invention, in the grinding burn evaluation step or grinding burn evaluation unit, the grinding burn of the metal part is evaluated based on a comparison between the average half-width value obtained from the intensity profile of the diffracted X-rays for each angle Ψ calculated by the half-width average calculation step or half-width average calculation unit and a grinding burn judgment threshold value that is preset to have a normal judgment region having a half-width lower than that of the white layer, which is a re-hardened layer affected by the heat of the grinding process, and a half-width higher than that of the black layer, which is a softened layer affected by the heat of the grinding process. Therefore, grinding burn occurring on the surface of the ground metal part can be evaluated more accurately.

According to the grinding burn evaluation device for metal parts of the second invention and the grinding burn evaluation method for metal parts of the seventh invention, the grinding burn judgment threshold is set to have the normal judgment region that is equal to or lower than the upper limit for removing rejects whose half-width is higher than the normal judgment region due to the presence of the white layer, and equal to or higher than the lower limit for removing rejects whose half-width is lower than the normal judgment region due to the presence of the black layer. Therefore, it is possible to determine defective products whose average half-width exceeds the upper limit of the grinding burn judgment threshold and defective products whose average half-width is lower than the lower limit of the grinding burn judgment threshold, so that grinding burns occurring on the surface of the ground metal part can be evaluated more accurately.

According to the grinding burn evaluation device for metal parts of the third invention and the grinding burn evaluation method for metal parts of the eighth invention, the angle Ψ can be changed to multiple angles within the range of 0° to 45°. This allows the size of the diffracted X-ray detection device that performs the X-ray irradiation process and the diffracted X-ray detection process to be reduced.

According to the grinding burn evaluation device for metal parts of the fourth invention and the grinding burn evaluation method for metal parts of the ninth invention, the grinding burn judgment threshold is a value set based on the average half-width calculated from the intensity profile of the diffracted X-rays by irradiating the surface of the metal parts that are evaluated as having normal grinding burns with incident X-rays at different angles Ψ in the same manner as in the X-ray irradiation process, detecting the diffracted X-rays diffracted on the surface of the metal parts in the same manner as in the diffracted X-ray detection process, and calculating the average half-width value in the same manner as in the half-width average calculation process. Since the grinding burn of the metal parts is evaluated based on a comparison between the grinding burn judgment threshold and the average half-width value, the grinding burn occurring on the surface of the ground metal parts can be evaluated more accurately.

According to the grinding burn evaluation device for metal parts of the fifth invention and the grinding burn evaluation method for metal parts of the tenth invention, the grinding burn judgment threshold is a value set based on the average value of the half-width calculated from the intensity profile of the diffracted X-rays in the same manner as in the X-ray irradiation process, by electrolytically polishing the surface of the ground metal part to a depth that is not affected by grinding heat, by irradiating the surface of the metal part polished by electrolytic polishing with incident X-rays while changing the angle Ψ in the same manner as in the X-ray diffraction detection process, and by detecting the diffracted X-rays on the surface of the metal part in the same manner as in the diffracted X-ray detection process. Since the grinding burn of the metal part is evaluated based on a comparison between such a grinding burn judgment threshold and the average value of the half-width, the grinding burn occurring on the surface of the ground metal part can be more accurately evaluated.

Here, preferably, the grinding burn judgment threshold is a value set so as to have a normal judgment region with a width of ±0.5° or a width of ±0.25° centered on the average value of the half-width for each angle Ψ.

Preferably, the incident optical axis of the incident X-ray is the central axis of the incident X-ray. The diffraction angle 2θ is the angle (including the angle after linear transformation) between the incident optical axis of the incident X-ray and the diffracted X-ray emitted from the portion of the surface of the metal part on which the incident X-ray is incident.

Preferably, the half-width of the diffracted X-rays is an angular width corresponding to half the height of a peak (mountain) formed in the intensity profile of the diffracted X-rays in a two-dimensional coordinate system with an axis representing the diffraction angle 2θ and an axis representing the intensity of the diffracted X-rays. The intensity of the diffracted X-rays is the number of X-ray gratings detected per unit time, or a physical quantity representing it (e.g., voltage, current, etc.).

1 is a block diagram illustrating an X-ray diffraction detector and an electronic control device used in a grinding burn determination method or grinding burn evaluation device for a metal part according to the present invention. FIG. 2 is a diagram illustrating a half-value width in an intensity profile of diffracted X-rays detected by the diffracted X-ray detection device of FIG. 1 . FIG. 2 is a process diagram illustrating a method for determining grinding burn on a metal part, which is an essential part of the control operation of the electronic control device in FIG. 1. 1 is a diagram showing the change in Vickers hardness Hv relative to the cross-sectional area of grinding allowance, comparing products with defects due to grinding burn with marks ● and normal products with marks ○. FIG. 13 is a diagram showing the change in half-value width HW relative to the grinding allowance cross-sectional area for a cylindrically ground metal part (SCr420), with abnormal parts due to grinding burn indicated by ● marks and normal parts indicated by ○ marks. FIG. 6 is a diagram showing the value of the half-value width HW with respect to the depth for the metal part shown in FIG. 5A. FIG. 6 is a diagram showing the value of the half-value width HW with respect to the depth for the metal part shown in FIG. 5B. FIG. 6 is a diagram showing the value of the half-value width HW with respect to the depth for the metal part shown in FIG. 5C. FIG. 6 is a structural diagram of the metal part shown in FIG. 5B taken by an optical microscope. FIG. 6 is a structural diagram of the metal part shown in FIG. 5C taken by an optical microscope. FIG. 6 is a diagram showing the half-value width measured for each grinding allowance cross-sectional area when the angle Ψ of the defective product in FIG. 5 is changed to six different values. FIG. 13 is a diagram showing the half-value widths HW obtained for six different angles Ψ for a surface-ground metal part (SCM440) from a depth of 0 μm to 100 μm. FIG. 13 is a diagram showing the half-value widths HW obtained for six different angles Ψ for the surface-ground metal component of FIG. 12 from a depth of 100 μm to 600 μm. FIG. 13 is a diagram showing the average half-width HWav (°) obtained by averaging the half-widths HW obtained for each of six different angles Ψ for the metal component subjected to surface grinding in FIG. 12, from a depth of 0 μm to 600 μm. FIG. 13 is a diagram for explaining a grinding burn determination threshold value TH for the surface-ground metal part of FIG. 12 . FIG. 13 is a diagram showing the relationship between the grinding allowance cross-sectional area (mm ² /mm) per unit circumferential length of the grinding wheel and the power consumption (kW) of the grinding wheel spindle when a metal part (SCr420) is cylindrically ground. FIG. 17 is a diagram showing the relationship between the grinding allowance cross-sectional area per unit circumferential length of the grinding wheel (mm ² /mm) and the amount of wear of the grinding wheel (μm) when a metal part is cylindrically ground under the same grinding conditions as in FIG. 16 . FIG. 17 is a diagram showing the relationship between the grinding allowance cross-sectional area per unit circumferential length of the grinding wheel (mm ² /mm) and the surface roughness Rz (μm) of the ground surface when a metal part is cylindrically ground under the same grinding conditions as in FIG. 16 .

An embodiment of the present invention will be described in detail below with reference to the drawings. Note that the drawings in the following embodiment are for explaining the main parts related to the invention, and the dimensions and shapes are not necessarily drawn accurately.

Figure 1 shows a grinding burn evaluation device 10 for evaluating grinding burn on a metal part 14, such as a bearing part made of hardened steel, that has been ground using a grinding wheel. The grinding burn evaluation device 10 includes an X-ray diffraction detector 12 that irradiates an incident X-ray onto a surface 16 of a metal part 14 made of steel, such as hardened steel, and detects diffracted X-rays emitted from the portion of the surface 16 of the metal part 14 on which the incident X-rays are incident, and an electronic control device 30 that controls the X-ray diffraction detector 12.

The diffracted X-ray detection device 12 includes an X-ray tube 18 that outputs incident X-rays along an incident optical axis Lx1 toward a point P on the surface (irradiation surface) 16 of the metal part 14, and an X-ray detector 20 that receives diffracted X-rays that are diffracted at point P and emitted along the diffracted optical axis Lx2 and detects the diffracted X-rays.

The X-ray tube 18 is movably supported by an arc-shaped guide 22 centered on point P. The X-ray tube 18 is provided with an actuator 18a that positions the X-ray tube 18 at multiple positions in the circumferential direction around point P along the guide 22 so that the angle Ψ, described below, is positioned at different positions. The X-ray detector 20 is movably supported by an arc-shaped guide 24 centered on point P. The X-ray detector 20 is provided with an actuator 20a that positions the X-ray detector 20 in the circumferential direction around point P along the guide 24 to obtain an intensity profile of the diffracted X-rays for each different angle Ψ, described below.

In FIG. 1, the X-ray tube 18 is positioned so that the incident optical axis Lx1 from the X-ray tube 18 coincides with the normal (irradiation surface normal) z1 of the surface 16 of the metal part 14. In other words, the X-ray tube 18 is positioned so that the angle Ψ, defined as the angle between the incident optical axis Lx1 from the X-ray tube 18 and the normal z1 of the surface 16 of the metal part 14, is 0°. In this embodiment, the angle Ψ is switched to multiple types, and the intensity profile of the diffracted X-rays is detected for each angle Ψ. The X-ray tube 18 shown by the dashed line in FIG. 1 indicates the position where the angle Ψ is switched from 0°.

The angle η shown in FIG. 1 is the angle between the incident optical axis Lx1 and the normal (crystal plane normal) z2 of the lattice plane 26, and is also the angle between the normal (crystal plane normal) z2 of the lattice plane 26 and the diffracted optical axis Lx2. This angle η is an angle derived by applying a linear transformation to the diffraction angle 2θ, and is one of the diffraction angles that can identify the positional relationship between the incident optical axis Lx1 and the diffracted optical axis Lx2.

The electronic control device 30 is a so-called microcomputer including a CPU, ROM, RAM, and an input/output interface, and processes input signals according to a program prestored in the ROM to perform an evaluation of grinding burn on the metal part 14. The electronic control device 30 functionally comprises an X-ray irradiation control unit 32, a diffracted X-ray detection control unit 34, an average half-value width calculation unit 36, and a grinding burn evaluation unit 38, and outputs the evaluation results by the grinding burn evaluation unit 38 and displays them on the display 40.

The X-ray irradiation control unit 32 changes the angle Ψ between the incident optical axis Lx1 of the X-rays incident on the metal part 14 and the normal z1 of the surface 16 of the metal part 14 to multiple types, for example, six types of 0°, 18.4°, 26.6°, 33.2°, 39.2°, and 45.0°, by moving the X-ray tube 18 along the guide 22 using the actuator 18a, and positions the X-ray tube 18, and irradiates the incident X-rays from the X-ray tube 18 to the surface 16 of the metal part 14.

The diffracted X-ray detection control unit 34 uses the actuator 20a to move the X-ray detector 20 in the circumferential direction around the point P along the guide 24 for each different angle Ψ to obtain an intensity profile of the diffracted X-rays for each different angle Ψ, and detects the intensity of the diffracted X-rays diffracted on the surface of the metal part 14 for each of multiple types of angle Ψ (six types in this embodiment) to detect the intensity profile.

The half-width average calculation unit 36 measures the half-width HW (°) of the intensity profile of the diffracted X-rays for each angle Ψ detected by the diffracted X-ray detection control unit 34 for each different angle Ψ, and calculates the average half-width HWav (°), which is the average of the half-widths HW (°). Figure 2 shows a part of the intensity profile of the diffracted X-rays in two-dimensional coordinates with an axis representing the diffraction angle 2θ and an axis representing the diffracted X-ray intensity I. This diffracted X-ray intensity I is the number of X-ray lattices detected per unit time, or a physical quantity representing it (for example, voltage, current, etc.). The diffraction angle corresponding to the maximum value Imax of the peak waveform in the intensity profile in Figure 2 is the multiple 2θ of the Bragg angle θ that satisfies the so-called Bragg formula (2dsinθ=nλ (where d is the lattice constant and λ is the wavelength)). The half-width HW is half the maximum value Imax of the peak waveform, i.e., the range of diffraction angles (°) corresponding to 1/2Imax.

The grinding burn evaluation unit 38 evaluates the grinding burn of the metal part 14 based on a comparison between the average half-width value HWav calculated by the average half-width value calculation unit 36 and a preset grinding burn judgment threshold value TH. For example, if the average half-width value HWav exceeds the grinding burn judgment threshold value TH, it makes a pass judgment, and if the average half-width value HWav is equal to or less than the grinding burn judgment threshold value TH, it makes a fail judgment due to grinding burn.

The above-mentioned preset grinding burn judgment threshold value TH is a value that is set based on the average half-width HWav calculated by, for example, irradiating incident X-rays to the surface of a ground metal part 14 that has been evaluated as having normal grinding burns, while changing the angle Ψ between the incident optical axis Lx1 of the X-rays incident on the metal part 14 and the normal z1 of the surface 16 of the metal part 14, as in the X-ray irradiation control unit 32, detecting the diffracted X-rays diffracted at the surface 16 of the metal part 14, as in the diffracted X-ray detection control unit 34, and calculating the average half-width HWav of the intensity profile of the diffracted X-rays for each angle Ψ, as in the average half-width calculation unit 36.

The above-mentioned preset grinding burn determination threshold value TH is a value that is set based on the following: for example, the surface of the ground metal part 14 is polished by electrolytic polishing to a depth that is not affected by grinding heat; the surface of the metal part 14 polished by electrolytic polishing is irradiated with incident X-rays by changing the angle Ψ between the incident optical axis Lx1 of the X-rays incident on the metal part 14 and the normal z1 of the surface 16 of the metal part 14, as with the X-ray irradiation control unit 32; the diffracted X-rays diffracted at the surface 16 of the metal part 14 are detected, as with the diffracted X-ray detection control unit 34; and the average half-width HWav of the intensity profile of the diffracted X-rays for each angle Ψ is calculated, as with the average half-width calculation unit 36;

The grinding burn judgment threshold value TH may be set to the average half-width value HWav as is, or may be set to a value that has a normal judgment range within a range of, for example, about ±10% based on the average half-width value HWav. It may also be set to a value that has a normal judgment range of, for example, a width of ±5° or a width of ±2.5° based on the average half-width value HWav.

Figure 3 is a flow chart explaining the main parts of the control operation of the electronic control device 30. In Figure 3, in the X-ray irradiation process of step S1 (hereinafter, step will be omitted) corresponding to the X-ray irradiation control unit 32, the angle Ψ between the incident optical axis Lx1 of the X-ray incident on the metal part 14 and the normal z1 of the surface 16 of the metal part 14 is changed to six types of 0°, 18.4°, 26.6°, 33.2°, 39.2°, and 45.0° by moving the X-ray tube 18 by the actuator 18a, and the X-ray tube 18 is positioned respectively, and the incident X-ray is irradiated from the X-ray tube 18 to the point P on the surface 16 of the metal part 14 every time it is positioned.

Next, in the diffracted X-ray detection process S2 corresponding to the diffracted X-ray detection control unit 34, in order to obtain the intensity profile of the diffracted X-ray for each different angle Ψ, the actuator 20a is used to move the X-ray detector 20 in the circumferential direction around point P along the guide 24 for each different angle Ψ, and the intensity of the diffracted X-ray diffracted at point P on the surface 16 of the metal part 14 is detected for each of the six different angles Ψ to obtain an intensity profile.

Next, in the average half-width calculation step S3 corresponding to the average half-width calculation unit 36, the half-widths HW(°) of the intensity profile of the diffracted X-rays for each angle Ψ detected in the diffracted X-ray detection step S2 are measured for each different angle Ψ, and the average half-width value HWav(°), which is the average of these half-widths HW(°), is calculated.

Then, in the grinding burn evaluation step S4 corresponding to the grinding burn evaluation unit 38, grinding burn of the metal part 14 is evaluated based on a comparison between the average half-width value HWav calculated in the average half-width value calculation step S3 and a preset grinding burn judgment threshold value TH. For example, if the average half-width value HWav exceeds the grinding burn judgment threshold value TH, it is judged as pass, and if the average half-width value HWav is equal to or less than the grinding burn judgment threshold value TH, it is judged as fail due to grinding burn.

The inventors conducted a test under the following measurement conditions to compare the Vickers hardness Hv (JIS Z 2244:2009) and half-value width HW of a normal part (accepted part) ground with grinding wheel 1 and an abnormal part (rejected part) with grinding burn that was ground with grinding wheel 2 out of the metal parts 14 made of carburized and hardened steel that were cylindrically ground under the grinding conditions shown below.

(Cylindrical grinding conditions)
Grindstone 1: PA 120 H + 10 V (no burn)
Grindstone 2: PA 80 H + 10 V (with burn marks)
Machine: Cylindrical grinding machine Grinding method: Wet cylindrical plunge grinding Grinding wheel peripheral speed: 60 m/s
Workpiece peripheral speed: 0.6m/s
Peripheral speed ratio: 100
Grinding efficiency: 2.0mm ³ /mm・s
Dresser: 0.8 square prism diamond single stone dresser Dress lead: 0.1mm/r.o.w.
Dressing depth: 0.01 mm/pass
Work material: SCr420 (carburized steel)

(Vickers hardness measurement conditions)
Magnification of observation lens: 10x Load holding time: 10 sec
Load: 10 kgf

(Conditions for measuring half-width)
Measurement site: surface layer 0μm
Diffraction X-ray measurement range: 140° to 170°
2θ: 156.40° (α-Fe(211))
Step: 0.20°
Angle Ψ: 45.0°
Tube: Cr
Characteristic X-rays: Kα rays

4 and 5 show the measured values of Vickers hardness Hv (HV10) and half-value width HW (°). In FIG. 4, in a two-dimensional coordinate system with the horizontal axis representing the grinding allowance cross-sectional area (mm ² /mm) per unit circumference length of the grinding wheel and the vertical axis representing the Vickers hardness, the Vickers hardness Hv of the abnormal product (rejected product) due to grinding burn is displayed with ● for each increase in the grinding allowance cross-sectional area, and the Vickers hardness Hv of the accepted product without grinding burn is displayed with ○ for each increase in the grinding allowance cross-sectional area. In FIG. 5, in a two-dimensional coordinate system with the horizontal axis representing the grinding allowance cross-sectional area (mm ² /mm) per unit circumference length of the grinding wheel and the vertical axis representing the half-value width HW (°), the Vickers hardness Hv of the abnormal product (rejected product) due to grinding burn is displayed with ● for each increase in the grinding allowance cross-sectional area, and the Vickers hardness Hv of the accepted product without grinding burn is displayed with ○ for each increase in the grinding allowance cross-sectional area. Here, the grinding allowance cross-sectional area per unit circumferential length of the grinding wheel (mm ² /mm) is a value obtained by dividing the grinding allowance cross-sectional area by the circumferential length of the grinding wheel.

As shown in FIG. 4, the Vickers hardness Hv of the abnormal product is equal to that of the normal product at the beginning of the increase in the grinding allowance cross-sectional area, but then drops sharply. In contrast, as shown in FIG. 5, the half-value width HW of the acceptable product shows a substantially constant value regardless of the increase in the grinding allowance cross-sectional area. The half-value width HW of the abnormal product decreases gradually from the beginning of the increase in the grinding allowance cross-sectional area, and shows a smaller value overall than that of the normal product from the beginning of the increase in the grinding allowance cross-sectional area. In other words, since a difference ΔHW is formed between the minimum value of the normal product marked with a circle and the maximum value of the abnormal product marked with a ●, it is possible to distinguish between an abnormal product and an acceptable product from the beginning of the increase in the grinding allowance cross-sectional area by using the half-value width HW.

Next, the inventors prepared samples of different depths by electrolytic polishing for three metal parts A, B, and C shown in A, B, and C of Figure 5, which were among the metal parts 14 (abnormal parts) with grinding burns that showed the same Vickers hardness Hv value as normal parts in the area with a small grinding allowance cross-sectional area, and measured the half-width HW for each different depth using the half-width measurement conditions described above. Figures 6, 7, and 8 show the half-width HW values versus depth for the three metal parts 14 shown in A, B, and C of Figure 5 above, respectively.

As shown in Figures 6 and 7, metal parts A and B, which showed values equivalent to those of normal products in Vickers hardness Hv, had a half-value width HW lower than the normal tissue layer NL at a depth within the measurement depth (indenter pressing depth at 10 kgf) at Vickers hardness HV10, but beyond the measurement depth at Vickers hardness HV10, it was within the range of the half-value width HW of the normal tissue layer NL. In the tissue diagrams of these metal parts A and B, a black layer (softened layer) BL, which will be described later, was present on the surface. Due to the influence of this black layer BL, the grinding burn on the surface layer, which cannot be evaluated by Vickers hardness HV10, is shown to have a half-value width HW lower than the normal tissue layer NL. This shows that the half-value width HW can be used to evaluate within the measurement depth at Vickers hardness HV10.

As shown in Figure 8, for metal part C, the half-value width HW at a position several μm below the surface layer is initially lower than the half-value width HW of the surface layer, and it was confirmed that it increases toward the normal range when the measurement depth at Vickers hardness HV10 is exceeded. Figure 9 is a structural diagram of metal part B photographed with an optical microscope. Also, Figure 10 is a structural diagram of metal part C photographed with an optical microscope, in which a black layer (softened layer) BL and a white layer (re-hardened layer) WL that have been affected by the grinding heat are formed in sequence on top of a normal structural layer NL that has not been affected by the grinding heat.

The value of the half-value width HW in FIG. 8 is a value that includes the white layer WL and the black layer BL in the structure diagram in FIG. 10. In this respect, the half-value width HW captures the presence of the white layer WL, and indicates that grinding burn in the surface layer (portion shallower than the measurement depth with Vickers hardness HV10) that cannot be evaluated with Vickers hardness HV10 can be measured using the half-value width HW. Here, the white layer WL is a re-hardened layer whose structure has changed due to the influence of grinding heat and whose Vickers hardness Hv is higher than that of the normal structure layer NL that is not affected by the grinding heat, while the black layer BL is a softened layer whose structure has changed due to the influence of grinding heat and whose Vickers hardness Hv is lower than that of the normal structure layer NL that is not affected by the grinding heat.

Next, the inventor measured the half-width HW for six angles Ψ (0°, 18.4°, 26.6°, 33.2°, 39.2°, and 45.0°) for four locations on the outer periphery of a metal part 14 (abnormal part) with grinding burn that showed the same Vickers hardness Hv value as a normal part in the area with a small grinding allowance cross-sectional area, under the same conditions as the half-width measurement conditions described above. When the measured values were plotted on a two-dimensional coordinate system similar to that shown in FIG. 5, it was found that the value of the half-width HW also fluctuated depending on the angle Ψ, as shown in FIG. 11. For this reason, the inventor discovered that by evaluating grinding burn using the half-width average value HWav, which is the average of the half-widths HW obtained from each angle Ψ, it is possible to evaluate grinding burn from information on multiple crystal planes of a metal part 14 to which grinding heat is not uniformly applied.

As is clear from Figures 6 to 8, the more strongly the metal part 14 is affected by the grinding heat, the more a white layer (re-hardened layer) WL is formed, and the half-width HW is detected as higher than that of a normal part, and the more weakly the metal part 14 is affected by the grinding heat, the more a black layer (softened layer) BL is formed, and the half-width HW is detected as lower than that of a normal part. In this respect, for the grinded carburized and hardened steel SCr420, a grinding burn determination threshold TH is used, which is set to have a normal determination region that is equal to or lower than the upper limit THmax of THmax1, THmax2, etc., and equal to or higher than the lower limit THmin of THmin1, THmin2, etc., obtained in the same manner as in the case shown in Figure 15 described later. Here, the median THC of the grinding burn determination threshold TH is set to 6.75, which is the same value as the half-width average value HWav, which is the average value of the half-width HW, as shown in Figure 5.

The upper limit THmax is a value for removing non-conforming products whose half-width HW is higher than that of normal products due to the presence of a white layer WL caused by grinding burns, and the lower limit THmin is a value for removing non-conforming products whose half-width HW is lower than that of normal products due to the presence of a black layer BL caused by grinding burns.

The upper limit THmax1 shown in Figures 5 to 8 is set to +0.5° (+7.40%) of the median THC, which is the average half-width HWav obtained by averaging the half-widths HW obtained from each angle Ψ, and the lower limit THmin1 is set to -0.5° (-7.40%) of the median THC, which is the average half-width HWav obtained by averaging the half-widths HW obtained from each angle Ψ. The absolute value of this ±0.5° is based on the difference between the maximum value of the abnormal product shown in Figure 5 and the median THC, which is the average half-width HWav.

The upper limit THmax2 shown in Figures 5 to 8 is set to +0.25° (+3.70%) of the median THC, which is the average half-width HWav obtained by averaging the half-width HW obtained from each angle Ψ, and the lower limit THmin2 is set to -0.25° (-3.70%) of the median THC, which is the average half-width HWav obtained by averaging the half-width HW obtained from each angle Ψ. The absolute value of ±0.25° is based on the difference between the value of the half-width HW when the black layer BL (depth of about 20 μm) is removed from the metal part B shown in Figures 7 and 9, and the median THC, which is the average half-width HWav.

Furthermore, the inventors conducted a test to clarify the relationship between the depth and the half-value width HW for a metal part 14 made of hardened steel (SCM440) that had been surface ground under the grinding conditions shown below, using the following measurement conditions.

(Surface grinding conditions)
Machine: Surface grinder Grinding method: Reciprocating grinding (down cut)
Grinding efficiency: 3.3mm ³ /mm・s
Dressing method: Traverse (down cut)
Dresser: SD 40 Q 75 M (rotary type)
Dresser peripheral speed: 28 m/s
Dresser cutting depth: 2.5 μm/pass
Dress lead: 0.1mm/r.o.w
Dresser peripheral speed ratio: 0.7
Grinding wheel (wheel specs): CBN 80 L 200 V
Wheel speed: 40 m/s
Left and right table speed: 4m/min
Work material: SCM440 (hardened steel)

(Conditions for measuring half-width)
Measurement site: surface layer 0μm
Diffraction X-ray measurement range: 140° to 170°
2θ: 156.40° (α-Fe(211))
Step: 0.20°
Angle Ψ: 0°, 18.4°, 26.6°, 33.2°,
39.2°, 45.0°
Tube: Cr
Characteristic X-rays: Kα rays

12 and 13 show the measured values of the half-width HW obtained for each of six different angles Ψ in a two-dimensional coordinate system with the horizontal axis showing the depth (μm) removed from the surface of the metal part 14 to the measurement surface by electrolytic polishing, and the vertical axis showing the half-width HW (°) obtained at each of six different angles Ψ for each depth of the surface layer. FIG. 12 shows the measured values of the half-width HW from a depth of 0 μm to 100 μm, showing the variation influenced by the six different angles Ψ. FIG. 13 shows the measured values of the half-width HW from a depth of 100 μm to 600 μm, showing smaller variation according to the six different angles Ψ compared to FIG. 12. In other words, it shows that the half-width HW differs under the influence of the angle Ψ up to a depth of 100 μm.

14 shows the average half-width HWav (°) obtained by averaging the half-width HW obtained for each of six different angles Ψ in a two-dimensional coordinate system with the horizontal axis indicating the depth (μm) of the surface layer removed from the surface of the metal part 14 by electrolytic polishing and the vertical axis indicating the average half-width HWav (°) obtained by averaging the half-width HW obtained for each of six different angles Ψ for each depth of the surface layer. As shown in FIG. 15, the median THC of the grinding burn determination threshold TH for a metal part 14 made of ground hardened steel SCM440 is set to 4.18, which is the same value as the average half-width HWav, which is the average value of the half-width HWav of the intensity profile of the diffracted X-rays for each angle Ψ, by irradiating the surface of the metal part 14 evaluated as having normal grinding burns with incident X-rays at six different angles Ψ, detecting the diffracted X-rays diffracted at the surface 16 of the metal part 14 in the same manner as the diffracted X-ray detection control unit 34, and calculating the average half-width HWav in the same manner as the average half-width calculation unit 36.

The upper limit THmax1 or THmax2 is a value for removing non-conforming products whose half-width HW is higher than that of normal products due to the presence of a white layer WL caused by grinding burn, and the lower limit THmin1 or THmin2 is a value for removing non-conforming products whose half-width HW is lower than that of normal products due to the presence of a black layer BL caused by grinding burn. For example, the upper limit THmax1 and the lower limit THmin1 are set to +0.5° (+11.96%) or -0.5° (-11.96%) of the median THC, which is the average half-width HWav obtained by averaging the half-widths HW obtained from each angle Ψ. The absolute value of ±0.5° corresponds to the difference between the average half-width HWav and the maximum value of the abnormal products. The upper limit THmax2 and the lower limit THmin2 are set to +0.25° (+5.98%) or -0.25° (-5.98%) of the median THC, which is the average half-width HWav obtained by averaging the half-widths HW obtained from each angle Ψ. The absolute value of ±0.25° corresponds to the difference between the average half-width HWav and the half-width HW when the softened layer, the black layer BL, is removed.

16, 17, and 18 show the relationship between the grinding allowance cross-sectional area (mm2/mm) per unit circumference length of the grinding wheel, the power consumption value (kW) of the grinding wheel shaft, the amount of wear of the grinding wheel (μm), and the surface roughness (maximum height roughness) Rz (μm) (JIS B 0601:2013) of the ground surface when a metal part 14 made of carburized and hardened steel ( ^SCr420 ) is ground under the same grinding conditions as the above-mentioned cylindrical grinding conditions. In Fig. 16, 17, and 18, ● indicates that the metal part 14 is ground under grinding conditions under which grinding burns are generated by using, for example, the above-mentioned grinding wheel 2, and ○ indicates that the metal part 14 is ground under grinding conditions under which grinding burns are generated by using, for example, the above-mentioned grinding wheel 1.

As shown in Figures 16, 17, and 18, the grinding burn products are superior to the products without grinding burn in terms of the power consumption (kW) of the grinding wheel spindle, the amount of grinding wheel wear (μm), and the surface roughness Rz (μm). In order to obtain such products without grinding burn, it is industrially important to use the grinding burn evaluation device 10 of this embodiment and the grinding burn evaluation method implemented by the grinding burn evaluation device 10 to develop grinding wheels, select grinding wheels, and select grinding conditions.

As described above, according to the grinding burn evaluation device 10 of this embodiment and the grinding burn evaluation method implemented by the grinding burn evaluation device 10, the X-ray irradiation control unit 32 or X-ray irradiation process S1 irradiates the surface 16 of the metal part 14 with the incident X-ray by changing the angle Ψ between the incident optical axis Lx1 of the X-ray incident on the metal part 14 and the normal z1 of the surface 16 of the metal part 14, the diffracted X-ray detection control unit 34 or diffracted X-ray detection process S2 detects the intensity profile of the diffracted X-ray diffracted on the surface 16 of the metal part 14 for each angle Ψ, the half-width average calculation unit 36 or half-width average calculation process S3 calculates the average value HWav of the half-width HW obtained from the intensity profile of the diffracted X-ray for each angle Ψ detected in the diffracted X-ray detection process S2, and the grinding burn evaluation unit 38 or grinding burn evaluation process S4 evaluates the grinding burn of the metal part 14 based on a comparison between the average value HWav of the half-width HW and a preset grinding burn judgment threshold TH. In this way, in the grinding burn evaluation unit 38 or grinding burn evaluation process S4, the grinding burn of the metal part 14 is evaluated based on a comparison between the average value HWav of the half-width HW of the diffracted X-ray intensity profile for each angle Ψ calculated by the half-width average calculation unit 36 or the half-width average calculation process S3 and the preset grinding burn judgment threshold value TH, so that grinding burns occurring on the surface of the ground metal part 14 can be accurately evaluated.

In addition, according to the grinding burn evaluation device 10 of this embodiment and the grinding burn evaluation method implemented by the grinding burn evaluation device 10, the grinding burn judgment threshold TH is set to have a normal judgment region that is equal to or less than the upper limit THmax and equal to or more than the lower limit THmin. Therefore, it is possible to judge defective products whose average value HWav of the half-width HW exceeds the upper limit THmax of the grinding burn judgment threshold TH and defective products whose average value HWav is below the lower limit THmin of the grinding burn judgment threshold TH, and therefore it is possible to accurately evaluate grinding burns that occur on the surface 16 of the ground metal part 14.

In addition, according to the grinding burn evaluation device 10 of this embodiment and the grinding burn evaluation method implemented by the grinding burn evaluation device 10, the angle Ψ can be changed to multiple angles within the range of 0° to 45°, so the diffracted X-ray detection device 12 for performing the X-ray irradiation process S1 and the diffracted X-ray detection process S2 can be made small.

In addition, according to the grinding burn evaluation device 10 of this embodiment and the grinding burn evaluation method implemented by the grinding burn evaluation device 10, the median THC of the grinding burn judgment threshold value TH is set by irradiating incident X-rays to the surface of a metal part 14 that has been ground and that has been evaluated as having normal grinding burns, while changing the angle Ψ between the incident optical axis Lx1 of the X-rays incident on the metal part 14 and the normal z1 of the surface 16 of the metal part 14, as in the X-ray irradiation process S1, detecting the diffracted X-rays diffracted at the surface 16 of the metal part 14, as in the diffracted X-ray detection process S2, and calculating the average value HWav of the half-width HWav of the intensity profile of the diffracted X-rays for each detected angle Ψ, as in the half-width average calculation process S3, and setting the median THC of the grinding burn judgment threshold value TH based on the average value HWav of the half-width for each angle Ψ. The grinding burn of the metal part 14 is evaluated based on a comparison between the grinding burn judgment threshold TH and the average half-width HWav of the intensity profile of the diffracted X-rays for each angle Ψ, so that the grinding burn occurring on the surface 16 of the ground metal part 14 can be accurately evaluated.

In addition, according to the grinding burn evaluation device 10 of this embodiment and the grinding burn evaluation method implemented by the grinding burn evaluation device 10, the median THC of the grinding burn judgment threshold value TH is set by electrolytically polishing the surface of the ground metal part 14 to a depth that is not affected by the grinding process, irradiating the electrolytically polished surface of the metal part 14 with incident X-rays by changing the angle Ψ between the incident optical axis Lx1 of the X-ray incident on the metal part 14 and the normal z1 of the surface 16 of the metal part 14, as in the X-ray irradiation process S1, detecting the diffracted X-rays diffracted at the surface 16 of the metal part 14, as in the diffracted X-ray detection process S2, calculating the average value HWav of the half-width HWav of the intensity profile of the diffracted X-rays for each detected angle Ψ, as in the half-width average calculation process S3, and setting the median THC based on the average value HWav of the half-width HW for each angle Ψ. The grinding burn of the metal part 14 is evaluated based on a comparison between the grinding burn judgment threshold TH and the average value HWav of the half-width HW of the intensity profile of the diffracted X-rays for each angle Ψ, so that the grinding burn occurring on the surface 16 of the ground metal part 14 can be accurately evaluated.

The above describes an embodiment of the present invention in detail with reference to the drawings, but the present invention can also be applied in other aspects.

For example, in the above embodiment, six types of angles Ψ were used to calculate the average half-width HWav, but multiple types of angles Ψ, such as five or less or seven or more, may be used.

In the above embodiment, the median THC of the grinding burn determination threshold TH was the same as the average value HWav of the half-width HW for each angle Ψ, but it may be a different value. For example, the median THC of the grinding burn determination threshold TH may be a value 0.1° larger than the average value HWav, and in that case, the normal determination region has a width of, for example, -0.4° to +0.6° based on the average value HWav.

In the above embodiment, the metal part 14 is made of carburized chrome steel SCr420 or hardened chrome molybdenum steel SCM440, but it may be made of other types of hardened steel. In short, it is sufficient for the part to be made of a material that can be hardened.

In addition, in the above-described embodiment, the metal part 14 is a part having a shape with an outer peripheral surface that is cylindrically ground, such as a cylindrical part, or a part having a shape with a surface that is flat ground, such as a plate-shaped part, but it may have other shapes.

Note that the above is merely one embodiment, and the present invention can be implemented in various forms with various modifications and improvements based on the knowledge of those skilled in the art.

10: Grinding burn evaluation device 12: Diffraction X-ray detection device 14: Metal part 16: Surface 32: X-ray irradiation control unit 34: Diffraction X-ray detection control unit 36: Half-value width average calculation unit 38: Grinding burn evaluation unit Lx1: Incident optical axis z1: Irradiation surface normal (normal to the surface of the metal part)
TH: grinding burn judgment threshold value THmax, THmax1, THmax2: upper limit value THmin, THmin1, THmin2: lower limit value HW: half width HWav: half width average value (average value of half width)

Claims (10)

A grinding burn evaluation device for evaluating grinding burns occurring on a surface of a ground metal part, comprising:
an X-ray irradiation control unit that changes an angle Ψ between an incident optical axis of the X-ray incident on the metal part and a normal line to a surface of the metal part, and irradiates the surface of the metal part with the incident X-ray;
a diffracted X-ray detection control unit that detects an intensity profile of diffracted X-rays diffracted at a surface of the metal part for each of the angles Ψ;
a half-width average calculation unit that calculates an average half-width obtained from the intensity profiles of the diffracted X-rays for each angle Ψ detected by the diffracted X-ray detection control unit;
and a grinding burn evaluation unit that evaluates the grinding burn of the metal part based on a comparison between the average half-value width and a grinding burn judgment threshold value that is preset to have a normal judgment region having a half-value width lower than that of a white layer, which is a re-hardened layer affected by the heat of grinding processing, and a half -value width higher than that of a black layer, which is a softened layer affected by the heat of grinding processing. 2. The grinding burn evaluation device for metal parts according to claim 1, characterized in that the grinding burn judgment threshold is set to have the normal judgment region equal to or lower than an upper limit value for removing reject products whose half -value width shows a higher value than the normal judgment region due to the presence of the white layer, and equal to or higher than a lower limit value for removing reject products whose half-value width shows a lower value than the normal judgment region due to the presence of the black layer. 3. The grinding burn evaluation device for a metal part according to claim 1, wherein the X-ray irradiation control unit irradiates the surface of the metal part with the incident X-rays by changing the angle Ψ to a plurality of angles within a range of 0° to 45°. 4. The device for evaluating grinding burn on a metal part according to claim 1, wherein the grinding burn judgment threshold is a value set based on an average value of the half-width calculated from an intensity profile of the diffracted X-rays in the same manner as the X-ray irradiation control unit, by irradiating incident X-rays at the angle Ψ to a surface of the metal part that has been evaluated as having normal grinding burn among the ground metal parts, and detecting diffracted X-rays on the surface of the metal part in the same manner as the diffracted X-ray detection control unit. 4. The device for evaluating grinding burn on a metal part according to claim 1, wherein the grinding burn determination threshold is a value set based on an average value of the half-width calculated from an intensity profile of the diffracted X-rays in a similar manner to the diffracted X-ray detection control unit, by electrolytically polishing the ground surface of the metal part to a depth that is not affected by grinding heat, by irradiating the surface of the metal part polished by electrolytic polishing with incident X-rays while changing the angle Ψ, by detecting diffracted X-rays on the surface of the metal part in a similar manner to the diffracted X-ray detection control unit, and by the average value of the half-width calculated from the intensity profile of the diffracted X-rays in a similar manner to the average half-width calculation unit. A method for evaluating grinding burns on a surface of a ground metal part, comprising:
an X-ray irradiation process in which an angle Ψ between an incident optical axis of the X-ray incident on the metal part and a normal line to the surface of the metal part is changed, and the incident X-ray is irradiated onto the surface of the metal part;
a diffracted X-ray detection step of detecting an intensity profile of diffracted X-rays diffracted at the surface of the metal component for each of the angles Ψ;
a half-width average calculation step of calculating an average half-width obtained from the intensity profiles of the diffracted X-rays for each angle Ψ detected in the diffracted X-ray detection step;
and a grinding burn evaluation step of evaluating the grinding burn of the metal part based on a comparison between the average half-value width and a grinding burn judgment threshold value that is preset so as to have a normal judgment region having a half-value width lower than that of a white layer, which is a re-hardened layer affected by the heat of grinding processing, and a half -value width higher than that of a black layer, which is a softened layer affected by the heat of grinding processing. The method for evaluating grinding burn on a metal part according to claim 6, characterized in that the grinding burn judgment threshold is set so as to have the normal judgment region that is equal to or lower than an upper limit value for removing reject products whose half -value width shows a higher value than the normal judgment region due to the presence of the white layer , and is equal to or higher than a lower limit value for removing reject products whose half-value width shows a lower value than the normal judgment region due to the presence of the black layer. The method for evaluating grinding burns on a metal part according to claim 6 or 7, characterized in that in the X-ray irradiation step, incident X-rays are irradiated onto the surface of the metal part by changing the angle Ψ to a plurality of angles within a range of 0° to 45°. 9. The method for evaluating grinding burns on a metal part according to claim 6, wherein the grinding burn judgment threshold is a value set based on an average value of the half-width calculated from an intensity profile of the diffracted X-rays in the same manner as in the diffracted X-ray detection step, by irradiating incident X-rays at the angle Ψ to a surface of the metal part, the surface of which is evaluated to have normal grinding burns among the ground metal parts, and detecting the diffracted X-rays on the surface of the metal part in the same manner as in the diffracted X-ray detection step. 9. The method for evaluating grinding burns on a metal part according to claim 6, wherein the grinding burn determination threshold is a value set based on an average value of the half-width calculated from an intensity profile of the diffracted X-rays in a similar manner to the X-ray diffracted detection step, the grinding burn evaluation method comprising the steps of: polishing the ground surface of the metal part by electrolytic polishing to a depth that is not affected by grinding heat; irradiating the surface of the metal part polished by electrolytic polishing with incident X-rays while changing the angle Ψ in a similar manner to the X-ray irradiation step; detecting diffracted X-rays diffracted on the surface of the metal part in a similar manner to the X-ray diffracted detection step; and calculating the average half-width value from the intensity profile of the diffracted X-rays in a similar manner to the FWHM calculation step.

Images