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An optimization scheme to improve measurement accuracy during the dimension measurement of hot workpiece
Measurement 59 (2015) 129–138
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An optimization scheme to improve measurement accuracy during the dimension measurement of hot workpiece Yucun Zhang, HaiBin Song ⇑, Bin Wei, Fuli Zhang Yanshan University, Qinhuangdao City 066004, China
a r t i c l e
i n f o
Article history: Received 6 December 2013 Received in revised form 21 August 2014 Accepted 16 September 2014 Available online 28 September 2014 Keywords: Hot workpiece Dimension measurement Fuzzy green laser Microscopic model of electromagnetic wave propagation Optimal width
a b s t r a c t In order to solve the problem that the fuzzy green laser stripe affects the measurement accuracy during the dimension measurement of hot workpiece, the underlying cause is analyzed and the green laser stripe width is optimized. The relationship between the elastic coefficient and Lame constants is derived by constructing a two-dimensional wave equation. By using this relationship, the microscopic model of electromagnetic wave propagation is presented to research the electromagnetic wave on hot workpiece. The electromagnetic wave is produced by the green laser stripe of the same power and the different width. Then on the premise of meeting the measurement accuracy, the optimal width of the green laser stripe is chosen to make the green laser stripe clearer on hot workpiece. And the accuracy of the green laser measurement system is improved. The microscopic model of electromagnetic wave propagation proposed in the paper is viable according to the experiment results. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction For the green laser measurement system, the clear green laser stripe is necessary for improving the measurement accuracy. In the past decades, a lot of researches have been conducted by domestic and foreign scholars. Images of hot workpiece are dealt with by Zhao through the image processing technology, such as selective filter, image enhancement and smoothing. Then the influence of part of ambient light on the image clarity of the green laser stripe is removed [1–3]. The influence of red thermal radiation on the image clarity is overcome by Liu using the combination of digital filter technology and physical filter technology [4,5]. The problem of the broken line of light-knife is solved by Wu based on the self adaptive threshold method for light-knife center acquiring and the 1D/2D data mending technique [6–8]. The edge values of the object are then
⇑ Corresponding author. E-mail address: [email protected] (H. Song). http://dx.doi.org/10.1016/j.measurement.2014.09.047 0263-2241/Ó 2014 Elsevier Ltd. All rights reserved.
obtained by Dr. Huang using the image processing procedures of sliding, stretching, edge enhancement and binary disposal [9–11]. The methods outlined above are proposed from the point of image. The following methods are from the point of the course of image acquisition. The spectrum selective method is proposed by Jia. The self-emitted radiation character and reflection character of hot workpiece are analyzed. The light intensity is supplemented by the illumination lamp to acquire images of hot workpiece. Using this method, radiation light captured by CCD is removed [12–14]. A free-electron model of metal is adopted by Ke. By using the model, the metal surface irradiated with green laser is simulated. The impact of angle of incidence on metal absorption rate is presented [15,16]. Bulygin and Kovalev think the quality of laser beams is measured by methods of Fourier optics. The quality of laser beams is improved by a Fourier transform [17–20]. The influence of angle of incidence, object color and measuring distance on the laser scanning process in computer are analyzed by Vukašinovic´ et al. [21]. Thereby it is necessary
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to find a more effective solution to the fuzzy green laser stripe for the green laser measurement system. The cause of the fuzzy green laser stripe is analyzed through using the electromagnetic wave theory. The microscopic model of electromagnetic wave propagation is constructed. By using the model, the electromagnetic wave produced by the laser stripe of the same power and the different width is simulated. According to the simulation result, the optimal laser stripe width is chosen to make the least energy loss of laser in electromagnetic wave and to meet the accuracy of the green laser measurement system. Thereby the green laser stripe of the optimal width on hot workpiece is clearer. And the accuracy of the green laser measurement system can be improved by using the presented method. In the end, the proposed model is capable of analyzing the cause of the fuzzy green laser stripe according to the experiment.
2. The mechanism why the green laser stripe on hot workpiece is fuzzy Schematic of the measurement system is shown in Fig. 1. As is shown in Fig. 1, the green laser device is installed on the linear guide rail. The green laser device is moved by the servo motor. Firstly, the green laser device is moved to the left edge of the hot workpiece. The location of the green laser device is recorded by the control computer. Then the green laser device is moved to the right edge of the hot workpiece. The location of the green laser device is recorded by the control computer. Finally, the measurement data of the length of the hot workpiece is received and displayed by the control computer. The image of the green laser stripe on hot workpiece captured by CCD is shown in Fig. 2. The green laser stripe is fuzzy in Fig. 2. To explore the mechanism why the green laser stripe on hot workpiece is fuzzy, the microscopic model of electromagnetic wave propagation is constructed. The analysis process is as follows. The coordinate system for the cross-section of hot workpiece is shown in Fig. 3. The laser source producing electromagnetic wave is the line source in parallel to the z axis. The spatial distribution of laser energy has identity on the z axis. The plane of xy is intercepted. The
Fig. 2. The green laser stripe on hot workpiece.
Cross-section
y
Green laser
z
x Fig. 3. The cross-section of hot workpiece.
microscopic model of electromagnetic wave propagation is established on the cross-section. Eq. (1) is the two-dimensional wave equation on the cross-section:
8 2 < q @ 2u ¼ C 11 @2 2u þ ðC 12 þ C 13 Þ @2 v þ C 33 @ 2 u2 þ F x @x@y @x @y @t : q @ 2 2v ¼ C 11 @ 2 v2 þ ðC 12 þ C 13 Þ @ 2 u þ C 33 @ 2 v2 þ F y @x@y @y @x @t
ð1Þ
where u is the displacement on the x axis, v is the displacement on the y axis, Fx is the force on the x axis, Fy is the force on the y axis, q is the density of material, and Cij is the stiffness matrix of each element. The relationship between Cij and Lame constants can be written as:
C 11 ¼ C 22 ¼ k þ 2l; C 12 ¼ k; C 33 ¼ l
ð2Þ
Fy in Eq. (1) is:
F y ¼ F 0 f ðrÞ
ð3Þ
where f(r) is the spatial distribution of power source. It is shown as follows:
Fig. 1. Schematic of the green laser measurement system.
2 r f ðrÞ ¼ exp 2 a0
ð4Þ
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where a0 is the width of the green laser stripe. And a0 is shown in Fig. 4. Through changing the width of the green laser stripe, the different spatial distribution of force source is produced. The plane of oxy is divided into a grid which is shown in Fig. 5. The width of the square is h, and h = 1 ns. Each node of the grid is modeled as particle. As is shown in Fig. 6, each particle is connected by an idealized tiny spring. In Fig. 6, k1 is the stretching elastic coefficient on the x axis, k3 is the stretching elastic coefficient on y axis, k2 is the synthetic elastic coefficient on the x and y axis, and ak2 is the twist elastic coefficient. To simplify the calculations, b = a/(2h2) is introduced. Through the force analysis of i and j particles, the kinetic equations of the particle are obtained:
For the isotropic material, k1, k2, k3, and b can be obtained according to Eqs. (2) and (7).
k1 ¼ k3 ¼ k þ l; k2 ¼
k þ 3l lk ; b¼ k þ 3l 4
ð8Þ
The initial conditions of the difference equations are shown as follows:
(
ujt¼0 ¼ 0; @u ¼ 0; @t t¼0
ð9Þ
The initial conditions of particles of the microscopic model of electromagnetic wave propagation can be written as:
uki;j ¼ 0; ðk ¼ 1; 0Þ
8 > k1 ukiþ1;j þuki1;j 2uki;j k2 ukiþ1;jþ1 þukiþ1;j1 þuki1;jþ1 þuki1;j1 4uki;j > > 2 @ u > > q 2 ¼ Fx þ þ 2 2 > > > @t h 2h > > > > k2 b ukiþ1;jþ1 þukiþ1;j1 þuki1;jþ1 þuki1;j1 4uki;j k2 v kiþ1;jþ1 v kiþ1;j1 v ki1;jþ1 þv ki1;j1 > > > þ þ > > 2h2 2h2 > > > > k k k k > k2 b v iþ1;jþ1 þv iþ1;j1 þv i1;jþ1 v i1;j1 > > < þ 2h2 > k k k > k3 v i;jþ1 þv i;j1 2v i;j k2 v kiþ1;jþ1 þv kiþ1;j1 þv ki1;jþ1 þv ki1;j1 4v ki;j > > @2 v > q @t2 ¼ F y þ þ > 2 2 > > h 2h > > > > k2 b v kiþ1;jþ1 þv kiþ1;j1 þv ki1;jþ1 þv ki1;j1 4v ki;j k2 ukiþ1;jþ1 ukiþ1;j1 uki1;jþ1 þuki1;j1 > > > > þ þ > 2h2 2h2 > > > > > k k k k k b u þu þu u > 2 > iþ1;jþ1 iþ1;j1 i1;jþ1 i1;j1 : þ 2h2
ð10Þ
ð5Þ
Eq. (6) is obtained through analyzing Eq. (5) by using the finite difference:
8 kþ1 > q ui;j þuk1 2uki;j k1 ukiþ1;j þuki1;j 2uki;j k2 ukiþ1;jþ1 þukiþ1;j1 þuki1;jþ1 þuki1;j1 4uki;j > i;j > > ¼ Fx þ þ > > ðDt Þ2 h2 2h2 > > > > > > k2 b ukiþ1;jþ1 þukiþ1;j1 þuki1;jþ1 þuki1;j1 4uki;j k2 v kiþ1;jþ1 v kiþ1;j1 v ki1;jþ1 þv ki1;j1 > > > þ þ > > 2h2 2h2 > > > > > k2 b v kiþ1;jþ1 þv kiþ1;j1 þv ki1;jþ1 v ki1;j1 > > < þ 2 2h > kþ1 k1 k > q v i;j þv i;j 2v i;j k3 v ki;jþ1 þv ki;j1 2v ki;j k2 v kiþ1;jþ1 þv kiþ1;j1 þv ki1;jþ1 þv ki1;j1 4v ki;j > > > ¼ Fy þ þ > > ðDt Þ2 h2 2h2 > > > > > > k2 b v kiþ1;jþ1 þv kiþ1;j1 þv ki1;jþ1 þv ki1;j1 4v ki;j k2 ukiþ1;jþ1 ukiþ1;j1 uki1;jþ1 þuki1;j1 > > > þ þ > > 2h2 2h2 > > > > k k k k > k b u þu þu u 2 > iþ1;jþ1 iþ1;j1 i1;jþ1 i1;j1 : þ 2h2
Eq. (1) is analyzed through the finite difference. Then the relationship between Cij and k1, k2, k3, b is obtained according to the comparison between Eq. (6) and finite difference scheme of Eq. (1). The relationship is shown as follows:
(
k1 ¼ C 11 C 33 ; k2 ¼ 3C334þC 12 C 33 C 12 k3 ¼ C 22 C 33 ; b ¼ 3C 33 þC 12
ð7Þ
ð6Þ
The boundary points on hot workpiece are shown in Fig. 7. Through the force analysis of the boundary points on hot workpiece, the difference equations of boundary points can be calculated: The difference equations of boundary point a are shown as follows:
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The difference equations of corner point B are shown as follows:
a0
Fig. 4. The schematic of the green laser stripe on hot workpiece.
8 kþ1 k1 k k k k k k k > > > q ui;j þui;j 2ui;j ¼ 2k1 ui1;j ui;j þ 2k2 ui1;j1 ui;j þ 2k2 b ui1;j1 4ui;j > > > ðDtÞ2 h2 h2 h2 > > > > > > 2k2 v ki1;j1 v ki;j 2k2 b v ki;j v ki1;j1 > > > < þ þ h2 h2 > > kþ1 k1 k k q v i;j þv i;j 2v i;j 2k3 v i;j1 v ki;j 2k2 v ki1;j1 v ki;j 2k2 b v ki1;j1 4v ki;j > > > > ¼ þ þ > ðDtÞ2 h2 h2 h2 > > > > > > > 2k2 uki1;j1 uki;j 2k2 b uki;j uki1;j1 > : þ þ h2 h2 ð13Þ
8 kþ1 > q ui;j þuk1 2uki;j k1 ukiþ1;j þuki1;j 2uki;j k2 ukiþ1;j1 þuki1;j1 2uki;j k2 b ukiþ1;j1 þuki1;j1 2uki;j > i;j > > ¼ þ þ > > ðDt Þ2 h2 h2 h2 > > > > > > > > k2 v kiþ1;j1 þv ki1;j1 k2 b v kiþ1;j1 v ki1;j1 > > > þ < þ 2 h h2
ð11Þ
> > kþ1 > q v i;j þv k1 2v ki;j 2k3 v kiþ1;j v ki;j k2 v kiþ1;j1 þv ki1;j1 2v ki;j k2 b v kiþ1;j1 þv ki1;j1 2v ki;j > i;j > > ¼ þ þ > > ðDt Þ2 h2 h2 h2 > > > > > > > > k2 ukiþ1;j1 þuki1;j1 k2 b ukiþ1;j1 uki1;j1 > : þ þ h2 h2
The difference equations of corner point A are shown as follows: 8 > q ukþ1 þuk1 2uki;j 2k1 ukiþ1;j uki;j 2k2 ukiþ1;j1 uki;j 2k2 b ukiþ1;j1 uki;j > i;j i;j > > ¼ þ þ > > ðDtÞ2 h2 h2 h2 > > > > > > 2k2 v ki;j v kiþ1;j1 2k2 b v kiþ1;j1 v ki;j > > > < þ þ h2 h2 > > kþ1 k1 k k q v i;j þv i;j 2v i;j 2k3 v i;j1 v ki;j 2k2 v kiþ1;j1 v ki;j 2k2 b v kiþ1;j1 v ki;j > > > > ¼ þ þ > ðDtÞ2 h2 h2 h2 > > > > > > > 2k2 uki;j ukiþ1;j1 2k2 b ukiþ1;j1 uki;j > : þ þ h2 h2
i-1,j+1
i,j+1
i+1,j+1 αk2
αk2 k3
k2
i-1,j
i,j
k1
i+1,j
k1
k2
k2
ð12Þ
k2
k3 y
h
αk2 h
x
Fig. 5. The grid of cross-section of hot workpiece.
i+1,j-1 αk2
Fig. 6. The microscopic model of electromagnetic wave propagation.
A 0
i,j-1
i-1,j-1
c
a
B d
Fig. 7. The schematic of the boundary points on hot workpiece.
Y. Zhang et al. / Measurement 59 (2015) 129–138
133
The difference equations of boundary point c are shown as follows:
8 kþ1 > q ui;j þuk1 2uki;j 2k1 ukiþ1;j uki;j k2 ukiþ1;jþ1 þukiþ1;j1 2uki;j k2 b ukiþ1;jþ1 þukiþ1;j1 2uki;j > i;j > > ¼ þ þ > > ðDt Þ2 h2 h2 h2 > > > > > k k k k > k2 v iþ1;jþ1 v iþ1;j1 k2 b v iþ1;jþ1 þv iþ1;j1 > > < þ þ h2 h2 > kþ1 > q v i;j þv k1 2v ki;j k3 v ki;jþ1 þv ki;j1 2v ki;j k2 v kiþ1;jþ1 þv kiþ1;j1 2v ki;j k2 b v kiþ1;jþ1 þv kiþ1;j1 2v ki;j > i;j > > ¼ þ þ > > ðDt Þ2 h2 h2 h2 > > > > > > k uk uk k b ukiþ1;jþ1 þukiþ1;j1 > : þ 2 iþ1;jþ1 iþ1;j1 þ 2 2 h h2
ð14Þ
The difference equations of boundary point d are shown as follows:
8 kþ1 > q ui;j þuk1 2uki;j 2k1 uki1;j uki;j k2 uki1;jþ1 þuki1;j1 2uki;j k2 b uki1;jþ1 þuki1;j1 2uki;j > i;j > > ¼ þ þ > > ðDt Þ2 h2 h2 h2 > > > > > > k2 v ki1;jþ1 v ki1;j1 k2 b v ki1;jþ1 þv ki1;j1 > > < þ þ h2 h2 > kþ1 k1 k k k > k3 v i;jþ1 þv i;j1 2v ki;j k2 v ki1;jþ1 þv ki1;j1 2v ki;j k2 b v ki1;jþ1 þv ki1;j1 2v ki;j > q v i;j þv i;j 2v i;j > > ¼ þ þ > > ðDt Þ2 h2 h2 h2 > > > > > > k2 uki1;jþ1 uki1;j1 k2 b uki1;jþ1 þuki1;j1 > : þ þ h2 h2
According to these difference equations, the factors affecting the electromagnetic wave produced by the laser stripe on hot workpiece are the power of the laser, the laser stripe width, and Lame constants of the workpiece. 3. The simulation In order to verify that electromagnetic wave produced by the laser stripe on hot workpiece is affected by the green laser stripe width and the experiment material, the cast iron and the 45#steel are chosen as the material of the simulation experiment. Some parameters of the cast iron and the 45#steel are shown in Table 1. According to equations of the center point, equations of the boundary points and the initial conditions, the waveform of the electromagnetic wave amplitude is simulated. The electromagnetic wave amplitude is produced by the laser stripe of the same power and the different widths. The laser stripe widths of 2.1 mm, 2.4 mm, 2.7 mm, and 3.0 mm are chosen as the train set. The model is obtained Table 1 The parameters of the cast iron and the 45#steel. Materials
The cast iron The steel
Young’s modulus (Pa)
Poisson’s ratio
Lame constants
l (Pa)
k (Pa)
10
10
10.5 10
0.28
8.6 10
19.5 1010
0.28
17.0 1010
4.4 1010 8.3 1010
ð15Þ
through analyzing the train set. The waveforms of the electromagnetic wave amplitudes produced by the laser stripes of the different widths on the cast iron workpiece are shown in Fig. 8. The waveforms are received at a distance of 0.5 mm from the laser stripe center. The waveform of the electromagnetic wave amplitude produced by the laser stripe width of 2.1 mm is shown in Fig. 8(a). The waveform of the electromagnetic wave amplitude produced by the laser stripe width of 2.4 mm is shown in Fig. 8(b). The waveform of the electromagnetic wave amplitude produced by the laser stripe width of 2.7 mm is shown in Fig. 8(c). The waveform of the electromagnetic wave amplitude produced by the laser stripe width of 3.0 mm is shown in Fig. 8(d). The energy loss of laser in electromagnetic wave on hot workpiece is proportional to the integral of the curve in the figure. Because the waveforms are received in the same position, these waveforms are at the same arrival time. So the energy loss of laser in electromagnetic wave on hot workpiece is reflected by the electromagnetic wave amplitude. In the application of the cast iron material, the electromagnetic wave amplitudes produced by the laser stripes of the different widths at 150 ns are shown in Fig. 9. As is shown in Fig. 9, the model is that as the width of the green laser stripe becomes large, the electromagnetic wave amplitude will increase with the decrease at first. The laser stripe widths of 2.2 mm, 2.5 mm, and 2.8 mm are chosen as the test set. The model is tested by the test
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Y. Zhang et al. / Measurement 59 (2015) 129–138
(a)
Nonmalized Displacement
Nonmalized Displacement
1.5 1 0.5 0 -0.5 -1 -1.5 -2
0
100
200
300
1.2 1.1 1 0.9 0.8 0.7 0.6 0.5
400
T/ns
Nonmalized Displacement
1.5
Fig. 9. The graph illustrating the conclusion in the application of the cast iron material.
1 0.5 0 -0.5 -1 -1.5 -2
0
100
200
300
400
T/ns
Nonmalized Displacement
(c)
1.5
1.1 1 0.9 0.8 0.7 0.6
2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9
3
Width/mm
0.5 Fig. 10. The graph testing the model.
0 -0.5 -1 -1.5 100
200
300
400
300
400
T/ns Nonmalized Displacement
1.2
0.5
1
-2 0
(d)
3
Width/mm
Nonmalized Displacement
(b)
2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9
1.5 1 0.5 0 -0.5 -1 -1.5 -2 0
100
200
T/ns Fig. 8. The cast iron material simulation waveform.
set. The electromagnetic wave amplitudes produced by the laser stripes of the different widths at 150 ns are tested. The graph testing the model is shown in Fig. 10.
As is shown in Fig. 10, the truthfulness of the model is confirmed. The laser stripe widths of 2.1 mm, 2.4 mm, 2.7 mm, and 3.0 mm are chosen as the train set. The model is obtained through analyzing the train set. The waveforms of the electromagnetic wave amplitudes produced by the laser stripes of the different widths on the 45#steel workpiece are shown in Fig. 11. The waveforms are received at a distance of 0.5 mm from the laser stripe center. The waveform of the electromagnetic wave amplitude produced by the laser stripe width of 2.1 mm is shown in Fig. 11(a). The waveform of the electromagnetic wave amplitude produced by the laser stripe width of 2.4 mm is shown in Fig. 11(b). The waveform of the electromagnetic wave amplitude produced by the laser stripe width of 2.7 mm is shown in Fig. 11(c). The waveform of the electromagnetic wave amplitude produced by the laser stripe width of 3.0 mm is shown in Fig. 11(d). In the application of the 45#steel material, the electromagnetic wave amplitudes produced by the laser stripes of the different widths at 150 ns are shown in Fig. 12. As is shown in Fig. 12, the model is that as the width of the green laser stripe becomes large, the electromagnetic wave amplitude will increase with the decrease at first. The laser stripe widths of 2.2 mm, 2.5 mm, and 2.8 mm are chosen as the test set. The model is tested by the test
135
Y. Zhang et al. / Measurement 59 (2015) 129–138
(a)
Nonmalized Displacement
Nonmalized Displacement
1.5
1 0.5
0 -0.5 -1
1 0.9 0.8 0.7 0.6 0.5
-1.5 -2 0
1.1
0.4 100
200
300
400
2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9
T/ns
(b)
Fig. 12. The graph illustrating the conclusion in the application of the 45#steel material.
Nonmalized Displacement
1.5 1
0 -0.5 -1 -1.5 0
100
200
300
400
T/ns
Nonmalized Displacement
0.5
-2
1.1 1 0.9 0.8 0.7 0.6 0.5 0.4
(c) Nonmalized Displacement
3
Width/mm
1.5
2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9
3
Width/mm
1 Fig. 13. The graph testing the model.
0.5 0 -0.5 -1 -1.5 -2
0
100
200
300
400
T/ns
Nonmalized Displacement
(d)
1.5 1 0.5 0 -0.5 -1 -1.5 -2 0
100
200
300
400
T/ns Fig. 11. 45#steel material simulation waveform.
set. The electromagnetic wave amplitudes produced by the laser stripes of the different widths at 150 ns are tested.
The graph testing the model is shown in Fig. 13. As is shown in Fig. 13, the truthfulness of the model is confirmed. The energy loss of the laser in electromagnetic wave on hot workpiece is reflected by the electromagnetic wave amplitude. Through the comparison between Figs. 8(a)– (d) and 11(a)–(d), a qualitative conclusion can be got: the cast iron or the 45#steel is chosen as the material. As the width of the green laser stripe becomes large, the electromagnetic wave amplitude will increase with the decrease at first. Then with green laser stripe width becoming large, the energy loss of laser in electromagnetic wave on hot workpiece decreases and then increases. In the application of the cast iron material, the width of the green laser stripe should be chosen between 2.4 mm and 3.0 mm to ensure the green laser stripe more clearly on hot workpiece. And in the application of the 45#steel material, the width of green laser stripe should be chosen between 2.1 mm and 2.7 mm to ensure the green laser stripe more clearly on the 45#steel workpiece. To illustrate the correctness of the conclusion, the other position is chosen to receive the vertical displacement of 45#steel body wave. It is shown in Fig. 14. As is shown in Fig. 14(a)–(d), the green laser stripe width is respectively 2.1 mm, 2.4 mm, 2.7 mm, and 3.0 mm. By comparing the amplitudes of waveforms, the same conclusion can be gotten: with green laser stripe width becoming
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Y. Zhang et al. / Measurement 59 (2015) 129–138
Nonmalized Displacement
(a)
Nonmalized Displacement
(b)
Nonmalized Displacement
(c)
0.4 0.2 0 0
200
400
0
200
400
0
200
400
T/ns
600
800
1000
600
800
1000
600
800
1000
600
800
1000
0.4 0.2 0
T/ns
0.4 0.2 0
T/ns Nonmalized Displacement
(d) 0.4 0.2 0 0
200
400
T/ns Fig. 14. The waveform of other location.
large, the energy loss of laser in electromagnetic wave on hot workpiece decreases and then increases. 4. Testing experiment In order to verify the validity of the theory and the accuracy of the green laser measurement system proposed in the paper, an experiment platform is designed by our research group. Equipments of the experiment platform are as follows: (1) The heating furnace used is model C19-P1600B2. The heating temperature range of the heating furnace is 500–1550 °C. (2) Industrial CCD camera used is of model MVVE078SM/SC. The maximum resolution is 1024 768, and the size of a pixel on the CCD array is 4.65 lm 4.65 lm. The focal length of the optical lens is 12 mm, and its back focal length is 9.7 mm. The relative aperture (F) is 1:1.4. The field of view is 2/300 . Working distance is 4000 mm.
Fig. 15. The image received by the control computer.
Y. Zhang et al. / Measurement 59 (2015) 129–138
(3) The line laser projector is of MGL-III type. The wavelength is 532 nm. The aperture angle is 30°. Servo motors and drives are of MR-J2S-10A/B type. The maximum error of the drive is ±10 turns. (4) High-precision guide used is BGXS45BE-type high precision linear guide rail, and the walking error is less than 40 lm. The experimental environment is no wind with dust. The ambient light is lamplight. The indoor temperature is 300 K and the humidity is 60%. The procedure of this experiment is given as follows: (1) The 45#steel workpiece is heated to 900 °C by the heating furnace. In order to overcome temperature variations and surface parameters variations, as far as possible, the measurement time should be cut. (2) The hot workpiece is placed at a distance of 4 m from the laser transmitter. (3) The measurement system is started, and the green laser stripe width is respectively set to 2.1 mm, Table 2 The experimental data. The actual length/mm
The measure length/mm
The error/mm
The width 2.1 mm
1 2 3 4 5 6 7 8 9 10
360.0 360.0 360.0 360.0 360.0 360.0 360.0 360.0 360.0 360.0
362.5 358.0 361.3 357.9 357.1 362.4 362.5 357.8 358.4 361.9
2.5 2.0 1.3 2.1 2.9 2.4 2.5 2.2 1.6 1.9
The width 2.4 mm
1 2 3 4 5 6 7 8 9 10
360.0 360.0 360.0 360.0 360.0 360.0 360.0 360.0 360.0 360.0
361.8 359.4 361.1 360.3 358.8 360.5 360.7 361.0 358.7 358.9
1.8 0.6 1.1 0.3 1.2 0.5 0.7 1.0 1.3 1.1
The width 2.7 mm
1 2 3 4 5 6 7 8 9 10
360.0 360.0 360.0 360.0 360.0 360.0 360.0 360.0 360.0 360.0
361.5 358.3 361.3 357.9 360.8 362.6 359.0 358.6 361.3 358.2
1.5 1.7 1.3 2.1 0.8 2.6 1.0 1.4 1.3 1.8
The width 3.0 mm
1 2 3 4 5 6 7 8 9 10
360.0 360.0 360.0 360.0 360.0 360.0 360.0 360.0 360.0 360.0
358.0 362.5 358.1 357.4 362.7 363.3 362.5 357.4 362.9 357.6
2.0 2.5 1.9 2.6 2.7 3.3 2.5 2.6 2.9 2.4
137
2.4 mm, 2.7 mm, and 3.0 mm. The green laser stripe is moved to the left edge of the hot workpiece by the control system. The image of the hot workpiece received by the control computer is shown in Fig. 15. Then the green laser stripe is moved to the right edge of the hot workpiece. The measurement data is received and displayed. The measurement process is repeated ten times and the measurement data are shown in Table 2. The image containing the laser stripe width of 2.1 mm is shown in Fig. 15(a). The image containing the laser stripe width of 2.4 mm is shown in Fig. 15(b). The image containing the laser stripe width of 2.7 mm is shown in Fig. 15(c). The image containing the laser stripe width of 3.0 mm is shown in Fig. 15(d). The laser stripe in Fig. 15(b) is certainly clearer than (a), (c) and (d) in Fig. 15. It verified that with choosing 45#steel workpiece, the green laser stripe width between 2.1 mm and 2.7 mm will make the laser stripe clearer on hot workpiece. In order to ensure that the result of proposed method is superior to conventional methods, the result of conventional methods is compared with proposed method in this paper. The images are shown in Figs. 16 and 17. The laser stripe in Fig. 17 is certainly clearer than that in Fig. 16. So the proposed method in this paper is a more
Fig. 16. The image under the image processing technology.
Fig. 17. The image based on proposed method in this paper.
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effective solution to the fuzzy green laser stripe for the green laser measurement system. The accuracy is shown as follows: d = (Dmax)/(Amax) 100%, where d is the accuracy, Dmax is the maximum error of the measurement, and Amax is the instrument range. The green laser device is installed on the linear guide rail. The instrument range is determined by the length of the linear guide rail. The length of the linear guide rail used is 500.0 cm. According to Table 2, when the green laser stripe width is set to 2.1 mm, the accuracy is: d = 2.9/ 500.0 100% = 0.58%. When the green laser stripe width is set to 2.4 mm, the accuracy is: d = 1.8/500.0 100% = 0.36%. When the green laser stripe width is set to 2.7 mm, the accuracy is: d = 2.6/500.0 100% = 0.52%. When the green laser stripe width is set to 3.0 mm, the accuracy is: d = 3.3/500.0 100% = 0.66%. When the green laser stripe width is set to 2.4 mm, the accuracy is improved to 0.36%. It can meet the requirement of accuracy. 5. Conclusions The cause of the fuzzy green laser stripe is found through the microscopic model of electromagnetic wave propagation in this paper. The electromagnetic wave is the major energy loss of laser on hot workpiece. With green laser stripe width becoming large, the energy loss of laser in electromagnetic wave on hot workpiece decreases and then increases. Then the optimal width of the laser stripe is chosen to make the green laser stripe clearer on hot workpiece. It suppresses the energy loss of laser in electromagnetic wave on hot workpiece effectively. The validity of the theory is verified through the simulation. This theory is applied to the measurement system, and the accuracy of measurement is improved to 0.36%. Acknowledgements This study is supported by the Natural Science Foundation of Hebei Province, China (Grant No.: E2014203070). References [1] L. Li, Z.P. Zhao, Research on dimensional measurement of heavy cylindrical forging at high temperature based on binocular stereo vision, Transducer Microsyst. Technol. 29 (4) (2010) 49–51.
[2] H. Zhang, L.Y. Zhang, Z.P. Zhao, Field calibration of binocular stereo system based on planar template and free snapping, Acta Aeronautica et Astronautica Sinica 28 (3) (2007) 695–701. [3] Y. Xue, Z.P. Zhao, Measurement system of tracking contour along edge based on visible technology, Mach. Des. Manuf. 7 (2009) 209– 210. [4] G.H. Liu, X.Y. Liu, Online measurement techniques of large forgings in thermal state, Opto-Electron. Eng. 37 (9) (2009) 91–97. [5] L.J. Wu, G.H. Liu, Shiny surfaces measurement based on structured light system, Tool Eng. 45 (3) (2011) 88–92. [6] J.B. Wu, Z. Cui, H. Zhao, An adaptive threshold method for light-knife center acquisition, Semicond. Optoelectron. 22 (1) (2001) 62–64. [7] J.B. Wu, H. Zhao, Y.S. Tan, A novel solving method to the broken line problem of light-knife, Opt. Tech. 27 (2) (2001) 189–191. [8] J.B. Wu, M.M. Li, H. Zhao, 3D data acquiring system fusing active projection and stereo vision, Proc. SPIE Int. Soc. Opt. Eng. WuHan (2001) 256–261. [9] D.S. Huang, Y.W. Lin, A prototype system of three-dimensional noncontact measurement, Int. J. Adv. Manuf. Technol. 11 (1996) 336–342. [10] D.S. Huang, Y.W. Lin, A three-dimensional non-contact measurement system, Int. J. Adv. Manuf. Technol. 13 (1997) 419–425. [11] X.B. Fu, Y.C. Zhang, An optical non-contact measurement method for hot-state size of cylindrical shell forging, Measurement 45 (6) (2012) 1343–1349. [12] Z.Y. Jia, B.G. Wang, W. Liu, An improved image acquiring method for machine vision measurement of hot formed parts, J. Mater. Process. Technol. 210 (2010) (2009) 267–271. [13] W. Liu, S.J. Liu, Z.Y. Jia, Study on online measurement system of dimension for hot heavy forging, Piezoelectrics Acoustooptics 33 (5) (2011) 831–836. [14] W. Liu, Z.Y. Jia, An improved online dimensional measurement method of large hot cylindrical forging, Measurement 45 (8) (2012) 2041–2051. [15] W.N. Ke, D.Q. Zhu, G.B. Cai, Simulation and analysis of spectral emissivity of metals, Acta Aeronautica et Astronautica Sinica 31 (11) (2010) 2139–2145. [16] V. Hajipour, A. Kazemi, M. Mousavi, A fuzzy expert system to increase accuracy and precision in measurement system analysis, Measurement 46 (8) (2013) 2770–2780. [17] F.V. Bulygin, A.A. Kovalev, K.D. Maramzin, Measurement of the quality of laser beams by methods of Fourier optics, Meas. Tech. 49 (10) (2006) 1015–1020. [18] M. Jouaneh, R.L. Lemaster, D.A. Dornfeld, Measuring workpiece dimensions using a non-contact laser detector system, Int. J. Adv. Manuf. Technol. 2 (1) (1987) 59–74. [19] C. Ferri, J. Faraway, E. Brousseau, Calibration of a white light interferometer for the measurement of micro-scale dimensions, Int. J. Adv. Manuf. Technol. 47 (1–4) (2010) 125–135. [20] Y. Hojjat, M.R. Karafi, M. Ghanbari, M. Lotfi, Development of an inductive encoder for simultaneous measurement of two dimensional displacement, Int. J. Adv. Manuf. Technol. 5–8 (2011) 681–688. [21] N. Vukašinovic´, D. Bracˇun, J. Mozˇina, J. Duhovnik, The influence of incident angle, object colour and distance on CNC laser scanning, Int. J. Adv. Manuf. Technol. 50 (2010) 265–274.
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Measurement journal homepage: www.elsevier.com/locate/measurement
An optimization scheme to improve measurement accuracy during the dimension measurement of hot workpiece Yucun Zhang, HaiBin Song ⇑, Bin Wei, Fuli Zhang Yanshan University, Qinhuangdao City 066004, China
a r t i c l e
i n f o
Article history: Received 6 December 2013 Received in revised form 21 August 2014 Accepted 16 September 2014 Available online 28 September 2014 Keywords: Hot workpiece Dimension measurement Fuzzy green laser Microscopic model of electromagnetic wave propagation Optimal width
a b s t r a c t In order to solve the problem that the fuzzy green laser stripe affects the measurement accuracy during the dimension measurement of hot workpiece, the underlying cause is analyzed and the green laser stripe width is optimized. The relationship between the elastic coefficient and Lame constants is derived by constructing a two-dimensional wave equation. By using this relationship, the microscopic model of electromagnetic wave propagation is presented to research the electromagnetic wave on hot workpiece. The electromagnetic wave is produced by the green laser stripe of the same power and the different width. Then on the premise of meeting the measurement accuracy, the optimal width of the green laser stripe is chosen to make the green laser stripe clearer on hot workpiece. And the accuracy of the green laser measurement system is improved. The microscopic model of electromagnetic wave propagation proposed in the paper is viable according to the experiment results. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction For the green laser measurement system, the clear green laser stripe is necessary for improving the measurement accuracy. In the past decades, a lot of researches have been conducted by domestic and foreign scholars. Images of hot workpiece are dealt with by Zhao through the image processing technology, such as selective filter, image enhancement and smoothing. Then the influence of part of ambient light on the image clarity of the green laser stripe is removed [1–3]. The influence of red thermal radiation on the image clarity is overcome by Liu using the combination of digital filter technology and physical filter technology [4,5]. The problem of the broken line of light-knife is solved by Wu based on the self adaptive threshold method for light-knife center acquiring and the 1D/2D data mending technique [6–8]. The edge values of the object are then
⇑ Corresponding author. E-mail address: [email protected] (H. Song). http://dx.doi.org/10.1016/j.measurement.2014.09.047 0263-2241/Ó 2014 Elsevier Ltd. All rights reserved.
obtained by Dr. Huang using the image processing procedures of sliding, stretching, edge enhancement and binary disposal [9–11]. The methods outlined above are proposed from the point of image. The following methods are from the point of the course of image acquisition. The spectrum selective method is proposed by Jia. The self-emitted radiation character and reflection character of hot workpiece are analyzed. The light intensity is supplemented by the illumination lamp to acquire images of hot workpiece. Using this method, radiation light captured by CCD is removed [12–14]. A free-electron model of metal is adopted by Ke. By using the model, the metal surface irradiated with green laser is simulated. The impact of angle of incidence on metal absorption rate is presented [15,16]. Bulygin and Kovalev think the quality of laser beams is measured by methods of Fourier optics. The quality of laser beams is improved by a Fourier transform [17–20]. The influence of angle of incidence, object color and measuring distance on the laser scanning process in computer are analyzed by Vukašinovic´ et al. [21]. Thereby it is necessary
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to find a more effective solution to the fuzzy green laser stripe for the green laser measurement system. The cause of the fuzzy green laser stripe is analyzed through using the electromagnetic wave theory. The microscopic model of electromagnetic wave propagation is constructed. By using the model, the electromagnetic wave produced by the laser stripe of the same power and the different width is simulated. According to the simulation result, the optimal laser stripe width is chosen to make the least energy loss of laser in electromagnetic wave and to meet the accuracy of the green laser measurement system. Thereby the green laser stripe of the optimal width on hot workpiece is clearer. And the accuracy of the green laser measurement system can be improved by using the presented method. In the end, the proposed model is capable of analyzing the cause of the fuzzy green laser stripe according to the experiment.
2. The mechanism why the green laser stripe on hot workpiece is fuzzy Schematic of the measurement system is shown in Fig. 1. As is shown in Fig. 1, the green laser device is installed on the linear guide rail. The green laser device is moved by the servo motor. Firstly, the green laser device is moved to the left edge of the hot workpiece. The location of the green laser device is recorded by the control computer. Then the green laser device is moved to the right edge of the hot workpiece. The location of the green laser device is recorded by the control computer. Finally, the measurement data of the length of the hot workpiece is received and displayed by the control computer. The image of the green laser stripe on hot workpiece captured by CCD is shown in Fig. 2. The green laser stripe is fuzzy in Fig. 2. To explore the mechanism why the green laser stripe on hot workpiece is fuzzy, the microscopic model of electromagnetic wave propagation is constructed. The analysis process is as follows. The coordinate system for the cross-section of hot workpiece is shown in Fig. 3. The laser source producing electromagnetic wave is the line source in parallel to the z axis. The spatial distribution of laser energy has identity on the z axis. The plane of xy is intercepted. The
Fig. 2. The green laser stripe on hot workpiece.
Cross-section
y
Green laser
z
x Fig. 3. The cross-section of hot workpiece.
microscopic model of electromagnetic wave propagation is established on the cross-section. Eq. (1) is the two-dimensional wave equation on the cross-section:
8 2 < q @ 2u ¼ C 11 @2 2u þ ðC 12 þ C 13 Þ @2 v þ C 33 @ 2 u2 þ F x @x@y @x @y @t : q @ 2 2v ¼ C 11 @ 2 v2 þ ðC 12 þ C 13 Þ @ 2 u þ C 33 @ 2 v2 þ F y @x@y @y @x @t
ð1Þ
where u is the displacement on the x axis, v is the displacement on the y axis, Fx is the force on the x axis, Fy is the force on the y axis, q is the density of material, and Cij is the stiffness matrix of each element. The relationship between Cij and Lame constants can be written as:
C 11 ¼ C 22 ¼ k þ 2l; C 12 ¼ k; C 33 ¼ l
ð2Þ
Fy in Eq. (1) is:
F y ¼ F 0 f ðrÞ
ð3Þ
where f(r) is the spatial distribution of power source. It is shown as follows:
Fig. 1. Schematic of the green laser measurement system.
2 r f ðrÞ ¼ exp 2 a0
ð4Þ
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where a0 is the width of the green laser stripe. And a0 is shown in Fig. 4. Through changing the width of the green laser stripe, the different spatial distribution of force source is produced. The plane of oxy is divided into a grid which is shown in Fig. 5. The width of the square is h, and h = 1 ns. Each node of the grid is modeled as particle. As is shown in Fig. 6, each particle is connected by an idealized tiny spring. In Fig. 6, k1 is the stretching elastic coefficient on the x axis, k3 is the stretching elastic coefficient on y axis, k2 is the synthetic elastic coefficient on the x and y axis, and ak2 is the twist elastic coefficient. To simplify the calculations, b = a/(2h2) is introduced. Through the force analysis of i and j particles, the kinetic equations of the particle are obtained:
For the isotropic material, k1, k2, k3, and b can be obtained according to Eqs. (2) and (7).
k1 ¼ k3 ¼ k þ l; k2 ¼
k þ 3l lk ; b¼ k þ 3l 4
ð8Þ
The initial conditions of the difference equations are shown as follows:
(
ujt¼0 ¼ 0; @u ¼ 0; @t t¼0
ð9Þ
The initial conditions of particles of the microscopic model of electromagnetic wave propagation can be written as:
uki;j ¼ 0; ðk ¼ 1; 0Þ
8 > k1 ukiþ1;j þuki1;j 2uki;j k2 ukiþ1;jþ1 þukiþ1;j1 þuki1;jþ1 þuki1;j1 4uki;j > > 2 @ u > > q 2 ¼ Fx þ þ 2 2 > > > @t h 2h > > > > k2 b ukiþ1;jþ1 þukiþ1;j1 þuki1;jþ1 þuki1;j1 4uki;j k2 v kiþ1;jþ1 v kiþ1;j1 v ki1;jþ1 þv ki1;j1 > > > þ þ > > 2h2 2h2 > > > > k k k k > k2 b v iþ1;jþ1 þv iþ1;j1 þv i1;jþ1 v i1;j1 > > < þ 2h2 > k k k > k3 v i;jþ1 þv i;j1 2v i;j k2 v kiþ1;jþ1 þv kiþ1;j1 þv ki1;jþ1 þv ki1;j1 4v ki;j > > @2 v > q @t2 ¼ F y þ þ > 2 2 > > h 2h > > > > k2 b v kiþ1;jþ1 þv kiþ1;j1 þv ki1;jþ1 þv ki1;j1 4v ki;j k2 ukiþ1;jþ1 ukiþ1;j1 uki1;jþ1 þuki1;j1 > > > > þ þ > 2h2 2h2 > > > > > k k k k k b u þu þu u > 2 > iþ1;jþ1 iþ1;j1 i1;jþ1 i1;j1 : þ 2h2
ð10Þ
ð5Þ
Eq. (6) is obtained through analyzing Eq. (5) by using the finite difference:
8 kþ1 > q ui;j þuk1 2uki;j k1 ukiþ1;j þuki1;j 2uki;j k2 ukiþ1;jþ1 þukiþ1;j1 þuki1;jþ1 þuki1;j1 4uki;j > i;j > > ¼ Fx þ þ > > ðDt Þ2 h2 2h2 > > > > > > k2 b ukiþ1;jþ1 þukiþ1;j1 þuki1;jþ1 þuki1;j1 4uki;j k2 v kiþ1;jþ1 v kiþ1;j1 v ki1;jþ1 þv ki1;j1 > > > þ þ > > 2h2 2h2 > > > > > k2 b v kiþ1;jþ1 þv kiþ1;j1 þv ki1;jþ1 v ki1;j1 > > < þ 2 2h > kþ1 k1 k > q v i;j þv i;j 2v i;j k3 v ki;jþ1 þv ki;j1 2v ki;j k2 v kiþ1;jþ1 þv kiþ1;j1 þv ki1;jþ1 þv ki1;j1 4v ki;j > > > ¼ Fy þ þ > > ðDt Þ2 h2 2h2 > > > > > > k2 b v kiþ1;jþ1 þv kiþ1;j1 þv ki1;jþ1 þv ki1;j1 4v ki;j k2 ukiþ1;jþ1 ukiþ1;j1 uki1;jþ1 þuki1;j1 > > > þ þ > > 2h2 2h2 > > > > k k k k > k b u þu þu u 2 > iþ1;jþ1 iþ1;j1 i1;jþ1 i1;j1 : þ 2h2
Eq. (1) is analyzed through the finite difference. Then the relationship between Cij and k1, k2, k3, b is obtained according to the comparison between Eq. (6) and finite difference scheme of Eq. (1). The relationship is shown as follows:
(
k1 ¼ C 11 C 33 ; k2 ¼ 3C334þC 12 C 33 C 12 k3 ¼ C 22 C 33 ; b ¼ 3C 33 þC 12
ð7Þ
ð6Þ
The boundary points on hot workpiece are shown in Fig. 7. Through the force analysis of the boundary points on hot workpiece, the difference equations of boundary points can be calculated: The difference equations of boundary point a are shown as follows:
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The difference equations of corner point B are shown as follows:
a0
Fig. 4. The schematic of the green laser stripe on hot workpiece.
8 kþ1 k1 k k k k k k k > > > q ui;j þui;j 2ui;j ¼ 2k1 ui1;j ui;j þ 2k2 ui1;j1 ui;j þ 2k2 b ui1;j1 4ui;j > > > ðDtÞ2 h2 h2 h2 > > > > > > 2k2 v ki1;j1 v ki;j 2k2 b v ki;j v ki1;j1 > > > < þ þ h2 h2 > > kþ1 k1 k k q v i;j þv i;j 2v i;j 2k3 v i;j1 v ki;j 2k2 v ki1;j1 v ki;j 2k2 b v ki1;j1 4v ki;j > > > > ¼ þ þ > ðDtÞ2 h2 h2 h2 > > > > > > > 2k2 uki1;j1 uki;j 2k2 b uki;j uki1;j1 > : þ þ h2 h2 ð13Þ
8 kþ1 > q ui;j þuk1 2uki;j k1 ukiþ1;j þuki1;j 2uki;j k2 ukiþ1;j1 þuki1;j1 2uki;j k2 b ukiþ1;j1 þuki1;j1 2uki;j > i;j > > ¼ þ þ > > ðDt Þ2 h2 h2 h2 > > > > > > > > k2 v kiþ1;j1 þv ki1;j1 k2 b v kiþ1;j1 v ki1;j1 > > > þ < þ 2 h h2
ð11Þ
> > kþ1 > q v i;j þv k1 2v ki;j 2k3 v kiþ1;j v ki;j k2 v kiþ1;j1 þv ki1;j1 2v ki;j k2 b v kiþ1;j1 þv ki1;j1 2v ki;j > i;j > > ¼ þ þ > > ðDt Þ2 h2 h2 h2 > > > > > > > > k2 ukiþ1;j1 þuki1;j1 k2 b ukiþ1;j1 uki1;j1 > : þ þ h2 h2
The difference equations of corner point A are shown as follows: 8 > q ukþ1 þuk1 2uki;j 2k1 ukiþ1;j uki;j 2k2 ukiþ1;j1 uki;j 2k2 b ukiþ1;j1 uki;j > i;j i;j > > ¼ þ þ > > ðDtÞ2 h2 h2 h2 > > > > > > 2k2 v ki;j v kiþ1;j1 2k2 b v kiþ1;j1 v ki;j > > > < þ þ h2 h2 > > kþ1 k1 k k q v i;j þv i;j 2v i;j 2k3 v i;j1 v ki;j 2k2 v kiþ1;j1 v ki;j 2k2 b v kiþ1;j1 v ki;j > > > > ¼ þ þ > ðDtÞ2 h2 h2 h2 > > > > > > > 2k2 uki;j ukiþ1;j1 2k2 b ukiþ1;j1 uki;j > : þ þ h2 h2
i-1,j+1
i,j+1
i+1,j+1 αk2
αk2 k3
k2
i-1,j
i,j
k1
i+1,j
k1
k2
k2
ð12Þ
k2
k3 y
h
αk2 h
x
Fig. 5. The grid of cross-section of hot workpiece.
i+1,j-1 αk2
Fig. 6. The microscopic model of electromagnetic wave propagation.
A 0
i,j-1
i-1,j-1
c
a
B d
Fig. 7. The schematic of the boundary points on hot workpiece.
Y. Zhang et al. / Measurement 59 (2015) 129–138
133
The difference equations of boundary point c are shown as follows:
8 kþ1 > q ui;j þuk1 2uki;j 2k1 ukiþ1;j uki;j k2 ukiþ1;jþ1 þukiþ1;j1 2uki;j k2 b ukiþ1;jþ1 þukiþ1;j1 2uki;j > i;j > > ¼ þ þ > > ðDt Þ2 h2 h2 h2 > > > > > k k k k > k2 v iþ1;jþ1 v iþ1;j1 k2 b v iþ1;jþ1 þv iþ1;j1 > > < þ þ h2 h2 > kþ1 > q v i;j þv k1 2v ki;j k3 v ki;jþ1 þv ki;j1 2v ki;j k2 v kiþ1;jþ1 þv kiþ1;j1 2v ki;j k2 b v kiþ1;jþ1 þv kiþ1;j1 2v ki;j > i;j > > ¼ þ þ > > ðDt Þ2 h2 h2 h2 > > > > > > k uk uk k b ukiþ1;jþ1 þukiþ1;j1 > : þ 2 iþ1;jþ1 iþ1;j1 þ 2 2 h h2
ð14Þ
The difference equations of boundary point d are shown as follows:
8 kþ1 > q ui;j þuk1 2uki;j 2k1 uki1;j uki;j k2 uki1;jþ1 þuki1;j1 2uki;j k2 b uki1;jþ1 þuki1;j1 2uki;j > i;j > > ¼ þ þ > > ðDt Þ2 h2 h2 h2 > > > > > > k2 v ki1;jþ1 v ki1;j1 k2 b v ki1;jþ1 þv ki1;j1 > > < þ þ h2 h2 > kþ1 k1 k k k > k3 v i;jþ1 þv i;j1 2v ki;j k2 v ki1;jþ1 þv ki1;j1 2v ki;j k2 b v ki1;jþ1 þv ki1;j1 2v ki;j > q v i;j þv i;j 2v i;j > > ¼ þ þ > > ðDt Þ2 h2 h2 h2 > > > > > > k2 uki1;jþ1 uki1;j1 k2 b uki1;jþ1 þuki1;j1 > : þ þ h2 h2
According to these difference equations, the factors affecting the electromagnetic wave produced by the laser stripe on hot workpiece are the power of the laser, the laser stripe width, and Lame constants of the workpiece. 3. The simulation In order to verify that electromagnetic wave produced by the laser stripe on hot workpiece is affected by the green laser stripe width and the experiment material, the cast iron and the 45#steel are chosen as the material of the simulation experiment. Some parameters of the cast iron and the 45#steel are shown in Table 1. According to equations of the center point, equations of the boundary points and the initial conditions, the waveform of the electromagnetic wave amplitude is simulated. The electromagnetic wave amplitude is produced by the laser stripe of the same power and the different widths. The laser stripe widths of 2.1 mm, 2.4 mm, 2.7 mm, and 3.0 mm are chosen as the train set. The model is obtained Table 1 The parameters of the cast iron and the 45#steel. Materials
The cast iron The steel
Young’s modulus (Pa)
Poisson’s ratio
Lame constants
l (Pa)
k (Pa)
10
10
10.5 10
0.28
8.6 10
19.5 1010
0.28
17.0 1010
4.4 1010 8.3 1010
ð15Þ
through analyzing the train set. The waveforms of the electromagnetic wave amplitudes produced by the laser stripes of the different widths on the cast iron workpiece are shown in Fig. 8. The waveforms are received at a distance of 0.5 mm from the laser stripe center. The waveform of the electromagnetic wave amplitude produced by the laser stripe width of 2.1 mm is shown in Fig. 8(a). The waveform of the electromagnetic wave amplitude produced by the laser stripe width of 2.4 mm is shown in Fig. 8(b). The waveform of the electromagnetic wave amplitude produced by the laser stripe width of 2.7 mm is shown in Fig. 8(c). The waveform of the electromagnetic wave amplitude produced by the laser stripe width of 3.0 mm is shown in Fig. 8(d). The energy loss of laser in electromagnetic wave on hot workpiece is proportional to the integral of the curve in the figure. Because the waveforms are received in the same position, these waveforms are at the same arrival time. So the energy loss of laser in electromagnetic wave on hot workpiece is reflected by the electromagnetic wave amplitude. In the application of the cast iron material, the electromagnetic wave amplitudes produced by the laser stripes of the different widths at 150 ns are shown in Fig. 9. As is shown in Fig. 9, the model is that as the width of the green laser stripe becomes large, the electromagnetic wave amplitude will increase with the decrease at first. The laser stripe widths of 2.2 mm, 2.5 mm, and 2.8 mm are chosen as the test set. The model is tested by the test
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Y. Zhang et al. / Measurement 59 (2015) 129–138
(a)
Nonmalized Displacement
Nonmalized Displacement
1.5 1 0.5 0 -0.5 -1 -1.5 -2
0
100
200
300
1.2 1.1 1 0.9 0.8 0.7 0.6 0.5
400
T/ns
Nonmalized Displacement
1.5
Fig. 9. The graph illustrating the conclusion in the application of the cast iron material.
1 0.5 0 -0.5 -1 -1.5 -2
0
100
200
300
400
T/ns
Nonmalized Displacement
(c)
1.5
1.1 1 0.9 0.8 0.7 0.6
2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9
3
Width/mm
0.5 Fig. 10. The graph testing the model.
0 -0.5 -1 -1.5 100
200
300
400
300
400
T/ns Nonmalized Displacement
1.2
0.5
1
-2 0
(d)
3
Width/mm
Nonmalized Displacement
(b)
2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9
1.5 1 0.5 0 -0.5 -1 -1.5 -2 0
100
200
T/ns Fig. 8. The cast iron material simulation waveform.
set. The electromagnetic wave amplitudes produced by the laser stripes of the different widths at 150 ns are tested. The graph testing the model is shown in Fig. 10.
As is shown in Fig. 10, the truthfulness of the model is confirmed. The laser stripe widths of 2.1 mm, 2.4 mm, 2.7 mm, and 3.0 mm are chosen as the train set. The model is obtained through analyzing the train set. The waveforms of the electromagnetic wave amplitudes produced by the laser stripes of the different widths on the 45#steel workpiece are shown in Fig. 11. The waveforms are received at a distance of 0.5 mm from the laser stripe center. The waveform of the electromagnetic wave amplitude produced by the laser stripe width of 2.1 mm is shown in Fig. 11(a). The waveform of the electromagnetic wave amplitude produced by the laser stripe width of 2.4 mm is shown in Fig. 11(b). The waveform of the electromagnetic wave amplitude produced by the laser stripe width of 2.7 mm is shown in Fig. 11(c). The waveform of the electromagnetic wave amplitude produced by the laser stripe width of 3.0 mm is shown in Fig. 11(d). In the application of the 45#steel material, the electromagnetic wave amplitudes produced by the laser stripes of the different widths at 150 ns are shown in Fig. 12. As is shown in Fig. 12, the model is that as the width of the green laser stripe becomes large, the electromagnetic wave amplitude will increase with the decrease at first. The laser stripe widths of 2.2 mm, 2.5 mm, and 2.8 mm are chosen as the test set. The model is tested by the test
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(a)
Nonmalized Displacement
Nonmalized Displacement
1.5
1 0.5
0 -0.5 -1
1 0.9 0.8 0.7 0.6 0.5
-1.5 -2 0
1.1
0.4 100
200
300
400
2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9
T/ns
(b)
Fig. 12. The graph illustrating the conclusion in the application of the 45#steel material.
Nonmalized Displacement
1.5 1
0 -0.5 -1 -1.5 0
100
200
300
400
T/ns
Nonmalized Displacement
0.5
-2
1.1 1 0.9 0.8 0.7 0.6 0.5 0.4
(c) Nonmalized Displacement
3
Width/mm
1.5
2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9
3
Width/mm
1 Fig. 13. The graph testing the model.
0.5 0 -0.5 -1 -1.5 -2
0
100
200
300
400
T/ns
Nonmalized Displacement
(d)
1.5 1 0.5 0 -0.5 -1 -1.5 -2 0
100
200
300
400
T/ns Fig. 11. 45#steel material simulation waveform.
set. The electromagnetic wave amplitudes produced by the laser stripes of the different widths at 150 ns are tested.
The graph testing the model is shown in Fig. 13. As is shown in Fig. 13, the truthfulness of the model is confirmed. The energy loss of the laser in electromagnetic wave on hot workpiece is reflected by the electromagnetic wave amplitude. Through the comparison between Figs. 8(a)– (d) and 11(a)–(d), a qualitative conclusion can be got: the cast iron or the 45#steel is chosen as the material. As the width of the green laser stripe becomes large, the electromagnetic wave amplitude will increase with the decrease at first. Then with green laser stripe width becoming large, the energy loss of laser in electromagnetic wave on hot workpiece decreases and then increases. In the application of the cast iron material, the width of the green laser stripe should be chosen between 2.4 mm and 3.0 mm to ensure the green laser stripe more clearly on hot workpiece. And in the application of the 45#steel material, the width of green laser stripe should be chosen between 2.1 mm and 2.7 mm to ensure the green laser stripe more clearly on the 45#steel workpiece. To illustrate the correctness of the conclusion, the other position is chosen to receive the vertical displacement of 45#steel body wave. It is shown in Fig. 14. As is shown in Fig. 14(a)–(d), the green laser stripe width is respectively 2.1 mm, 2.4 mm, 2.7 mm, and 3.0 mm. By comparing the amplitudes of waveforms, the same conclusion can be gotten: with green laser stripe width becoming
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Nonmalized Displacement
(a)
Nonmalized Displacement
(b)
Nonmalized Displacement
(c)
0.4 0.2 0 0
200
400
0
200
400
0
200
400
T/ns
600
800
1000
600
800
1000
600
800
1000
600
800
1000
0.4 0.2 0
T/ns
0.4 0.2 0
T/ns Nonmalized Displacement
(d) 0.4 0.2 0 0
200
400
T/ns Fig. 14. The waveform of other location.
large, the energy loss of laser in electromagnetic wave on hot workpiece decreases and then increases. 4. Testing experiment In order to verify the validity of the theory and the accuracy of the green laser measurement system proposed in the paper, an experiment platform is designed by our research group. Equipments of the experiment platform are as follows: (1) The heating furnace used is model C19-P1600B2. The heating temperature range of the heating furnace is 500–1550 °C. (2) Industrial CCD camera used is of model MVVE078SM/SC. The maximum resolution is 1024 768, and the size of a pixel on the CCD array is 4.65 lm 4.65 lm. The focal length of the optical lens is 12 mm, and its back focal length is 9.7 mm. The relative aperture (F) is 1:1.4. The field of view is 2/300 . Working distance is 4000 mm.
Fig. 15. The image received by the control computer.
Y. Zhang et al. / Measurement 59 (2015) 129–138
(3) The line laser projector is of MGL-III type. The wavelength is 532 nm. The aperture angle is 30°. Servo motors and drives are of MR-J2S-10A/B type. The maximum error of the drive is ±10 turns. (4) High-precision guide used is BGXS45BE-type high precision linear guide rail, and the walking error is less than 40 lm. The experimental environment is no wind with dust. The ambient light is lamplight. The indoor temperature is 300 K and the humidity is 60%. The procedure of this experiment is given as follows: (1) The 45#steel workpiece is heated to 900 °C by the heating furnace. In order to overcome temperature variations and surface parameters variations, as far as possible, the measurement time should be cut. (2) The hot workpiece is placed at a distance of 4 m from the laser transmitter. (3) The measurement system is started, and the green laser stripe width is respectively set to 2.1 mm, Table 2 The experimental data. The actual length/mm
The measure length/mm
The error/mm
The width 2.1 mm
1 2 3 4 5 6 7 8 9 10
360.0 360.0 360.0 360.0 360.0 360.0 360.0 360.0 360.0 360.0
362.5 358.0 361.3 357.9 357.1 362.4 362.5 357.8 358.4 361.9
2.5 2.0 1.3 2.1 2.9 2.4 2.5 2.2 1.6 1.9
The width 2.4 mm
1 2 3 4 5 6 7 8 9 10
360.0 360.0 360.0 360.0 360.0 360.0 360.0 360.0 360.0 360.0
361.8 359.4 361.1 360.3 358.8 360.5 360.7 361.0 358.7 358.9
1.8 0.6 1.1 0.3 1.2 0.5 0.7 1.0 1.3 1.1
The width 2.7 mm
1 2 3 4 5 6 7 8 9 10
360.0 360.0 360.0 360.0 360.0 360.0 360.0 360.0 360.0 360.0
361.5 358.3 361.3 357.9 360.8 362.6 359.0 358.6 361.3 358.2
1.5 1.7 1.3 2.1 0.8 2.6 1.0 1.4 1.3 1.8
The width 3.0 mm
1 2 3 4 5 6 7 8 9 10
360.0 360.0 360.0 360.0 360.0 360.0 360.0 360.0 360.0 360.0
358.0 362.5 358.1 357.4 362.7 363.3 362.5 357.4 362.9 357.6
2.0 2.5 1.9 2.6 2.7 3.3 2.5 2.6 2.9 2.4
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2.4 mm, 2.7 mm, and 3.0 mm. The green laser stripe is moved to the left edge of the hot workpiece by the control system. The image of the hot workpiece received by the control computer is shown in Fig. 15. Then the green laser stripe is moved to the right edge of the hot workpiece. The measurement data is received and displayed. The measurement process is repeated ten times and the measurement data are shown in Table 2. The image containing the laser stripe width of 2.1 mm is shown in Fig. 15(a). The image containing the laser stripe width of 2.4 mm is shown in Fig. 15(b). The image containing the laser stripe width of 2.7 mm is shown in Fig. 15(c). The image containing the laser stripe width of 3.0 mm is shown in Fig. 15(d). The laser stripe in Fig. 15(b) is certainly clearer than (a), (c) and (d) in Fig. 15. It verified that with choosing 45#steel workpiece, the green laser stripe width between 2.1 mm and 2.7 mm will make the laser stripe clearer on hot workpiece. In order to ensure that the result of proposed method is superior to conventional methods, the result of conventional methods is compared with proposed method in this paper. The images are shown in Figs. 16 and 17. The laser stripe in Fig. 17 is certainly clearer than that in Fig. 16. So the proposed method in this paper is a more
Fig. 16. The image under the image processing technology.
Fig. 17. The image based on proposed method in this paper.
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effective solution to the fuzzy green laser stripe for the green laser measurement system. The accuracy is shown as follows: d = (Dmax)/(Amax) 100%, where d is the accuracy, Dmax is the maximum error of the measurement, and Amax is the instrument range. The green laser device is installed on the linear guide rail. The instrument range is determined by the length of the linear guide rail. The length of the linear guide rail used is 500.0 cm. According to Table 2, when the green laser stripe width is set to 2.1 mm, the accuracy is: d = 2.9/ 500.0 100% = 0.58%. When the green laser stripe width is set to 2.4 mm, the accuracy is: d = 1.8/500.0 100% = 0.36%. When the green laser stripe width is set to 2.7 mm, the accuracy is: d = 2.6/500.0 100% = 0.52%. When the green laser stripe width is set to 3.0 mm, the accuracy is: d = 3.3/500.0 100% = 0.66%. When the green laser stripe width is set to 2.4 mm, the accuracy is improved to 0.36%. It can meet the requirement of accuracy. 5. Conclusions The cause of the fuzzy green laser stripe is found through the microscopic model of electromagnetic wave propagation in this paper. The electromagnetic wave is the major energy loss of laser on hot workpiece. With green laser stripe width becoming large, the energy loss of laser in electromagnetic wave on hot workpiece decreases and then increases. Then the optimal width of the laser stripe is chosen to make the green laser stripe clearer on hot workpiece. It suppresses the energy loss of laser in electromagnetic wave on hot workpiece effectively. The validity of the theory is verified through the simulation. This theory is applied to the measurement system, and the accuracy of measurement is improved to 0.36%. Acknowledgements This study is supported by the Natural Science Foundation of Hebei Province, China (Grant No.: E2014203070). References [1] L. Li, Z.P. Zhao, Research on dimensional measurement of heavy cylindrical forging at high temperature based on binocular stereo vision, Transducer Microsyst. Technol. 29 (4) (2010) 49–51.
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