- Email: [email protected]
Photoelastic fringe pattern analysis by real-time phase-shifting method
Optics and Lasers in Engineering 39 (2003) 1–13
Photoelastic fringe pattern analysis by real-time phase-shifting method S. Yoneyama*, Y. Morimoto, R. Matsui Department of Opto-Mechatronics, Wakayama University, 930 Sakaedani, Wakayama 640-8510, Japan Received 3 April 2002; received in revised form 7 May 2002; accepted 22 May 2002
Abstract This paper describes high-speed phase-shifting method for the real-time analysis of isochromatic and isoclinic parameters in photoelasticity. By rotating an analyzer at a constant rate and an output quarter-wave plate at a double rate of the analyzer and recording images by a CCD camera continuously, sequential phase-shifted images whose brightness is integrated by sensors are obtained. Then, the distributions of the isochromatic and isoclinic parameters are obtained at a video rate using the proposed phase-shifting algorithm. The phase distributions of the isochromatic and isoclinic parameters of a disk under the static and increasing loadings are simulated. Then, an application of the proposed method to a problem under increasing load is demonstrated and the time variation of the isochromatic parameter is evaluated. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Photoelasticity; Phase-shifting method; Real time; Isochromatics; Isoclinics
1. Introduction Recently, several phase-shifting methods have been proposed and widely used for the analysis of photoelastic fringe patterns. These techniques are summarized in recent review articles [1–3]. In order to apply these techniques to time-varying phenomena, it is requested to develop more rapid analysis method. For this purpose, the authors [4] proposed the integrated phase-shifting method for photoelasticity, which provides the phase distributions of isochromatics and isoclinics at a video rate (30 frames/s in case of a NTSC CCD camera). In the proposed method, by rotating *Corresponding author. Tel.: +81-73-457-8188; fax: +81-73-457-8213. E-mail address: [email protected] (S. Yoneyama). 0143-8166/03/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 3 - 8 1 6 6 ( 0 2 ) 0 0 0 9 8 - 2
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S. Yoneyama et al. / Optics and Lasers in Engineering 39 (2003) 1–13
an analyzer at a constant rate and an output quarter-wave plate at a double rate of the analyzer and recording images by a CCD camera continuously, sequential phaseshifting images whose brightness is integrated by sensors are obtained. Then, the distributions of the retardation (isochromatic parameter) and the principal direction (isoclinic parameter) are obtained at a video rate using the proposed phase-shifting algorithm. Since the accuracy of the phase distributions depends on the movement of the fringe pattern in case of a time-varying problem even if the fringe moves slowly, the relationship between the movement of the fringe pattern and the accuracy should be clear. In the present paper, the phase distributions of the isochromatic and isoclinic parameters of a disk under static and increasing loadings are simulated. Then, an application of the proposed method to the same problem with the simulation under the increasing load is demonstrated and the time variation of the isochromatic parameter is evaluated. The simulation and the experimental results show that the isochromatic parameter is evaluated accurately by the proposed method even if the load changes slowly with time. On the other hand, the fringe movement influences the isoclinics clearly.
2. Integrated phase-shifting method for photoelasticity For an arrangement of a circular polariscope shown in Fig. 1, the emerging light intensity I of photoelastic fringe pattern is expressed as a a I ¼ þ fsin 2b cos d sin 2ðy 2bÞcos 2b sin dg þ b; ð1Þ 2 2 where a and b are the amplitude and the background bias of the fringe pattern, b is the angle between a reference axis and the slow axis of an analyzer and y is the direction of the maximum principal stress of a specimen, respectively. d denotes the phase retardation given by d ¼ 2pN ¼
2pCs d ðs1 s2 Þ; l
ð2Þ
y S S
F 2β
β
S
F 3π/4
θ
Light source L π/2 x
Polarizer P
1/4 wave plate Q1 Specimen T
CCD 1/4 wave plate Q2 Camera Analyzer A
Fig. 1. Optical arrangement for high-speed phase analysis.
S. Yoneyama et al. / Optics and Lasers in Engineering 39 (2003) 1–13
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where N is the fringe order, Cs is the stress-optic coefficient, d is the thickness of the specimen, l is the wavelength of the monochromatic light, and s1 and s2 are the principal stresses, respectively. Eq. (1) can be easily obtained by matrix calculation such as Mueller or Jones calculus [3,5]. In the proposed method, the angles between the reference axis and the axes of an analyzer and a quarter-wave plate in a circular polariscope shown in Fig. 1 are changed at constant rates of 5p and 10p rad/s, respectively, while a CCD camera records fringe patterns. That means that the analyzer and the quarter-wave plate rotate at p=6 and p=3 rad, respectively, during the acquiring of a single photoelastic image since the exposure time of an ordinary NTSC CCD camera is 1/30 s. The output signal of the brightness from the CCD camera is obtained by calculating the integral of the variation of the light intensity during the exposure [6]. When the rotation angles b and 2b of the analyzer and the quarter-wave plate are synchronized with the frame rate of the CCD camera, the brightness I of the fringe pattern is obtained by integrating Eq. (1) by the time t or by the angle b using the relation b ¼ 5pt as [4] Z b2 n o a a I ¼ þ ½sin 2b cos d sin 2ðy 2bÞcos 2b sin d þ b db 2 2 b1 Z t2 n o a a ¼ 5p þ ½sin 10pt cos d sin 2ðy 10ptÞcos 10pt sin d þ bg dt; ð3Þ 2 2 t1 where t1 and t2 are the beginning and ending time of the exposure and b1 and b2 are the initial and final angular position of the analyzer at t1 and t2 ; respectively. For the first frame of an exposure, the brightness I1 can be expressed by calculating Eq. (3) from 0 to 1/30 s as pffiffiffi a p ð4Þ I1 ¼ ð4p 6cos d 3 3sin d sin 2y þ 7sin d cos 2yÞ þ b: 48 6 Since a and b are constants, Eq. (4) can be rewritten as pffiffiffi Ia I1 ¼ ð4p 6cos d 3 3sin d sin 2y þ 7sin d cos 2yÞ þ Ib ; 8p
ð5Þ
where Ia is the proportionality constant and Ib is background/stray light intensity. The sequential equations of the brightness I2 ; I3 y of the fringe pattern can be obtained as the same manner. Using the six brightness values, the isochromatic parameter (retardation) d and the isoclinic parameter (principal stress direction) y can be obtained as follows: pffiffiffi 3 3pIa sin d sin 2y 2ðI4 I1 I6 þ I3 Þ ¼ pffiffiffi tan 2y ¼ pffiffiffi 3 3pIa sin d cos 2y 3 3ðI5 I2 I4 þ I1 I6 þ I3 Þ for sin da0; ð6Þ 18pIa sin d cos 2y 6ðI5 I2 I4 þ I1 I6 þ I3 Þ ¼ 18pIa cos d cos 2y f7ðI5 I2 Þ I4 þ I1 I6 þ I3 gcos 2y for cos 2ya0:
tan d ¼
ð7Þ
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Thus, a series of six phase-shifted images that is obtained continuously within 6/30 s provides the distributions of the isochromatic and isoclinic parameters d and y: The feature of the proposed method is the phase-shifting algorithm with rotating the analyzer and the quarter-wave plate continuously at constant rates. By using this algorithm, the procedure for the phase-shifting becomes easier and the processing speed becomes faster.
3. Apparatus for integrated phase-shifting method In order to utilize the proposed method to practical photoelastic experiment and to perform analysis in real time, an experimental setup including a device for the rotation of the analyzer and the quarter-wave plate has been developed as shown in Fig. 2. A laser light of l ¼ 514:5 nm passes through a polarizer, a quarter-wave plate and a birefringent specimen. Then, the light passes through the continuously rotating quarter-wave plate and analyzer. A CCD camera records photoelastic fringe patterns focused on a screen. In order to reduce the effect of speckle noise, the screen rotates continuously. The rotational angles of the analyzer and the quarter-wave plate (b and 2b) and the horizontal and vertical synchronizing signals of the CCD camera (HD and VD) are synchronized by external signals controlled by a computer. A
Polarizer
Specimen
1/4 wave plate
Analyzer CCD Camera
Light source 1/4 wave plate
Stepping Motors
Loading device Pulse
Pulse
Camera adaptor
Motor driver Control signal (RS-232C)
Pulse
Synchronous circuit
HD VD
Trigger Images
Fig. 2. Schematic diagram of experimental setup.
S. Yoneyama et al. / Optics and Lasers in Engineering 39 (2003) 1–13
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computer processes the sequential images recorded by the CCD camera and the distributions of isochromatic and isoclinic parameters are obtained. In order to obtain the phase distributions immediately, look-up tables are used for the calculation.
4. Influence of the change of load during phase-shifting It is possible for the proposed method to provide the phase distributions of isochromatics and isoclinics at a video rate [4]. Then, the time variations of these photoelastic parameters can be obtained. However, since the accuracy of the phase distributions depends on the moving rate of the fringe pattern, the relationship between the movement of the fringe pattern and the accuracy of analysis should be clear. For simplification, a constant rate increasing loading problem is considered here. That is, it is assumed that the isochromatic parameter varies at a constant rate during the acquisition of six images, i.e., 6/30 s. It can be considered that this assumption is also valid for the randomly changing load if the load varies slowly. When the isochromatic parameter d changes during an acquisition, Eq. (3) can be rewritten as Z d2 Z b2 n o a a I¼ þ ½sin 2b cos d sin 2ðy 2bÞcos 2b sin d þ b db dd; ð8Þ 2 2 d1 b1 where d1 and d2 are the initial and the final values of the isochromatic parameter. The equations for the isoclinic and isochromatic parameters (corresponding to Eqs. (6) and (7)) cannot be obtained from Eq. (8) because the six light intensity equations with the phase-shift contain different values of the isochromatic parameter. It is difficult to show the influence of the change of the load through theoretical consideration because mathematical treatment is too complicated. Thus, the influence is investigated through simulations. Fig. 3(a) shows the simulated differences of the isoclinic parameters between the static and constant rate increasing load problems as functions of the isochromatic parameter d and the variation rate of the isochromatic parameter dd=dt: Fig. 3(b) also shows the differences of the isochromatic parameters. In this simulation, the isoclinic parameter y is set to 0 rad. Similarly, Fig. 4 shows the simulated differences of the isoclinic parameters between the static and constant rate increasing load problems as functions of the isoclinic parameter y and the variation rate of the isochromatic parameter dd=dt in the case that the isochromatic parameter d is p=12 rad. These plots are obtained from Eqs. (3) and (6)–(8). In Fig. 3(a), the errors in the isoclinic parameter where the isochromatic parameter is 0 or p rad are very high. The main cause of these errors is considered as isochromatic–isoclinic interaction [3]. However, it is observed that the errors in the isoclinic parameter not only near the region where isochromatic parameter is 0 or p rad but also the other regions increase rapidly with the increase of the variation rate of the isochromatic parameter. On the other hand, it is seen from Fig. 3(b) that the errors in the isochromatic parameter are very small. In Fig. 3(b), the maximum error is estimated
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Fig. 3. Influence of fringe movement on (a) the isoclinic parameter and (b) the isoclinic parameter as functions of the isochromatic parameter and the variation rate of isochromatic parameter (y ¼ 0 rad).
about 0.26 rad, i.e., 0.04 fringe orders even if the variation rate of the isochromatic parameter is 2p rad/s (p=15 frames/s), i.e., 1 fringe order/s (0.033 fringe orders/ frame). Similar results are observed in Fig. 4. Through the errors in the isoclinic parameter around the regions where the isoclinic parameter is p=8 rad are quite small, the errors near the region where the isoclinic parameter is 0 or p=4 rad are very high. The maximum error in the Fig. 4(a) is estimated about p=4 rad. However, the errors in the isochromatic parameter are quite small in Fig. 4(b). Similar results are also obtained for the other combinations of the isochromatic and isoclinic parameters. From these simulations, it can be considered that the change of the
S. Yoneyama et al. / Optics and Lasers in Engineering 39 (2003) 1–13
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Fig. 4. Influence of fringe movement on (a) the isoclinic parameter and (b) the isoclinic parameter as functions of the isoclinic parameter and the variation rate of isochromatic parameter (d ¼ p=15 rad).
load during phase-shifting influences the results of the isoclinic parameter. On the other hand, the influence on the isochromatic parameter is very small though Eq. (7) contains the results of the isoclinic parameter. In order to make it more intelligible, a disk under compression of a constant rate increasing load is simulated. The disk specimen for the simulation is shown in Fig. 5. Fig. 6 shows the simulated phase distributions of the isochromatics and the isoclinics obtained by the proposed method under the static and constant rate increasing loads. Here, the wavelength l is set to 514.5 nm and the stress-optic coefficient Cs is 82.8 1012 m2/N in this simulation. In Figs. 6(c) and (d), the phase distributions at the moment of the load of P ¼ 150 N under the loading rate of P ¼ 96 N/s (3.2 N/ frame) are shown. It is noted that the isochromatics close to the top and bottom
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y
Load
A
2R
0 =5
mm
B
13.8 mm
x
Thickness d = 5.6 mm Load Fig. 5. Specimen geometry.
Fig. 6. Simulated phase distributions of (a) the isoclinic and (b) isochromatic parameters under the static load and (c) the isoclinic and (d) isochromatic parameters under the constant rate increasing load.
S. Yoneyama et al. / Optics and Lasers in Engineering 39 (2003) 1–13
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Fig. 7. Simulated phase distributions of (a) the isoclinic parameter and (b) the isochromatic parameter along the y ¼ 0:5R line.
areas of the disk contain incorrect regions where the isochromatic parameter has the wrong mathematical sign [3]. Fig. 7 shows the simulated phase distributions of the isochromatics and the isoclinics along the y ¼ 0:5R line perpendicular to the load. It is seen from Figs. 6 and 7 that the isoclinic parameters are affected by the fringe movement clearly but the influence on the isochromatic parameters is very small as predicted in Figs. 3 and 4. For phase-shifting and phase-stepping methods in photoelasticity, it is known that though the results of isoclinics influence the calculation of the isochromatics, its influence is very small [3]. The same phenomenon is observed in the proposed method. Originally the results of the isoclinics contained many errors near the region of d ¼ 0; p and 2p because Eq. (6) contains sin d in both the numerator and the denominator. In case the load changes during phase-shifting, this error increases as
S. Yoneyama et al. / Optics and Lasers in Engineering 39 (2003) 1–13
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shown in Figs. 3, 6 and 7. The small change of the measured light intensities affects the isoclinics near these regions because the digitized values both the numerator and the denominator are very small. In Eq. (7), on the other hand, both of the numerator and the denominator contain cos 2y for the calculation of the isochromatic parameter. Thus, the effect of the isoclinic parameter is canceled and the error is not propagating to the calculation of the isochromatics. In addition, the range of the calculated results of the isochromatics is 2p rad whereas the range of the isoclinics is limited to p=2 rad. Thus, the errors in the isoclinic parameter become relatively large again compared with those of the isochromatics. The influence of the changes of the principal direction (isoclinic parameter) with time on the results is the topic of interest for viscoelastic nonproportional loading problems [7,8]. This will be investigated later.
5. Analysis using integrated phase-shifting method The integrated phase-shifting photoelastic technique is applied to a problem under constant rate increasing load in order to evaluate the effectiveness of the proposed method and the apparatus. A same disk with the simulation made of polycarbonate; 50 mm in diameter and 5.6 mm in thickness as shown in Fig. 5 is subjected to diametric compression.
55/30 s
56/30 s
57/30 s
58/30 s
59/30 s
60/30 s
(a)
(b)
(c) Fig. 8. Series of (a) the photoelastic fringe patterns, (b) the isoclinics and (c) the isochromatics obtained by the proposed method.
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Fig. 8(a) shows an example series of the photoelastic fringe patterns obtained by use of the aforementioned apparatus within 6/30 s. The sequential numbers shown in this figure express the recording time. The differences of the fringe patterns among six images due to the differences of the angular positions of the analyzer and the quarter-wave plate are easily recognized. Figs. 8(b) and (c) show a series of the phase-distributions of the isochromatics and the isoclinics obtained by the proposed method. The distributions of the isoclinic and isochromatic parameters along the y ¼ 0:5R line perpendicular to the load are shown in Fig. 9. Almost same results with the simulation in Figs. 6 and 7 are observed in Figs. 8 and 9. It was shown in a previous paper [4] that in case of a static problem, the phase distributions of the isochromatic and isoclinic parameters obtained by the proposed method are almost same with the results obtained by the conventional phase-shifting method. On the
Fig. 9. Phase distributions of (a) the isoclinic parameter and (b) the isochromatic parameter along the y ¼ 0:5R line.
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Fig. 10. Time variations of the isochromatic parameter at the points A and B.
other hand, it is observed that the errors due to the isochromatic–isoclinic interaction and the fringe movement appear in the phase distribution of isoclinics [3] as predicted in the simulation. It reaches the conclusion that the variation of the fringe pattern remarkably affects the isoclinics in the proposed method. Fig. 10 shows the time variation of the unwrapped isochromatics at the points A and B in Fig. 5 obtained by the proposed method. In order to observe the difference of the errors in various conditions of fringe motion, the tests are carried out under two different loading rates of p ¼ 58 and 96 N/s. In these figures, solid lines indicate the theoretical results [9]. Though the time variations of isochromatic parameter contain some noises, it is observed that these parameters increase lineally. The average of the absolute errors in Fig. 10 is about 0.17 fringe orders and the standard deviation is evaluated as 0.13 fringe orders.
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6. Conclusions In the present paper, the real-time phase-shifting method for photoelasticity is described. Then, the error of the phase distributions in case of the disk under increasing compression is simulated. The results of the simulation and the experiment indicate that the phase distribution of the isochromatics can be obtained accurately even if the load varies slowly with time. On the other hand, phase distribution of isoclinics contains more errors than the isochromatics under the same condition. Presently, the proposed method cannot analyze the isoclinic parameter accurately when the load is varying with time. However, the time-variation of the isochromatic parameter can be analyzed when the load changes slowly. The proposed method and the apparatus can also be used for high-speed inspection of birefringence in glass products since the phase distribution of the isochromatics is obtainable at a video rate.
Acknowledgements The authors appreciate the financial support by the Grant-in-Aid for Encouragement of Young Scientists from the Japan Society for the Promotion of Science. Our gratitude is extended to Drs. T. Nomura and M. Fujigaki for their helpful discussions.
References [1] Ajovalasit A, Barone S, Petrucci G. A review of automated methods for the collection and analysis of photoelastic data. J Strain Anal Eng Des 1998;33:75–91. [2] Chen TY. Digital photoelasticity. In: Rastogi PK, editor. Photomechanics. Berlin: Springer, 1999. p. 197–232. [3] Ramesh K. Digital photoelasticity. Berlin: Springer, 2000. [4] Yoneyama S, Morimoto Y, Nomura T, Fujigaki M, Matsui R. Real-time analysis of isochromatics and isoclinics using phase-shifting method. Proc SEM Annu Conf Exp Mech 2001:117–20. [5] Theocaris PS, Gdoutos EE. Matrix theory of photoelasticity. Berlin: Springer, 1979. [6] Morimoto Y, Fujigaki M, Toda H. Real-time shape measurement by integrated phase-shifting method. In: Kujawinska M, Takeda M, editors. Interferometry 99. Proceedings of the SPIE, vol. 3744, 1999. p. 118–25. [7] Yoneyama S, Gotoh J, Takashi M. Tricolor photoviscoelastic technique and its application to moving contact. Exp Mech 1998;38:211–7. [8] Yoneyama S, Gotoh J, Takashi M. Experimental analysis of rolling contact stresses in a viscoelastic strip. Exp Mech 2000;40:203–10. [9] Timoshenko SP, Goodier JN. Theory of elasticity. New York: McGraw-Hill, 1970.
Photoelastic fringe pattern analysis by real-time phase-shifting method S. Yoneyama*, Y. Morimoto, R. Matsui Department of Opto-Mechatronics, Wakayama University, 930 Sakaedani, Wakayama 640-8510, Japan Received 3 April 2002; received in revised form 7 May 2002; accepted 22 May 2002
Abstract This paper describes high-speed phase-shifting method for the real-time analysis of isochromatic and isoclinic parameters in photoelasticity. By rotating an analyzer at a constant rate and an output quarter-wave plate at a double rate of the analyzer and recording images by a CCD camera continuously, sequential phase-shifted images whose brightness is integrated by sensors are obtained. Then, the distributions of the isochromatic and isoclinic parameters are obtained at a video rate using the proposed phase-shifting algorithm. The phase distributions of the isochromatic and isoclinic parameters of a disk under the static and increasing loadings are simulated. Then, an application of the proposed method to a problem under increasing load is demonstrated and the time variation of the isochromatic parameter is evaluated. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Photoelasticity; Phase-shifting method; Real time; Isochromatics; Isoclinics
1. Introduction Recently, several phase-shifting methods have been proposed and widely used for the analysis of photoelastic fringe patterns. These techniques are summarized in recent review articles [1–3]. In order to apply these techniques to time-varying phenomena, it is requested to develop more rapid analysis method. For this purpose, the authors [4] proposed the integrated phase-shifting method for photoelasticity, which provides the phase distributions of isochromatics and isoclinics at a video rate (30 frames/s in case of a NTSC CCD camera). In the proposed method, by rotating *Corresponding author. Tel.: +81-73-457-8188; fax: +81-73-457-8213. E-mail address: [email protected] (S. Yoneyama). 0143-8166/03/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 3 - 8 1 6 6 ( 0 2 ) 0 0 0 9 8 - 2
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S. Yoneyama et al. / Optics and Lasers in Engineering 39 (2003) 1–13
an analyzer at a constant rate and an output quarter-wave plate at a double rate of the analyzer and recording images by a CCD camera continuously, sequential phaseshifting images whose brightness is integrated by sensors are obtained. Then, the distributions of the retardation (isochromatic parameter) and the principal direction (isoclinic parameter) are obtained at a video rate using the proposed phase-shifting algorithm. Since the accuracy of the phase distributions depends on the movement of the fringe pattern in case of a time-varying problem even if the fringe moves slowly, the relationship between the movement of the fringe pattern and the accuracy should be clear. In the present paper, the phase distributions of the isochromatic and isoclinic parameters of a disk under static and increasing loadings are simulated. Then, an application of the proposed method to the same problem with the simulation under the increasing load is demonstrated and the time variation of the isochromatic parameter is evaluated. The simulation and the experimental results show that the isochromatic parameter is evaluated accurately by the proposed method even if the load changes slowly with time. On the other hand, the fringe movement influences the isoclinics clearly.
2. Integrated phase-shifting method for photoelasticity For an arrangement of a circular polariscope shown in Fig. 1, the emerging light intensity I of photoelastic fringe pattern is expressed as a a I ¼ þ fsin 2b cos d sin 2ðy 2bÞcos 2b sin dg þ b; ð1Þ 2 2 where a and b are the amplitude and the background bias of the fringe pattern, b is the angle between a reference axis and the slow axis of an analyzer and y is the direction of the maximum principal stress of a specimen, respectively. d denotes the phase retardation given by d ¼ 2pN ¼
2pCs d ðs1 s2 Þ; l
ð2Þ
y S S
F 2β
β
S
F 3π/4
θ
Light source L π/2 x
Polarizer P
1/4 wave plate Q1 Specimen T
CCD 1/4 wave plate Q2 Camera Analyzer A
Fig. 1. Optical arrangement for high-speed phase analysis.
S. Yoneyama et al. / Optics and Lasers in Engineering 39 (2003) 1–13
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where N is the fringe order, Cs is the stress-optic coefficient, d is the thickness of the specimen, l is the wavelength of the monochromatic light, and s1 and s2 are the principal stresses, respectively. Eq. (1) can be easily obtained by matrix calculation such as Mueller or Jones calculus [3,5]. In the proposed method, the angles between the reference axis and the axes of an analyzer and a quarter-wave plate in a circular polariscope shown in Fig. 1 are changed at constant rates of 5p and 10p rad/s, respectively, while a CCD camera records fringe patterns. That means that the analyzer and the quarter-wave plate rotate at p=6 and p=3 rad, respectively, during the acquiring of a single photoelastic image since the exposure time of an ordinary NTSC CCD camera is 1/30 s. The output signal of the brightness from the CCD camera is obtained by calculating the integral of the variation of the light intensity during the exposure [6]. When the rotation angles b and 2b of the analyzer and the quarter-wave plate are synchronized with the frame rate of the CCD camera, the brightness I of the fringe pattern is obtained by integrating Eq. (1) by the time t or by the angle b using the relation b ¼ 5pt as [4] Z b2 n o a a I ¼ þ ½sin 2b cos d sin 2ðy 2bÞcos 2b sin d þ b db 2 2 b1 Z t2 n o a a ¼ 5p þ ½sin 10pt cos d sin 2ðy 10ptÞcos 10pt sin d þ bg dt; ð3Þ 2 2 t1 where t1 and t2 are the beginning and ending time of the exposure and b1 and b2 are the initial and final angular position of the analyzer at t1 and t2 ; respectively. For the first frame of an exposure, the brightness I1 can be expressed by calculating Eq. (3) from 0 to 1/30 s as pffiffiffi a p ð4Þ I1 ¼ ð4p 6cos d 3 3sin d sin 2y þ 7sin d cos 2yÞ þ b: 48 6 Since a and b are constants, Eq. (4) can be rewritten as pffiffiffi Ia I1 ¼ ð4p 6cos d 3 3sin d sin 2y þ 7sin d cos 2yÞ þ Ib ; 8p
ð5Þ
where Ia is the proportionality constant and Ib is background/stray light intensity. The sequential equations of the brightness I2 ; I3 y of the fringe pattern can be obtained as the same manner. Using the six brightness values, the isochromatic parameter (retardation) d and the isoclinic parameter (principal stress direction) y can be obtained as follows: pffiffiffi 3 3pIa sin d sin 2y 2ðI4 I1 I6 þ I3 Þ ¼ pffiffiffi tan 2y ¼ pffiffiffi 3 3pIa sin d cos 2y 3 3ðI5 I2 I4 þ I1 I6 þ I3 Þ for sin da0; ð6Þ 18pIa sin d cos 2y 6ðI5 I2 I4 þ I1 I6 þ I3 Þ ¼ 18pIa cos d cos 2y f7ðI5 I2 Þ I4 þ I1 I6 þ I3 gcos 2y for cos 2ya0:
tan d ¼
ð7Þ
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Thus, a series of six phase-shifted images that is obtained continuously within 6/30 s provides the distributions of the isochromatic and isoclinic parameters d and y: The feature of the proposed method is the phase-shifting algorithm with rotating the analyzer and the quarter-wave plate continuously at constant rates. By using this algorithm, the procedure for the phase-shifting becomes easier and the processing speed becomes faster.
3. Apparatus for integrated phase-shifting method In order to utilize the proposed method to practical photoelastic experiment and to perform analysis in real time, an experimental setup including a device for the rotation of the analyzer and the quarter-wave plate has been developed as shown in Fig. 2. A laser light of l ¼ 514:5 nm passes through a polarizer, a quarter-wave plate and a birefringent specimen. Then, the light passes through the continuously rotating quarter-wave plate and analyzer. A CCD camera records photoelastic fringe patterns focused on a screen. In order to reduce the effect of speckle noise, the screen rotates continuously. The rotational angles of the analyzer and the quarter-wave plate (b and 2b) and the horizontal and vertical synchronizing signals of the CCD camera (HD and VD) are synchronized by external signals controlled by a computer. A
Polarizer
Specimen
1/4 wave plate
Analyzer CCD Camera
Light source 1/4 wave plate
Stepping Motors
Loading device Pulse
Pulse
Camera adaptor
Motor driver Control signal (RS-232C)
Pulse
Synchronous circuit
HD VD
Trigger Images
Fig. 2. Schematic diagram of experimental setup.
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computer processes the sequential images recorded by the CCD camera and the distributions of isochromatic and isoclinic parameters are obtained. In order to obtain the phase distributions immediately, look-up tables are used for the calculation.
4. Influence of the change of load during phase-shifting It is possible for the proposed method to provide the phase distributions of isochromatics and isoclinics at a video rate [4]. Then, the time variations of these photoelastic parameters can be obtained. However, since the accuracy of the phase distributions depends on the moving rate of the fringe pattern, the relationship between the movement of the fringe pattern and the accuracy of analysis should be clear. For simplification, a constant rate increasing loading problem is considered here. That is, it is assumed that the isochromatic parameter varies at a constant rate during the acquisition of six images, i.e., 6/30 s. It can be considered that this assumption is also valid for the randomly changing load if the load varies slowly. When the isochromatic parameter d changes during an acquisition, Eq. (3) can be rewritten as Z d2 Z b2 n o a a I¼ þ ½sin 2b cos d sin 2ðy 2bÞcos 2b sin d þ b db dd; ð8Þ 2 2 d1 b1 where d1 and d2 are the initial and the final values of the isochromatic parameter. The equations for the isoclinic and isochromatic parameters (corresponding to Eqs. (6) and (7)) cannot be obtained from Eq. (8) because the six light intensity equations with the phase-shift contain different values of the isochromatic parameter. It is difficult to show the influence of the change of the load through theoretical consideration because mathematical treatment is too complicated. Thus, the influence is investigated through simulations. Fig. 3(a) shows the simulated differences of the isoclinic parameters between the static and constant rate increasing load problems as functions of the isochromatic parameter d and the variation rate of the isochromatic parameter dd=dt: Fig. 3(b) also shows the differences of the isochromatic parameters. In this simulation, the isoclinic parameter y is set to 0 rad. Similarly, Fig. 4 shows the simulated differences of the isoclinic parameters between the static and constant rate increasing load problems as functions of the isoclinic parameter y and the variation rate of the isochromatic parameter dd=dt in the case that the isochromatic parameter d is p=12 rad. These plots are obtained from Eqs. (3) and (6)–(8). In Fig. 3(a), the errors in the isoclinic parameter where the isochromatic parameter is 0 or p rad are very high. The main cause of these errors is considered as isochromatic–isoclinic interaction [3]. However, it is observed that the errors in the isoclinic parameter not only near the region where isochromatic parameter is 0 or p rad but also the other regions increase rapidly with the increase of the variation rate of the isochromatic parameter. On the other hand, it is seen from Fig. 3(b) that the errors in the isochromatic parameter are very small. In Fig. 3(b), the maximum error is estimated
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Fig. 3. Influence of fringe movement on (a) the isoclinic parameter and (b) the isoclinic parameter as functions of the isochromatic parameter and the variation rate of isochromatic parameter (y ¼ 0 rad).
about 0.26 rad, i.e., 0.04 fringe orders even if the variation rate of the isochromatic parameter is 2p rad/s (p=15 frames/s), i.e., 1 fringe order/s (0.033 fringe orders/ frame). Similar results are observed in Fig. 4. Through the errors in the isoclinic parameter around the regions where the isoclinic parameter is p=8 rad are quite small, the errors near the region where the isoclinic parameter is 0 or p=4 rad are very high. The maximum error in the Fig. 4(a) is estimated about p=4 rad. However, the errors in the isochromatic parameter are quite small in Fig. 4(b). Similar results are also obtained for the other combinations of the isochromatic and isoclinic parameters. From these simulations, it can be considered that the change of the
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Fig. 4. Influence of fringe movement on (a) the isoclinic parameter and (b) the isoclinic parameter as functions of the isoclinic parameter and the variation rate of isochromatic parameter (d ¼ p=15 rad).
load during phase-shifting influences the results of the isoclinic parameter. On the other hand, the influence on the isochromatic parameter is very small though Eq. (7) contains the results of the isoclinic parameter. In order to make it more intelligible, a disk under compression of a constant rate increasing load is simulated. The disk specimen for the simulation is shown in Fig. 5. Fig. 6 shows the simulated phase distributions of the isochromatics and the isoclinics obtained by the proposed method under the static and constant rate increasing loads. Here, the wavelength l is set to 514.5 nm and the stress-optic coefficient Cs is 82.8 1012 m2/N in this simulation. In Figs. 6(c) and (d), the phase distributions at the moment of the load of P ¼ 150 N under the loading rate of P ¼ 96 N/s (3.2 N/ frame) are shown. It is noted that the isochromatics close to the top and bottom
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y
Load
A
2R
0 =5
mm
B
13.8 mm
x
Thickness d = 5.6 mm Load Fig. 5. Specimen geometry.
Fig. 6. Simulated phase distributions of (a) the isoclinic and (b) isochromatic parameters under the static load and (c) the isoclinic and (d) isochromatic parameters under the constant rate increasing load.
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Fig. 7. Simulated phase distributions of (a) the isoclinic parameter and (b) the isochromatic parameter along the y ¼ 0:5R line.
areas of the disk contain incorrect regions where the isochromatic parameter has the wrong mathematical sign [3]. Fig. 7 shows the simulated phase distributions of the isochromatics and the isoclinics along the y ¼ 0:5R line perpendicular to the load. It is seen from Figs. 6 and 7 that the isoclinic parameters are affected by the fringe movement clearly but the influence on the isochromatic parameters is very small as predicted in Figs. 3 and 4. For phase-shifting and phase-stepping methods in photoelasticity, it is known that though the results of isoclinics influence the calculation of the isochromatics, its influence is very small [3]. The same phenomenon is observed in the proposed method. Originally the results of the isoclinics contained many errors near the region of d ¼ 0; p and 2p because Eq. (6) contains sin d in both the numerator and the denominator. In case the load changes during phase-shifting, this error increases as
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shown in Figs. 3, 6 and 7. The small change of the measured light intensities affects the isoclinics near these regions because the digitized values both the numerator and the denominator are very small. In Eq. (7), on the other hand, both of the numerator and the denominator contain cos 2y for the calculation of the isochromatic parameter. Thus, the effect of the isoclinic parameter is canceled and the error is not propagating to the calculation of the isochromatics. In addition, the range of the calculated results of the isochromatics is 2p rad whereas the range of the isoclinics is limited to p=2 rad. Thus, the errors in the isoclinic parameter become relatively large again compared with those of the isochromatics. The influence of the changes of the principal direction (isoclinic parameter) with time on the results is the topic of interest for viscoelastic nonproportional loading problems [7,8]. This will be investigated later.
5. Analysis using integrated phase-shifting method The integrated phase-shifting photoelastic technique is applied to a problem under constant rate increasing load in order to evaluate the effectiveness of the proposed method and the apparatus. A same disk with the simulation made of polycarbonate; 50 mm in diameter and 5.6 mm in thickness as shown in Fig. 5 is subjected to diametric compression.
55/30 s
56/30 s
57/30 s
58/30 s
59/30 s
60/30 s
(a)
(b)
(c) Fig. 8. Series of (a) the photoelastic fringe patterns, (b) the isoclinics and (c) the isochromatics obtained by the proposed method.
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Fig. 8(a) shows an example series of the photoelastic fringe patterns obtained by use of the aforementioned apparatus within 6/30 s. The sequential numbers shown in this figure express the recording time. The differences of the fringe patterns among six images due to the differences of the angular positions of the analyzer and the quarter-wave plate are easily recognized. Figs. 8(b) and (c) show a series of the phase-distributions of the isochromatics and the isoclinics obtained by the proposed method. The distributions of the isoclinic and isochromatic parameters along the y ¼ 0:5R line perpendicular to the load are shown in Fig. 9. Almost same results with the simulation in Figs. 6 and 7 are observed in Figs. 8 and 9. It was shown in a previous paper [4] that in case of a static problem, the phase distributions of the isochromatic and isoclinic parameters obtained by the proposed method are almost same with the results obtained by the conventional phase-shifting method. On the
Fig. 9. Phase distributions of (a) the isoclinic parameter and (b) the isochromatic parameter along the y ¼ 0:5R line.
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Fig. 10. Time variations of the isochromatic parameter at the points A and B.
other hand, it is observed that the errors due to the isochromatic–isoclinic interaction and the fringe movement appear in the phase distribution of isoclinics [3] as predicted in the simulation. It reaches the conclusion that the variation of the fringe pattern remarkably affects the isoclinics in the proposed method. Fig. 10 shows the time variation of the unwrapped isochromatics at the points A and B in Fig. 5 obtained by the proposed method. In order to observe the difference of the errors in various conditions of fringe motion, the tests are carried out under two different loading rates of p ¼ 58 and 96 N/s. In these figures, solid lines indicate the theoretical results [9]. Though the time variations of isochromatic parameter contain some noises, it is observed that these parameters increase lineally. The average of the absolute errors in Fig. 10 is about 0.17 fringe orders and the standard deviation is evaluated as 0.13 fringe orders.
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6. Conclusions In the present paper, the real-time phase-shifting method for photoelasticity is described. Then, the error of the phase distributions in case of the disk under increasing compression is simulated. The results of the simulation and the experiment indicate that the phase distribution of the isochromatics can be obtained accurately even if the load varies slowly with time. On the other hand, phase distribution of isoclinics contains more errors than the isochromatics under the same condition. Presently, the proposed method cannot analyze the isoclinic parameter accurately when the load is varying with time. However, the time-variation of the isochromatic parameter can be analyzed when the load changes slowly. The proposed method and the apparatus can also be used for high-speed inspection of birefringence in glass products since the phase distribution of the isochromatics is obtainable at a video rate.
Acknowledgements The authors appreciate the financial support by the Grant-in-Aid for Encouragement of Young Scientists from the Japan Society for the Promotion of Science. Our gratitude is extended to Drs. T. Nomura and M. Fujigaki for their helpful discussions.
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