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Simultaneous in-plane and out-of-plane deformation measurement by speckle multi-recording method
Accepted Manuscript Simultaneous in-plane and out-of-plane deformation measurement by speckle multi-recording method Yasuhiko Arai PII: DOI: Reference:
S0263-2241(16)30183-X http://dx.doi.org/10.1016/j.measurement.2016.05.037 MEASUR 4053
To appear in:
Measurement
Received Date: Revised Date: Accepted Date:
7 August 2015 1 March 2016 10 May 2016
Please cite this article as: Y. Arai, Simultaneous in-plane and out-of-plane deformation measurement by speckle multi-recording method, Measurement (2016), doi: http://dx.doi.org/10.1016/j.measurement.2016.05.037
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Title page Paper title: Simultaneous in-plane and out-of-plane deformation measurement by speckle multi-recording method
Name of author: Yasuhiko Arai Affiliation: Department of Mechanical Engineering, Faculty of Engineering Science, Kansai University e-mail: [email protected] Mailing address: 3-3-35, Yamate-cho, Suita, Osaka, Japan 564-8680 Phone number: +81-6-6368-0778 Fax number: +81-6-6388-8785
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Highlight ・In-plane and out-of-plane simultaneous deformation measurement method based on speckle interferometry is proposed. ・Multi-recording technology in speckle interferometry is also proposed. ・The problems in the multi-recording technology are discussed. ・The new method is applied to the analysis of phenomenon of buckling in a beam.
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Abstract
An in-plane and out-of-plane deformation simultaneous measurement method using only two speckle patterns grabbed before and after deformation of an object with rough surfaces using two cameras has been proposed. However, there are some problems concerning the setup of optical system, the aspect ratio of grabbed image data and so on in the conventional method. To solve these problems, the new optical system is proposed
by
using
the
multi-recording
technology.
Though
multi-recording
technologies have been already employed conveniently in the off-axis digital holography, the technologies have not been used functionally enough in speckle interferometry. In this paper, an in-plane and out-of-plane deformation simultaneous measurement is performed by the proposed method based on multi-recording technology in speckle interferometry. From experimental results, it can be confirmed that the proposed optical system can measure simultaneously in-plane and out-of-plane deformations in high resolution by one camera.
Keywords: ESPI, multi-recording technology, in-plane and out-of-plane speckle deformation measurement, and buckling.
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1. Introduction
Deformation measurement methods based on the speckle phenomenon were developed as speckle interferometry in the 1960s[1-4]. The technologies have been employed in the field of vibration analysis. ESPI (electronic speckle pattern interferometry) was also developed by introducing the TV-camera technology into speckle interferometry [5]. The fringe scanning technologies which were developed in the 1980s were also introduced to the speckle interferometry for improving the measurement accuracy [4]. Then, the resolving power of the speckle interferometry was improved to about 1/100 of the wavelength of the light source [6]. A deformation analysis by a new speckle interferometer that is based on a similar idea of the off-axes digital holography [7] has been performed using only two speckle patterns [8-10]. At the present time, the out-of-plane and in-plane deformations can be measured by the optical system based on two-beam speckle interferometry [10]. Then, the optical system was constructed by a pair of the new speckle interferometers. The high resolution deformation measurement was also realized by using fringe scanning technologies and two cameras. However, the speckle interferometer using two cameras requires the pre-processing of the measurement [10], for example; the exact alignment of the optical axes of two cameras and the adjustment of the magnification of each camera. Furthermore, because the axes 4
of the cameras do not set up in the normal direction of the measured object, the aspect ratio of the image that is grabbed by this optical system is not one. Therefore, the vertical and horizontal scales of the measured phase map are different. This disagreement of scales in both directions is a large trouble in practical uses in the case of high resolution and high precise measurement [11].
In this paper, the method which can measure simultaneously in-plane and out-of-plane deformations in high resolution is proposed by using the novel optical system that uses only one camera. In this optical system, because a measured object is viewed from the normal direction by one camera, the aspect ratio of the image is one. However, a multi-recording technology as a new technology in speckle interferometry is required, because the information of two directions’ images must be grabbed by one camera. The required feature of new optical system is discussed. And the multi-recording technology [12-14] for speckle interferometry is discussed.
In the experiments for confirming the principle, the experimental apparatus [6, 10] which can produce some optional deformations or displacements is employed. In-plane and out-of-plane displacements are produced precisely by the apparatus. The measured results concerning the in-plane and out-of-plane displacements are discussed. Furthermore, the in-plane and out-of-plane deformations in the buckling of a 5
mechanical beam are measured by the proposed optical system as one of samples of applications of the method. Then, it confirms that the problem depending on the multi-recording technology in the speckle interferometry exists as a low S/N ratio image of grabbed speckle patterns. The solution of the problem is also discussed.
As the results, the phenomena of in-plane and out-of-plane deformations and/or displacements are simultaneously analyzed by the proposed optical system and some filtering technologies. It can be confirmed that the optical system based on the novel speckle interferometry which can solve some troubles in conventional optical systems is constructed.
2. Optical system
When the new optical system [8-10] shown in Fig. 1 (a) is employed, the speckle patterns are grabbed as shown in Fig. 1 (b). Furthermore, when the speckle pattern is transformed by Fourier transform, the intensity of the speckle pattern in frequency domain is shown in Fig. 1 (c). The bias and signal components can be separated in the frequency domain [7-10].The signal surrounded by a broken-line circle shown in Fig. 1 (c) is extracted by a 2-dimensional filter in the frequency domain. Consequently, an
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Measured object O
y
Prism-1 Laser beam Lens
Mirror-1
x (b) Bias Signal
Mirror-2
Prism-2
Pitching angle:θ2
CCD
Yawing angle:θ1
Fy
Fx (c)
(a)
Fig. 1 Speckle interferometry using only two speckle patterns: (a) Optical system (b) Speckle pattern (c) Signal of speckle pattern in frequency domain inverse Fourier Transform is applied. Then, the signal that is separated from a bias component can be extracted. This technology based on the similar idea of the off-axis digital holography [7] is also useful in speckle interferometry. The deformation measurement of the object with rough surfaces can be performed by using this technology.
Furthermore, this technology is introduced to in-plane deformation measurement by adopting the technology of the two-beam interferometer [6]. An in-plane deformation can generally be measured by the conventional two-beam interferometer. In this interferometer, the axis of the camera is right angle with the measured surface, and two beams are irradiated from both sides with the same angle. However, the two-beam
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Out of plane deformation In-plane deformation
y
x
z
Object
θ1 θ2 Prism Camera-A Reference beam-A
M-2
M-1 Prism Beam
Camera-B
Reference beam-B
ND
ND
Laser
Half mirror
Fig.2 In-plane and out-of-plane deformations measurement speckle interferometer using two cameras
interferometer can detect only in-plane deformation because the out-of-plane deformation of the measured object can be canceled by the symmetry of two beams against the optical axis of the camera. The analyzing principle of this interferometer can be explained by the change of optical path distance (OPD) of the laser beam by the deformation of the object.
Using the new optical system shown in Fig. 1 (a) and the idea of two-beam interferometer, a novel optical system [10] which can perform an in-plane and out-of-plane deformation simultaneous measurement by using two cameras can be constructed as shown in Fig.2. An in-plane and out-of-plane deformation measurement
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method by this optical system has been reported [10]. This system can measure precisely deformation. However, there are some difficulties in practical uses as follows.
Laser-A Laser-B P-4
P-3
Reference-A Reference-B
Camera
P-2
P-1
θ2 θ1
Illumination-A
Illumination-B
Measured Out-of-plane deformation Object In-plane deformation Fig.3 New optical system using multi-recording technology
At the time of the setup of the optical system, the exact alignment of the optical axes of the two cameras was required. And, the adjustment of the magnification of each camera is also required. Furthermore, the aspect ratio of the grabbed image is not one, because the camera does not face up to the measured object. In the case of a precise
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measurement, this problem concerning the aspect ratio of an image is a large trouble. To solve such problems, a new optical system as shown in Fig.3 is proposed in this paper.
This optical system is based on the idea of the conventional two-beam interferometer [6] and the new speckle interferometry [8-10] based on off-axis digital holography [7]. In the optical system shown in Fig.3, the laser beams are irradiated to the object surface at θ1 and θ2 from both sides of the camera. These angles are not always required to be same in this optical system, because the feature of the symmetry of two beams against the optical axis of the camera is not employed in this method. That is, this optical system is constructed by two independent optical systems which are set up by each laser and one camera.
Because the optical system is constructed with one camera, this system does not require the exact alignment between optical axes of plural cameras, and the adjustment of the magnification of each camera. Furthermore, the trouble concerning the aspect ratio of the image is saved. However, the multi-recording [13, 14] as a new technology for speckle interferometry is required, because this system must record plural information by using only one camera.
3. Multi-recording technology 10
When plural information must be recorded in one speckle pattern, multi-recording technology is required. Though the multi-recording technology has been already employed conveniently in the off-axis digital holography [14], the technology has not been used functionally enough in speckle interferometry. Both the digital holography and the speckle interferometry are technologies based on the interferometry in the optics. However, the functional purposes between holography and speckle interferometry are different in the practical uses. Therefore, it is thought that the multi-recording technology in digital holography cannot be employed directly in speckle interferometry without any discussions. Generally, digital holography is a technology for recording a three-dimensional image. In the processing of off-axis digital holography, one image is treated, except in-line holography using fringe scanning technologies. Then, a small size speckle is required for recording a high spatial frequency. On the other hand, speckle interferometry is a technology for detecting a deformation of an object. So, plural images which change by a deformation are always treated. Then, speckles in a speckle pattern as an image move by the deformation. And, the shape of speckles also changes by the deformation. Therefore, a large size speckle is employed for keeping spatially overlap of speckle together before and after deformation.
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Under this situation, the multi-recording technology that is suitable for the purpose of the function must be used in each situation of measurement. That is, the multi-recording technology, which is generally used in digital holography [7], may not be able to always use in the speckle interferometry. In the speckle interferometry, a large size speckle is generally used. And, the speckle size in speckle interferometry is not always small as like situation of digital
Fig.4 Relationship between specklesize and accuracy of measured results: (a) 7.5μm
(b)
9.5μm (c) 13.5μm
holography, because the speckle size is required as a suitable size for keeping a high measuring accuracy in speckle interferometry. It has to be understood that the discussion based on the functional purpose is required even if both technologies are based on the same interferometry.
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In Fig.4, one experimental example that is the relationship between measured accuracy and speckle size in the speckle interferometry is shown. Here, when a flat plane is rotated by 1.82×10-4 rad in the optical system shown in Fig.1(a), the difference between the phase change by the real displacement that is given by the PZT and the measured phase map is detected. The relationship between the standard deviation of the distribution of the difference and the diameter of speckle is shown in Fig.4. From this result, in each case of large speckle size (: 13.5 μm) shown in(c) and small speckle size (: 7.5 μm) shown in (a), a high accuracy is not realized. It can be confirmed that there is the optimum size shown in (b) for the high accuracy of measurement in the speckle interferometry. The optimum speckle size for performing a high measuring accuracy is required, because the speckle interferometry is a deformation measurement method. Generally, the small size speckles are favorable in off-axis digital holography in order to get signals including high frequency spatial signals. However, it can be confirmed that it is not enough to set up a small size as speckles for the realization of a high accuracy in the measurement in speckle interferometry. That is, the best condition for the digital holography is not always the best condition for the speckle interferometry.
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In this experiment, the suitable speckle size is set by using experimental results. Furthermore, the central frequency of the signal in the frequency domain must be also set at the best frequency by using experimental results. Generally, the central frequency of the signal in the frequency domain is defined in order to set a wide frequency bandwidth of the signal in digital holography. However, in experimental results of speckle interferometry, it is confirmed that a high frequency component of the signal is decreased by the frequency response function of the camera system. As the results, a high accuracy measurement cannot be realized in the case of the high frequency area. It can be also confirmed that the optimum condition exists for the central frequency of the signal in the speckle interferometry. Because the feature of deformation analysis is quantitatively evaluated as a measuring accuracy in the speckle interferometry, the optimum condition or each feature must be quantitatively discussed. Under such discussions, the optimum condition for each measurement firstly is always checked quantitatively. Optical system based on the optimum condition must be constructed in the speckle interferometry for a high accuracy measurement using the quantitative evaluation. This procedure in the speckle interferometry is the most different important point from the idea of the digital holography.
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To execute the multi-recording for speckle interferometry, the area surrounded by the broken line in Fig.3 is constructed with mirrors and prisms. The object beam from the measured object focuses on the image sensor through the lens and the prism, P-1. On the other hand, reference beams (Reference-A and Reference-B in Fig.3) reach to the camera. Then, the wave front of Reference-B is tilted against the object beam to produce the carriers in the signals. The speckle pattern grabbed by the novel optical system is shown in Fig.5 (a). The speckle pattern in frequency domain after Fourier
Signal-1
Signal-2 (a) Speckle patterns (b) Speckle patterns in frequency (b) domain (a)
Fig.5 Speckle pattern grabbed by the new optical system: (a) Speckle pattern (b) Speckle patterns in frequency domain
transform is also shown in Fig.5 (b). It can be confirmed that the signals concerning Laser-A and Laser-Bare multi-recorded in the frequency domain.
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4.
Separation-analysis
of
in-plane
and
out-of-plane
deformations
from
multi-recorded speckle pattern In Fig.6, assuming that the point-P0 on the measured object moves to the point-P1 by the deformation of the object, the change of the optical path distance of the light taken by the Laser-A corresponds to the sum of length of P1P3 and P1P2. Here, point-P3 is located on the line at right angle with the optical axis of laser-A. Then, the line also includes Point-P0 which is the original point before the deformation.
Δx
Deformed position
Original position
P0
y z
P3
x
θ1
P1 P4
Δy
P2
θ2 Laser-B
Laser-A Camera
Fig.6 Separation of in-plane and out-of-plane deformations Furthermore, when it is defined that Δx is the in-plane component of the deformation and that Δy is the out-of-plane component of the deformation, the total changed optical path distance of laser-A is Δy +Δy/cosθ1-(tanθ1Δy -Δx) sinθ1. 16
On the other hand, the change of the optical path distance of Laser-B is P1P4+P2P1. That is, the total changed optical path distance at Laser-B is Dy+Dy/cosq2-(Dx+ tanq2Dy) sinq2. From the relationship between the change of optical path distance (laser-A: DA and laser-B: DB) and the deformation of Dx and Dy, the simultaneous equations shown in Eq. (1) are given. DA= sinq1Dx+(1+cosq1) Dy
(1)
DB =-sinq2Dx+(1+cosq2) Dy When this simultaneous equation is solved concerning Dx and Dy, Dx and Dy can be defined as Eq.(2). Dx=(DA´b- DB´a)/(b´sinq1+a´sinq2)
(2)
Dy=(DA´sinq2+DB´sinq1)/(b´sinq1+a´sinq2)
Here, a=1+cosq1, b=1+cosq2.
The in-plane and out-of-plane deformations of the measured object can be simultaneously measured from DA and DB which are phase maps of Laser-A and -B by using the Eq. (2). In this paper, the phase maps of Laser-A and -B are detected in high resolution by speckle interferometry using only two speckle patterns shown in Fig.1 (a). 17
5. Experimental results 5.1 Optical system and measured object The optical system shown in Fig.3 is constructed in order to confirm the validity of this method as shown in Fig.7 (a). In Fig.7 (a), one camera and two lasers are employed. θ1 and θ2 are set at 30 degrees. The lasers (output: 100mW, wavelength: 532nm) are used. In the optical system, the illuminating beam to the measured object and the reference beam are divided from each laser. Then, each beam is collimated to
Laser-B
Laser-A Camera
Measured object (b)
(a) Fig.7 Optical system and measured object: (a) Optical system (b) Displacement generator
the parallel beams in the same manner of the speckle interferometer as shown in Fig.1(a). The illuminating beam is emitted to the measured object. The reference beam is guided to two prisms (P-2 and P-1) and the camera as shown in Fig.3. In this 18
process guiding the beams, the fringe carriers are given by setting the mirror angle of each reference beam. As the results, the deformation signals in the frequency domain are multi-recorded as shown in Fig.5 (b).
In the optical system shown in Fig.7 (a), because the camera faces up to the measured object, the image of which the aspect ratio corresponds to one can be taken by the camera. The problems concerning the alignment of the optical axes of two cameras and the adjustment of the magnification of each camera are able to be solved in the new system.
In this paper, the apparatus shown in Fig.7 (b) as the displacement generator produces the in-plane and out-of-plane displacements as a standard reference displacement. The validity of the principle of the proposed method is confirmed by using the reference displacements by the apparatus [6]. The displacement generator is constructed by the rotating table for generating an out-of-plane displacement and the rotating white plate for generating an in-plane displacement as the reference measuring object. The rotating table can be precisely rotated by piezo-actuator (PZT) for generating an out-of-plane displacement. On the other hand, the rotatable white plate that is set up on the rotating table is supported by the horizontal axis with two bearings. Furthermore, the white plate is also precisely rotated by another PZT for generating an 19
in-plane displacement. As the results, the optional out-of-plane and in-plane displacements can be precisely generated by this displacement generator.
5.2 In-plane and out-of-plane displacement measurement
Firstly, only in-plane displacement is given to the measured object. The measured results are shown in Fig.8 (a) and (b). Furthermore, the measured results concerning only out-of-plane displacement are shown in Fig.8(c) and (d). The difference distributions between the real displacements given by the PZTs shown in Fig.7(b) and the measured results shown in Fig.8(a) and (b) are shown in Fig.9. It is clear that the errors of these results both of the in-plane and the out-of-plane measurements are very small. Then, the standard deviations of the in-plane and the out-of-plane measurements are 4.49 nm and 1.44 nm, respectively. From these results, it can be confirmed that the measuring accuracy of the proposed in-plane and out-of-plane displacement measurements method using only one camera is high.
5 532nm
y [mm]
y [mm]
5 2.5 0
2.5 x [mm]
2.5
5
(a)In-plane deformation (a)
0
5 y [mm]
y [mm]
5
(b)Out-of-plane (b) deformation
5 2.5 0
2.5 x [mm]
2.5 x [mm]
5
(c)In-plane deformation (c)
532nm
2.5 0
2.5 20x [mm]
5
(d)Out-of-plane (d) deformation
Fig.8 Measured results of in-plane and out-of-plane displacement: (a) In-plane displacement by only in-plane displacement (b) Out-of-plane displacement by only in-plane displacement (c)In-plane displacement by only out-of-plane displacement (d) Out-of-plane displacement by only out-of-plane displacement
85 0 -85 0
85 0 -85
5 2.5 x [mm]
5
0
2.5 y [mm]
5 2.5 y [mm]
0 2.5 x [mm]
50
(b)
(a)
Fig.9 Error distribution of in-plane and out-of-plane displacement shown in Fig.8 (a) and (d): (a) Error distribution of in-plane displacement shown in Fig. 8(a) (b) Error distribution of out-of-plane displacement shown in Fig. 8(d)
Furthermore, the results of the case of giving simultaneously the in-plane and out-of-plane displacements are shown in Fig.10 (a) and (b). The in-plane and 21
out-of-plane displacement distributions are also detected as shown in Fig.10 (a) and (b) by the separating operation based on Eq. (2). The displacement measurement results shown in Fig.10are performed by the same quantity of displacements of the cases of Fig. 8(a), (b) and Fig. 8(c), (d). It can be thought that the results shown in Fig. 10 (a) and (b) should be same as the sum of two kinds of results shown in Fig.8, because the in-plane and out-of-plane displacements that are same as the displacements shown in Fig.8 are also given to the measured results shown in Fig.10 as the reference displacement.
5 532nm
2.5 0
2.5 x [mm]
532nm
y [mm]
y [mm]
5
2.5
5
0
5
5 y [mm]
y [mm]
5
(b)Out-of-plane (b) deformation
(a)In-plane deformation (a)
2.5 0
2.5 x [mm]
2.5 x [mm]
5
(c)In-plane deformation (c)
532nm
2.5 0
2.5 x [mm]
5
(d) deformation (d)Out-of-plane
Fig.10 Measured results of in-plane and out-of-plane simultaneous displacements and resultant displacement: (a)In-plane displacement by in-plane and out-of-plane displacement (b)Out-of-plane displacement by in-plane and out-of-plane displacement (c)In-plane displacement by resultant of Fig.8 (a) and (c) (d) Out-of-plane displacement by resultant of Fig.8 (b) and (d)
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In this case, if each measurement of the in-plane displacement and the out-of-plane displacement is independently performed, the sum of each in-plane and out-of-plane displacement results shown in Fig.8 (a), (b) and Fig.8 (c), (d), respectively must be same as the results shown in Fig.10 (a) and (b). In Fig.10 (c), the sum of the in-plane displacement results of Fig. 8 (a) and (c) is shown. And, the sum of the out-of-plane displacement results of Fig. 8 (b) and (d) is also shown in Fig.10 (d). It can be confirmed that Fig.10 (a) agrees with Fig.10 (c) and that Fig.10 (b) agrees with Fig.10 (d).
From these results, it can be confirmed that the proposed method can measure simultaneously and independently the in-plane and out-of-plane displacement distributions. Then, the new optical system that uses only one camera doesn't require to alignment of the plural cameras’ optical axes and the adjustment of magnification of the cameras. Furthermore, the problem concerning the aspect ratio of the image is also solved in the new optical system. And, it can be confirmed that the multi-recording technology functions well in the speckle interferometry.
5.3 The deformation analysis in buckling of a mechanical beam
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The buckling phenomenon of a mechanical beam is analyzed by the proposed optical system in order to investigate the validity and problems of this system. Because the buckling phenomenon of a beam simultaneously has the in-plane and the out-of-plane deformations as shown in previous paper [10], it is one of useful measured samples for checking some features of the new optical measurement system.
A measured object is the beam made of phosphor bronze as shown in Fig. 11(a). The left side of the beam is fixed. And, the right side of the beam is set rotatable on the support. Then, the beam is compressed by the axial load of the PZT. When the axial load is over the Euler’s buckling load [16], the buckling phenomenon can be generally
Mechanical beam 22mm
Force PZT
5mm
y
t=0.08mm
x
(a)
(b) Fig.11 Experiment of buckling phenomenon: (a) Apparatus for producing a buckling (b) Specklegram
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observed. The specklegram of the beam under a buckling phenomenon is shown in Fig. 11(b). The specification of the beam is as follows; the width is 5mm, length is 22 mm, and thickness is 0.08mm. The Euler’s buckling load can be calculated as 0.478 N [16], then the Young modulus of phosphor bronze is 110×109Pa.
The in-plane and out-of-plane deformations are shown in Fig. 12 (c) and (d) by the same manner of the analysis method shown in the previous section. Though the out-of-plane deformation shown in Fig.12(d) can be observed as the same manner as the results in the previous paper [10], the out-of-plane deformation includes some distortion. On the other hand, the in-plane deformation is shown in Fig.12(c). However, the in-plane deformation is too noisy to confirm the real deformation. The fluctuation in the deformation results in both A-direction and B-direction shown in Fig.12(a) and (b) causes these problems. Furthermore, the deterioration of the ratio of signal and noise of the speckle pattern that is grabbed by the multi-recording technology causes the deterioration of the result shown in Fig.12(c) and (d). Such problems by the deterioration of S/N ratio were not able to be observed in the results that were grabbed by single recording technology in the previous paper [10]. It is thought that this problem is caused by the multi-recording technology.
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In Fig.13, two speckle patterns that are grabbed by using the multi-recording and the single recording technologies are shown in the frequency domain.
In the case of only single illumination of this optical system, the signal in frequency domain is strongly recorded as shown in Fig.13(d). On the other hand, it can be confirmed that the signal is weak when both right and left lasers are simultaneously emitted to the mechanical beam in the case of Fig.13(b), because one camera grabs two images in two directions. It can be thought that the dynamic range of the camera is decreased corresponding to only one bit of the A/D transform. However, the influence that is much more than one bit of A/D transform would be observed in the
-0.5
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2 0
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Deformation [nm]
Deformation [μm]
multi-recording technology in practical use of speckle interferometry.
22 0
0
1.5
-0.5 0
y [mm] x [mm] (d)
2 0
Fig.12 Measured results by conventional processing: (a) Deformation in A-direction (b) Deformation in B-direction (c) In-plane deformation (d) Out-of-plane deformation 26
Signal (b)
(d)
Fy
Fy
Fx
Fx
(c)
(a)
Fig.13 S/N ratio in multi-recording technology: (a) Multi-recording (b) Multi-recording (Magnification) (c) Single-recording (d) Single-recording (Magnification)
That is, it is confirmed that the S/N ratio of the speckle pattern decreases strongly in the multi-recording technology.
Area-A Fx [period/pixel]
Fy Area-B [period/pixel] Fx [period/pixel]
(b)
1/256 3/512
Fx [period/pixel] (c)
Fy [period/pixel]
(a) Fy [period/pixel]
Fy [period/pixel]
1/32 35/512
Fx [period/pixel] (d)
Fig.14 Signal of specklegram in frequency domain: (a) Rotation of plane (b) Buckling of beam (c) Magnification of Area-A (d) Magnification of Area-B 27
In particular, the in-plane deformation shown in Fig.8 and 10 by rotating the flat plane has only simple modulation in the frequency domain as shown in Fig.14 (a). Fig.14(c) shows the magnification of Area-Ain Fig.14(a). The signal in the frequency domain is simple. Therefore, the influence of the deterioration of the ratio of signal and noise of the speckle pattern can be neglected. However, it can be confirmed that the influence of the deterioration of S/N in the complex modulation that happens in the buckling phenomenon as shown in Fig.14(b) cannot be neglected.
The signal in
frequency domain is very complex as shown in Fig.14(d). The signal widely distributes in the frequency domain. And, it includes a lot of noise. Though the multi-recording technology is very useful, it can be also confirmed that the additional processing for
1.5 y [mm]
0 x [mm] (a)
Amplitude [-]
Amplitude [-]
Fluctuations
1.5 y [mm]
0 x [mm]
2 0
(b)
2 0
Fig.15 Fluctuation in specklegram: (a) Without averaging filter (b) With averaging filter
avoiding such influences is required when the technology is employed in the measurement concerning some complex shapes.
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As the results, the specklegram includes some fluctuations as shown in Fig.15(a). The influence of the fluctuation cannot be removed by using only filtering based on Fourier transform. To reduce this influence, the area filter based on averaging processing (processing area: 50 pixels ×20pixels) is employed to the specklegram. The result is shown in Fig.15(b). The influence by the low S/N ratio is reduced effectively by the filter.
Finally, the in-plane and out-of-plane deformations in the buckling phenomenon of the beam are shown in Fig.16 (c) and (d). Though the in-plane deformation shown in Fig.16(c) includes somewhat fluctuation, almost proportional relationship between the
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(b)
A
40
Deformation [μm]
0
2 0
Deformation [μm]
Deformation [μm]
deformation and the position of the beam can be observed.
0 B -0.5 0
(c)
x [mm]
1.5 B
y [mm]
20
(d)
Fig.16 Measured results of buckling phenomenon by proposed optical system: (a) Deformation in A-direction 29 (b) Deformation in B-direction (c) In-plane deformation (d) Out-of-plane deformation
That is, it can be thought that the axial strain of the beam is almost constant.
The results in A- and B- directions are also smooth as shown in Fig.16(a) and (b).The section A-A and B-B in Fig.16(c) and (d) are shown in Fig.17 (a) and (b), respectively.
In Fig.17(a), the value of the in-plane deformation is shown as zero at the point at which the out-of-plane deformation is maximum. As the practical value, however, the in-plane deformation is not zero at the point of the maximum deflection of the beam. In this experiment, the data of beam over whole area were not recorded in one fringe pattern image at the present optical system in the same manner as previous paper [10]. Therefore, the quantity of the in-plane deformation at the point of the maximum deflection is also defined as zero in this paper.
From section A-A shown in Fig.17(a), the axial strain in the beam can be calculated as 11.5×10-6. The axial load can be also calculated as 0.506 N. That is, it can be confirmed that the real axial load of the beam is larger than the Euler’s buckling load. Therefore, it can be thought that the buckling phenomenon happens in the beam.
30
In this experiment, though the quantity of the in-plane deformation is very small, it is confirmed that both the in-plane and out-of-plane deformations can be detected by the
Deformation [nm]
new speckle interferometer. 25 0 0 -25
2 1 x [mm]
Deformation [μm]
(a) : Measured result : Euler buckling theory
0 0 -0.5
1 x [mm]
2
(b)
Fig.17 Measured results of in-plane and out-of-plane deformations: (a) In-plane deformation (section A-A) (b) Out-of-plane deformation (section B-B)
6. Conclusion
In this paper, the measurement method that can measure simultaneously in-plane and out-of-plane deformations in high resolution by using multi-recording method is proposed. In this method, the deformation analysis by using only two speckle patterns before and after a deformation by only one camera can be performed. From experimental results, it can be also confirmed that the in-plane and out-of-plane displacements can be detected in high resolution by this system. Furthermore, it can be confirmed that the 31
in-plane and out-of-plane deformations in the buckling phenomenon can be detected by using the optimum measurement condition concerning the multi-recording and the filtering technologies for the speckle interferometry.
REFERENCE
[1] R. S. Sirohi, Speckle Metrology, Marcel Dekker, New York,1993, pp.99-234. [2] B. J. Thompson, Selected papers on Electronic Speckle Pattern Interferometry Principles and Practice, SPIE Optical Engineering Press, Bellingham, Washington, 1996,pp.1-518. [3] G. Cloud, Optical Methods of Engineering Analysis, Cambridge University Press, New York, NY,1995, pp.395-476. [4] D. Malacara, Optical Shop Testing, John Wiley &Sons, NY, 1992, pp.501-652. [5] O.J. Lokberg, G.A.Slettemoen, Basic Electronic Speckle-pattern Interferometry,In Applied Optics and Optical Engineering, Vol.X Academic Press, San Diego, 1987, pp.455-504. [6] Y. Arai, S. Yokozeki, In-plane displacement measurement using electronic speckle pattern interferometry based on spatial fringe analysis method, Opt. Eng., 43 32
(2004)2168-2174. [7] N. Pavillon, S. C.Seelamantula, J. Kuhn, M. Unser, C. Depeursinge, Suppression of the zero-oder term in off-axis digital holography through nonlinear filtering, Appl. Opt., 48 (2009)H186-H195. [8] Y. Arai, Electronic Speckle Pattern Interferometry based on spatial information using only two sheets of speckle patterns, Modern Optics., 61 (2014)297-306. [9] Y. Arai, Improvement of measuring accuracy of spatial fringe analysis method using only two speckle patterns in electronic speckle pattern interferometry, Opt. Eng. 53(2014) 034107. [10] Y. Arai, Development of in-plane and out-of-plane deformation simultaneous measurement method for the analysis of buckling, Opt. Eng. 54,(2015)024102. [11] Y. Arai, Measuring accuracy of in-plane and out-of-plane deformation simultaneous measurement of speckle interferometry using multi-recording method, Proceedings of SPIE Vol.9489, Baltimore, (2015) 9489-14. [12] A. W.Lohmann, Reconstruction of vectorial wave fronts, Appl. Opt., 4(1965)1667-1668. [13] A.A. Friesem, J. Fedorowicz, Multicolor wavefront reconstruction, Appl. Opt., 6 (1967)529-536. [14]J.Kuhn, T. Colomb, F. Montfort, F. Charriere, Y. Emery, E. Cuche, P. Marquet, C. Depeursinge, Real-time dual-wavelength digital holographic microscopy with a single hologram acquisition, Opt. Exp. ,15 (2007)7231-7242. [15] M. Takeda, H. Ina, and S. Kobayashi, Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry, J. Opt. Soc. Am., 72(1982)156-160. [16] F. Bleich, Buckling strength of metal structures, McGRAW-HILL, New York, NY, 1952, pp.1-54
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Caption Fig. 1 Speckle interferometry using only two speckle patterns: (a)Optical system (b) Speckle pattern (c) Signal of speckle pattern in frequency domain Fig.2 In-plane and out-of-plane deformations measurement speckle interferometer using two cameras Fig.3 New optical system using multi-recording technology Fig.4 Relationship between specklesize and accuracy of measured results: (a) 7.5μm (b) 9.5μm (c) 13.5μm Fig.5 Speckle pattern grabbed by the new optical system: (a) Speckle pattern (b) Speckle patterns in frequency domain Fig.6 Separation of in-plane and out-of-plane deformations Fig.7 Optical system and measured object: (a) Optical system (b) Displacement generator Fig.8 Measured results of in-plane and out-of-plane displacement: (a)In-plane displacement by only in-plane displacement (b) Out-of-plane displacement by only in-plane displacement (c) In-plane displacement by only out-of-plane displacement (d) Out-of-plane displacement by only out-of-plane displacement Fig.9 Error distribution of in-plane and out-of-plane displacement shown in Fig.8 (a) and (d): (a) Error distribution of in-plane displacement shown in Fig. 8(a) (b) Error distribution of out-of-plane displacement shown in Fig. 8(d)
34
Fig.10 Measured results of in-plane and out-of-plane simultaneous displacements and resultant displacement: (a)In-plane displacement by in-plane and out-of-plane displacement (b)Out-of-plane displacement by in-plane and out-of-plane displacement (c)In-plane displacement by resultant of Fig.8 (a) and (c) (d)Out-of-plane displacement by resultant of Fig.8 (b) and (d) Fig.11 Experiment of buckling phenomenon (a)Apparatus for producing a buckling (b)Specklegram Fig.12 Measured results by conventional processing (a)Deformation in A-direction (b) Deformation in B-direction (c) In-plane deformation (d) Out-of-plane deformation Fig.13 S/N ratio in multi-recording technology (a)Multi-recording (b)Multi-recording (Magnification) (c)Single-recording (d)Single-recording (Magnification) Fig.14 Signal of specklegram in frequency domain (a)Rotation of plane (b)Buckling of beam (c)Magnification of Area-A (d)Magnification of Area-B Fig.15 Fluctuation in specklegram (a)Without averaging filter (b)With averaging filter Fig.16 Measured results of buckling phenomenon by proposed optical system (a) Deformation in A-direction (b) Deformation in B-direction (c) In-plane deformation (d) Out-of-plane deformation 35
Fig.17 Measured results of in-plane and out-of-plane deformations (a)In-plane deformation (section A-A) (b)Out-of-plane deformation (section B-B)
36
S0263-2241(16)30183-X http://dx.doi.org/10.1016/j.measurement.2016.05.037 MEASUR 4053
To appear in:
Measurement
Received Date: Revised Date: Accepted Date:
7 August 2015 1 March 2016 10 May 2016
Please cite this article as: Y. Arai, Simultaneous in-plane and out-of-plane deformation measurement by speckle multi-recording method, Measurement (2016), doi: http://dx.doi.org/10.1016/j.measurement.2016.05.037
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Title page Paper title: Simultaneous in-plane and out-of-plane deformation measurement by speckle multi-recording method
Name of author: Yasuhiko Arai Affiliation: Department of Mechanical Engineering, Faculty of Engineering Science, Kansai University e-mail: [email protected] Mailing address: 3-3-35, Yamate-cho, Suita, Osaka, Japan 564-8680 Phone number: +81-6-6368-0778 Fax number: +81-6-6388-8785
1
Highlight ・In-plane and out-of-plane simultaneous deformation measurement method based on speckle interferometry is proposed. ・Multi-recording technology in speckle interferometry is also proposed. ・The problems in the multi-recording technology are discussed. ・The new method is applied to the analysis of phenomenon of buckling in a beam.
2
Abstract
An in-plane and out-of-plane deformation simultaneous measurement method using only two speckle patterns grabbed before and after deformation of an object with rough surfaces using two cameras has been proposed. However, there are some problems concerning the setup of optical system, the aspect ratio of grabbed image data and so on in the conventional method. To solve these problems, the new optical system is proposed
by
using
the
multi-recording
technology.
Though
multi-recording
technologies have been already employed conveniently in the off-axis digital holography, the technologies have not been used functionally enough in speckle interferometry. In this paper, an in-plane and out-of-plane deformation simultaneous measurement is performed by the proposed method based on multi-recording technology in speckle interferometry. From experimental results, it can be confirmed that the proposed optical system can measure simultaneously in-plane and out-of-plane deformations in high resolution by one camera.
Keywords: ESPI, multi-recording technology, in-plane and out-of-plane speckle deformation measurement, and buckling.
3
1. Introduction
Deformation measurement methods based on the speckle phenomenon were developed as speckle interferometry in the 1960s[1-4]. The technologies have been employed in the field of vibration analysis. ESPI (electronic speckle pattern interferometry) was also developed by introducing the TV-camera technology into speckle interferometry [5]. The fringe scanning technologies which were developed in the 1980s were also introduced to the speckle interferometry for improving the measurement accuracy [4]. Then, the resolving power of the speckle interferometry was improved to about 1/100 of the wavelength of the light source [6]. A deformation analysis by a new speckle interferometer that is based on a similar idea of the off-axes digital holography [7] has been performed using only two speckle patterns [8-10]. At the present time, the out-of-plane and in-plane deformations can be measured by the optical system based on two-beam speckle interferometry [10]. Then, the optical system was constructed by a pair of the new speckle interferometers. The high resolution deformation measurement was also realized by using fringe scanning technologies and two cameras. However, the speckle interferometer using two cameras requires the pre-processing of the measurement [10], for example; the exact alignment of the optical axes of two cameras and the adjustment of the magnification of each camera. Furthermore, because the axes 4
of the cameras do not set up in the normal direction of the measured object, the aspect ratio of the image that is grabbed by this optical system is not one. Therefore, the vertical and horizontal scales of the measured phase map are different. This disagreement of scales in both directions is a large trouble in practical uses in the case of high resolution and high precise measurement [11].
In this paper, the method which can measure simultaneously in-plane and out-of-plane deformations in high resolution is proposed by using the novel optical system that uses only one camera. In this optical system, because a measured object is viewed from the normal direction by one camera, the aspect ratio of the image is one. However, a multi-recording technology as a new technology in speckle interferometry is required, because the information of two directions’ images must be grabbed by one camera. The required feature of new optical system is discussed. And the multi-recording technology [12-14] for speckle interferometry is discussed.
In the experiments for confirming the principle, the experimental apparatus [6, 10] which can produce some optional deformations or displacements is employed. In-plane and out-of-plane displacements are produced precisely by the apparatus. The measured results concerning the in-plane and out-of-plane displacements are discussed. Furthermore, the in-plane and out-of-plane deformations in the buckling of a 5
mechanical beam are measured by the proposed optical system as one of samples of applications of the method. Then, it confirms that the problem depending on the multi-recording technology in the speckle interferometry exists as a low S/N ratio image of grabbed speckle patterns. The solution of the problem is also discussed.
As the results, the phenomena of in-plane and out-of-plane deformations and/or displacements are simultaneously analyzed by the proposed optical system and some filtering technologies. It can be confirmed that the optical system based on the novel speckle interferometry which can solve some troubles in conventional optical systems is constructed.
2. Optical system
When the new optical system [8-10] shown in Fig. 1 (a) is employed, the speckle patterns are grabbed as shown in Fig. 1 (b). Furthermore, when the speckle pattern is transformed by Fourier transform, the intensity of the speckle pattern in frequency domain is shown in Fig. 1 (c). The bias and signal components can be separated in the frequency domain [7-10].The signal surrounded by a broken-line circle shown in Fig. 1 (c) is extracted by a 2-dimensional filter in the frequency domain. Consequently, an
6
Measured object O
y
Prism-1 Laser beam Lens
Mirror-1
x (b) Bias Signal
Mirror-2
Prism-2
Pitching angle:θ2
CCD
Yawing angle:θ1
Fy
Fx (c)
(a)
Fig. 1 Speckle interferometry using only two speckle patterns: (a) Optical system (b) Speckle pattern (c) Signal of speckle pattern in frequency domain inverse Fourier Transform is applied. Then, the signal that is separated from a bias component can be extracted. This technology based on the similar idea of the off-axis digital holography [7] is also useful in speckle interferometry. The deformation measurement of the object with rough surfaces can be performed by using this technology.
Furthermore, this technology is introduced to in-plane deformation measurement by adopting the technology of the two-beam interferometer [6]. An in-plane deformation can generally be measured by the conventional two-beam interferometer. In this interferometer, the axis of the camera is right angle with the measured surface, and two beams are irradiated from both sides with the same angle. However, the two-beam
7
Out of plane deformation In-plane deformation
y
x
z
Object
θ1 θ2 Prism Camera-A Reference beam-A
M-2
M-1 Prism Beam
Camera-B
Reference beam-B
ND
ND
Laser
Half mirror
Fig.2 In-plane and out-of-plane deformations measurement speckle interferometer using two cameras
interferometer can detect only in-plane deformation because the out-of-plane deformation of the measured object can be canceled by the symmetry of two beams against the optical axis of the camera. The analyzing principle of this interferometer can be explained by the change of optical path distance (OPD) of the laser beam by the deformation of the object.
Using the new optical system shown in Fig. 1 (a) and the idea of two-beam interferometer, a novel optical system [10] which can perform an in-plane and out-of-plane deformation simultaneous measurement by using two cameras can be constructed as shown in Fig.2. An in-plane and out-of-plane deformation measurement
8
method by this optical system has been reported [10]. This system can measure precisely deformation. However, there are some difficulties in practical uses as follows.
Laser-A Laser-B P-4
P-3
Reference-A Reference-B
Camera
P-2
P-1
θ2 θ1
Illumination-A
Illumination-B
Measured Out-of-plane deformation Object In-plane deformation Fig.3 New optical system using multi-recording technology
At the time of the setup of the optical system, the exact alignment of the optical axes of the two cameras was required. And, the adjustment of the magnification of each camera is also required. Furthermore, the aspect ratio of the grabbed image is not one, because the camera does not face up to the measured object. In the case of a precise
9
measurement, this problem concerning the aspect ratio of an image is a large trouble. To solve such problems, a new optical system as shown in Fig.3 is proposed in this paper.
This optical system is based on the idea of the conventional two-beam interferometer [6] and the new speckle interferometry [8-10] based on off-axis digital holography [7]. In the optical system shown in Fig.3, the laser beams are irradiated to the object surface at θ1 and θ2 from both sides of the camera. These angles are not always required to be same in this optical system, because the feature of the symmetry of two beams against the optical axis of the camera is not employed in this method. That is, this optical system is constructed by two independent optical systems which are set up by each laser and one camera.
Because the optical system is constructed with one camera, this system does not require the exact alignment between optical axes of plural cameras, and the adjustment of the magnification of each camera. Furthermore, the trouble concerning the aspect ratio of the image is saved. However, the multi-recording [13, 14] as a new technology for speckle interferometry is required, because this system must record plural information by using only one camera.
3. Multi-recording technology 10
When plural information must be recorded in one speckle pattern, multi-recording technology is required. Though the multi-recording technology has been already employed conveniently in the off-axis digital holography [14], the technology has not been used functionally enough in speckle interferometry. Both the digital holography and the speckle interferometry are technologies based on the interferometry in the optics. However, the functional purposes between holography and speckle interferometry are different in the practical uses. Therefore, it is thought that the multi-recording technology in digital holography cannot be employed directly in speckle interferometry without any discussions. Generally, digital holography is a technology for recording a three-dimensional image. In the processing of off-axis digital holography, one image is treated, except in-line holography using fringe scanning technologies. Then, a small size speckle is required for recording a high spatial frequency. On the other hand, speckle interferometry is a technology for detecting a deformation of an object. So, plural images which change by a deformation are always treated. Then, speckles in a speckle pattern as an image move by the deformation. And, the shape of speckles also changes by the deformation. Therefore, a large size speckle is employed for keeping spatially overlap of speckle together before and after deformation.
11
Under this situation, the multi-recording technology that is suitable for the purpose of the function must be used in each situation of measurement. That is, the multi-recording technology, which is generally used in digital holography [7], may not be able to always use in the speckle interferometry. In the speckle interferometry, a large size speckle is generally used. And, the speckle size in speckle interferometry is not always small as like situation of digital
Fig.4 Relationship between specklesize and accuracy of measured results: (a) 7.5μm
(b)
9.5μm (c) 13.5μm
holography, because the speckle size is required as a suitable size for keeping a high measuring accuracy in speckle interferometry. It has to be understood that the discussion based on the functional purpose is required even if both technologies are based on the same interferometry.
12
In Fig.4, one experimental example that is the relationship between measured accuracy and speckle size in the speckle interferometry is shown. Here, when a flat plane is rotated by 1.82×10-4 rad in the optical system shown in Fig.1(a), the difference between the phase change by the real displacement that is given by the PZT and the measured phase map is detected. The relationship between the standard deviation of the distribution of the difference and the diameter of speckle is shown in Fig.4. From this result, in each case of large speckle size (: 13.5 μm) shown in(c) and small speckle size (: 7.5 μm) shown in (a), a high accuracy is not realized. It can be confirmed that there is the optimum size shown in (b) for the high accuracy of measurement in the speckle interferometry. The optimum speckle size for performing a high measuring accuracy is required, because the speckle interferometry is a deformation measurement method. Generally, the small size speckles are favorable in off-axis digital holography in order to get signals including high frequency spatial signals. However, it can be confirmed that it is not enough to set up a small size as speckles for the realization of a high accuracy in the measurement in speckle interferometry. That is, the best condition for the digital holography is not always the best condition for the speckle interferometry.
13
In this experiment, the suitable speckle size is set by using experimental results. Furthermore, the central frequency of the signal in the frequency domain must be also set at the best frequency by using experimental results. Generally, the central frequency of the signal in the frequency domain is defined in order to set a wide frequency bandwidth of the signal in digital holography. However, in experimental results of speckle interferometry, it is confirmed that a high frequency component of the signal is decreased by the frequency response function of the camera system. As the results, a high accuracy measurement cannot be realized in the case of the high frequency area. It can be also confirmed that the optimum condition exists for the central frequency of the signal in the speckle interferometry. Because the feature of deformation analysis is quantitatively evaluated as a measuring accuracy in the speckle interferometry, the optimum condition or each feature must be quantitatively discussed. Under such discussions, the optimum condition for each measurement firstly is always checked quantitatively. Optical system based on the optimum condition must be constructed in the speckle interferometry for a high accuracy measurement using the quantitative evaluation. This procedure in the speckle interferometry is the most different important point from the idea of the digital holography.
14
To execute the multi-recording for speckle interferometry, the area surrounded by the broken line in Fig.3 is constructed with mirrors and prisms. The object beam from the measured object focuses on the image sensor through the lens and the prism, P-1. On the other hand, reference beams (Reference-A and Reference-B in Fig.3) reach to the camera. Then, the wave front of Reference-B is tilted against the object beam to produce the carriers in the signals. The speckle pattern grabbed by the novel optical system is shown in Fig.5 (a). The speckle pattern in frequency domain after Fourier
Signal-1
Signal-2 (a) Speckle patterns (b) Speckle patterns in frequency (b) domain (a)
Fig.5 Speckle pattern grabbed by the new optical system: (a) Speckle pattern (b) Speckle patterns in frequency domain
transform is also shown in Fig.5 (b). It can be confirmed that the signals concerning Laser-A and Laser-Bare multi-recorded in the frequency domain.
15
4.
Separation-analysis
of
in-plane
and
out-of-plane
deformations
from
multi-recorded speckle pattern In Fig.6, assuming that the point-P0 on the measured object moves to the point-P1 by the deformation of the object, the change of the optical path distance of the light taken by the Laser-A corresponds to the sum of length of P1P3 and P1P2. Here, point-P3 is located on the line at right angle with the optical axis of laser-A. Then, the line also includes Point-P0 which is the original point before the deformation.
Δx
Deformed position
Original position
P0
y z
P3
x
θ1
P1 P4
Δy
P2
θ2 Laser-B
Laser-A Camera
Fig.6 Separation of in-plane and out-of-plane deformations Furthermore, when it is defined that Δx is the in-plane component of the deformation and that Δy is the out-of-plane component of the deformation, the total changed optical path distance of laser-A is Δy +Δy/cosθ1-(tanθ1Δy -Δx) sinθ1. 16
On the other hand, the change of the optical path distance of Laser-B is P1P4+P2P1. That is, the total changed optical path distance at Laser-B is Dy+Dy/cosq2-(Dx+ tanq2Dy) sinq2. From the relationship between the change of optical path distance (laser-A: DA and laser-B: DB) and the deformation of Dx and Dy, the simultaneous equations shown in Eq. (1) are given. DA= sinq1Dx+(1+cosq1) Dy
(1)
DB =-sinq2Dx+(1+cosq2) Dy When this simultaneous equation is solved concerning Dx and Dy, Dx and Dy can be defined as Eq.(2). Dx=(DA´b- DB´a)/(b´sinq1+a´sinq2)
(2)
Dy=(DA´sinq2+DB´sinq1)/(b´sinq1+a´sinq2)
Here, a=1+cosq1, b=1+cosq2.
The in-plane and out-of-plane deformations of the measured object can be simultaneously measured from DA and DB which are phase maps of Laser-A and -B by using the Eq. (2). In this paper, the phase maps of Laser-A and -B are detected in high resolution by speckle interferometry using only two speckle patterns shown in Fig.1 (a). 17
5. Experimental results 5.1 Optical system and measured object The optical system shown in Fig.3 is constructed in order to confirm the validity of this method as shown in Fig.7 (a). In Fig.7 (a), one camera and two lasers are employed. θ1 and θ2 are set at 30 degrees. The lasers (output: 100mW, wavelength: 532nm) are used. In the optical system, the illuminating beam to the measured object and the reference beam are divided from each laser. Then, each beam is collimated to
Laser-B
Laser-A Camera
Measured object (b)
(a) Fig.7 Optical system and measured object: (a) Optical system (b) Displacement generator
the parallel beams in the same manner of the speckle interferometer as shown in Fig.1(a). The illuminating beam is emitted to the measured object. The reference beam is guided to two prisms (P-2 and P-1) and the camera as shown in Fig.3. In this 18
process guiding the beams, the fringe carriers are given by setting the mirror angle of each reference beam. As the results, the deformation signals in the frequency domain are multi-recorded as shown in Fig.5 (b).
In the optical system shown in Fig.7 (a), because the camera faces up to the measured object, the image of which the aspect ratio corresponds to one can be taken by the camera. The problems concerning the alignment of the optical axes of two cameras and the adjustment of the magnification of each camera are able to be solved in the new system.
In this paper, the apparatus shown in Fig.7 (b) as the displacement generator produces the in-plane and out-of-plane displacements as a standard reference displacement. The validity of the principle of the proposed method is confirmed by using the reference displacements by the apparatus [6]. The displacement generator is constructed by the rotating table for generating an out-of-plane displacement and the rotating white plate for generating an in-plane displacement as the reference measuring object. The rotating table can be precisely rotated by piezo-actuator (PZT) for generating an out-of-plane displacement. On the other hand, the rotatable white plate that is set up on the rotating table is supported by the horizontal axis with two bearings. Furthermore, the white plate is also precisely rotated by another PZT for generating an 19
in-plane displacement. As the results, the optional out-of-plane and in-plane displacements can be precisely generated by this displacement generator.
5.2 In-plane and out-of-plane displacement measurement
Firstly, only in-plane displacement is given to the measured object. The measured results are shown in Fig.8 (a) and (b). Furthermore, the measured results concerning only out-of-plane displacement are shown in Fig.8(c) and (d). The difference distributions between the real displacements given by the PZTs shown in Fig.7(b) and the measured results shown in Fig.8(a) and (b) are shown in Fig.9. It is clear that the errors of these results both of the in-plane and the out-of-plane measurements are very small. Then, the standard deviations of the in-plane and the out-of-plane measurements are 4.49 nm and 1.44 nm, respectively. From these results, it can be confirmed that the measuring accuracy of the proposed in-plane and out-of-plane displacement measurements method using only one camera is high.
5 532nm
y [mm]
y [mm]
5 2.5 0
2.5 x [mm]
2.5
5
(a)In-plane deformation (a)
0
5 y [mm]
y [mm]
5
(b)Out-of-plane (b) deformation
5 2.5 0
2.5 x [mm]
2.5 x [mm]
5
(c)In-plane deformation (c)
532nm
2.5 0
2.5 20x [mm]
5
(d)Out-of-plane (d) deformation
Fig.8 Measured results of in-plane and out-of-plane displacement: (a) In-plane displacement by only in-plane displacement (b) Out-of-plane displacement by only in-plane displacement (c)In-plane displacement by only out-of-plane displacement (d) Out-of-plane displacement by only out-of-plane displacement
85 0 -85 0
85 0 -85
5 2.5 x [mm]
5
0
2.5 y [mm]
5 2.5 y [mm]
0 2.5 x [mm]
50
(b)
(a)
Fig.9 Error distribution of in-plane and out-of-plane displacement shown in Fig.8 (a) and (d): (a) Error distribution of in-plane displacement shown in Fig. 8(a) (b) Error distribution of out-of-plane displacement shown in Fig. 8(d)
Furthermore, the results of the case of giving simultaneously the in-plane and out-of-plane displacements are shown in Fig.10 (a) and (b). The in-plane and 21
out-of-plane displacement distributions are also detected as shown in Fig.10 (a) and (b) by the separating operation based on Eq. (2). The displacement measurement results shown in Fig.10are performed by the same quantity of displacements of the cases of Fig. 8(a), (b) and Fig. 8(c), (d). It can be thought that the results shown in Fig. 10 (a) and (b) should be same as the sum of two kinds of results shown in Fig.8, because the in-plane and out-of-plane displacements that are same as the displacements shown in Fig.8 are also given to the measured results shown in Fig.10 as the reference displacement.
5 532nm
2.5 0
2.5 x [mm]
532nm
y [mm]
y [mm]
5
2.5
5
0
5
5 y [mm]
y [mm]
5
(b)Out-of-plane (b) deformation
(a)In-plane deformation (a)
2.5 0
2.5 x [mm]
2.5 x [mm]
5
(c)In-plane deformation (c)
532nm
2.5 0
2.5 x [mm]
5
(d) deformation (d)Out-of-plane
Fig.10 Measured results of in-plane and out-of-plane simultaneous displacements and resultant displacement: (a)In-plane displacement by in-plane and out-of-plane displacement (b)Out-of-plane displacement by in-plane and out-of-plane displacement (c)In-plane displacement by resultant of Fig.8 (a) and (c) (d) Out-of-plane displacement by resultant of Fig.8 (b) and (d)
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In this case, if each measurement of the in-plane displacement and the out-of-plane displacement is independently performed, the sum of each in-plane and out-of-plane displacement results shown in Fig.8 (a), (b) and Fig.8 (c), (d), respectively must be same as the results shown in Fig.10 (a) and (b). In Fig.10 (c), the sum of the in-plane displacement results of Fig. 8 (a) and (c) is shown. And, the sum of the out-of-plane displacement results of Fig. 8 (b) and (d) is also shown in Fig.10 (d). It can be confirmed that Fig.10 (a) agrees with Fig.10 (c) and that Fig.10 (b) agrees with Fig.10 (d).
From these results, it can be confirmed that the proposed method can measure simultaneously and independently the in-plane and out-of-plane displacement distributions. Then, the new optical system that uses only one camera doesn't require to alignment of the plural cameras’ optical axes and the adjustment of magnification of the cameras. Furthermore, the problem concerning the aspect ratio of the image is also solved in the new optical system. And, it can be confirmed that the multi-recording technology functions well in the speckle interferometry.
5.3 The deformation analysis in buckling of a mechanical beam
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The buckling phenomenon of a mechanical beam is analyzed by the proposed optical system in order to investigate the validity and problems of this system. Because the buckling phenomenon of a beam simultaneously has the in-plane and the out-of-plane deformations as shown in previous paper [10], it is one of useful measured samples for checking some features of the new optical measurement system.
A measured object is the beam made of phosphor bronze as shown in Fig. 11(a). The left side of the beam is fixed. And, the right side of the beam is set rotatable on the support. Then, the beam is compressed by the axial load of the PZT. When the axial load is over the Euler’s buckling load [16], the buckling phenomenon can be generally
Mechanical beam 22mm
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(b) Fig.11 Experiment of buckling phenomenon: (a) Apparatus for producing a buckling (b) Specklegram
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observed. The specklegram of the beam under a buckling phenomenon is shown in Fig. 11(b). The specification of the beam is as follows; the width is 5mm, length is 22 mm, and thickness is 0.08mm. The Euler’s buckling load can be calculated as 0.478 N [16], then the Young modulus of phosphor bronze is 110×109Pa.
The in-plane and out-of-plane deformations are shown in Fig. 12 (c) and (d) by the same manner of the analysis method shown in the previous section. Though the out-of-plane deformation shown in Fig.12(d) can be observed as the same manner as the results in the previous paper [10], the out-of-plane deformation includes some distortion. On the other hand, the in-plane deformation is shown in Fig.12(c). However, the in-plane deformation is too noisy to confirm the real deformation. The fluctuation in the deformation results in both A-direction and B-direction shown in Fig.12(a) and (b) causes these problems. Furthermore, the deterioration of the ratio of signal and noise of the speckle pattern that is grabbed by the multi-recording technology causes the deterioration of the result shown in Fig.12(c) and (d). Such problems by the deterioration of S/N ratio were not able to be observed in the results that were grabbed by single recording technology in the previous paper [10]. It is thought that this problem is caused by the multi-recording technology.
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In Fig.13, two speckle patterns that are grabbed by using the multi-recording and the single recording technologies are shown in the frequency domain.
In the case of only single illumination of this optical system, the signal in frequency domain is strongly recorded as shown in Fig.13(d). On the other hand, it can be confirmed that the signal is weak when both right and left lasers are simultaneously emitted to the mechanical beam in the case of Fig.13(b), because one camera grabs two images in two directions. It can be thought that the dynamic range of the camera is decreased corresponding to only one bit of the A/D transform. However, the influence that is much more than one bit of A/D transform would be observed in the
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multi-recording technology in practical use of speckle interferometry.
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Fig.12 Measured results by conventional processing: (a) Deformation in A-direction (b) Deformation in B-direction (c) In-plane deformation (d) Out-of-plane deformation 26
Signal (b)
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Fig.13 S/N ratio in multi-recording technology: (a) Multi-recording (b) Multi-recording (Magnification) (c) Single-recording (d) Single-recording (Magnification)
That is, it is confirmed that the S/N ratio of the speckle pattern decreases strongly in the multi-recording technology.
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Fig.14 Signal of specklegram in frequency domain: (a) Rotation of plane (b) Buckling of beam (c) Magnification of Area-A (d) Magnification of Area-B 27
In particular, the in-plane deformation shown in Fig.8 and 10 by rotating the flat plane has only simple modulation in the frequency domain as shown in Fig.14 (a). Fig.14(c) shows the magnification of Area-Ain Fig.14(a). The signal in the frequency domain is simple. Therefore, the influence of the deterioration of the ratio of signal and noise of the speckle pattern can be neglected. However, it can be confirmed that the influence of the deterioration of S/N in the complex modulation that happens in the buckling phenomenon as shown in Fig.14(b) cannot be neglected.
The signal in
frequency domain is very complex as shown in Fig.14(d). The signal widely distributes in the frequency domain. And, it includes a lot of noise. Though the multi-recording technology is very useful, it can be also confirmed that the additional processing for
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Fig.15 Fluctuation in specklegram: (a) Without averaging filter (b) With averaging filter
avoiding such influences is required when the technology is employed in the measurement concerning some complex shapes.
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As the results, the specklegram includes some fluctuations as shown in Fig.15(a). The influence of the fluctuation cannot be removed by using only filtering based on Fourier transform. To reduce this influence, the area filter based on averaging processing (processing area: 50 pixels ×20pixels) is employed to the specklegram. The result is shown in Fig.15(b). The influence by the low S/N ratio is reduced effectively by the filter.
Finally, the in-plane and out-of-plane deformations in the buckling phenomenon of the beam are shown in Fig.16 (c) and (d). Though the in-plane deformation shown in Fig.16(c) includes somewhat fluctuation, almost proportional relationship between the
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Fig.16 Measured results of buckling phenomenon by proposed optical system: (a) Deformation in A-direction 29 (b) Deformation in B-direction (c) In-plane deformation (d) Out-of-plane deformation
That is, it can be thought that the axial strain of the beam is almost constant.
The results in A- and B- directions are also smooth as shown in Fig.16(a) and (b).The section A-A and B-B in Fig.16(c) and (d) are shown in Fig.17 (a) and (b), respectively.
In Fig.17(a), the value of the in-plane deformation is shown as zero at the point at which the out-of-plane deformation is maximum. As the practical value, however, the in-plane deformation is not zero at the point of the maximum deflection of the beam. In this experiment, the data of beam over whole area were not recorded in one fringe pattern image at the present optical system in the same manner as previous paper [10]. Therefore, the quantity of the in-plane deformation at the point of the maximum deflection is also defined as zero in this paper.
From section A-A shown in Fig.17(a), the axial strain in the beam can be calculated as 11.5×10-6. The axial load can be also calculated as 0.506 N. That is, it can be confirmed that the real axial load of the beam is larger than the Euler’s buckling load. Therefore, it can be thought that the buckling phenomenon happens in the beam.
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In this experiment, though the quantity of the in-plane deformation is very small, it is confirmed that both the in-plane and out-of-plane deformations can be detected by the
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Fig.17 Measured results of in-plane and out-of-plane deformations: (a) In-plane deformation (section A-A) (b) Out-of-plane deformation (section B-B)
6. Conclusion
In this paper, the measurement method that can measure simultaneously in-plane and out-of-plane deformations in high resolution by using multi-recording method is proposed. In this method, the deformation analysis by using only two speckle patterns before and after a deformation by only one camera can be performed. From experimental results, it can be also confirmed that the in-plane and out-of-plane displacements can be detected in high resolution by this system. Furthermore, it can be confirmed that the 31
in-plane and out-of-plane deformations in the buckling phenomenon can be detected by using the optimum measurement condition concerning the multi-recording and the filtering technologies for the speckle interferometry.
REFERENCE
[1] R. S. Sirohi, Speckle Metrology, Marcel Dekker, New York,1993, pp.99-234. [2] B. J. Thompson, Selected papers on Electronic Speckle Pattern Interferometry Principles and Practice, SPIE Optical Engineering Press, Bellingham, Washington, 1996,pp.1-518. [3] G. Cloud, Optical Methods of Engineering Analysis, Cambridge University Press, New York, NY,1995, pp.395-476. [4] D. Malacara, Optical Shop Testing, John Wiley &Sons, NY, 1992, pp.501-652. [5] O.J. Lokberg, G.A.Slettemoen, Basic Electronic Speckle-pattern Interferometry,In Applied Optics and Optical Engineering, Vol.X Academic Press, San Diego, 1987, pp.455-504. [6] Y. Arai, S. Yokozeki, In-plane displacement measurement using electronic speckle pattern interferometry based on spatial fringe analysis method, Opt. Eng., 43 32
(2004)2168-2174. [7] N. Pavillon, S. C.Seelamantula, J. Kuhn, M. Unser, C. Depeursinge, Suppression of the zero-oder term in off-axis digital holography through nonlinear filtering, Appl. Opt., 48 (2009)H186-H195. [8] Y. Arai, Electronic Speckle Pattern Interferometry based on spatial information using only two sheets of speckle patterns, Modern Optics., 61 (2014)297-306. [9] Y. Arai, Improvement of measuring accuracy of spatial fringe analysis method using only two speckle patterns in electronic speckle pattern interferometry, Opt. Eng. 53(2014) 034107. [10] Y. Arai, Development of in-plane and out-of-plane deformation simultaneous measurement method for the analysis of buckling, Opt. Eng. 54,(2015)024102. [11] Y. Arai, Measuring accuracy of in-plane and out-of-plane deformation simultaneous measurement of speckle interferometry using multi-recording method, Proceedings of SPIE Vol.9489, Baltimore, (2015) 9489-14. [12] A. W.Lohmann, Reconstruction of vectorial wave fronts, Appl. Opt., 4(1965)1667-1668. [13] A.A. Friesem, J. Fedorowicz, Multicolor wavefront reconstruction, Appl. Opt., 6 (1967)529-536. [14]J.Kuhn, T. Colomb, F. Montfort, F. Charriere, Y. Emery, E. Cuche, P. Marquet, C. Depeursinge, Real-time dual-wavelength digital holographic microscopy with a single hologram acquisition, Opt. Exp. ,15 (2007)7231-7242. [15] M. Takeda, H. Ina, and S. Kobayashi, Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry, J. Opt. Soc. Am., 72(1982)156-160. [16] F. Bleich, Buckling strength of metal structures, McGRAW-HILL, New York, NY, 1952, pp.1-54
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Caption Fig. 1 Speckle interferometry using only two speckle patterns: (a)Optical system (b) Speckle pattern (c) Signal of speckle pattern in frequency domain Fig.2 In-plane and out-of-plane deformations measurement speckle interferometer using two cameras Fig.3 New optical system using multi-recording technology Fig.4 Relationship between specklesize and accuracy of measured results: (a) 7.5μm (b) 9.5μm (c) 13.5μm Fig.5 Speckle pattern grabbed by the new optical system: (a) Speckle pattern (b) Speckle patterns in frequency domain Fig.6 Separation of in-plane and out-of-plane deformations Fig.7 Optical system and measured object: (a) Optical system (b) Displacement generator Fig.8 Measured results of in-plane and out-of-plane displacement: (a)In-plane displacement by only in-plane displacement (b) Out-of-plane displacement by only in-plane displacement (c) In-plane displacement by only out-of-plane displacement (d) Out-of-plane displacement by only out-of-plane displacement Fig.9 Error distribution of in-plane and out-of-plane displacement shown in Fig.8 (a) and (d): (a) Error distribution of in-plane displacement shown in Fig. 8(a) (b) Error distribution of out-of-plane displacement shown in Fig. 8(d)
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Fig.10 Measured results of in-plane and out-of-plane simultaneous displacements and resultant displacement: (a)In-plane displacement by in-plane and out-of-plane displacement (b)Out-of-plane displacement by in-plane and out-of-plane displacement (c)In-plane displacement by resultant of Fig.8 (a) and (c) (d)Out-of-plane displacement by resultant of Fig.8 (b) and (d) Fig.11 Experiment of buckling phenomenon (a)Apparatus for producing a buckling (b)Specklegram Fig.12 Measured results by conventional processing (a)Deformation in A-direction (b) Deformation in B-direction (c) In-plane deformation (d) Out-of-plane deformation Fig.13 S/N ratio in multi-recording technology (a)Multi-recording (b)Multi-recording (Magnification) (c)Single-recording (d)Single-recording (Magnification) Fig.14 Signal of specklegram in frequency domain (a)Rotation of plane (b)Buckling of beam (c)Magnification of Area-A (d)Magnification of Area-B Fig.15 Fluctuation in specklegram (a)Without averaging filter (b)With averaging filter Fig.16 Measured results of buckling phenomenon by proposed optical system (a) Deformation in A-direction (b) Deformation in B-direction (c) In-plane deformation (d) Out-of-plane deformation 35
Fig.17 Measured results of in-plane and out-of-plane deformations (a)In-plane deformation (section A-A) (b)Out-of-plane deformation (section B-B)
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