The Design and Commissioning of a Novel Tomographic Polariscope

Transcription

1 The Design and Commissioning of a Novel Tomographic Polariscope Rachel A Tomlinson*, Hui Yang*, David Szotten +, and William R B Lionheart + * Department of Mechanical Engineering, University of Sheffield, S1 3JD, UK + University of Manchester, UK ABSTRACT An experimental method has been proposed which allows the non-destructive determination of 3-D stress distributions in birefringent materials thus eliminating the necessity to section stress frozen models. The problem of recovering the full stress tensor field from photoelastic measurements has been addressed mathematically and a new approach devised. This paper describes the experimental issues which were involved in the designing and commissioning of our Tomographic Polariscope which will implement this mathematical approach. A standard transmission polariscope is used through which to view the model with the addition of a component positioning device. The need to take photoelastic measurements around all spherical angles has been eliminated and the model is just rotated around two axes independently inside the immersion tank using a precision rotation stage. A Fourier polarimetry method has been developed to obtain the characteristic photoelastic parameters for each view. The polarizers and rotation stage are controlled by the same system and thus the rotation of model and birefringent data collection is synchronized and automated. 1. INTRODUCTION In photoelasticity, the birefringent effect has been used widely to determine stresses in models of engineering components. However in order to evaluate the internal stresses in three dimensional components, stress freezing [1] is currently the only experimental technique routinely used. This method requires a model to be loaded and subject to a thermal cycle to lock in strains; and the model to be sectioned for two dimensional analysis, consequently destroying the model. The combination of photoelasticity with tomographic techniques could be the solution to this problem. In traditional hard field tomography, for the determination of the internal structure of an object, a certain radiation is passed through a section of the body and a property of this radiation, after it has passed through the body, is measured. This property may be intensity, phase, polarisation, etc. The internal structure of the object can be reconstructed from data measured at many different views from around the body, in terms of the Radon transform. This mathematics is well understood for scalar fields, however the strain field which produces the birefringent effect is a tensor quantity rather than a scalar quantity as usually encountered in tomography. Moreover the integral equation involved is non-linear and couples the components of the tensor. In a research collaboration between the Universities of Manchester and Sheffield, a tomographic photoelasticity method has been proposed which is able to provide three-dimensional stress distributions in birefringent materials [2,3,4]. In [3] we described a method employing a two-dimensional inverse Radon transformation to reconstruct one component of the dielectric tensor in the direction as the tomographic rotation axis. The method of that paper relied on a method of determination a global phase which we have subsequently found to be unstable in the presence of noise. However the same method can be used to determine the anisotropic part of the dielectric tensor using data collected for five rotation axes, and hence the deviatoric stress can be determined from the stress-optic law. In [4] we show how the deviatoric stress can be obtained by an algebraic reconstruction method and if required the hydrostatic stress recovered from the equilibrium condition. Simulation exercises with the new mathematical approach established that highly accurate experimental data was needed for successful implementation. Therefore this paper describes the experimental issues involved in the

2 design and commissioning of the apparatus to obtain such data. 2. IMPROVED FOURIER POLARIMETRY In order to use a two-dimensional Radon inverse transformation to reconstruct the element of the dielectric tensor, it is required to measure three characteristic parameters of a 3D object at different scanning angles to calculate the unitary transfer matrices [ 5]. Several methods have been offered to determine the three characteristic parameters in integrated photoelasticity. When contrasted to the phase-stepping [6] and phase shifting [7] methods it has been shown that Fourier methods [8] are more accurate, particularly when measuring the characteristic directions [9]. However, the Fourier method is not as efficient as other methods since it requires that a minimum of nine intensity images be collected during a whole revolution of a polarizer while the phase-stepping method only needs six intensity images [6]. In practise many more images are recorded to achieve greater accuracy. If the technique is to be used for tomographic photoelasticity, the efficiency needs to be improved since is has been established [4] that many images are required for successful implementation. Therefore we have developed recently an advanced Fourier method which is more efficient for the automated measurement of the characteristic parameters [ 1 ]. Three characteristic parameters are given through the magnitudes of the spectrum of the output periodic intensity at the angular frequencies of 2(n-1) and 2(n+1): 1 arctan[ B2( n 1) / A2( n 1) ] 2 1 arctan[ B2( n 1) / A2( n 1) ] / arctan{[ A B ]/[ A B ]} ( n 1) 2( n 1) 2( n 1) 2( n 1) 1/4 (1) where n is the angular rotation ratio between the analyzer and the polarizer, A 2(n-1), B 2(n-1), A 2(n+1) and B 2(n+1) are the magnitudes of the Fourier components at the angular frequencies of 2(n-1) and 2(n+1) and 2, and are the characteristic retardation, the primary characteristic direction and the characteristic angle, respectively. As the absolute values of 2(n-1) and 2(n+1) should be the smallest integers to satisfy the requirements of efficiency and proper data accuracy, the angular rotation ratio of 1:2 is therefore the optimized value. In this way, the polarizer is still stepwise rotated over 36 degrees but the analyzer is only rotated over 18 degrees and the minimum number of the intensity images to be collected is now only five images. Even with the large number of images required for tomographic photoelasticity, this optimized angular rotation ratio results in data collection about 3 times faster than the angular rotation ratio suggested by previous researchers. The accuracy in terms of noise measured by the angular rotation ratios 1:2 and 3:1 was also investigated. Simulated data showed that the noise levels using the angular rotation ratios 1:2 and 3:1 are very similar for all three characteristic parameters. The characteristic retardation has better performance against the noise when the fringe order is not close to m /2 (m=, 1, 2 ) even when the SNR of the light intensity is 3dB. Accuracy of the primary characteristic direction is heavily affected in the area of around /4 even when the SNR is 1dB. Accuracy of the characteristic angle is independent of its magnitude. Noise in the Fourier method mainly comes from the stability of the light source and the quality of the polarizer and so these were considered when designing the apparatus. The use of a fast Fourier transform means that many images can be sampled which reduces the effect of noise and increases the accuracy of the method. 3. EXPERIMENTAL APPARATUS DESIGN Figure 1 shows the configuration of the experimental apparatus. The experimental apparatus consists of four main parts: the imaging system, the Fourier Polarimetry system, the tomographic system and the GUI (Graphic User Interface). The imaging system is composed of the light source, a spatial filter, a collimating lens, imaging lens and a CCD camera. The CW He-Ne laser with the wavelength of 632nm is used as the light source because the laser beam is much easier to be collimated. A 5 m pinhole and a 6x microscopic objective are combined to work as a low-pass spatial filter to eliminate the laser beam high frequency noise. A 4 diameter convex lens is used to expand the laser beam diameter to 9mm and an imaging lens and a CCD camera (SONY XCD71) are used to record the images. Sony XCD71 is a machine version standard camera which its properties can be controlled by the computer through its API. Its maximum resolution is 124x768pixels and its maximum speed is 3 frames per second.

3 Fourier Polarimetry system is composed of a polarizer, an analyzer, two rotation stages and a motion controller. The analyzer and polarizer are assembled within the rotation stages to enable automatic control. The rotation stages provide rotations with a resolution of.1 degrees, a repeatability of.1 degrees, an absolute accuracy of.23 degrees and a maximum speed of 2 degrees per second. The analyzer and polarizer are rotated step by step simultaneously at the fixed ratios of 1:2 over one 36 revolution of the polarizer, so a series of images will be recorded for Fourier analysis. So for 36 images recording, the rotation interval of the polarizer is 1 degrees. Lens Analyzer in Rotation stage Polarizer in Rotation stage Motion controller Spatial filter Laser CCD camera GUI in PC Specimen in immersion tank rotated by the rotation stage below Figure 1a) Photograph of the experimental apparatus Lens Analyzer Tank Specimen Polarizer Lens Spatial filter CCD Laser Rotation stage Rotation stage Rotation stage Computer Motion controller/driver Laser controller/driver Figure 1b) Configuration of the experimental apparatus The tomographic system includes a rotation stage, a tank with immersion fluid and the specimen positioning devices. It is more convenient to rotate the specimen than the imaging system for tomographic scanning and this is achieved by the rotating stage inside the tank which is coupled via a sealed bearing to a precision rotation stage mounted under the tank. The specimen is rotated at discrete intervals (5 degrees) and therefore 36 sets of characteristic parameters are obtained. The specimen is immersed in the fluid, which refractive index is exactly same as that of the specimen to avoid refraction of the light beam. The tank is made of stress free glass and has parallel sides so that the light is not refracted as it passes through the tank. The specimen positioning devices, which are made of the same transparent materials as the specimen, are used to mount and fix the specimen at different views. One-view data can provide one component of the dielectric tensor and five elements could be obtained by five specified views. The specimen is tilted 45 degrees and 9 degrees manually to achieve full data required by the tomographic reconstruction approach [2]. The GUI (Figure 2) was developed using Matlab to control image acquisition, configure the CCD camera and

4 drive the motion controller. The camera and motion controller, which controls the rotation of analyzer, polarizer and specimen, are all connected to a computer. By using the GUI, it is able to control the motion of rotation, set the properties of CCD camera and furthermore synchronously carry out the whole image acquisition procedure with limited operator input. 4. EXPERIMENTAL RESULTS 4.1 Disc in Compression Figure 2 Graphic User Interface, GUI The following experimental procedure was implemented for the commissioning of the experimental apparatus with the Fourier Polarimetry method. A stress-frozen disc was placed in the tomographic polariscope (Figure 1) and the analyzer and polarizer were rotated simultaneously at the fixed ratios of 1:2 and 3:1 with respect to each other over one 36º revolution of the polarizer. 18 images for each of n=1:2 and n=3:1 were recorded at discrete intervals by the CCD camera connected to the PC. These images were then processed to find the three characteristic parameters. Figure 3 shows experimental data obtained at the rotation ratios of 1:2 and 3:1 respectively. It is shown that the three characteristic parameters have similar distributions at the angular rotation ratios of 1:2 and 3:1. Figure 4 shows a comparison of the characteristic retardation measured by the angular rotation ratios 1:2 and 3:1 along the lines shown in Fig. 3. The two rotation ratios provide similar fringe orders with comparable accuracy and noise levels. Considering the efficiency of the data collection (one third of the time for the same number of images) and the accuracy of the data analysis, the rotation ratio of 1:2 is the optimized choice for the Fourier method. It is noted that there are some undefined regions in the primary characteristic direction and the characteristic angle, which appears when the characteristic retardation =m /2 (m=, 1, 2 ). This is a known phenomenon discussed by Tomlinson [6].

5 Characteristic parameters (n=1:2) Characteristic parameters (n=3:1) Characteristic retardation (fringes) Primary characteristic direction (degrees) Characteristic angle (degrees) Figure 3 Full field maps of the characteristic parameters for the disc in compression.9.8 Characteristic Retardation (fringes) n=3:1 n=1: Pixel Figure 4 A comparison of the characteristic retardation measured by the angular rotation ratios 1:2 and 3:1 along the lines shown in Figure Cube with an offset point load A three-dimensional birefringent cubic specimen was designed for the experimental verification of tomographic algorithms developed by Szotten [4]. For the application of this novel tomographic method, the linear approximation of integrated photoelasticity was assumed. This means that tomographic algorithms are valid only when the rotation of the principal stress axes is small or optical retardation is low. A specimen with low retardation (smaller than 1 fringe order) was used in our experimental test: an epoxy resin cube with dimensions 25mmx25mmx25mm. A three-dimensional stress field was created by applying a point load offset from the surface centre during a stress freezing thermal cycle. For each view of the specimen 36 images (1º intervals of the polarizer) and 72 images (5º intervals of the polarizer) were recorded with n=1:2 and then the three periodic characteristic parameters were determined using the Fourier method. Figure 5 shows the results for one view of the specimen. The fringes around the loading point are very clear although the noise from the background is still a little high. Figure 6 shows a comparison of the characteristic retardation measured by 36 and 72 images along the lines shown in Figure 5. It can be seen that the noise in the data produced with 72 images is much smaller than that with 36 images. A typical full tomographic measurement cycle (72x36x6 images = images, i.e. 72 images for the Fourier polarimetry, 5º

6 interval of the specimen rotation stage and 6 views) is expected to take 4.5 hours excluding the time taken for manual re-positioning about horizontal axes. These images are then processed to obtain 36x6 sets of three characteristic parameters, which will be used for tomographic reconstruction. 36 Images Images Characteristic Retardation (fringes) Primary Characteristic Direction (degrees) Characteristic Angle (degrees) Figure 5 The results of one view of the cube specimen.5 Characteristic Retardation (fringes) images 36 images Figure 6 A comparison of the characteristic retardation measured by 36 and 72 images along the lines shown in Figure 5 Pixels

7 5. DISCUSSION The issues which were identified when assembling the tomographic polariscope and developing a methodology to collect sufficient data for the tomographic reconstruction were speed of data collection, accuracy of data collected, and accuracy of image position within the field of view. With possibly a minimum of images to record, the speed at which these can be recorded needs to be reduced further while still maintaining quality data. However by changing the rotation ratio to 1:2 means full data collection within 4.5 hours rather than 13.5 hours under alternative ratios. The total automation and synchronization of the recording and rotation systems via the GUI also means little operator input is necessary. The majority of time has been spent in trying to reduce the noise in the system, since the reconstruction approach ideally needs noise free data. Issues which have been addressed include superior spatial filtering, improving polarizer quality, and careful alignment of the optical components. In 2-D transmission photoelasticity alignment of the immersion tank perfectly perpendicular to the light source is desirable but not absolutely essential as with the tomographic method. The fact that we wanted a full-field technique meant some optical components had to be relatively large, such as the imaging lenses. Quality over price is an issue here since high quality large lenses are difficult to acquire within a tight budget. Accuracy of positioning of the specimen is more about knowledge of its location. The tilt angle of the specimen is critical since each view gives one component of the tensor field, but as long as this angle is known it can be included in the processing. Therefore accurate measurement is necessary, which was achieved by accurate machining of the positioning devices. The rotation stages were selected for their precision and repeatability. Additionally it is essential that the specimen is accurately located within the field of view. To achieve this, a target was placed on each surface of the test specimen, four images taken and a correlation method used to determined the relative position of the marker to the rotation axes. 6. CONCLUSION In this paper we have reported the design and commissioning of an advanced tomographic polariscope. The experimental apparatus integrated with a new Fourier polarimetry method has been constructed and tested. The experimental results have shown that the system is able to provide the characteristic parameters efficiently and accurately. These results are currently being analyzed using the tomographic reconstruction method and it is hope to present further data at the conference. 7. ACKNOWLEDGEMENTS The authors gratefully acknowledge the support of the Engineering and Physical Research Council (Grant Number GR/R8627/1). 8. REFERENCES 1 Fessler, H., (1993),An assessment of frozen stress photoelasticity, J. Strain Analysis, 27 (3), Hammer, H, Lionheart, W R B, and Tomlinson, R A, Recent Advances In Tomographic Photoelasticity, (24) Proceedings of the SEM X International Congress & Exposition on Experimental and Applied Mechanics, Costa Mesa, California, USA, June 7-1, 24, Paper No Hammer, H. and Lionheart, W. R. B., (25), Reconstruction of spatially inhomogeneous dielectric tensors via optical tomography,j. OSA (A), 22 (2), Szotten, D, Tomlinson, R A, and Lionheart, W R B, (26), Tomographic Reconstruction of Stress from Photoelastic Measurements using Elastic Regularization, submitted to Inverse Problems 5 Hammer, H., (24)Characteristic parameters in integrated photoelasticity: an application of Poincare s equivalence theorem, J. Modern Optics, 51, Tomlinson, R. A. and Patterson, E. A., (22) The use of phase stepping for the measurement of characteristic parameters in integrated photoelasticity, Experimental Mechanics, 42 (1), Mangal, S.K. and Ramesh, K., (1999),Determination of Characteristic Parameters in Integrated Photoelasticity by Phase-shifting Techniques. Optics and Lasers in Engineering, 31, Berezhna, S., Berezhnyy, I. and Takashi, M., (21), Integrated photoelasticity through imaging Fourier polarimetry of an elliptic retarder, Applied Optics, 4 (5), Gibson, S, Tomlinson, R A, and Patterson, E A,(23) Towards Accurate Full-field Collection of Characteristic Parameters, Proceedings of the SEM Annual Conf. on Experimental and Applied Mechanics, 2-4 June, 23, Charlotte, USA, Paper No Yang, H. and Tomlinson, R. A., (26), Improvement of Fourier Polarimetry for Applications in Tomographic Photoelasticity, Submitted to Experimental Mechanics