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Application of digital speckle pattern interferometry and wavelet transform in measurement of transverse vibrations in square plate
ARTICLE IN PRESS
Optics and Lasers in Engineering 42 (2004) 585–602
Application of digital speckle pattern interferometry and wavelet transform in measurement of transverse vibrations in square plate Rajesh Kumar, Chandra Shakher* Laser Applications and Holography Laboratory, Instrument Design Development Centre, Indian Institute of Technology, New Delhi 110 016, India Received 12 May 2003; accepted 30 December 2003
Abstract In this paper the study of out-of-plane or transverse vibrations in a square plate using digital speckle pattern interferometry (DSPI) is presented. To improve the measurement accuracy, we have implemented a new filtering scheme based on combination of average/median filtering and Symlet wavelet filtering which enhances the signal to noise ratio in the speckle interferogram obtained from DSPI. A large number of fringe patterns are shown for square plate with two different boundary conditions. Experimentally obtained resonance frequencies for the square plate for the boundary condition one edge fixed and other edges being free, the resonance frequencies obtained from DSPI show good agreement with that obtained from classical theory for thin plates. r 2004 Elsevier Ltd. All rights reserved. Keywords: Digital speckle pattern interferometry (DSPI); Vibration; Mode shape; Signal-to-noise ratio (SNR); Histogram equalization; Symlet wavelet
1. Introduction Investigation of plate vibration has received considerable attention for academic research and engineering applications for almost two centuries. The vast literature *Corresponding author. Tel.: +91-11-659-1432; fax: +91-11-686-2037. E-mail address: [email protected], [email protected], [email protected] (C. Shakher). 0143-8166/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlaseng.2003.12.006
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exists on the analytical solutions for the free vibration of plates [1–4]. For the plate having two opposite edges simply supported (with combination of simply supported, clamped and free for the remaining two edges), available exact solutions are extension of Voigt’s [5] early work. Completely clamped case is also investigated in detail and is used frequently as a standard problem for analytical methods because of simplicity of boundary conditions. Ritz used the completely free problem for obtaining upper bounds on vibration frequencies [6]. Analytical solutions for cantilever plate have also been investigated in detail. Free vibrations of rectangular plate with the possible 21 combinations of classical boundary conditions are available in the work reported by Liessa [2]. For complicated boundary conditions, analytical solutions to the vibration problems are difficult and time consuming. Digital speckle pattern interferometry (DSPI) technique [7–19] is a full field, noncontact, and real-time method to measure the vibrations of structures subjected to various kinds of loading. DSPI is faster in operation and more insensitive to environment than holographic interferometry. However, this method cannot attain the high image quality as that of holographic interferometry due to the low resolution of video camera system. To reduce the DC noise in the speckle interferogram, coming from the environment, the subtraction method was developed [7,8] in which reference frame is recorded before vibration and the incoming frames are continuously subtracted from the reference frame. In order to increase the visibility of the fringe pattern and reduce the environmental noise simultaneously, an amplitude-fluctuation ESPI method was proposed [9–11]. In this paper, a study of out-of-plane or transverse vibrations in square plate using DSPI is presented. Time-average sequential subtraction DSPI method is implemented to record the speckle interferograms. In time-average sequential subtraction DSPI method, the reference frame of the vibrating object is continuously refreshed, which has added advantage of recording the speckle interferogram with improved signal-to-noise ratio (SNR) even in the low-frequency fluctuating lighting conditions. For further enhancement in SNR of the speckle interferograms, a filtering scheme based on average or median filtering followed by Symlet wavelet filtering is implemented [12,20,21] which is very effective at edges of the fringes as well. The case of cantilever plate was taken up as it has practical importance in aerospace industry.
2. Theory of vibration of plate For small deflections of a flat plate of uniform thickness made of homogeneous isotropic material and subjected to normal and shear forces in the plane of the plate as shown in Fig. 1, the following equation relates the lateral deflection w to the loading on the plate [22]: 4 qw q4 w q4 w q2 w q2 w q2 w Dr4 w ¼ D þ N þ 2 þ þ 2N ; ð1Þ ¼ P þ N x xy y qx4 qx2 qy2 qy4 qx2 qx@y qy2
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Fig. 1. An element of plate showing bending moments, normal and shear forces.
where D ¼ Eh3 =12ð1 n2 Þ is the plate stiffness, E is modulus of elasticity, h is the thickness of the plate and n is Poisson’s ratio of the material of the plate and x and y are spatial variable for the plate. The parameter P is the loading intensity, Nx is normal (in-plane) loading in the X-direction per unit length, Ny (in-plane) is normal loading in Y-direction per unit length, and Nxy the shear load parallel to the plate surface in the X - and Y -directions, per unit length. The boundary conditions at an edge parallel to Y-axis, for fixed edge are w ¼ 0 and qw=qx ¼ 0: Boundary conditions for the free edge are bending moment ¼ Dðq2 w=qy2 þ nðq2 w=qx2 ÞÞ ¼ 0; twisting moment ¼ Dð1 nÞq2 w=qxqy ¼ 0; and the shearing force per unit length normal to surface of the plate ¼ Dððq3 w=qx3 Þ þ nðq3 w=qy2 qxÞÞ ¼ 0: We have demonstrated the transverse vibration of square plate with two different boundary conditions. For one plate, one of the edges is fixed and other edges are free. For another plate, one of the edges and a corner formed by intersection of two free edges are fixed. If a system vibrates in a natural mode, all parts of the system execute simple ( yÞ cosðon t þ harmonic motion about the equilibrium position and hence w ¼ AWðx; ( yÞ; where W is a function of x and y satisfying the necessary boundary conditions. For the case of free transverse vibration, the in-plane forces Nx ; Ny ; Nxy are set equal to zero and the loading intensity P is replaced by inertial force. The equation of motion (Eq. (1)), then reduces to ( ¼ rho2 W; ( Dr4 W ð2Þ n
where r is the mass density of the plate. The modes of vibration of plate with all edges simply supported are such that the deflection of each section of the plate parallel to an edge is of the same form as the deflection of a beam with both ends simply supported. For other combinations of edge conditions this does not hold true. By using an admissible mode function in Rayleigh’s method, an approximate expression for natural frequency is obtained. This can be improved by using Ritz’s method [23] in which the natural mode shape function is assumed in series form. The Rayleigh’s method gives the following
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frequency equation: o2n ¼ rh R R 2
Vmax ; ( 2 dx dy W
ð3Þ
A
where V is strain energy and is given by [22] " 2 2 #) 2 Z Z ( 2 D d w d2 w q2 w q2 w qw þ 2 2ð1 nÞ V¼ dx dy: 2 2 dy2 2 dx dy dx qxqy S
ð4Þ
( 1 ðx; yÞ þ ( to be of the form W ( ¼ a1 W The Ritz method involves assuming W ( ( ( a2 W2 ðx; yÞ þ ?: In which W1 ; W2 ; y all satisfy at least the natural boundary conditions, and a1 ; a2 ; y are adjusted to give a minimum frequency. ( 1; W ( 2 ; y etc. for a square plate with one edge fixed The admissible functions W and other edges free are available in literature [2,22,23]. The first few natural frequencies obtained using these functions are given in Table 1. However, it is difficult to calculate the natural frequency of a plate with a complicated boundary condition, for example the square plate fixed at one edge and at one of the remaining corners along with other edges free. Classical mode shapes of vibrations of a cantilever plate are shown in Fig. 2. In general the natural mode in a rectangular plate is denoted by m=n; where ‘m’ is number of nodal lines perpendicular to the fixed edge and ‘n’ is number of nodal lines parallel to the fixed edge. The mode shape of the plate may be the bending, torsional or plate type. In some of the cases the observed mode shape may be a combination of different classical mode shapes giving rise to a complex mode. In the case of vibration measurement using DSPI, let us assume that the frequency of vibration ‘o’ of the harmonically vibrating object is greater than the frame rate of CCD camera used to record image of the object. We have recorded two timeaveraged specklegrams of the vibrating plate. The intensity, if two time-averaged specklegrams are subtracted, is given by [24,25] Iðx; yÞ ¼ 2Ao Ar jJ0 ½ð2p=lÞgw0 ðx; yÞ j jcos½2ðfo fr Þ j;
ð5Þ
where, A0 and Ar are amplitude of object and reference wavefronts, respectively; l is the wavelength of laser light used to illuminate the object (plate); fr is phase of reference beam and fo is a position-dependent phase of the object beam which corresponds to the original state of the object; w0 ðx; yÞ is
Table 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Frequency parameter on = D=rha4 for different modes for a square plate fixed at one edge and other edges being free [23] Mode sequence pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi on = D=rha4
1 m=n ¼ 0=1
2 m=n ¼ 1=1
3 m=n ¼ 0=2
4 m=n ¼ 2=1
5 m=n ¼ 1=2
3.49
8.54
21.44
27.46
31.17
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Fig. 2. Schematic of classical mode shape of vibration of plate fixed at one edge and free at other edges.
phase-dependent out-of-plane displacement of the harmonically vibrating object with respect to some reference position; g is geometric factor which depends on the angle of illumination and the angle of observation, and J0 is a zero-order Bessel function. The term jcos½2ðfo fr Þ j represents phase-dependent high-frequency speckle information. The Bessel function J0 spatially modulates brightness of the speckle pattern.
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3. Filtering scheme for removal of speckle noise in DSPI fringes DSPI fringe pattern has inherent speckle noise. Presence of undesirable bright speckles in the area of dark fringes and in the other dark region and similarly the dark speckles in the bright region make inaccuracy in measurement from speckle interferogram. To improve the SNR in the speckle interferogram we have implemented a filtering scheme based on average/median filtering followed by Symlet wavelet filtering. In comparison to the Daubechies wavelet filtering scheme, the Symlet wavelet filtering is also effective at the edges of the fringes. If the number of speckles in the fringes is more, a combination of median filtering and Symlet wavelet filtering is more effective. If the number of speckles in the fringes is moderate, a combination of average filtering and Symlet wavelet filtering is more effective. Details of the filtering scheme are given in our earlier work [12].
4. Experiment on square plates and analysis of results The DSPI set-up for recording the fringe patterns of out-of-plane/transverse vibration of the square plate excited by a shaker (model number: EX 6/6.4, make: Prodera, France) is shown in Fig. 3. A beam of 30 mW He–Ne laser of wavelength 632.8 nm is split into two beams by a beam splitter BS1. One of the beams illuminates the surface of the plate under study and the other beam is used as the reference beam. The value of g for our experimental setup is 1.938. The object beam is combined with the reference beam to form a speckle interferogram that is converted into a video signal by a CCD camera. The video analog output from HTC-550B/W CCIR CCD camera is fed to the PC-based image-processing system developed using National Instrument’s IMAQ PCI-1408 card. LabVIEW 5.0 based program [26] in a graphical programming language which was developed to acquire, process and display the interferogram. The program implements accumulated linear histogram equalization after subtraction of the interferograms. The IMAQ PCI-1408 card is set to process the images of interferogram at the rate of 30 images/s. The time 1 average interferogram of the vibrating plate over the frame acquisition period 30 s is grabbed and subtracted from the just previous time-average interferogram in sequential subtraction manner. The subtracted interferogram so obtained is displayed continuously on the computer screen. Experiments were conducted on square plates made of aluminium (Young’s modulus=70 GPa, Density=2700 kg/ m3, and Poisson’s ratio=0.3). Surface of the plate was made flat on the optical polishing machine. Function generator (model number: HP 33120A) regulates the frequency and magnitude of the force of the exciter. The function generator was set to generate the sinusoidal signal. Three different studies were made. In the first study, experiments were conducted on the plate of thickness 0.8 mm fixed at one edge and other edges were free. The plate was initially of nominal thickness 1.0 mm and was polished on optical grinding machine to get the required flatness and smoothness of the surface. Dimensions, point of loading and boundary condition for the plate fixed at one edge and other edges free used in the first study
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Fig. 3. Schematic of DSPI setup for measurement of out-of-plane vibrations.
are shown in Fig. 4(a). Large number of speckle interferograms for different excitation force and frequency were recorded. To improve the SNR in the recorded speckle interferograms, average/median filtering followed by Symlet wavelet filtering was implemented which is found to be very effective in removing speckle noise. As the noises in the recorded interferograms were low, average filtering followed by
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Fig. 4. Sketch of square plates with dimensions, point of loading and boundary conditions for the condition (a) one edge fixed and other edges free, and (b) one edge and one of the corner is fixed.
Symlet wavelet filtering was implemented. Some typical filtered speckle interferograms at different excitation frequencies and amplitudes of force are shown in Figs. 5(a)–(k). Resonance were identified at lower excitation force. Refinement in the speckle interferograms for getting more details was done by increasing the amplitude at the same resonance frequency. The various modes obtained from DSPI can be interpreted in terms of classical modes shown in Fig. 2. Speckle interferograms shown in Figs. 5(a) is first torsional mode. Interferograms in Figs. 5(c) and (d)
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Fig. 5. Filtered speckle interferograms at different excitation frequencies and amplitudes of force for square plate having dimension (50 mm 50 mm 0.8 mm) fixed at one edge and other edges being free.
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Fig. 5 (continued).
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Fig. 6. Interpretation of a typical combined mode of vibration shown in Fig. 5(e) of a square plate fixed at one edge and free at other edges obtained by DSPI with the classical modes.
represent second torsional and first plate modes, respectively. Interferogram in Fig. 5(g) is third torsional mode. Modes shown in Figs. 5(h) and (i) are clearly higher plate modes. A typical combined mode obtained from DSPI shown in Fig. 5(e) is interpreted with the help of classical modes and is shown in Fig. 6. The speckle interferogram in Fig. 5(e) is a complex mode from the combination of first plate and third bending modes. In a similar manner other combined modes obtained from natural modes can also be interpreted. It was observed that if the nodal line, for example in the first torsional mode, is very close to the point of excitation, the mode shape gets slightly distorted. It was also observed that effect of this distortion can be minimized by reducing the amplitude of excitation force. Significant distortions in mode shape were observed by changing the excitation frequency slightly away (710 Hz in case of first torsion mode) from the natural frequency. In the second study, the effects of variation in magnitude of excitation force for the same frequency were investigated. To visualize the effect of change in magnitude of force in a better manner, experiments were conducted on a square plate of higher
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Fig. 7. Filtered speckle interferograms at excitation frequency of 8.9 kHz (frequency parameter: 53.29) and at different amplitudes of force for square plate having dimension (50 mm 50 mm 1.7 mm) fixed at one edge and other edges free.
stiffness than the plate used in first study (by increasing thickness of the plate). Thickness of the plate (initially of 2 mm) after polishing flat was 1.7 mm. Other parameters such as boundary conditions and point of loading were kept the same as shown in Fig. 4(a). The four filtered speckle interferograms shown in Figs. 7(a)–(d) are at a excitation frequency of 8.9 kHz. The excitation forces for the interferograms shown in Figs. 7(a)–(d) are 2.9 103, 5.5 103, 7.0 103 and 15.2 103 N, respectively. From the above figures it is demonstrated that at the same excitation frequency, with increase in excitation force there is thinning and separation in nodal lines of the vibrating plate. In the third study, the experiments were conducted on the plate having dimensions 50 mm 50 mm 0.8 mm, fixed at one end AB and also at one of the corners D as shown in Fig. 4(b). Corner D of the plate is fixed by epoxy. Cyclic load is applied at point P. Figs. 8(a)–(l) show some of the typical selective filtered interferograms at different excitation frequencies and amplitude of sinusoidal force. As in this case one corner is fixed, the modes have become asymmetrical. These can not be explained on the basis of experiments alone. To explain these modes, rigorous theoretical modeling is required.
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Fig. 8. Some typical filtered speckle interferograms at different excitation frequencies and amplitudes of force for square plate having dimension (50 mm 50 mm 0.8 mm) fixed at one edge and also at one of the remaining corners.
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Fig. 8 (continued).
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For the mode shape and resonance frequency of vibration shown in Fig. 5 of the 50 mm 50 mm 0.8 mm plate (fixed at one edge and other edges free), the dimensionless frequency parameter for the respective mode sequence were obtained and compared with the results obtained from the classical theory [23]. A comparison of measured resonance frequencies for classical modes with theoretical values from literature for the plate fixed at one edge and other edges free is given in Table 2. Small variation in resonance frequencies may be due to the reason that the fixed edge of the plate could not be fixed in an ideal way. The boundary conditions which could not be applied in ideal way also result in some distortion in mode shapes as well as formation of some complex modes. Vibration amplitude is calculated from the speckle interferogram using Eq. (5). Points on the maximum intensity fringe have zero amplitude. On both the sides of the maximum intensity fringe, points on the second and third maximum intensity fringes (for bright fringes intensity decreases with increase in amplitude) have vibration amplitudes 0.202 and 0.363 mm, respectively. To check the validity of results, measurements for frequency and amplitude at different points on the surface of the plate are made by piezo-electric accelerometer (Model No. 4374 DELTA SHEARs accelerometer of Bruel . & Kjaer, Denmark). Amplitude obtained from accelerometer and experimental data of interferogram are in good agreement. Variation in results obtained for accelerometer and the
Table 2 Comparison of measured resonance frequencies for classical modes with theoretical values from literature for a square plate (50 mm 50 mm 0.8 mm) fixed at one edge and other edges being free pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Sl. no. Dimensionless frequency parameter, ‘on = D=rha4 ’ and the Remark on observed corresponding resonance frequency mode shapes Observed 1 2 3 4
— 8.52 26.37 24.63
From literature [23] (670 Hz) (2.074 kHz) (1.937 kHz)
3.49 8.54 27.46 31.17
(274 Hz) (672 Hz) (2.158 kHz) (2.450 kHz)
— First torsional mode First plate mode Second torsional mode
Fig. 9. Line profile for the speckle interferogram shown in Fig. 5(a).
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Fig. 10. Map of the deformed surface of the square plate obtained from evaluating the amplitude at different points for the interferogram shown in Fig. 5(a).
interferogram is less than 5.0%. Some error might have occurred in measuring the position of accelerometer, as it can not be exactly represented by a point. The measurement for amplitude was also verified with the portable laser Doppler vibrometer (Make: Polytec) and variation in results is found to be less than 3.0%. Line profile for the intensity distribution of speckle interferogram in Fig. 5(a) is shown in Fig. 9. Deformed surface of the plate for the speckle interferogram shown in Fig. 5(a) is shown in Fig. 10.
5. Discussion and conclusion Failure in structure and machine components subjected to vibration generally takes place at resonance frequencies at which amplitude of vibration become high. For failure analysis it is therefore important to understand the behavior of a component at resonance frequencies. It is often found that the higher frequencies contain information on faults developing well before they influence the actual ability of the machine/component to do its job, whereas the lower frequencies show the fault when they have occurred. Experimental investigations reveal that DSPI can be effectively used to monitor/measure vibrations in structures and machine components almost in real time. Speckle noise in the DSPI fringes is removed by implementing a filtering scheme based on combination of either average and Symlet wavelet filtering or median and Symlet wavelet filtering. The choice of filtering scheme depends upon noise level present in the speckle interferogram. It was observed that the filtering scheme implemented in the present work is effective in removing speckle noise in simple lower mode patterns as well as complicated higher mode patterns.
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In speckle interferograms, all points on a particular fringe have the same amplitude. From the interferograms shown in Figs. 5(a)–(k), the shift in nodal points and high stress zones can be identified. It is also observed that in the case of transverse/out-of-plane vibration in the plate, the normal modes are in general not harmonically related. Speckle interferograms in Fig. 5 show the presence of classical modes as well as the complex combination modes. From Figs. 7(a)–(d), it can be observed that the fringes become thinner at higher excitation force. Also with increase in excitation force, formation of bright fringes of lesser intensity is taking place in some portion of the plate, which show that the amplitude in this area has increased. With further increase in the force, these less intensity fringes disappear which implies that the amplitude in these zone have increased to further higher level which well satisfies Eq. (5). From Figs. 8(a)–(l) we can observe the effect of introduction of additional boundary condition on the interferogram and the corresponding change in mode shape. It was also observed that at resonance frequencies, there is slight variation in mode pattern with change in position of point of application of excitation force. Experimental investigations show that excessive tightening of the plate results in an increase in the resonance frequency of the plate. The technique is able to measure/ monitor vibrations under any boundary conditions online without changing any physical property of the component. Values of amplitudes obtained from the accelerometer and experimental data of interferogram are in good agreement. The package program, based on finite element method, can give only the approximate solutions. As DSPI technique for measurement of vibration is non-contact type and very accurate, it can be used as a bench mark for calibrating other techniques for measurement of vibration.
Acknowledgements Fruitful discussions with Prof. Kshitij Gupta, Head, Mechanical Engineering Department of IIT Delhi is gratefully acknowledged. Financial assistance from Propulsion panel of Aeronautical Research & Development Board (AR & DB), Government of India is gratefully acknowledged. Rajesh Kumar wishes to acknowledge All India Council for Technical Education (AICTE), New Delhi for the financial assistance.
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[5] Voigt W. Bemerkungene zu de problem der transversalen schwingungen rechteckiger platen. Nachr . Ges Wiss (Gottingen) 1893;6:225–30. [6] Ritz W. Theorie der transversalschwingungen einer quadratischen platte mit freien r.andern. Annalen der Physik 1909;28:737–86. [7] Pouet B, Chatters T, Krishnaswamy S. Synchronized reference updating technique for electronic speckle interferometry. J Nondestruct Eval 1993;12:133–8. [8] Creath K, Slettemoen GA. Vibration-observation techniques for digital speckle-pattern interferometry. J Opt Soc Am A 1985;2:1629–36. [9] Wang WC, Hwang CH, Lin SY. Vibration measurement by the time-averaged electronic speckle pattern interferometry. Appl Opt 1996;35:4502–9. [10] Huang CH. Experimental measurements by an optical method of resonance frequencies and mode shapes for square plates with rounded corners and chamfers. J Sound Vib 2002;253:571–83. [11] Huang CH, Ma CC. Vibration characteristics for piezoelectric cylinders using amplitude-fluctuation electronic speckle pattern interferometry. Am Inst Aeronaut Astronaut J 1998;36:2262–8. [12] Kumar R, Singh IP, Shakher C. Measurement of out-of-plane static and dynamic deformations by processing digital speckle pattern interferometry using wavelet transform. Opt Lasers Eng 2003;41:81–93. [13] Chaudhari UM, Ganeshan AR, Shakher C, Godbole PB, Sirohi RS. Investigation on in plane stresses on bolted flanged joints using digital speckle pattern interferometry. Opt Lasers Eng 1989;11:257–64. [14] More AJ, Tyrer JR. Two-dimensional strain measurement with ESPI. Opt Lasers Eng 1996;24: 381–402. [15] Petzing JN, Tyrer JR. Recent developments and application in electronic speckle pattern interferometry. J Strain Analysis Eng Des 1998;33:153–69. [16] Chen F, Griffen CT, Alien TE. Digital speckle interferometry: some developments and applications for vibration measurement in the automotive industry. Opt Eng 1998;37:1390–7. [17] Rastogi PK, editor. Digital speckle pattern interferometry and related techniques. Chichester: Wiley; 2001. p. 141–55. [18] Shakher C, Matsuda K, Tenjimbayashi K, Sundarajan N, Ravi D. Image processing and analysis of digital interferometric images for monitoring surface vibration/tilt. Proceedings ICICS, Singapore, 9–12 September 1997. p. 953–6. [19] Kaufmann GH, Galizzi GE. Speckle noise reduction in television holography fringes using wavelet thresholding. Opt Eng 1996;35:9–14. [20] Kumar R, Singh SK, Shakher C. Analysis of small vibrations of computer hard disk surface using digital speckle pattern interferometry and wavelet thresholding. Opt Laser Technol 2001;33:567–71. [21] Shakher C, Kumar R, Singh SK, Kazmi SA. Application of wavelet filtering for vibration analysis using digital speckle pattern interferometry. Opt Eng 2002;41:176–80. [22] Stokey WF. Vibrations of systems having distributed mass and elasticity. In: Harris CM, editor. Shock and vibration handbook. New York: McGraw-Hill Book Comp; 1961. p. 7.1–7.37. [23] Young D. Vibration of rectangular plates by the Ritz method. J Appl Mech 1950;17:448–53. [24] Lu B, Yang X, Abendroth H, Eggers H. Time-average subtraction method in electronic speckle pattern interferometry. Opt Commun 1989;70:177–80. [25] Lu B, Hu Z, Abendroth H, Eggers H, Ziolkowski E. Improvement of time-average subtraction technique applied to vibration analysis with TV-holography. Opt Commun 1990;78:217–21. [26] National Instruments Corporation. IMAQt Vision for G Reference Manual. Austin: National Instruments Corporation; 1997.
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Application of digital speckle pattern interferometry and wavelet transform in measurement of transverse vibrations in square plate Rajesh Kumar, Chandra Shakher* Laser Applications and Holography Laboratory, Instrument Design Development Centre, Indian Institute of Technology, New Delhi 110 016, India Received 12 May 2003; accepted 30 December 2003
Abstract In this paper the study of out-of-plane or transverse vibrations in a square plate using digital speckle pattern interferometry (DSPI) is presented. To improve the measurement accuracy, we have implemented a new filtering scheme based on combination of average/median filtering and Symlet wavelet filtering which enhances the signal to noise ratio in the speckle interferogram obtained from DSPI. A large number of fringe patterns are shown for square plate with two different boundary conditions. Experimentally obtained resonance frequencies for the square plate for the boundary condition one edge fixed and other edges being free, the resonance frequencies obtained from DSPI show good agreement with that obtained from classical theory for thin plates. r 2004 Elsevier Ltd. All rights reserved. Keywords: Digital speckle pattern interferometry (DSPI); Vibration; Mode shape; Signal-to-noise ratio (SNR); Histogram equalization; Symlet wavelet
1. Introduction Investigation of plate vibration has received considerable attention for academic research and engineering applications for almost two centuries. The vast literature *Corresponding author. Tel.: +91-11-659-1432; fax: +91-11-686-2037. E-mail address: [email protected], [email protected], [email protected] (C. Shakher). 0143-8166/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlaseng.2003.12.006
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exists on the analytical solutions for the free vibration of plates [1–4]. For the plate having two opposite edges simply supported (with combination of simply supported, clamped and free for the remaining two edges), available exact solutions are extension of Voigt’s [5] early work. Completely clamped case is also investigated in detail and is used frequently as a standard problem for analytical methods because of simplicity of boundary conditions. Ritz used the completely free problem for obtaining upper bounds on vibration frequencies [6]. Analytical solutions for cantilever plate have also been investigated in detail. Free vibrations of rectangular plate with the possible 21 combinations of classical boundary conditions are available in the work reported by Liessa [2]. For complicated boundary conditions, analytical solutions to the vibration problems are difficult and time consuming. Digital speckle pattern interferometry (DSPI) technique [7–19] is a full field, noncontact, and real-time method to measure the vibrations of structures subjected to various kinds of loading. DSPI is faster in operation and more insensitive to environment than holographic interferometry. However, this method cannot attain the high image quality as that of holographic interferometry due to the low resolution of video camera system. To reduce the DC noise in the speckle interferogram, coming from the environment, the subtraction method was developed [7,8] in which reference frame is recorded before vibration and the incoming frames are continuously subtracted from the reference frame. In order to increase the visibility of the fringe pattern and reduce the environmental noise simultaneously, an amplitude-fluctuation ESPI method was proposed [9–11]. In this paper, a study of out-of-plane or transverse vibrations in square plate using DSPI is presented. Time-average sequential subtraction DSPI method is implemented to record the speckle interferograms. In time-average sequential subtraction DSPI method, the reference frame of the vibrating object is continuously refreshed, which has added advantage of recording the speckle interferogram with improved signal-to-noise ratio (SNR) even in the low-frequency fluctuating lighting conditions. For further enhancement in SNR of the speckle interferograms, a filtering scheme based on average or median filtering followed by Symlet wavelet filtering is implemented [12,20,21] which is very effective at edges of the fringes as well. The case of cantilever plate was taken up as it has practical importance in aerospace industry.
2. Theory of vibration of plate For small deflections of a flat plate of uniform thickness made of homogeneous isotropic material and subjected to normal and shear forces in the plane of the plate as shown in Fig. 1, the following equation relates the lateral deflection w to the loading on the plate [22]: 4 qw q4 w q4 w q2 w q2 w q2 w Dr4 w ¼ D þ N þ 2 þ þ 2N ; ð1Þ ¼ P þ N x xy y qx4 qx2 qy2 qy4 qx2 qx@y qy2
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Fig. 1. An element of plate showing bending moments, normal and shear forces.
where D ¼ Eh3 =12ð1 n2 Þ is the plate stiffness, E is modulus of elasticity, h is the thickness of the plate and n is Poisson’s ratio of the material of the plate and x and y are spatial variable for the plate. The parameter P is the loading intensity, Nx is normal (in-plane) loading in the X-direction per unit length, Ny (in-plane) is normal loading in Y-direction per unit length, and Nxy the shear load parallel to the plate surface in the X - and Y -directions, per unit length. The boundary conditions at an edge parallel to Y-axis, for fixed edge are w ¼ 0 and qw=qx ¼ 0: Boundary conditions for the free edge are bending moment ¼ Dðq2 w=qy2 þ nðq2 w=qx2 ÞÞ ¼ 0; twisting moment ¼ Dð1 nÞq2 w=qxqy ¼ 0; and the shearing force per unit length normal to surface of the plate ¼ Dððq3 w=qx3 Þ þ nðq3 w=qy2 qxÞÞ ¼ 0: We have demonstrated the transverse vibration of square plate with two different boundary conditions. For one plate, one of the edges is fixed and other edges are free. For another plate, one of the edges and a corner formed by intersection of two free edges are fixed. If a system vibrates in a natural mode, all parts of the system execute simple ( yÞ cosðon t þ harmonic motion about the equilibrium position and hence w ¼ AWðx; ( yÞ; where W is a function of x and y satisfying the necessary boundary conditions. For the case of free transverse vibration, the in-plane forces Nx ; Ny ; Nxy are set equal to zero and the loading intensity P is replaced by inertial force. The equation of motion (Eq. (1)), then reduces to ( ¼ rho2 W; ( Dr4 W ð2Þ n
where r is the mass density of the plate. The modes of vibration of plate with all edges simply supported are such that the deflection of each section of the plate parallel to an edge is of the same form as the deflection of a beam with both ends simply supported. For other combinations of edge conditions this does not hold true. By using an admissible mode function in Rayleigh’s method, an approximate expression for natural frequency is obtained. This can be improved by using Ritz’s method [23] in which the natural mode shape function is assumed in series form. The Rayleigh’s method gives the following
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frequency equation: o2n ¼ rh R R 2
Vmax ; ( 2 dx dy W
ð3Þ
A
where V is strain energy and is given by [22] " 2 2 #) 2 Z Z ( 2 D d w d2 w q2 w q2 w qw þ 2 2ð1 nÞ V¼ dx dy: 2 2 dy2 2 dx dy dx qxqy S
ð4Þ
( 1 ðx; yÞ þ ( to be of the form W ( ¼ a1 W The Ritz method involves assuming W ( ( ( a2 W2 ðx; yÞ þ ?: In which W1 ; W2 ; y all satisfy at least the natural boundary conditions, and a1 ; a2 ; y are adjusted to give a minimum frequency. ( 1; W ( 2 ; y etc. for a square plate with one edge fixed The admissible functions W and other edges free are available in literature [2,22,23]. The first few natural frequencies obtained using these functions are given in Table 1. However, it is difficult to calculate the natural frequency of a plate with a complicated boundary condition, for example the square plate fixed at one edge and at one of the remaining corners along with other edges free. Classical mode shapes of vibrations of a cantilever plate are shown in Fig. 2. In general the natural mode in a rectangular plate is denoted by m=n; where ‘m’ is number of nodal lines perpendicular to the fixed edge and ‘n’ is number of nodal lines parallel to the fixed edge. The mode shape of the plate may be the bending, torsional or plate type. In some of the cases the observed mode shape may be a combination of different classical mode shapes giving rise to a complex mode. In the case of vibration measurement using DSPI, let us assume that the frequency of vibration ‘o’ of the harmonically vibrating object is greater than the frame rate of CCD camera used to record image of the object. We have recorded two timeaveraged specklegrams of the vibrating plate. The intensity, if two time-averaged specklegrams are subtracted, is given by [24,25] Iðx; yÞ ¼ 2Ao Ar jJ0 ½ð2p=lÞgw0 ðx; yÞ j jcos½2ðfo fr Þ j;
ð5Þ
where, A0 and Ar are amplitude of object and reference wavefronts, respectively; l is the wavelength of laser light used to illuminate the object (plate); fr is phase of reference beam and fo is a position-dependent phase of the object beam which corresponds to the original state of the object; w0 ðx; yÞ is
Table 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Frequency parameter on = D=rha4 for different modes for a square plate fixed at one edge and other edges being free [23] Mode sequence pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi on = D=rha4
1 m=n ¼ 0=1
2 m=n ¼ 1=1
3 m=n ¼ 0=2
4 m=n ¼ 2=1
5 m=n ¼ 1=2
3.49
8.54
21.44
27.46
31.17
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Fig. 2. Schematic of classical mode shape of vibration of plate fixed at one edge and free at other edges.
phase-dependent out-of-plane displacement of the harmonically vibrating object with respect to some reference position; g is geometric factor which depends on the angle of illumination and the angle of observation, and J0 is a zero-order Bessel function. The term jcos½2ðfo fr Þ j represents phase-dependent high-frequency speckle information. The Bessel function J0 spatially modulates brightness of the speckle pattern.
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3. Filtering scheme for removal of speckle noise in DSPI fringes DSPI fringe pattern has inherent speckle noise. Presence of undesirable bright speckles in the area of dark fringes and in the other dark region and similarly the dark speckles in the bright region make inaccuracy in measurement from speckle interferogram. To improve the SNR in the speckle interferogram we have implemented a filtering scheme based on average/median filtering followed by Symlet wavelet filtering. In comparison to the Daubechies wavelet filtering scheme, the Symlet wavelet filtering is also effective at the edges of the fringes. If the number of speckles in the fringes is more, a combination of median filtering and Symlet wavelet filtering is more effective. If the number of speckles in the fringes is moderate, a combination of average filtering and Symlet wavelet filtering is more effective. Details of the filtering scheme are given in our earlier work [12].
4. Experiment on square plates and analysis of results The DSPI set-up for recording the fringe patterns of out-of-plane/transverse vibration of the square plate excited by a shaker (model number: EX 6/6.4, make: Prodera, France) is shown in Fig. 3. A beam of 30 mW He–Ne laser of wavelength 632.8 nm is split into two beams by a beam splitter BS1. One of the beams illuminates the surface of the plate under study and the other beam is used as the reference beam. The value of g for our experimental setup is 1.938. The object beam is combined with the reference beam to form a speckle interferogram that is converted into a video signal by a CCD camera. The video analog output from HTC-550B/W CCIR CCD camera is fed to the PC-based image-processing system developed using National Instrument’s IMAQ PCI-1408 card. LabVIEW 5.0 based program [26] in a graphical programming language which was developed to acquire, process and display the interferogram. The program implements accumulated linear histogram equalization after subtraction of the interferograms. The IMAQ PCI-1408 card is set to process the images of interferogram at the rate of 30 images/s. The time 1 average interferogram of the vibrating plate over the frame acquisition period 30 s is grabbed and subtracted from the just previous time-average interferogram in sequential subtraction manner. The subtracted interferogram so obtained is displayed continuously on the computer screen. Experiments were conducted on square plates made of aluminium (Young’s modulus=70 GPa, Density=2700 kg/ m3, and Poisson’s ratio=0.3). Surface of the plate was made flat on the optical polishing machine. Function generator (model number: HP 33120A) regulates the frequency and magnitude of the force of the exciter. The function generator was set to generate the sinusoidal signal. Three different studies were made. In the first study, experiments were conducted on the plate of thickness 0.8 mm fixed at one edge and other edges were free. The plate was initially of nominal thickness 1.0 mm and was polished on optical grinding machine to get the required flatness and smoothness of the surface. Dimensions, point of loading and boundary condition for the plate fixed at one edge and other edges free used in the first study
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Fig. 3. Schematic of DSPI setup for measurement of out-of-plane vibrations.
are shown in Fig. 4(a). Large number of speckle interferograms for different excitation force and frequency were recorded. To improve the SNR in the recorded speckle interferograms, average/median filtering followed by Symlet wavelet filtering was implemented which is found to be very effective in removing speckle noise. As the noises in the recorded interferograms were low, average filtering followed by
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Fig. 4. Sketch of square plates with dimensions, point of loading and boundary conditions for the condition (a) one edge fixed and other edges free, and (b) one edge and one of the corner is fixed.
Symlet wavelet filtering was implemented. Some typical filtered speckle interferograms at different excitation frequencies and amplitudes of force are shown in Figs. 5(a)–(k). Resonance were identified at lower excitation force. Refinement in the speckle interferograms for getting more details was done by increasing the amplitude at the same resonance frequency. The various modes obtained from DSPI can be interpreted in terms of classical modes shown in Fig. 2. Speckle interferograms shown in Figs. 5(a) is first torsional mode. Interferograms in Figs. 5(c) and (d)
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Fig. 5. Filtered speckle interferograms at different excitation frequencies and amplitudes of force for square plate having dimension (50 mm 50 mm 0.8 mm) fixed at one edge and other edges being free.
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Fig. 5 (continued).
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Fig. 6. Interpretation of a typical combined mode of vibration shown in Fig. 5(e) of a square plate fixed at one edge and free at other edges obtained by DSPI with the classical modes.
represent second torsional and first plate modes, respectively. Interferogram in Fig. 5(g) is third torsional mode. Modes shown in Figs. 5(h) and (i) are clearly higher plate modes. A typical combined mode obtained from DSPI shown in Fig. 5(e) is interpreted with the help of classical modes and is shown in Fig. 6. The speckle interferogram in Fig. 5(e) is a complex mode from the combination of first plate and third bending modes. In a similar manner other combined modes obtained from natural modes can also be interpreted. It was observed that if the nodal line, for example in the first torsional mode, is very close to the point of excitation, the mode shape gets slightly distorted. It was also observed that effect of this distortion can be minimized by reducing the amplitude of excitation force. Significant distortions in mode shape were observed by changing the excitation frequency slightly away (710 Hz in case of first torsion mode) from the natural frequency. In the second study, the effects of variation in magnitude of excitation force for the same frequency were investigated. To visualize the effect of change in magnitude of force in a better manner, experiments were conducted on a square plate of higher
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Fig. 7. Filtered speckle interferograms at excitation frequency of 8.9 kHz (frequency parameter: 53.29) and at different amplitudes of force for square plate having dimension (50 mm 50 mm 1.7 mm) fixed at one edge and other edges free.
stiffness than the plate used in first study (by increasing thickness of the plate). Thickness of the plate (initially of 2 mm) after polishing flat was 1.7 mm. Other parameters such as boundary conditions and point of loading were kept the same as shown in Fig. 4(a). The four filtered speckle interferograms shown in Figs. 7(a)–(d) are at a excitation frequency of 8.9 kHz. The excitation forces for the interferograms shown in Figs. 7(a)–(d) are 2.9 103, 5.5 103, 7.0 103 and 15.2 103 N, respectively. From the above figures it is demonstrated that at the same excitation frequency, with increase in excitation force there is thinning and separation in nodal lines of the vibrating plate. In the third study, the experiments were conducted on the plate having dimensions 50 mm 50 mm 0.8 mm, fixed at one end AB and also at one of the corners D as shown in Fig. 4(b). Corner D of the plate is fixed by epoxy. Cyclic load is applied at point P. Figs. 8(a)–(l) show some of the typical selective filtered interferograms at different excitation frequencies and amplitude of sinusoidal force. As in this case one corner is fixed, the modes have become asymmetrical. These can not be explained on the basis of experiments alone. To explain these modes, rigorous theoretical modeling is required.
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Fig. 8. Some typical filtered speckle interferograms at different excitation frequencies and amplitudes of force for square plate having dimension (50 mm 50 mm 0.8 mm) fixed at one edge and also at one of the remaining corners.
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Fig. 8 (continued).
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For the mode shape and resonance frequency of vibration shown in Fig. 5 of the 50 mm 50 mm 0.8 mm plate (fixed at one edge and other edges free), the dimensionless frequency parameter for the respective mode sequence were obtained and compared with the results obtained from the classical theory [23]. A comparison of measured resonance frequencies for classical modes with theoretical values from literature for the plate fixed at one edge and other edges free is given in Table 2. Small variation in resonance frequencies may be due to the reason that the fixed edge of the plate could not be fixed in an ideal way. The boundary conditions which could not be applied in ideal way also result in some distortion in mode shapes as well as formation of some complex modes. Vibration amplitude is calculated from the speckle interferogram using Eq. (5). Points on the maximum intensity fringe have zero amplitude. On both the sides of the maximum intensity fringe, points on the second and third maximum intensity fringes (for bright fringes intensity decreases with increase in amplitude) have vibration amplitudes 0.202 and 0.363 mm, respectively. To check the validity of results, measurements for frequency and amplitude at different points on the surface of the plate are made by piezo-electric accelerometer (Model No. 4374 DELTA SHEARs accelerometer of Bruel . & Kjaer, Denmark). Amplitude obtained from accelerometer and experimental data of interferogram are in good agreement. Variation in results obtained for accelerometer and the
Table 2 Comparison of measured resonance frequencies for classical modes with theoretical values from literature for a square plate (50 mm 50 mm 0.8 mm) fixed at one edge and other edges being free pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Sl. no. Dimensionless frequency parameter, ‘on = D=rha4 ’ and the Remark on observed corresponding resonance frequency mode shapes Observed 1 2 3 4
— 8.52 26.37 24.63
From literature [23] (670 Hz) (2.074 kHz) (1.937 kHz)
3.49 8.54 27.46 31.17
(274 Hz) (672 Hz) (2.158 kHz) (2.450 kHz)
— First torsional mode First plate mode Second torsional mode
Fig. 9. Line profile for the speckle interferogram shown in Fig. 5(a).
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Fig. 10. Map of the deformed surface of the square plate obtained from evaluating the amplitude at different points for the interferogram shown in Fig. 5(a).
interferogram is less than 5.0%. Some error might have occurred in measuring the position of accelerometer, as it can not be exactly represented by a point. The measurement for amplitude was also verified with the portable laser Doppler vibrometer (Make: Polytec) and variation in results is found to be less than 3.0%. Line profile for the intensity distribution of speckle interferogram in Fig. 5(a) is shown in Fig. 9. Deformed surface of the plate for the speckle interferogram shown in Fig. 5(a) is shown in Fig. 10.
5. Discussion and conclusion Failure in structure and machine components subjected to vibration generally takes place at resonance frequencies at which amplitude of vibration become high. For failure analysis it is therefore important to understand the behavior of a component at resonance frequencies. It is often found that the higher frequencies contain information on faults developing well before they influence the actual ability of the machine/component to do its job, whereas the lower frequencies show the fault when they have occurred. Experimental investigations reveal that DSPI can be effectively used to monitor/measure vibrations in structures and machine components almost in real time. Speckle noise in the DSPI fringes is removed by implementing a filtering scheme based on combination of either average and Symlet wavelet filtering or median and Symlet wavelet filtering. The choice of filtering scheme depends upon noise level present in the speckle interferogram. It was observed that the filtering scheme implemented in the present work is effective in removing speckle noise in simple lower mode patterns as well as complicated higher mode patterns.
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In speckle interferograms, all points on a particular fringe have the same amplitude. From the interferograms shown in Figs. 5(a)–(k), the shift in nodal points and high stress zones can be identified. It is also observed that in the case of transverse/out-of-plane vibration in the plate, the normal modes are in general not harmonically related. Speckle interferograms in Fig. 5 show the presence of classical modes as well as the complex combination modes. From Figs. 7(a)–(d), it can be observed that the fringes become thinner at higher excitation force. Also with increase in excitation force, formation of bright fringes of lesser intensity is taking place in some portion of the plate, which show that the amplitude in this area has increased. With further increase in the force, these less intensity fringes disappear which implies that the amplitude in these zone have increased to further higher level which well satisfies Eq. (5). From Figs. 8(a)–(l) we can observe the effect of introduction of additional boundary condition on the interferogram and the corresponding change in mode shape. It was also observed that at resonance frequencies, there is slight variation in mode pattern with change in position of point of application of excitation force. Experimental investigations show that excessive tightening of the plate results in an increase in the resonance frequency of the plate. The technique is able to measure/ monitor vibrations under any boundary conditions online without changing any physical property of the component. Values of amplitudes obtained from the accelerometer and experimental data of interferogram are in good agreement. The package program, based on finite element method, can give only the approximate solutions. As DSPI technique for measurement of vibration is non-contact type and very accurate, it can be used as a bench mark for calibrating other techniques for measurement of vibration.
Acknowledgements Fruitful discussions with Prof. Kshitij Gupta, Head, Mechanical Engineering Department of IIT Delhi is gratefully acknowledged. Financial assistance from Propulsion panel of Aeronautical Research & Development Board (AR & DB), Government of India is gratefully acknowledged. Rajesh Kumar wishes to acknowledge All India Council for Technical Education (AICTE), New Delhi for the financial assistance.
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