- Email: [email protected]
measurement of out-of-plane vibrations using shearing interferometry and interferometric grating
Optics and Lasers in Engineering 38 (2002) 269–277
Monitoring/measurement of out-of-plane vibrations using shearing interferometry and interferometric grating Chandra Shakher*, Shashi Prakash1 Laser applications and Holography Laboratory, Instrument Design Development Centre, Indian Institute of Technology, New Delhi 110016, India Received 12 June 2001; received in revised form 18 August 2001; accepted 19 October 2001
Abstract In this paper, an application of shear plate interferometry combined with moir!e readout to monitor/measure out-of-plane vibrations is presented. Moire! fringes are produced between the fringe pattern from the shear plate and interferometric grating recorded by photographing the interference pattern generated from the shear plate. It is demonstrated that the method can be used to study periodic and non-periodic vibrations. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Out-of-plane vibration measurement; Shearing interferometry; Interferometric grating; moir!e
1. Introduction A number of optical methods have been investigated for vibration analysis. Prominent among them are classical interferometric techniques [1–5], moire! technique [6–8], holographic [9–13] and speckle- based [14–17] techniques, Talbot interferometric techniques [18,19]. Lateral shearing interferometry has been extensively used in diverse applications such as optical testing [20,21], collimation testing [22], determination of refractive index of lens and lens parameters [23], determination of the linear thermal expansion coefficients of metallic bars [24] and flame diagnostics [25] etc. In this *Corresponding author. Tel.: +91-11-659-1432; fax: +91-11-686-2037. E-mail address: [email protected] (C. Shakher). 1 On leave from - Institute of Instrumentation, Khandwa Road Campus, Devi Ahilya University, Indore 452017, India. 0143-8166/02/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 3 - 8 1 6 6 ( 0 1 ) 0 0 1 6 7 - 1
270
C. Shakher, S. Prakash / Optics and Lasers in Engineering 38 (2002) 269–277
communication, we present an application of shear plate shearing interferometry and moire! readout for monitoring/measurement of out-of-plane vibrations. In this arrangement, the moire! fringes are produced between the fringe pattern from shear plate and interferometric grating recorded by photographing the interference pattern generated from the shear plate.
2. Analysis An overview of the analytical description of lateral shearing interferometry is given by Mantravadi [21]. If a collimated beam of light with wavefront aberration W ðx; yÞ is assumed as an input to the interferometer, the shearing device divides the incoming beam into two outgoing beams with a lateral displacement S in the direction of the shear. The shearing interferogram indicates the resulting path difference DW : If S is assumed to be in the x-direction then DW ¼ ½W ðx; yÞ W ðx S; yÞ:
ð1Þ
Usually S is small, and one can use the approximation DW ¼ ð@W =@xÞS: This means that one can measure the partial derivative of the wavefront in the direction of the shear. If the wavefront error due to defocusing of collimating lens is present, one observes the straight-line fringes, which are equally spaced, and perpendicular to the direction of shear. The straight-line fringes appear in the common area of the two interfering wavefronts. A schematic of the experimental set-up for measurement/monitoring of vibrations using shear plate interferometer is shown in Fig. 1. The fringe pattern formed due to defocusing of collimating lens in lateral shearing interferometer falls on a lightweight plane mirror attached on the loudspeaker membrane. As the mirror used is lightweight, it is assumed that it does not modify the characteristics of the
Fig. 1. Schematic of experimental setup to measure/monitor vibrations using shearing interferometry.
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loudspeaker membrane. The fringe pattern generated by the shear plate is reflected from the mirror and falls on the interferometric grating G, which produces an infinite fringe moire! pattern. The collimator is set in such a manner that the fringe spacing is of the same order as that of the grating pitch. When the loudspeaker membrane vibrates in response to the driving signal, the displacement of the fringes is proportional to the out-of-plane motion of the speaker membrane. For timedependent displacement of membrane, a time-dependent monitoring of the fringe pattern results. At any given instant of time, the increment of out-of-plane displacement between two fringes is given by Dh ¼ d tan b; where d is the pitch of the grating and b the angle of tilt of grating G. Moire! fringe pattern has a triangular profile. If a small-area (compared with fringe width) photo-detector is placed in the moire! fringe pattern, the current produced by the light falling on the detector is proportional to the motion of the fringes.
3. Experimental The schematic of experimental arrangement is shown in Fig. 1. Beam from He–Ne laser is expanded by using a 40 microscope objective and a pinhole of 5 mm diameter. A diverging beam from microscope objective and pinhole arrangement is incident on the collimating lens of focal length 250 mm, mounted on a precision translation stage to translate the lens along the optic axis. Initially, the lens position is adjusted to get the collimated beam. This collimated beam is incident on the plane parallel plate. Light is reflected from the front and the back surface of the plate. The thickness of the shear plate produces lateral shear between the two reflected wavefronts from the plate. If the collimating lens is defocused, a typical straight-line fringe pattern results. This fringe pattern is recorded on Slavich PFG-01 emulsion and developed, fixed following the standard procedure supplied by the manufacturer. The pitch of the recorded pattern was measured under a high-power microscope. It was 0.4 mm. This recorded pattern then acts as a sinusoidal grating ‘G’ of pitch 0.4 mm for the experiment. The resulting fringe pattern due to defocusing of collimating lens is incident on the mirror, pasted at the center of the loudspeaker diaphragm (290 g nickel magnetic speaker having 145 mm paper cone). The beam incident on the loudspeaker is reflected back by the vibrating mirror attached to its diaphragm and subsequently it falls on sinusoidal grating G. The position of collimating lens is adjusted, so that the resulting fringe pattern has the same pitch as that of the grating G. In such a condition, one obtains infinite fringe moire! pattern on grating G. Grating G is tilted by an angle ‘b’ as shown in Fig. 1. The Hewlett–Packard frequency generator HP 33120-A is used to excite the loudspeaker diaphragm. A silicon photo-detector, much smaller in area than moire! fringe width, monitors the variation in incident intensity as a result of vibrating fringe pattern. Photocurrent from the detector is fed to X–Y plotter and recorded.
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4. Results Fig. 2 is the output of X–Y plotter. The lower and upper curves in Fig. 2 show the signal fed by the function generator to the loudspeaker and corresponding output voltage signal generated in the photodiode due to motion of fringe pattern as a result of out-of-plane vibration of the loudspeaker membrane, respectively. The temporal phase lag between the photo-detector output and the input driving signal from the function generator can be estimated from Fig. 2. The variation of phase lag with frequency is shown in Fig. 3. The phase lag occurs due to the inertia of the membrane of loudspeaker. The phase lag increases with increase in frequency for almost the full range for which measurements were made. However, it can be seen that the increase is not linear. The variation of amplitude of speaker membrane at its center as a function of frequency was also studied. The loudspeaker is driven by the frequency generator. After fixed frequency intervals, we measure vibration amplitude in terms of peak-to-peak output voltage. Fig. 4 shows the variation of membrane amplitude at its center as a function of frequency. From the figure, it can be observed that the resonance peaks corresponding to vibration amplitudes are observed at frequencies of 117, 142, 274 and 425 Hz, for a constant drive voltage of 400 mV.
Fig. 2. The output of X–Y plotter. The lower and upper curves shows the signal fed by the function generator to the loudspeaker and the corresponding output voltage signal generated in the photodiode due to motion of fringe pattern as a result of out-of-plane vibration of the loudspeaker membrane.
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Fig. 3. Graph showing the variation of phase lag between the driving oscillation and the photo-detector output with frequency for a constant drive voltage of 400 mV.
Fig. 4. Graph showing the variation of the membrane amplitude at its center with driving frequency at a constant driving voltage of 400 mV.
Fig. 5 shows the folding effect of phenomenon inherent to moire! readout method. When the driving voltage is increased, the frequency of the membrane appears to increase. This occurs when the fringe shift d tan b becomes greater than the grating
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Fig. 5. Plot recorded by using the X–Y plotter showing folding effect as observed in the output signal in the upper curve. The lower curve corresponds to the driving voltage.
pitch and a new infinite moire! fringe begins to appear. This effect has been used to advantage in determining the amplitude of displacement since the full fringe shift corresponds to a displacement Dh equal to d tan b: One can eliminate the appearance of the effect by adjusting the angle b so that the fringe shift p ¼ Dh=tan bpd (pitch of the grating). Fig. 6 shows the graph between the amplitude of the membrane at its center and driving voltage when frequency (117 Hz) is kept constant. The membrane amplitude increases linearly with the increase in drive voltage in the limit under which folding effect is observed. The kinks in the curve are observed whenever the fringe shifts by an amount equal to grating pitch or an integral multiple of it, indicating the onset of folding effect. The method can also be used to monitor the vibration of the membrane when the loudspeaker is driven by a non-periodic input signal. Fig. 7 gives the output of the X–Y plotter. The lower and upper curves in Fig. 7 show the driving signal in the form of music applied by a record player and photo-detector output resulting due to oscillation of the membrane. Because of the non-linearity in the amplitude response and the phase lag, the functional form of two curves is quite dissimilar, though some correlation persists. It can be seen that high frequencies are damped.
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Fig. 6. Graph showing the variation of membrane displacement at its center as a function of driving voltage at a constant frequency of 117 Hz.
Fig. 7. The output of the X–Y plotter. The lower and upper curves show the deriving signal in the form of music applied by a record player and photo-detector output resulting due to oscillation of the membrane.
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5. Conclusion We have demonstrated application of lateral shearing interferometry combined with moire! readout, to study periodic and non-periodic vibrations. Straight-line fringes generated by the combination of shear plate and defocusing the collimated wavefront are projected on to the object. The moire! fringe pattern is formed between the reflected fringe pattern from the object and the recorded sinusoidal grating as shown in Fig. 1.The contrast of the moire! fringes is very good. The method provides variable sensitivity and is more appropriate for measurement of vibration of mechanical objects as compared to holography, if lower sensitivities are required. The method is simple and easy to implement, and fully quantitative when used in point-wise mode.
Acknowledgements The authors greatefully acknowledge the comments and suggestions made by referees to improve the manuscript. The financial support from structures and propulsion panel of AR&DB, Govt. of India is gratefully acknowledged. Mr. Shashi Prakash gratefully acknowledge the financial assistance from Council of Scientific and Industrial Research, New Delhi.
References [1] Pickering CJD, Halliwell NA, Wilmshurst TA. The laser vibrometer: a portable instrument. J Sound Vib 1986;107:471–85. [2] Baker JR, Laming RI, Wilmshurst TA, Halliwell NA. A new high sensitivity laser vibrometer. Opt Laser Technol 1990;22:241–4. [3] Weir K, Boyle WJO, Palmer AW, Grattan KTV. Low coherence, interferometric fiber optic vibrometer using novel optical signal processing scheme. Electron lett 1991;27:1658–60. [4] Weir K, Boyle WJO, Meggitt BT, Palmer AW, Grattan KTV. A novel adaptation of the Michelson interferometer for the measurement of vibration. J Lightwave Technol 1992;10:700–3. [5] Shakher C, Vyas AL, Seth A. Real time monitoring of vibrations using interferometric sensor. Proceedings of International Conference on Information, Communications and Signal Processing, Singapore, 9–12 September, 1997: 1185–8, Organised by School of Electrical & Electronic Engineering, Nanyang Technological University, IEEE Singapore Section. [6] Hovanesian JD, Hung YY. Moir!e contour-sum, contour-difference and vibration analysis of arbitrary objects. Appl Opt 1972;10:2734–8. [7] Ritter R, Meyer HJ. Vibration analysis of plates by a real time-averaged projection moir!e method. Appl Opt 1980;19:1630–3. [8] Kafri O, Band YB, Chin T, Heller DF, Walling JC. Real time moir!e vibration analysis of diffusive objects. Appl Opt 1985;24:240–2. [9] Stetson K. Holographic vibration analysis. In: Erf RK, editor. Holographic non-destructive testing. New York: Academic Press, 1974. p. 181–220. [10] Boane AD, Shakher C, Rao GV. Holographic vibration analysis of steam turbine blade. Proceeding of International Conference on Advances in Structural Testing, Analysis and Design, Indian Institute of Science, Bangalore, India, July 29–August 3, 1990.
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[11] Ellingsrud S, Rosvold GO. Analysis of a data based holography system used to measure small vibration amplitudes. J Opt Soc Am A 1992;9:237–51. [12] Dirksen D, Bally GV. Holographic double-exposure interferometry in near real time with photorefractive crystals. J Opt Soc Am B 1994;11:1858–63. [13] Nakadate S, Ishiki M. Real time vibration measurement by a spatial phase-shifting technique with a tilted holographic interferogram. Appl Opt 1997;36:281–4. [14] Chang FP, Jung RM. Vibrational analysis of plate and shell by laser speckle interferometry. Opt Acta 1976;23:997–1007. [15] Koyuncu B, Cookson J. Semi-automatic measurements of small high frequency vibrations using timeaveraged electronic speckle pattern interferometry. J Phys E 1980;13:206–8. [16] Shakher C, Matsuda K, Tenjimbayashi K, Sundarajan N, Ravi D. Image processing and analysis of digital speckle pattern interferometric images for monitoring surface vibrations/tilt. Proceedings of International Conference on Information, Communications and Signal Processing, Singapore, 9–12 September, 1997.pp. 953–6. [17] Steinchen YW, et al. Vibrational analysis by means of digital shearography. Opt Lasers Eng 1998;30:199–212. [18] Kaijun H, Jahns J, Lohman AW. Talbot interferometry with vibrating phase object. Opt Commun 1983;45:295–300. [19] Prakash S, Upadhyay S, Shakher C. Real time out-of-plane vibration measurement using Talbot interferometry. Opt Lasers Eng 2000;34:251–9. [20] Bryngdahl O. Applications of shearing interferometry. In: Wolf E, editor. Progress in Optics, vol. 4. Amsterdam: North-Holland, 1964. p. 39–83.(Chapter II) [21] Mantravadi MV. Lateral shearing interferometry. In: Malcara D, editor. Optical shoptesting. 2nd ed. New York: Wiley, 1992. p. 123–72.(Chapter IV) [22] Murty MVRK. The use of a single parallel plate as a lateral shearing interferometry with a visible gas laser source. Appl Opt 1964;3:531–4. [23] Kasana RS, Rosenbruch KJ. Determination of the refractive index of a lens using Murty shearing interferometer. Appl Opt 1983;22:3526–31. [24] Srivastava SN, Tomar MS, Kasana RS. Determination of the linear thermal expansion coefficient of long metallic bars by Murty shearing interferometry. Opt Laser Technol 1990;22:283–6. [25] Philipp H, et al. Flame diagnostics by light sheet imaging and by shearing interferometry. Opt Eng 1993;32:1025–32.
Monitoring/measurement of out-of-plane vibrations using shearing interferometry and interferometric grating Chandra Shakher*, Shashi Prakash1 Laser applications and Holography Laboratory, Instrument Design Development Centre, Indian Institute of Technology, New Delhi 110016, India Received 12 June 2001; received in revised form 18 August 2001; accepted 19 October 2001
Abstract In this paper, an application of shear plate interferometry combined with moir!e readout to monitor/measure out-of-plane vibrations is presented. Moire! fringes are produced between the fringe pattern from the shear plate and interferometric grating recorded by photographing the interference pattern generated from the shear plate. It is demonstrated that the method can be used to study periodic and non-periodic vibrations. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Out-of-plane vibration measurement; Shearing interferometry; Interferometric grating; moir!e
1. Introduction A number of optical methods have been investigated for vibration analysis. Prominent among them are classical interferometric techniques [1–5], moire! technique [6–8], holographic [9–13] and speckle- based [14–17] techniques, Talbot interferometric techniques [18,19]. Lateral shearing interferometry has been extensively used in diverse applications such as optical testing [20,21], collimation testing [22], determination of refractive index of lens and lens parameters [23], determination of the linear thermal expansion coefficients of metallic bars [24] and flame diagnostics [25] etc. In this *Corresponding author. Tel.: +91-11-659-1432; fax: +91-11-686-2037. E-mail address: [email protected] (C. Shakher). 1 On leave from - Institute of Instrumentation, Khandwa Road Campus, Devi Ahilya University, Indore 452017, India. 0143-8166/02/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 3 - 8 1 6 6 ( 0 1 ) 0 0 1 6 7 - 1
270
C. Shakher, S. Prakash / Optics and Lasers in Engineering 38 (2002) 269–277
communication, we present an application of shear plate shearing interferometry and moire! readout for monitoring/measurement of out-of-plane vibrations. In this arrangement, the moire! fringes are produced between the fringe pattern from shear plate and interferometric grating recorded by photographing the interference pattern generated from the shear plate.
2. Analysis An overview of the analytical description of lateral shearing interferometry is given by Mantravadi [21]. If a collimated beam of light with wavefront aberration W ðx; yÞ is assumed as an input to the interferometer, the shearing device divides the incoming beam into two outgoing beams with a lateral displacement S in the direction of the shear. The shearing interferogram indicates the resulting path difference DW : If S is assumed to be in the x-direction then DW ¼ ½W ðx; yÞ W ðx S; yÞ:
ð1Þ
Usually S is small, and one can use the approximation DW ¼ ð@W =@xÞS: This means that one can measure the partial derivative of the wavefront in the direction of the shear. If the wavefront error due to defocusing of collimating lens is present, one observes the straight-line fringes, which are equally spaced, and perpendicular to the direction of shear. The straight-line fringes appear in the common area of the two interfering wavefronts. A schematic of the experimental set-up for measurement/monitoring of vibrations using shear plate interferometer is shown in Fig. 1. The fringe pattern formed due to defocusing of collimating lens in lateral shearing interferometer falls on a lightweight plane mirror attached on the loudspeaker membrane. As the mirror used is lightweight, it is assumed that it does not modify the characteristics of the
Fig. 1. Schematic of experimental setup to measure/monitor vibrations using shearing interferometry.
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loudspeaker membrane. The fringe pattern generated by the shear plate is reflected from the mirror and falls on the interferometric grating G, which produces an infinite fringe moire! pattern. The collimator is set in such a manner that the fringe spacing is of the same order as that of the grating pitch. When the loudspeaker membrane vibrates in response to the driving signal, the displacement of the fringes is proportional to the out-of-plane motion of the speaker membrane. For timedependent displacement of membrane, a time-dependent monitoring of the fringe pattern results. At any given instant of time, the increment of out-of-plane displacement between two fringes is given by Dh ¼ d tan b; where d is the pitch of the grating and b the angle of tilt of grating G. Moire! fringe pattern has a triangular profile. If a small-area (compared with fringe width) photo-detector is placed in the moire! fringe pattern, the current produced by the light falling on the detector is proportional to the motion of the fringes.
3. Experimental The schematic of experimental arrangement is shown in Fig. 1. Beam from He–Ne laser is expanded by using a 40 microscope objective and a pinhole of 5 mm diameter. A diverging beam from microscope objective and pinhole arrangement is incident on the collimating lens of focal length 250 mm, mounted on a precision translation stage to translate the lens along the optic axis. Initially, the lens position is adjusted to get the collimated beam. This collimated beam is incident on the plane parallel plate. Light is reflected from the front and the back surface of the plate. The thickness of the shear plate produces lateral shear between the two reflected wavefronts from the plate. If the collimating lens is defocused, a typical straight-line fringe pattern results. This fringe pattern is recorded on Slavich PFG-01 emulsion and developed, fixed following the standard procedure supplied by the manufacturer. The pitch of the recorded pattern was measured under a high-power microscope. It was 0.4 mm. This recorded pattern then acts as a sinusoidal grating ‘G’ of pitch 0.4 mm for the experiment. The resulting fringe pattern due to defocusing of collimating lens is incident on the mirror, pasted at the center of the loudspeaker diaphragm (290 g nickel magnetic speaker having 145 mm paper cone). The beam incident on the loudspeaker is reflected back by the vibrating mirror attached to its diaphragm and subsequently it falls on sinusoidal grating G. The position of collimating lens is adjusted, so that the resulting fringe pattern has the same pitch as that of the grating G. In such a condition, one obtains infinite fringe moire! pattern on grating G. Grating G is tilted by an angle ‘b’ as shown in Fig. 1. The Hewlett–Packard frequency generator HP 33120-A is used to excite the loudspeaker diaphragm. A silicon photo-detector, much smaller in area than moire! fringe width, monitors the variation in incident intensity as a result of vibrating fringe pattern. Photocurrent from the detector is fed to X–Y plotter and recorded.
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4. Results Fig. 2 is the output of X–Y plotter. The lower and upper curves in Fig. 2 show the signal fed by the function generator to the loudspeaker and corresponding output voltage signal generated in the photodiode due to motion of fringe pattern as a result of out-of-plane vibration of the loudspeaker membrane, respectively. The temporal phase lag between the photo-detector output and the input driving signal from the function generator can be estimated from Fig. 2. The variation of phase lag with frequency is shown in Fig. 3. The phase lag occurs due to the inertia of the membrane of loudspeaker. The phase lag increases with increase in frequency for almost the full range for which measurements were made. However, it can be seen that the increase is not linear. The variation of amplitude of speaker membrane at its center as a function of frequency was also studied. The loudspeaker is driven by the frequency generator. After fixed frequency intervals, we measure vibration amplitude in terms of peak-to-peak output voltage. Fig. 4 shows the variation of membrane amplitude at its center as a function of frequency. From the figure, it can be observed that the resonance peaks corresponding to vibration amplitudes are observed at frequencies of 117, 142, 274 and 425 Hz, for a constant drive voltage of 400 mV.
Fig. 2. The output of X–Y plotter. The lower and upper curves shows the signal fed by the function generator to the loudspeaker and the corresponding output voltage signal generated in the photodiode due to motion of fringe pattern as a result of out-of-plane vibration of the loudspeaker membrane.
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Fig. 3. Graph showing the variation of phase lag between the driving oscillation and the photo-detector output with frequency for a constant drive voltage of 400 mV.
Fig. 4. Graph showing the variation of the membrane amplitude at its center with driving frequency at a constant driving voltage of 400 mV.
Fig. 5 shows the folding effect of phenomenon inherent to moire! readout method. When the driving voltage is increased, the frequency of the membrane appears to increase. This occurs when the fringe shift d tan b becomes greater than the grating
274
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Fig. 5. Plot recorded by using the X–Y plotter showing folding effect as observed in the output signal in the upper curve. The lower curve corresponds to the driving voltage.
pitch and a new infinite moire! fringe begins to appear. This effect has been used to advantage in determining the amplitude of displacement since the full fringe shift corresponds to a displacement Dh equal to d tan b: One can eliminate the appearance of the effect by adjusting the angle b so that the fringe shift p ¼ Dh=tan bpd (pitch of the grating). Fig. 6 shows the graph between the amplitude of the membrane at its center and driving voltage when frequency (117 Hz) is kept constant. The membrane amplitude increases linearly with the increase in drive voltage in the limit under which folding effect is observed. The kinks in the curve are observed whenever the fringe shifts by an amount equal to grating pitch or an integral multiple of it, indicating the onset of folding effect. The method can also be used to monitor the vibration of the membrane when the loudspeaker is driven by a non-periodic input signal. Fig. 7 gives the output of the X–Y plotter. The lower and upper curves in Fig. 7 show the driving signal in the form of music applied by a record player and photo-detector output resulting due to oscillation of the membrane. Because of the non-linearity in the amplitude response and the phase lag, the functional form of two curves is quite dissimilar, though some correlation persists. It can be seen that high frequencies are damped.
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Fig. 6. Graph showing the variation of membrane displacement at its center as a function of driving voltage at a constant frequency of 117 Hz.
Fig. 7. The output of the X–Y plotter. The lower and upper curves show the deriving signal in the form of music applied by a record player and photo-detector output resulting due to oscillation of the membrane.
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5. Conclusion We have demonstrated application of lateral shearing interferometry combined with moire! readout, to study periodic and non-periodic vibrations. Straight-line fringes generated by the combination of shear plate and defocusing the collimated wavefront are projected on to the object. The moire! fringe pattern is formed between the reflected fringe pattern from the object and the recorded sinusoidal grating as shown in Fig. 1.The contrast of the moire! fringes is very good. The method provides variable sensitivity and is more appropriate for measurement of vibration of mechanical objects as compared to holography, if lower sensitivities are required. The method is simple and easy to implement, and fully quantitative when used in point-wise mode.
Acknowledgements The authors greatefully acknowledge the comments and suggestions made by referees to improve the manuscript. The financial support from structures and propulsion panel of AR&DB, Govt. of India is gratefully acknowledged. Mr. Shashi Prakash gratefully acknowledge the financial assistance from Council of Scientific and Industrial Research, New Delhi.
References [1] Pickering CJD, Halliwell NA, Wilmshurst TA. The laser vibrometer: a portable instrument. J Sound Vib 1986;107:471–85. [2] Baker JR, Laming RI, Wilmshurst TA, Halliwell NA. A new high sensitivity laser vibrometer. Opt Laser Technol 1990;22:241–4. [3] Weir K, Boyle WJO, Palmer AW, Grattan KTV. Low coherence, interferometric fiber optic vibrometer using novel optical signal processing scheme. Electron lett 1991;27:1658–60. [4] Weir K, Boyle WJO, Meggitt BT, Palmer AW, Grattan KTV. A novel adaptation of the Michelson interferometer for the measurement of vibration. J Lightwave Technol 1992;10:700–3. [5] Shakher C, Vyas AL, Seth A. Real time monitoring of vibrations using interferometric sensor. Proceedings of International Conference on Information, Communications and Signal Processing, Singapore, 9–12 September, 1997: 1185–8, Organised by School of Electrical & Electronic Engineering, Nanyang Technological University, IEEE Singapore Section. [6] Hovanesian JD, Hung YY. Moir!e contour-sum, contour-difference and vibration analysis of arbitrary objects. Appl Opt 1972;10:2734–8. [7] Ritter R, Meyer HJ. Vibration analysis of plates by a real time-averaged projection moir!e method. Appl Opt 1980;19:1630–3. [8] Kafri O, Band YB, Chin T, Heller DF, Walling JC. Real time moir!e vibration analysis of diffusive objects. Appl Opt 1985;24:240–2. [9] Stetson K. Holographic vibration analysis. In: Erf RK, editor. Holographic non-destructive testing. New York: Academic Press, 1974. p. 181–220. [10] Boane AD, Shakher C, Rao GV. Holographic vibration analysis of steam turbine blade. Proceeding of International Conference on Advances in Structural Testing, Analysis and Design, Indian Institute of Science, Bangalore, India, July 29–August 3, 1990.
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[11] Ellingsrud S, Rosvold GO. Analysis of a data based holography system used to measure small vibration amplitudes. J Opt Soc Am A 1992;9:237–51. [12] Dirksen D, Bally GV. Holographic double-exposure interferometry in near real time with photorefractive crystals. J Opt Soc Am B 1994;11:1858–63. [13] Nakadate S, Ishiki M. Real time vibration measurement by a spatial phase-shifting technique with a tilted holographic interferogram. Appl Opt 1997;36:281–4. [14] Chang FP, Jung RM. Vibrational analysis of plate and shell by laser speckle interferometry. Opt Acta 1976;23:997–1007. [15] Koyuncu B, Cookson J. Semi-automatic measurements of small high frequency vibrations using timeaveraged electronic speckle pattern interferometry. J Phys E 1980;13:206–8. [16] Shakher C, Matsuda K, Tenjimbayashi K, Sundarajan N, Ravi D. Image processing and analysis of digital speckle pattern interferometric images for monitoring surface vibrations/tilt. Proceedings of International Conference on Information, Communications and Signal Processing, Singapore, 9–12 September, 1997.pp. 953–6. [17] Steinchen YW, et al. Vibrational analysis by means of digital shearography. Opt Lasers Eng 1998;30:199–212. [18] Kaijun H, Jahns J, Lohman AW. Talbot interferometry with vibrating phase object. Opt Commun 1983;45:295–300. [19] Prakash S, Upadhyay S, Shakher C. Real time out-of-plane vibration measurement using Talbot interferometry. Opt Lasers Eng 2000;34:251–9. [20] Bryngdahl O. Applications of shearing interferometry. In: Wolf E, editor. Progress in Optics, vol. 4. Amsterdam: North-Holland, 1964. p. 39–83.(Chapter II) [21] Mantravadi MV. Lateral shearing interferometry. In: Malcara D, editor. Optical shoptesting. 2nd ed. New York: Wiley, 1992. p. 123–72.(Chapter IV) [22] Murty MVRK. The use of a single parallel plate as a lateral shearing interferometry with a visible gas laser source. Appl Opt 1964;3:531–4. [23] Kasana RS, Rosenbruch KJ. Determination of the refractive index of a lens using Murty shearing interferometer. Appl Opt 1983;22:3526–31. [24] Srivastava SN, Tomar MS, Kasana RS. Determination of the linear thermal expansion coefficient of long metallic bars by Murty shearing interferometry. Opt Laser Technol 1990;22:283–6. [25] Philipp H, et al. Flame diagnostics by light sheet imaging and by shearing interferometry. Opt Eng 1993;32:1025–32.