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Evaluation of impact-induced transient deformations using double-pulsed electronic speckle pattern interferometry and finite elements
Optics and Lasers in Engineering 32 (2000) 473} 484
Evaluation of impact-induced transient deformations using double-pulsed electronic speckle pattern interferometry and "nite elements P.D. Ruiz , G.H. Kaufmann *, O. MoK ller, G.E. Galizzi Instituto de Fn& sica Rosario (CONICET-UNR) 2000 Rosario, Argentina Departamento de Fn& sica, Facultad de Ciencias Exactas, Ingeniern& a y Agrimensura, Universidad Nacional de Rosario, 2000 Rosario, Argentina Instituto de Meca& nica Aplicada y Estructuras, Facultad de Ciencias Exactas, Ingeniern& a y Agrimensura, Universidad Nacional de Rosario, 2000 Rosario, Argentina Received 2 December 1999; accepted 4 February 2000
Abstract Transient out-of-plane displacements generated in a steel cantilever beam by an impact load are measured using a double-pulsed electronic speckle pattern interferometry system. The state of deformation of the object is freezed using a ruby laser and the pulse emission is synchronised with the impact load. Correlation fringes for di!erent times after the start of the impact are obtained using a digital image system. For each time, transient displacements are evaluated by digital analysis of the fringes. In order to test the experimental results, the time response of the beam subjected to the impact load is evaluated using a 3-D "nite element analysis. A close agreement is found between the experimental and numerical results, which prove the reliability of the optical technique to measure high-speed transient deformations. 2000 Elsevier Science Ltd. All rights reserved.
1. Introduction Nowadays, electronic speckle pattern interferometry (ESPI) [1] is a well-developed optical technique for full "eld measurement of surface displacements, deformations * Correspondence address: Instituto de Fisica Rosario, Consejo Nac de Investigaciones, Universidad Nacional de Rosario, Bv. 27 De Febrero 210 BIS, 2000 Rosario, Argentina. Tel.: #54-341-485-3200/3222; fax: #54-341-482-1772. E-mail address: guille@i"r.i"r.edu.ar (G.H. Kaufmann). 0143-8166/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 3 - 8 1 6 6 ( 0 0 ) 0 0 0 1 7 - 8
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and vibrations [2]. Its noncontacting nature, the possibility of further automatic fringe analysis and the ability to use real-time digital image processing techniques, make it a valuable tool in engineering applications. The advent of pulsed lasers in the last decades [3] has extended the range of applications of this technique allowing the analysis of vibrations, high-speed phenomena and transient events among other applications. Pulsed lasers can produce powerful coherent pulses of 10}40 ns duration, which e!ectively freeze object motion. Two pulsed-ESPI recordings separated by a certain time interval can be subtracted on a pixel by pixel basis, leading to speckle correlation fringes. These fringes carry information about the phase changes occurred between both recordings in the optical path of the interferometer. With the laser source operating in the double-pulse mode, pulse separations on the order of 1}500 ls can be achieved. This short time between pulses prevents that thermal convective currents, building vibrations, or rigid body motion of the object could lead to spurious correlation fringes or speckle decorrelation. The generation of pulsed correlation fringes by subtraction requires that the primary interferograms were recorded in separate TV "elds, and afterwards be digitised and subtracted on a pixel by pixel basis. The subtraction process reduces the "xed dc noise contribution in the resulting fringes. Also, pulse to pulse intensity #uctuations can be overcome by rescaling the primary interferograms by appropriate constants. Double-pulsed subtraction ESPI combines the environmental immunity of double-pulsed techniques with the fringe quality of subtractive methods [4]. When the temporal evolution of high-speed transient phenomena is measured by means of pulsed ESPI, the low "ring rate of laser sources (from 60 Hz for an Nd : YAG laser to about six "rings per minute for a ruby laser) restricts the analysis to repetitive phenomena. In this way, various tests must be performed. Between each one of these, the time delay from the start of the transient event to "ring the second laser pulse is varied. In the last few years, various groups have been studying repetitive transient phenomena using pulsed ESPI [4}11]. Several systems were reported based on di!erent laser sources, cameras, synchronization schemes and study cases. The obtained results, both qualitative and quantitative, mainly consist of fringe patterns and displacement distributions for di!erent times after the start of the load. Qualitative results were also used to detect inhomogeneities or faults in the material where a transient wave is propagating [6]. However, most of the results have not been tested in order to prove the accuracy and reliability of pulsed ESPI techniques to study high-speed transient events. This can be due to the fact that the analytical solution of transient problems are very di$cult or even impossible to obtain, even for very simple objects. In this paper, a system based on a double-pulsed ruby laser and a standard interline transfer CCD camera was used to measure transient out-of-plane deformations of a steel cantilever beam generated by an impact load. The approach used to synchronise the emission of the laser pulses with image acquisition and the impact loading is described. The correlation fringes obtained with the system were used to calculate the optical phase distribution and the displacement "elds for di!erent times after the object impact. The numerical processing used to automate the evaluation of
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the correlation fringes is also discussed. In order to assess the experimental results, the deformation of the beam is numerically evaluated with a 3-D "nite element analysis. Finally, both the experimental and numerical results are presented and compared.
2. Experimental 2.1. Double-pulsed ESPI system A diagram of the double-pulsed ESPI system used to measure out-of-plane transient displacement "elds is shown in Fig. 1. The light source, which for clarity is not shown in the "gure, is a ruby laser (Laser Photonics 22HD30) which can emit two 40 ns pulses of approximately 200 mJ per pulse and a variable time separation that
Fig. 1. Out-of-plane interferometer used to measure transient deformations in a cantilever beam subjected to impact loading; W1, W2: wedge plates, M1, M2: mirrors, L1, L2: negative lenses, L3: SLR zoom objective, C: CCD camera, ND: neutral density "lter, O: steel cantilever beam, EM: electromagnet, P: steel pendulum, p: polarizing sheet; BS: beam splitter cube, Ph: photodetector, HD, VD: horizontal and vertical driving signals.
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ranges from 100 to 400 ls. The pulses are divided into two beams by an uncoated wedge plate (W1) in order to reduce light intensity and to avoid interference by multiple re#ections in the reference beam. The object beam is expanded by a divergent lens (L1) to avoid ionizing the air. A second wedge plate (W2) and a neutral density "lter (ND) further reduce light intensity before expanding the reference beam into the beam splitter (BS) in front of the CCD sensor. A high-speed photodetector (Ph) senses the occurrence of the light pulses. The image acquisition was carried out by an interline-transfer CCD camera (C) (Pulnix TM765) externally driven to work in the non-interlaced mode and a zoom objective (L3). In order to stop the light depolarised by surface re#ections, a polarising sheet (p) with its transmission axis parallel to the output laser polarisation vector was placed in front of the objective lens. The test object (O) was a 40 mm;90 mm;9 mm steel cantilever beam with its base welded to a heavy support. A pendulum (P) was used to generate the transient load by impacting on the back surface of the beam at its center of mass. In order to avoid the introduction of torsional vibration modes, the pendulum impacted on the beam half-width. An electromagnet (EM) loosened the pendulum when the coil current was switched o! to make the experiment repeatable with respect to the starting condition. A lateral view of the cantilever and the pendulum-electromagnet system is also shown in Fig. 1. The pendulum was designed to have minimum friction losses and the magnetic clamp maintains a minimum super"cial contact in order to obtain a repetitive time of #ight (the time between the interruption of the coil current and the impact). When the pendulum made contact with the cantilever beam, a voltage appeared across the 1 k) resistance. This voltage variation was used to determine the start of the transient event. Low voltages (below 3 V) were preferable in order to avoid premature sensing of the impact due to arc currents. 2.2. Synchronization In double-pulse subtraction ESPI, the primary interferograms need to be recorded in separate TV "elds. Also, the impact loading must occur between laser pulses in order to record an undeformed reference state of the object in a TV "eld, and the transient loading state in the next one. Interline transfer CCD cameras have a readout period of approximately 1 ls, which is short enough to allow the emission of the "rst laser pulse at the end of a "eld and the second pulse at the beginning of the following one [4]. As the standard output video signal of the CCD camera used in this work was "eld interlaced CCIR, horizontal HD and vertical VD driving signals were externally supplied to the camera by a signal generator specially constructed for obtaining non-interlaced video. In this way, only the even "elds are scanned out from the CCD sensor, improving speckle correlation but reducing the spatial resolution to half [4,12]. To obtain a non-#ickering display of the correlation fringes on the video monitor, the even "elds were copied over the odd ones after the subtraction of the primary speckle interferograms.
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Fig. 2. Timing diagram showing the synchronisation of the video signal, impact loading and laser pulse emission; t is the time that takes the deformation to evolve from impact to the second pulse emission.
The HD and VD signals supplied the timing to scan out the CCD sensor. The non-interlaced output of the camera was fed to a frame grabber (Imaging Technology AFG), which was controlled by a personal computer. Once the laser capacitors were ready for discharge, the frame grabber waited for the beginning of a vertical blanking period of the video signal. When this voltage was detected, the frame grabber triggered a delay circuit and sent a start signal to the computer. The delay circuit switched o! the electromagnet coil current after waiting a variable time delay t . As a result, the pendulum was loosened and some time later it impacted on the steel cantilever beam. The integer ratio between the time of #ight and the TV "eld period (20 ms) gives the number n of TV "elds that have passed during the pendulum movement. After receiving the start signal, the computer software waited for (n!1) TV "elds and then it sent a step signal to a second delay circuit which triggered the laser pulse emission some time later denoted by t . The delay was set to accurately locate the "rst laser pulse at the end of the "eld n and the second one at the beginning of the "eld (n#1), leaving the readout period of "eld n in between. Finally, the delay t was adjusted to locate the start of the impact between both the laser pulses. The time elapsed from the start of the impact to the second pulse emission is the time t that takes the transient phenomena to evolve from rest (see Fig. 2). This time was measured with a digital oscilloscope. Inasmuch as the impact is assumed repetitive, various tests must be performed varying the delay time t , thus enabling the temporal variation of the transient deformation to be measurable.
3. Fringe analysis The intensity distribution of a subtractive speckle correlation fringe pattern can be expressed as [1] "I !I """4(I I [sin(U#* /2) sin(* /2)]", M P
(1)
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Fig. 3. Sequence of out-of-plane correlation fringes for different times after the start of the impact: (a) t"40 ls, (b) t"60 ls, (c) t"126 ls and (d) t"138 ls.
where I and I are the speckle intensity distributions before and after object deformation, respectively, I and I are the intensity distributions of the object and M P reference beams, U is the random phase of the speckle distribution and * is the phase variation due to the object deformation, which for our out-of-plane interferometer can be expressed as 2p * (x, y, t)" w(x, y, t)(1#cos h), j
(2)
where j"694.3 nm is the wavelength of the ruby laser, w(x, y, t) is the component of the displacement normal to the surface in the point (x, y) at time t, and h"6.33 is the angle between the illuminating and viewing directions. Using the optical setup and the synchronization scheme described above, near 40 fringe patterns of the steel beam were recorded for di!erent times after the start of the impact (0(t(200 ls). For simplicity, four of the most signi"cant fringe patterns are displayed in Fig. 3. The slight asymmetry shown by the fringe pattern recorded for t"126 ls is caused by the excitation of a torsion mode. These modes are produced when the point of impact is not exactly at the half-width of the beam, and it is very di$cult to avoid them completely. From these raw data, the speckle modulation term 4(I I sin(U#* /2) was M P reduced by means of a "lter based on Markov "elds and the regularization theory [13]. The smoothed fringes can be considered as an estimate of the low-pass component sin (* /2) of the fringe pattern. They are obtained as the minimizer of a cost function and the amount of smoothing is controled by a regularization parameter. This "lter, as opposed to classical convolution or Fourier low-pass "lters, produces no
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Fig. 4. (a) fringe pattern of Fig. 3(b) after smoothing with the regularization "lter, (b) wrapped phase map obtained from Fig. 4(a) with the sign inverted over the upper half of the image.
artifacts or smoothing on the boundaries of the image. Fig. 4(a) shows a smoothed version of the fringe pattern corresponding to 60 ls. The optical phase distribution * was obtained by using a Fourier transform method similar to the one proposed by Kreis for the analysis of holographic interferograms [14]. This fringe analysis method is fully automatic and uses a single interferogram. It can be shown that the Fourier transform of a vertical cosinusoidal fringe pattern yields a zero frequency peak and two components of opposite sign that carry the phase information of the fringes. By band pass "ltering the amplitude spectrum in the #v half-plane, the zero frequency peak and the negative frequency component are "ltered out. As a result, the remaining spectrum is no longer Hermitean, and its inverse Fourier transform is a complex function. A wrapped phase distribution of the correlation fringes is obtained by calculating pointwisely the arctangent of the ratio between the imaginary and real parts of that complex valued function. This algorithm cannot specify the sign of the phase gradient and always leads to a monotonically increasing or decreasing phase distribution (in this case in the vertical direction). For this reason, non-monotonic phase distributions such as those corresponding to Figs. 3(a) and (b) have a phase inversion crossing the closed fringe centers, because an ambiguity for the direction of the displacements remains. In this work, the sign of the phase corresponding to the fringe patterns shown in Figs. 3(a) and (b) was reversed over the bottom-half of the beam. Fig. 4(b) shows the wrapped phase map obtained from Fig. 4(a). The continuous phase distribution was calculated using an iterative algorithm based on a least-squares formulation of the unwrapping problem whose solution is provided by a fast discrete cosine transform [15,16]. This algorithm deals with inconsistent pixels (points of erroneous phase discontinuities usually caused by noise)
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Fig. 5. (a) weighting matrix used for phase unwrapping (the algorithm interpolates over the regions in black), (b) unwrapped phase distribution.
Fig. 6. Out-of-plane transient displacement along the axis x"0 of the beam for 40, 60, 126 and 138 ls after the start of the impact.
by using a weighting matrix which assigns a weight of zero to bad data and then e!ectively interpolates the phase data at those points. The weighting matrix used in this work (see Fig. 5(a)) forces the unwrapping algorithm to interpolate the phase over the inconsistent pixels, over areas of fringe cuts or discontinuities, and over the horizontal line which separates the zones of di!erent phase sign. In this way, unwrapped phase distributions as the one shown in Fig. 5(b) were obtained. The transient out-of-plane displacements were obtained straightforward from the continuous phase distributions determined for each time using Eq. (2). Fig. 6 shows a plot of the displacement w(0, y, t) along the axis x"0 of the beam for t"40,60,126 and 138 ls after the start of the impact.
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4. Numerical calculation of the impact-induced displacements The accuracy of the transient displacements measured for di!erent times using the double-pulsed ESPI technique was determined by evaluating a numerical solution of the impact problem. This evaluation was carried out through a 3-D "nite element analysis. The cantilever beam was modelled as a set of 10;30;4 solid "nite elements (see Fig. 7). Each element is of eight-noded isoparametric type, with three translational degrees of freedom per node. The material constants used in the analysis were the steel density o"7.85;10 kg/m, the longitudinal elastic modulus E"2.05;10 MPa and the Poisson's ratio e"0.29. Displacement restrictions were applied over the clamping nodes where y"0. Because the contact area during the impact is less than 1 mm, a single joint of the "nite element mesh was submitted to the impact force (which is indicated with an arrow in Fig. 7). The impact process was modelled in a simpli"ed way, retaining its main physical characters, i.e. total momentum and duration [17]. As the temporal variation of the force during the impact was not known, it was assumed to be a triangular function
ht/t for 0)t)t , F F F(t)" h![ht/(¹!t )] for t (t)¹, F F 0 otherwise,
(3)
where t is the time in which F(t) has its maximum. The constant h"2G/¹ was F determined from the impulse G"14.38;10\ Ns transferred to the beam during the impact time ¹"90 ls. The impulse G was determined from a rigid body calculation
Fig. 7. 3-D "nite element mesh used for the calculation of the transient displacements. The arrow indicates the point where the dynamical force was applied.
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of the pendulum angular momentum variation during the impact. The duration of the impact was measured using the circuit shown in Fig. 1 and a digital oscilloscope. The STAAD-III "nite element package [18] was employed to perform the numerical analysis. The step-by-step modal superposition method was applied to integrate the movement equations, using the "rst 12 natural frecuencies and mode shapes, and a time increment *t"2 ls. The out-of-plane component w(0, y, t) of the transient displacement was computed for t "¹/3, ¹/2 and 2¹/3, and every 2 ls after the start F of the impact. For di!erent values of t , it was observed that the time response of the F out-of-plane displacements di!erred in magnitude and temporal phase, i.e. they were slow or fast, with respect to the experimental measurements shown in Fig. 6. For t "¹/3, the impulse is rapidly transferred to the beam and the computed displaceF ments become larger than the measured ones for shorter times after the impact. The opposite occurred for t "2¹/3 and an intermediate situation was found for t "¹/2. F F Inasmuch as the objective of this work was not the study of the impact process itself and therefore the shape of F(t), but an evaluation of the pulsed ESPI technique, only those results which resembled better the experimental data are presented. Fig. 8 shows the transient out-of-plane displacements measured using the double-pulsed ESPI technique and the "nite element results for t "2¹/3 and for t"40,60,126 and F 138 ls. It is seen that the experimental and numerical displacement pro"les are similar in shape. The average relative error between the experimental and numerical data gives a more quantitative description of this agreement. These values go from 9% for the data corresponding to t"60 ls to 26% for t"138 ls. Since the impact duration was 90$2 ls, the pendulum was still in contact with the beam during the "rst two times shown in Fig. 8. It can be seen that at the beginning of the impact, the free end of the beam goes backwards because of the inertia forces. Then, it goes forwards. It is also observed that the numerical results are still a few microseconds slow at the beginning of the impact and fast for t'120 ls, if compared with the experimental measurements for w(0, y, t). This discrepancy might be a consequence of a di!erence between the boundary conditions of the real problem and the
Fig. 8. Comparison between the measurements obtained with the double-pulsed ESPI system and the "nite element results for di!erent times after the start of the impact.
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ones assumed for the numerical problem. Even though the metal beam was welded to a heavy support, some deformation was transmitted to it, as it was con"rmed afterwards by "nding a fringe across the specimen base. Moreover, the numerical model assumes that the nodes with y"0 are perfectly "xed. This explains the di!erence of slope in the numerical and experimental curves for y"0. But the most remarkable cause of discrepancy is the particular shape of the applied force F(t). An exact "tting of the temporal evolution of the displacement "elds would only be expected if F(t) was completely de"ned.
5. Conclusions In this paper, double-pulsed ESPI is used for measuring the out-of-plane transient displacement of a steel cantilever beam submitted to impact loading. The scheme used to synchronise the emission of the laser pulses with image acquisition and the impact loading system is also described. The transient displacement "eld is determined by automatic analysis of correlation fringes obtained with the double-pulsed ESPI system. The experimental results evaluated for di!erent times after the start of the impact are compared with a numerical solution obtained using a 3-D "nite element analysis. The numerical solution shows a sti!er behavior and temporal phase shifts with respect to the experimental measurement of displacements. However, a relatively good agreement is achieved, obtaining values for the average relative error from 9% for t"60 ls to 26% for t"138 ls. The discrepancies encountered can be ascribed to di!erent causes. First, there was a di!erence between the boundary conditions of the real beam (actually a deformable welded joint) and the ones assumed for the numerical problem (perfectly "xed nodes). Also, the temporal response of the out-of-plane displacements was observed to be very sensitive to the variation of the contact force F(t) during the impact. It has shown the appropriateness of the pendulum as a simple, economic and repetitive impact device. For future analysis of transient events some considerations should be taken into account. Boundary conditions must be carefully de"ned in order to resemble the actual system in the numerical model. Another possibility is to modelise both the beam and its support as a single object. Even the impact system should also be modelised. It should be noted that the contact force F(t) must be known in advance in order to obtain an accurate numerical solution of the problem. Also, double-pulsed ESPI could be potentially useful to study the impact process by itself. This study could be carried out by comparing the experimental and numerical results and by using the shape of F(t) as a "tting parameter. To conclude, this work demonstrates the reliability of double-pulsed ESPI to study high speed transient deformations of metalic objects submitted to impact loading. It also proves that nowadays this technique is mature enough to help the improvement of the numerical modelization of mechanical systems submitted to dynamical loads.
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References [1] Jones R, Wykes C. Holographic and speckle interferometry. Cambridge: Cambridge University Press, 1989. p. 165. [2] L+kberg O. Video speckle interferometry. In: Sirohi RS, editor. Speckle metrology. New York: Marcel Dekker, 1993. p. 157. [3] Cookson TJ, Butters JN, Pollard HC. Pulse lasers in electronic speckle pattern interferometry. Opt Laser Tech 1978;10:119}24. [4] Spooren R. Double-pulse subtraction TV holography. Opt Eng 1992;31:1000}6. [5] Tong J, Zhang D, Li H, Li L. Study on in-plane displacement measurement under impact loading using digital speckle pattern interferometry. Opt Eng 1996;35:1080}3. [6] FernaH ndez A, Moore AJ, PeH rez LoH pez C, Doval AF, Blanco-GarcmH a J. Study of transient deformations with pulsed TV holography: application to crack detection. Appl Opt 1997;36:2058}65. [7] Pedrini G, P"ster B, Tiziani HJ. Double pulse-electronic speckle interferometry. J Mod Opt 1993;40:89}96. [8] Kaufmann GH, DmH az FV, Galizzi GE, Moore AJ. Transient in-plane deformation analysis by means of pulsed TV holography. Optik 1998;108:43}7. [9] Moore AJ, Hand DP, Barton JS, Jones JDC. Transient deformation measurement with electronic speckle pattern interferometry and a high-speed camera. Appl Opt 1999;38:1159}62. [10] FernaH ndez A, Doval AF, DaH vila A, Blanco-GarcmH a J, PeH rez-LoH pez C, FernaH ndez JL. Double-pulsed carrier speckle-shearing pattern interferometry for transient deformation analysis. Proc SPIE 1998;3478:353}8. [11] Farrant DI, Kaufmann GH, Petzing JN, Tyrer JR, Oreb BF, Kerr D. Transient deformation measurement using dual-pulse addition ESPI. Appl Opt 1998;37:7259}67. [12] Ruiz PD, Kaufmann GH, Galizzi GE. Operation modes of a CCD interline-transfer camera in double pulsed electronic speckle pattern interferometry. Opt Eng 1999;38:1603}8. [13] Marroquin JL, Rivera M, Botello S, RodrmH guez-Vera R, ServmH n M. Regularization methods for processing fringe-pattern images. Appl Opt 1999;38:788}94. [14] Kreis, T. Computer-aided evaluation of holographic interferograms. In: Rastogi PK, editor. Holographic interferometry. Berlin: Springer, 1994. [15] Ghiglia DC, Romero LA. Robust two-dimensional weighted and unweigthed phase unwrapping thag uses fast transforms and iterative methods. J Opt Soc Am 1994;11:107}17. [16] Kaufmann GH, Galizzi G, Ruiz PD. Unwrapping of ESPI phase maps using a preconditioned conjugate gradient algorithm. Appl Opt 1998;37:3076}84. [17] Goldsmith W. Impact. London: Edward Arnold, 1960. [18] STAAD-III, structural analysis and design software, user manual, Sacramento, 1996.
Evaluation of impact-induced transient deformations using double-pulsed electronic speckle pattern interferometry and "nite elements P.D. Ruiz , G.H. Kaufmann *, O. MoK ller, G.E. Galizzi Instituto de Fn& sica Rosario (CONICET-UNR) 2000 Rosario, Argentina Departamento de Fn& sica, Facultad de Ciencias Exactas, Ingeniern& a y Agrimensura, Universidad Nacional de Rosario, 2000 Rosario, Argentina Instituto de Meca& nica Aplicada y Estructuras, Facultad de Ciencias Exactas, Ingeniern& a y Agrimensura, Universidad Nacional de Rosario, 2000 Rosario, Argentina Received 2 December 1999; accepted 4 February 2000
Abstract Transient out-of-plane displacements generated in a steel cantilever beam by an impact load are measured using a double-pulsed electronic speckle pattern interferometry system. The state of deformation of the object is freezed using a ruby laser and the pulse emission is synchronised with the impact load. Correlation fringes for di!erent times after the start of the impact are obtained using a digital image system. For each time, transient displacements are evaluated by digital analysis of the fringes. In order to test the experimental results, the time response of the beam subjected to the impact load is evaluated using a 3-D "nite element analysis. A close agreement is found between the experimental and numerical results, which prove the reliability of the optical technique to measure high-speed transient deformations. 2000 Elsevier Science Ltd. All rights reserved.
1. Introduction Nowadays, electronic speckle pattern interferometry (ESPI) [1] is a well-developed optical technique for full "eld measurement of surface displacements, deformations * Correspondence address: Instituto de Fisica Rosario, Consejo Nac de Investigaciones, Universidad Nacional de Rosario, Bv. 27 De Febrero 210 BIS, 2000 Rosario, Argentina. Tel.: #54-341-485-3200/3222; fax: #54-341-482-1772. E-mail address: guille@i"r.i"r.edu.ar (G.H. Kaufmann). 0143-8166/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 3 - 8 1 6 6 ( 0 0 ) 0 0 0 1 7 - 8
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and vibrations [2]. Its noncontacting nature, the possibility of further automatic fringe analysis and the ability to use real-time digital image processing techniques, make it a valuable tool in engineering applications. The advent of pulsed lasers in the last decades [3] has extended the range of applications of this technique allowing the analysis of vibrations, high-speed phenomena and transient events among other applications. Pulsed lasers can produce powerful coherent pulses of 10}40 ns duration, which e!ectively freeze object motion. Two pulsed-ESPI recordings separated by a certain time interval can be subtracted on a pixel by pixel basis, leading to speckle correlation fringes. These fringes carry information about the phase changes occurred between both recordings in the optical path of the interferometer. With the laser source operating in the double-pulse mode, pulse separations on the order of 1}500 ls can be achieved. This short time between pulses prevents that thermal convective currents, building vibrations, or rigid body motion of the object could lead to spurious correlation fringes or speckle decorrelation. The generation of pulsed correlation fringes by subtraction requires that the primary interferograms were recorded in separate TV "elds, and afterwards be digitised and subtracted on a pixel by pixel basis. The subtraction process reduces the "xed dc noise contribution in the resulting fringes. Also, pulse to pulse intensity #uctuations can be overcome by rescaling the primary interferograms by appropriate constants. Double-pulsed subtraction ESPI combines the environmental immunity of double-pulsed techniques with the fringe quality of subtractive methods [4]. When the temporal evolution of high-speed transient phenomena is measured by means of pulsed ESPI, the low "ring rate of laser sources (from 60 Hz for an Nd : YAG laser to about six "rings per minute for a ruby laser) restricts the analysis to repetitive phenomena. In this way, various tests must be performed. Between each one of these, the time delay from the start of the transient event to "ring the second laser pulse is varied. In the last few years, various groups have been studying repetitive transient phenomena using pulsed ESPI [4}11]. Several systems were reported based on di!erent laser sources, cameras, synchronization schemes and study cases. The obtained results, both qualitative and quantitative, mainly consist of fringe patterns and displacement distributions for di!erent times after the start of the load. Qualitative results were also used to detect inhomogeneities or faults in the material where a transient wave is propagating [6]. However, most of the results have not been tested in order to prove the accuracy and reliability of pulsed ESPI techniques to study high-speed transient events. This can be due to the fact that the analytical solution of transient problems are very di$cult or even impossible to obtain, even for very simple objects. In this paper, a system based on a double-pulsed ruby laser and a standard interline transfer CCD camera was used to measure transient out-of-plane deformations of a steel cantilever beam generated by an impact load. The approach used to synchronise the emission of the laser pulses with image acquisition and the impact loading is described. The correlation fringes obtained with the system were used to calculate the optical phase distribution and the displacement "elds for di!erent times after the object impact. The numerical processing used to automate the evaluation of
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475
the correlation fringes is also discussed. In order to assess the experimental results, the deformation of the beam is numerically evaluated with a 3-D "nite element analysis. Finally, both the experimental and numerical results are presented and compared.
2. Experimental 2.1. Double-pulsed ESPI system A diagram of the double-pulsed ESPI system used to measure out-of-plane transient displacement "elds is shown in Fig. 1. The light source, which for clarity is not shown in the "gure, is a ruby laser (Laser Photonics 22HD30) which can emit two 40 ns pulses of approximately 200 mJ per pulse and a variable time separation that
Fig. 1. Out-of-plane interferometer used to measure transient deformations in a cantilever beam subjected to impact loading; W1, W2: wedge plates, M1, M2: mirrors, L1, L2: negative lenses, L3: SLR zoom objective, C: CCD camera, ND: neutral density "lter, O: steel cantilever beam, EM: electromagnet, P: steel pendulum, p: polarizing sheet; BS: beam splitter cube, Ph: photodetector, HD, VD: horizontal and vertical driving signals.
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ranges from 100 to 400 ls. The pulses are divided into two beams by an uncoated wedge plate (W1) in order to reduce light intensity and to avoid interference by multiple re#ections in the reference beam. The object beam is expanded by a divergent lens (L1) to avoid ionizing the air. A second wedge plate (W2) and a neutral density "lter (ND) further reduce light intensity before expanding the reference beam into the beam splitter (BS) in front of the CCD sensor. A high-speed photodetector (Ph) senses the occurrence of the light pulses. The image acquisition was carried out by an interline-transfer CCD camera (C) (Pulnix TM765) externally driven to work in the non-interlaced mode and a zoom objective (L3). In order to stop the light depolarised by surface re#ections, a polarising sheet (p) with its transmission axis parallel to the output laser polarisation vector was placed in front of the objective lens. The test object (O) was a 40 mm;90 mm;9 mm steel cantilever beam with its base welded to a heavy support. A pendulum (P) was used to generate the transient load by impacting on the back surface of the beam at its center of mass. In order to avoid the introduction of torsional vibration modes, the pendulum impacted on the beam half-width. An electromagnet (EM) loosened the pendulum when the coil current was switched o! to make the experiment repeatable with respect to the starting condition. A lateral view of the cantilever and the pendulum-electromagnet system is also shown in Fig. 1. The pendulum was designed to have minimum friction losses and the magnetic clamp maintains a minimum super"cial contact in order to obtain a repetitive time of #ight (the time between the interruption of the coil current and the impact). When the pendulum made contact with the cantilever beam, a voltage appeared across the 1 k) resistance. This voltage variation was used to determine the start of the transient event. Low voltages (below 3 V) were preferable in order to avoid premature sensing of the impact due to arc currents. 2.2. Synchronization In double-pulse subtraction ESPI, the primary interferograms need to be recorded in separate TV "elds. Also, the impact loading must occur between laser pulses in order to record an undeformed reference state of the object in a TV "eld, and the transient loading state in the next one. Interline transfer CCD cameras have a readout period of approximately 1 ls, which is short enough to allow the emission of the "rst laser pulse at the end of a "eld and the second pulse at the beginning of the following one [4]. As the standard output video signal of the CCD camera used in this work was "eld interlaced CCIR, horizontal HD and vertical VD driving signals were externally supplied to the camera by a signal generator specially constructed for obtaining non-interlaced video. In this way, only the even "elds are scanned out from the CCD sensor, improving speckle correlation but reducing the spatial resolution to half [4,12]. To obtain a non-#ickering display of the correlation fringes on the video monitor, the even "elds were copied over the odd ones after the subtraction of the primary speckle interferograms.
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Fig. 2. Timing diagram showing the synchronisation of the video signal, impact loading and laser pulse emission; t is the time that takes the deformation to evolve from impact to the second pulse emission.
The HD and VD signals supplied the timing to scan out the CCD sensor. The non-interlaced output of the camera was fed to a frame grabber (Imaging Technology AFG), which was controlled by a personal computer. Once the laser capacitors were ready for discharge, the frame grabber waited for the beginning of a vertical blanking period of the video signal. When this voltage was detected, the frame grabber triggered a delay circuit and sent a start signal to the computer. The delay circuit switched o! the electromagnet coil current after waiting a variable time delay t . As a result, the pendulum was loosened and some time later it impacted on the steel cantilever beam. The integer ratio between the time of #ight and the TV "eld period (20 ms) gives the number n of TV "elds that have passed during the pendulum movement. After receiving the start signal, the computer software waited for (n!1) TV "elds and then it sent a step signal to a second delay circuit which triggered the laser pulse emission some time later denoted by t . The delay was set to accurately locate the "rst laser pulse at the end of the "eld n and the second one at the beginning of the "eld (n#1), leaving the readout period of "eld n in between. Finally, the delay t was adjusted to locate the start of the impact between both the laser pulses. The time elapsed from the start of the impact to the second pulse emission is the time t that takes the transient phenomena to evolve from rest (see Fig. 2). This time was measured with a digital oscilloscope. Inasmuch as the impact is assumed repetitive, various tests must be performed varying the delay time t , thus enabling the temporal variation of the transient deformation to be measurable.
3. Fringe analysis The intensity distribution of a subtractive speckle correlation fringe pattern can be expressed as [1] "I !I """4(I I [sin(U#* /2) sin(* /2)]", M P
(1)
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Fig. 3. Sequence of out-of-plane correlation fringes for different times after the start of the impact: (a) t"40 ls, (b) t"60 ls, (c) t"126 ls and (d) t"138 ls.
where I and I are the speckle intensity distributions before and after object deformation, respectively, I and I are the intensity distributions of the object and M P reference beams, U is the random phase of the speckle distribution and * is the phase variation due to the object deformation, which for our out-of-plane interferometer can be expressed as 2p * (x, y, t)" w(x, y, t)(1#cos h), j
(2)
where j"694.3 nm is the wavelength of the ruby laser, w(x, y, t) is the component of the displacement normal to the surface in the point (x, y) at time t, and h"6.33 is the angle between the illuminating and viewing directions. Using the optical setup and the synchronization scheme described above, near 40 fringe patterns of the steel beam were recorded for di!erent times after the start of the impact (0(t(200 ls). For simplicity, four of the most signi"cant fringe patterns are displayed in Fig. 3. The slight asymmetry shown by the fringe pattern recorded for t"126 ls is caused by the excitation of a torsion mode. These modes are produced when the point of impact is not exactly at the half-width of the beam, and it is very di$cult to avoid them completely. From these raw data, the speckle modulation term 4(I I sin(U#* /2) was M P reduced by means of a "lter based on Markov "elds and the regularization theory [13]. The smoothed fringes can be considered as an estimate of the low-pass component sin (* /2) of the fringe pattern. They are obtained as the minimizer of a cost function and the amount of smoothing is controled by a regularization parameter. This "lter, as opposed to classical convolution or Fourier low-pass "lters, produces no
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Fig. 4. (a) fringe pattern of Fig. 3(b) after smoothing with the regularization "lter, (b) wrapped phase map obtained from Fig. 4(a) with the sign inverted over the upper half of the image.
artifacts or smoothing on the boundaries of the image. Fig. 4(a) shows a smoothed version of the fringe pattern corresponding to 60 ls. The optical phase distribution * was obtained by using a Fourier transform method similar to the one proposed by Kreis for the analysis of holographic interferograms [14]. This fringe analysis method is fully automatic and uses a single interferogram. It can be shown that the Fourier transform of a vertical cosinusoidal fringe pattern yields a zero frequency peak and two components of opposite sign that carry the phase information of the fringes. By band pass "ltering the amplitude spectrum in the #v half-plane, the zero frequency peak and the negative frequency component are "ltered out. As a result, the remaining spectrum is no longer Hermitean, and its inverse Fourier transform is a complex function. A wrapped phase distribution of the correlation fringes is obtained by calculating pointwisely the arctangent of the ratio between the imaginary and real parts of that complex valued function. This algorithm cannot specify the sign of the phase gradient and always leads to a monotonically increasing or decreasing phase distribution (in this case in the vertical direction). For this reason, non-monotonic phase distributions such as those corresponding to Figs. 3(a) and (b) have a phase inversion crossing the closed fringe centers, because an ambiguity for the direction of the displacements remains. In this work, the sign of the phase corresponding to the fringe patterns shown in Figs. 3(a) and (b) was reversed over the bottom-half of the beam. Fig. 4(b) shows the wrapped phase map obtained from Fig. 4(a). The continuous phase distribution was calculated using an iterative algorithm based on a least-squares formulation of the unwrapping problem whose solution is provided by a fast discrete cosine transform [15,16]. This algorithm deals with inconsistent pixels (points of erroneous phase discontinuities usually caused by noise)
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Fig. 5. (a) weighting matrix used for phase unwrapping (the algorithm interpolates over the regions in black), (b) unwrapped phase distribution.
Fig. 6. Out-of-plane transient displacement along the axis x"0 of the beam for 40, 60, 126 and 138 ls after the start of the impact.
by using a weighting matrix which assigns a weight of zero to bad data and then e!ectively interpolates the phase data at those points. The weighting matrix used in this work (see Fig. 5(a)) forces the unwrapping algorithm to interpolate the phase over the inconsistent pixels, over areas of fringe cuts or discontinuities, and over the horizontal line which separates the zones of di!erent phase sign. In this way, unwrapped phase distributions as the one shown in Fig. 5(b) were obtained. The transient out-of-plane displacements were obtained straightforward from the continuous phase distributions determined for each time using Eq. (2). Fig. 6 shows a plot of the displacement w(0, y, t) along the axis x"0 of the beam for t"40,60,126 and 138 ls after the start of the impact.
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4. Numerical calculation of the impact-induced displacements The accuracy of the transient displacements measured for di!erent times using the double-pulsed ESPI technique was determined by evaluating a numerical solution of the impact problem. This evaluation was carried out through a 3-D "nite element analysis. The cantilever beam was modelled as a set of 10;30;4 solid "nite elements (see Fig. 7). Each element is of eight-noded isoparametric type, with three translational degrees of freedom per node. The material constants used in the analysis were the steel density o"7.85;10 kg/m, the longitudinal elastic modulus E"2.05;10 MPa and the Poisson's ratio e"0.29. Displacement restrictions were applied over the clamping nodes where y"0. Because the contact area during the impact is less than 1 mm, a single joint of the "nite element mesh was submitted to the impact force (which is indicated with an arrow in Fig. 7). The impact process was modelled in a simpli"ed way, retaining its main physical characters, i.e. total momentum and duration [17]. As the temporal variation of the force during the impact was not known, it was assumed to be a triangular function
ht/t for 0)t)t , F F F(t)" h![ht/(¹!t )] for t (t)¹, F F 0 otherwise,
(3)
where t is the time in which F(t) has its maximum. The constant h"2G/¹ was F determined from the impulse G"14.38;10\ Ns transferred to the beam during the impact time ¹"90 ls. The impulse G was determined from a rigid body calculation
Fig. 7. 3-D "nite element mesh used for the calculation of the transient displacements. The arrow indicates the point where the dynamical force was applied.
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of the pendulum angular momentum variation during the impact. The duration of the impact was measured using the circuit shown in Fig. 1 and a digital oscilloscope. The STAAD-III "nite element package [18] was employed to perform the numerical analysis. The step-by-step modal superposition method was applied to integrate the movement equations, using the "rst 12 natural frecuencies and mode shapes, and a time increment *t"2 ls. The out-of-plane component w(0, y, t) of the transient displacement was computed for t "¹/3, ¹/2 and 2¹/3, and every 2 ls after the start F of the impact. For di!erent values of t , it was observed that the time response of the F out-of-plane displacements di!erred in magnitude and temporal phase, i.e. they were slow or fast, with respect to the experimental measurements shown in Fig. 6. For t "¹/3, the impulse is rapidly transferred to the beam and the computed displaceF ments become larger than the measured ones for shorter times after the impact. The opposite occurred for t "2¹/3 and an intermediate situation was found for t "¹/2. F F Inasmuch as the objective of this work was not the study of the impact process itself and therefore the shape of F(t), but an evaluation of the pulsed ESPI technique, only those results which resembled better the experimental data are presented. Fig. 8 shows the transient out-of-plane displacements measured using the double-pulsed ESPI technique and the "nite element results for t "2¹/3 and for t"40,60,126 and F 138 ls. It is seen that the experimental and numerical displacement pro"les are similar in shape. The average relative error between the experimental and numerical data gives a more quantitative description of this agreement. These values go from 9% for the data corresponding to t"60 ls to 26% for t"138 ls. Since the impact duration was 90$2 ls, the pendulum was still in contact with the beam during the "rst two times shown in Fig. 8. It can be seen that at the beginning of the impact, the free end of the beam goes backwards because of the inertia forces. Then, it goes forwards. It is also observed that the numerical results are still a few microseconds slow at the beginning of the impact and fast for t'120 ls, if compared with the experimental measurements for w(0, y, t). This discrepancy might be a consequence of a di!erence between the boundary conditions of the real problem and the
Fig. 8. Comparison between the measurements obtained with the double-pulsed ESPI system and the "nite element results for di!erent times after the start of the impact.
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ones assumed for the numerical problem. Even though the metal beam was welded to a heavy support, some deformation was transmitted to it, as it was con"rmed afterwards by "nding a fringe across the specimen base. Moreover, the numerical model assumes that the nodes with y"0 are perfectly "xed. This explains the di!erence of slope in the numerical and experimental curves for y"0. But the most remarkable cause of discrepancy is the particular shape of the applied force F(t). An exact "tting of the temporal evolution of the displacement "elds would only be expected if F(t) was completely de"ned.
5. Conclusions In this paper, double-pulsed ESPI is used for measuring the out-of-plane transient displacement of a steel cantilever beam submitted to impact loading. The scheme used to synchronise the emission of the laser pulses with image acquisition and the impact loading system is also described. The transient displacement "eld is determined by automatic analysis of correlation fringes obtained with the double-pulsed ESPI system. The experimental results evaluated for di!erent times after the start of the impact are compared with a numerical solution obtained using a 3-D "nite element analysis. The numerical solution shows a sti!er behavior and temporal phase shifts with respect to the experimental measurement of displacements. However, a relatively good agreement is achieved, obtaining values for the average relative error from 9% for t"60 ls to 26% for t"138 ls. The discrepancies encountered can be ascribed to di!erent causes. First, there was a di!erence between the boundary conditions of the real beam (actually a deformable welded joint) and the ones assumed for the numerical problem (perfectly "xed nodes). Also, the temporal response of the out-of-plane displacements was observed to be very sensitive to the variation of the contact force F(t) during the impact. It has shown the appropriateness of the pendulum as a simple, economic and repetitive impact device. For future analysis of transient events some considerations should be taken into account. Boundary conditions must be carefully de"ned in order to resemble the actual system in the numerical model. Another possibility is to modelise both the beam and its support as a single object. Even the impact system should also be modelised. It should be noted that the contact force F(t) must be known in advance in order to obtain an accurate numerical solution of the problem. Also, double-pulsed ESPI could be potentially useful to study the impact process by itself. This study could be carried out by comparing the experimental and numerical results and by using the shape of F(t) as a "tting parameter. To conclude, this work demonstrates the reliability of double-pulsed ESPI to study high speed transient deformations of metalic objects submitted to impact loading. It also proves that nowadays this technique is mature enough to help the improvement of the numerical modelization of mechanical systems submitted to dynamical loads.
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