Open Access
Add to BibSonomy
Abstract
Vibration measurement, particularly mode shape measurement, is an important aspect of structural dynamic analysis since it can validate finite element or analytical vibration models. Scanning laser Doppler vibrometry (LDV) and high-speed digital image correlation have become dominant methods for experimental mode shape measurement. However, these methods have high equipment costs and several disadvantages regarding spatial or temporal performance. This paper proposes a laser Doppler vibrometer induced stroboscopic digital image correction for non-contact mode shape and operational deflection shape measurement. Our results verify that single-point LDV and normal rate cameras can be used obtain high spatial resolution mode shape and operational deflection shape. Measurement frequency range is much higher than the camera capturing rate. We also show that the proposed approach coincides well with time-averaged electronic speckle pattern interferometry.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Ongoing requirements for higher performance and reliability for high-end intelligent manufacturing has placed stringent demands on evaluation and measurement methods used in their development. Vibration measurement, particularly mode shape measurement, is an important component of structural dynamic analysis since it can validate finite element, analytical vibration, or digital twin models. Optical dynamic measurement is non-contact and highly valued and has developed rapidly since it does not change structural dynamic behavior during measurement. Laser, digital camera, and computer developments have promoted digital interferometric techniques for structure mode shape evaluation based on time-averaging, such as electronic speckle pattern interferometry (ESPI) [1–3], digital holography [4], and digital shearing speckle interferometry [5,6]. The obtained interferometric fringe patterns can be treated as full-field displacement or strain mode contours [7]. Interferometry based on pulsed laser [8], stroboscopic illumination [9] and high-speed cameras [10,11] has been applied for deformation or strain measurement between two states, using spatial or temporal phase shifting algorithms [12–14] to precisely calculate the phases. Unfortunately, interferometric data can be easily contaminated by environmental disturbances and most such techniques perform well only under vibration isolation in laboratory environments.
Laser Doppler vibrometry (LDV) [15] has become a dominant method for optical vibration measurement. Although this photodetector based interferometric method is more robust compared with camera-based interferometry discussed above, it can only offer pointwise measurement. Thus, scanning laser Doppler vibrometry (SLDV) and continuous scanning laser Doppler vibrometry (CSLDV) methods have been developed for full-field steady-state vibration measurement [16]. However, SLDV has redundant measurement data in time domain, and relatively low spatial resolution (the maximum spatial resolution of commercial product is 256×256 points). On the other hand, CSLDV specimens are scanned continuously by the laser beam with specifically designed scanning speed and path, offering full-field vibration measurement with high spatial (millions of points) and temporal (25 MHz) resolution [17]. However, two drawbacks prevent CSLDV from wide applications:
- (1) speckle noise due to surface roughness causes pseudo-vibration in results [18], and
- (2) large measurement errors can be generated on some scanning points due to defocusing where the object contour causes large optical path changes during scanning, and real-time focusing during high-speed scanning is impossible using hardware.
Retro-reflective tape or paint is commonly applied on the specimen surface to avoid signal-noise ratio (SNR) drops due to defocusing, but it will change structural dynamic behaviors [16]. Multi-point LDV is another potential solution, but is generally only used to evaluated unsteady state vibration and the number of measuring points are insufficient for full-field measurement [19].
Recent developments in charge-coupled devices, complementary metal oxide semiconductor cameras, and computer capabilities have enabled camera-based measurement techniques to become more widely applied, particularly for industrial applications. Digital image correlation (DIC) is a deformation detection algorithm based on gray scale gradient template matching and widely used in experimental mechanics [20]. DIC can measure out-of-plane displacement to several micrometer accuracy using stereo vision (i.e., two cameras) [21]. Although including high-speed cameras in binocular systems can extend the capability to vibration measurement [22], high camera costs and technical issues, such as high-speed synchronization, has obstructed high-speed imaging-based optical vibration measurement for research and industrial applications.
Consequently, various techniques have been proposed for stereo measurement using single high-speed cameras. Pan and Durand introduced a 4-mirror adapter to form a pseudo-stereo vision system [23,24], but the proposed system was very sensitive to environment noise [25], and half the spatial resolution was sacrificed. Industrial cameras with normal frame rates tend to have larger pixel sizes (typically > 10 μm, whereas normal-rate cameras typically have 3–5 μm pixels) and limited dynamic ranges (typically 8 bit whereas normal-rate cameras support 12–16 bit), hence high-speed cameras could dramatically reduce measurement accuracy [26]. Some high-speed cameras rely on fan cooling [27], which can also introduce a frequency component related to fan speed in vibration measurements. Barone proposed a non-harmonic Fourier analysis method to reconstruct vibration mode shapes based on down sampling images in the time axis captured by normal-rate cameras [28]. However, this can cause serious data redundancy and much longer post-processing time.
This study proposes a single-point LDV-induced stroboscopic digital image correlation method using two normal rate stereo cameras for high spatial resolution and high accuracy vibration measurement. Vibration phase angles are calculated by a field programmable gate array (FPGA) board using analog signal output from single-point LDV. Synchronizing the normal-rate cameras is solved by simultaneously triggering several LED at different phase angles without delay during a long exposure time. Measurement frequency is up to 1204 Hz with 4 fps capture rate. We obtained different orders of vibration mode shapes for a cantilever beam and a rectangular aluminum plate, and compared these mode shapes with those from simulation and time-averaged ESPI. The proposed approach provided an efficient steady-state vibration measurement method with high spatial and sufficient temporal resolution. The proposed approach has considerably lower cost than current SLDV or high-speed imaging systems, and is suitable for many applications in various industrial areas.
2. Theoretical analysis
2.1 Structural vibration
The equation of motion for a multiple-degrees of freedom system with proportional damping can be expressed as [29]:
The exciting frequency $\omega $, i.e., displacement at a specific point on the structure, can be expressed as
2.2 Vibration measurement using LDV
Laser Doppler vibrometry measures vibrational displacement or velocity using heterodyne interferometry. LDV generates a frequency modulated (FM) signal on the photodetector using an acousto-optic frequency shifter. The laser beam is aimed at the vibrating object and scattered back, and the object’s velocity or displacement generates a frequency or phase modulation due to the Doppler effect. The returned beam exhibits a small frequency shift, i.e., the Doppler frequency fD, which is a function of the object’s velocity component v in the direction of the object beam,
where λ is the laser wavelength.The acousto-optic frequency shifter also generates an optical frequency shift fB in the reference beam (normally 40 MHz). Superimposing measurement and reference beams generates an FM signal on the photodetector with instantaneous frequency
and measurement point displacement, velocity, and acceleration can be extracted by demodulating the FM signal digitally. Displacement measurement resolution for LDV can reach several picometers in the frequency domain if the SNR is sufficiently high, and vibration amplitude resolution can achieve several nanometers in the time domain. However, LDV can only provide pointwise measurements.2.3 Full field displacement measurement using DIC
Digital image correlation is a widely used matching strategy for stereo vision to measure full-field small relative displacements. DIC provides calculation accuracy, speed and reliability for structures based on Zero-mean normalized sum of squared difference and the inverse compositional Gauss–Newton algorithm, which can be expressed as
Object out-of-displacement can be measured using stereo DIC. Measurement accuracy for out-of-plane displacement is in micrometers for normal-rate 12-bit cameras, compared with typically considerably worse accuracies for dynamic measurement using high-speed cameras. However, DIC enables full-field out-of-plane displacement measurement. This study first measures vibration signal for a single point using LDV, and then calculates the vibrational phase fluctuation. LEDs are triggered at specific phase values, producing two images by stereo DIC using normal-rate cameras. Thus, combing LDV and stereo DIC provides full-field vibration mode and operational deflection shape (ODS) with high spatial resolution.
3. Experimental illustration
3.1 Experimental setup and procedure
Figure 1 shows the experimental setup, including signal generator (SDG6052X, SIGLENT), shaker (Model V201, B&K) and amplifier (Model. LDS LPA100, B&K), test samples and fixtures, FPGA development board (based on EP4CE10F17C8, Cyclone IV E, Altera), LEDs and light amplifier, normal-rate industrial camera pair (3370CP-M-GL, IDS), and single point LDV (FNV-R1D-VD1, Holobright). Vibration measurement proceeded as follows.
- (1) Prepare the binocular vision system. Calibrate internal and external binocular camera parameters following Zhang’s method [33], and paint test sample surfaces with artificial speckles.
- (2) Obtain object’s natural frequency by shock test using a hammer. Transient vibration signals are collected by LDV during excitation and the signal spectrum extracted to identify main peaks as natural frequencies.
- (3) Acquire LDV-induced image. Excite the specimen using a shaker with known frequency, then use LDV to obtain the vibration signal for a point on its surface. We then digitize the analog vibration velocity from LDV using the FPGA board, and calculate vibration phase in real time. The cameras and LEDs are triggered successively at pre-selected phase steps by pulses generated from the FPGA board, and two images are captured at various instants corresponding to different vibration phases. The capturing process may vary across different vibration periods due to the camera imaging rate.
- (4) A sequence of images with pre-set phase steps are captured by each camera in the binocular vision system. In principle, any instant can be selected as the reference point, hence we chose images captured at maximum velocity as reference images to simplify understanding. This instant is also the balance point for the displacement curve, and conventional stereo DIC was applied to obtain instantaneous displacements at different vibration phase values. Figure 2 shows the image acquisition and processing procedure.
3.2 Synchronization and time delay compensation
Accurately estimating time delay due to each component is essential to ensure the LEDs flash at the correct pre-selected phases. Reaction time for each device must be considered when vibration frequency is high. Figure 3 shows the synchronization and time delay compensation process, with details as follows:
- (1) Time delay between vibration and LDV analog output is fixed and typical < 500 ns for the HoloBright LDV. This delay can be measured accurately and compensated easily in process, and can normally be ignored when vibration frequency < 10 kHz.
- (2) Time delay due to connections between components < 20 ns, which can also be ignored for normal cases.
- (3) Calculating phase steps from the vibration signal was realized using the FPGA board with ordinary 50 MHz crystal oscillation, requiring 5 clock cycles or 100 ns. This is a fixed value and can be compensated in process. The FPGA board generates two separate pulses for cameras (Pulse A) and LEDs (Pulse B).
- (4) Reaction time T1 for normal-rate cameras varies even with identical trigger signals, with difference = 3–50 μs depending on camera model. T2 is the time interval between pulses A and B.
- (5) The LED power amplifier time delay, T3, is the time interval between Pulse B and the instant the driving power reaches maximum; T4 is LED rise time; and T5 is flash duration. T3 is about 150 ns, which will be considered as necessary, depending on amplifying circuit; whereas T4 is about 1–2 ns and can be ignored.
- (6) Flash duration should be as short as possible (we set T5 = 1 μs) to minimize image blur due to vibration, ensuring good quality images for subsequent calculation. The proposed setup has camera exposure time ≫ LED flash duration.
- (7) The normal-rate camera reaction times are usually different, but synchronization can be guaranteed using the LED flashes within overlapping camera exposure times. We preset T2 = camera exposure time / 2, to ensure both cameras recorded images at the same instant (see Fig. 4).
- (8) We set T6 according to the number of phase steps, which can be any value from 2 to several hundred, depending on the frequency resolution. For this application, we set T6 = 28, i.e., 28 image pairs were recorded to reconstruct one vibration cycle.
Fig. 3. Procedure and component delays: External trigger camera delay, T1 = 3–15μs; preset time interval, T2 = camera exposure time / 2; amplifier time delay, T3 = 150 ns; LED electro-optical conversion (rise time), T4 = 1–2 ns; flash duration, T5 = 1 μs.
3.3 Test specimens and device settings
We selected an aluminum cantilever beam and a rectangular aluminum plate as test specimens to verify the proposed stroboscopic DIC method for vibration measurement. Cantilever beam dimensions = 250 × 10 × 2 mm, and rectangular plate dimensions = 175 × 150 × 0.8 mm (length × width × thickness). Figure 1 shows their boundary conditions. Power spectra for the specimens were obtained by shock test using a hammer, as shown in Fig. 5, identifying different natural frequency orders. We used a 12 bit AD module on the FPGA board to digitize the LDV analog output, with LED flash duration T5 = 1 μs and trigger interval T6 = 0.5 s. Camera frame rate and exposure time = 4 fps and 1 ms, respectively. Two images with 2048 × 2048 pixel spatial resolution were captured at π/14 rad phase steps, and 28 image pairs were captured to reconstruct one vibration cycle. We selected a 39×39 pixel window with 1 pixel step for stereo DIC.
4. Results and discussion
Figure 6 shows mode shape measurement results for the cantilever beam. The results presented are ODS when the excitation frequencies are the second to forth order of nature frequencies. Figure 7 compares normalized mode shapes with simulation results [34], and Table 1 compares node positions, relative errors, and Modal Assurance Criterion (MAC) between experimental results and simulation. MAC is a tool for quantifying the correlation between mode shapes [29]. MAC can be expressed as:
where ${U_i}$ and ${U_j}$ are vectors of mode shapes. When the MAC value approaches 1, it represents a strong correlation between two vectors; and approaching to 0 indicates no similarity. We measured cantilever beam ODS at arbitrarily selected frequencies to highlight the proposed method capability for structural vibration measurement. Excitation frequency (200 Hz) was between the second and third natural frequencies. Figure 8(a) shows temporal displacement for point K (see Fig. 8(b)) obtained by the proposed method. Displacement consistency at point K verifies the proposed approach achieved vibration amplitude accuracy > 10 μm.Fig. 8. Measured operational deflection shape for the cantilever beam under 200 Hz: (a) out-of-plane displacement at point K in time domain; and (b) full field out-of-plane displacement at maximum amplitude.
Table 1. Error estimation of mode shape
Figure 9 shows measured vibration for the rectangular plate fixed at the left side using the proposed stroboscopic DIC. Left columns show out-of-plane displacement distribution when the plate was excited by second to eleventh order natural frequencies. The right two columns show ESPI fringes at different natural frequencies applying conventional time-averaged ESPI to the same plate [3]. Mode shapes obtained by the two methods are very consistent across the different orders. Only twenty-eight image pairs were obtained, requiring < 300 MB total storage. Thus, the proposed method ensures high spatial and temporal vibration measurement without redundant data.
Fig. 9. Second to eleventh order mode shape for the rectangular plate measured by (a)–(j): proposed LDV-induced stroboscopic DIC; and (A)–(J) time-averaged ESPI (He-Ne Laser, 632.8 nm).
Total DIC calculation time < 10 minutes with DIC step size = 1 pixel, to obtain full-field displacement for any instant. Figure 10 shows media files containing videos for individual cycle vibration reconstructed by image sequences with known phase values when the plate was excited by fourth, fifth, sixth, and ninth order natural frequencies.
Fig. 10. Dynamic rectangular plate deformations at 227, 368, 411, and 823 Hz excitation (see Visualization 1, Visualization 2, Visualization 3, Visualization 4).
We selected six points (Fig. 11 (A), (B), (C), (D), (E), and (F)) for the fourth order mode shape to evaluate temporal axis accuracy. Figure 11 shows displacement variations for these points through the phase sequence, with a sinusoidal displacement curve for each position obtained using least squares fitting. Table 2 shows measurement values and errors for each position. Root mean square error (RMSE) and relative error for points A, B, E, and F are acceptable since vibration amplitudes for these points are high; whereas large errors occurred for points C and D because vibration amplitudes were close to DIC resolution limits.
Table 2. Fitted results and errors for out of plane displacement sequence
Spatial resolution for the proposed stroboscopic DIC method depends on camera sensor size and imaging system, which is normally much higher than scanning laser Doppler vibrometry. This will be helpful when comparing mode shapes (not only displacement mode, but also strain and curvature mode) to identify structural defects [35].
Temporal resolution for the proposed method depends on how many instants are measured within one vibration cycle, typically10–30 measurement points per cycle are sufficient. Thus, we can avoid data redundancy for SLDV, shortening both measurement and processing time. This method is totally different from the down-sampling strategy in Ref. [28]. In down-sampling strategy, the sampling rate does not satisfy Nyquist theorem, but the signal still can be reconstructed by performing a non-harmonic Fourier analysis, which mainly consists of a least square fit of the down-sampled signal with a sinusoidal function. However, it may also generate some errors in measurement. In our proposed method, the sampling rate in time axis still meets the requirement of Nyquist theorem. In the experiment, the vibration phase can be calculated in real time so that it is possible to determine how many points per cycle in measurement. In our test, 28 points per cycle is selected so that a smooth sinusoidal curve can be presented. However, it is easy to change the number of points per cycle in programming. Compared with down-sampling methods, although more hardware, including FPGA and single-point LDV are required, it can offer a vibration measurement with true high-temporal and high-spatial resolutions.
The only disadvantage for the proposed stroboscopic DIC approach is relatively low resolution for out-of-plane displacement. Stereo DIC can only measure out-of-plane displacement in micrometers, whereas LDV resolution is better than 1 nm. Thus, the proposed method performance is relatively poor for vibrations with amplitude < several micrometers. The low amplitude vibration measurement needs to be completed by SLDV, which is irreplaceable in this application scenario.
5. Concluding remarks
This paper proposed an LDV-induced stroboscopic digital image correlation method using normal-rate cameras for steady-state vibration measurement with high spatial resolution. We compared the proposed approach with simulation and time-average ESPI for a cantilever beam and rectangular plate to obtain ODS at 28 phase values within one vibration cycle. The different methods exhibited good agreement across excitation ranges and samples.
Thus, the proposed method successfully obtained full-field vibration measurement with high spatial resolution using two normal-rate cameras. This will greatly reduce measurement costs, allowing wider research and industrial application.
We are collaborating with, HoloBright, the LDV manufacturer, to modify the FPGA board and software in their LDV system, allowing the LDV to output a trigger signal directly according to the user selection of number of points per cycle. Future studies will extend the proposed technique beyond one-directional vibration measurement to full-field measurement in three directions, and/or rotating objects.
Funding
National Natural Science Foundation of China (11972235).
Disclosures
The authors declare no conflicts of interest.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
References
1. J. A. Leendertz, “Interferometric displacement measurement on scattering surfaces utilizing speckle effect,” J. Phys. E: Sci. Instrum. 3(3), 214–218 (1970). [CrossRef]
2. K. Creath and GA Slettemoen, “Vibration-observation techniques for digital speckle-pattern interferometry,” J. Opt. Soc. Am. 2(10), 1629–1636 (1985). [CrossRef]
3. W. C. Wang, C. H. Hwang, and S. Y. Lin, “Vibration measurement by the time-averaged electronic speckle pattern interferometry methods,” Appl. Opt. 35(22), 4502–4509 (1996). [CrossRef]
4. A. D. Wilson, “Characteristic functions for time-average holography,” J. Opt. Soc. Am. 60(8), 1068–1071 (1970). [CrossRef]
5. Y. Y. Hung, “Shearography: a new optical method for strain measurement and nondestructive testing,” Opt. Eng. 21(3), 213391 (1982). [CrossRef]
6. C. J. Tay and Y. Fu, “Determination of curvature and twist by digital shearography and wavelet transforms,” Opt. Lett. 30(21), 2873–2875 (2005). [CrossRef]
7. K. M. Qian, “Two-dimensional windowed Fourier transform for fringe pattern analysis: principles, applications and implementations,” Opt. Lasers Eng. 45(2), 304–317 (2007). [CrossRef]
8. S. Schedin, G Pedrini, H. J. Tiziani, and F. M. Santoyo, “Simultaneous three-dimensional dynamic deformation measurements with pulsed digital holography,” Appl. Opt. 38(34), 7056–7062 (1999). [CrossRef]
9. G. Pedrini, H. J. Tiziani, and Y. Zou, “Digital double pulse-TV-holography,” Opt. Lasers Eng. 26(2-3), 199–219 (1997). [CrossRef]
10. Y. Huang, F. Janabi-Sharifi, Y. Liu, and Y. Y. Hung, “Dynamic phase measurement in shearography by clustering method and Fourier filtering,” Opt. Express 19(2), 606–615 (2011). [CrossRef]
11. Y. Fu, R. M. Groves, G. Pedrini, and W. Osten, “Kinematic and deformation parameter measurement by spatiotemporal analysis of an interferogram sequence,” Appl. Opt. 46(36), 8645–8655 (2007). [CrossRef]
12. J. M. Huntley, G. H. Kaufmann, and D. Kerr, “Phase-shifted dynamic speckle pattern interferometry at 1 kHz,” Appl. Opt. 38(31), 6556–6563 (1999). [CrossRef]
13. Y. Ishii, “Time-average holographic interferometry with a sinusoidally modulated laser diode,” Opt. Lasers Eng. 36(5), 515–526 (2001). [CrossRef]
14. Y. Fu, C. J. Tay, C. Quan, and H. Miao, “Wavelet analysis of speckle patterns with a temporal carrier,” Appl. Opt. 44(6), 959–965 (2005). [CrossRef]
15. P. Castellini, M. Martarelli, and E. P. Tomasini, “Laser Doppler Vibrometry: Development of advanced solutions answering to technology's needs,” Mech. Syst. Signal. Process. 20(6), 1265–1285 (2006). [CrossRef]
16. S. J. Rothberg, M. S. Allen, P. Castellini, D. Di Maio, J. J. J. Dirckx, D. J. Ewins, B. J. Halkon, P. Muyshondt, N. Paone, T. Ryan, H. Steger, E. P. Tomasiniand, S. Vanlanduit, and J. F. Vignola, “An international review of laser Doppler vibrometry: Making light work of vibration measurement,” Opt. Lasers Eng. 99, 11–22 (2017). [CrossRef]
17. D. Di Maio, P. Castellini, M. Martarelli, S. Rothberg, M. S. Allen, W. D. Zhu, and D. J. Ewins, “Continuous Scanning Laser Vibrometry: A raison d’être and applications to vibration measurements,” Mech. Syst. Sig. Process. 156, 107573 (2021). [CrossRef]
18. M. Martarelli and D. J. Ewins, “Continuous scanning laser Doppler vibrometry and speckle noise occurrence,” Mech. Syst. Signal. Process. 20(8), 2277–2289 (2006). [CrossRef]
19. Y. Fu, M. Guo, and P B. Phua, “Multipoint laser Doppler vibrometry with single detector: principles, implementations, and signal analyses,” Appl. Opt. 50(10), 1280–1288 (2011). [CrossRef]
20. J. Baqersad, P Poozesh, C. Niezrecki, and P. Avitabile, “Photogrammetry and optical methods in structural dynamics–a review,” Mech. Syst. Signal. Process. 86, 17–34 (2017). [CrossRef]
21. T. J. Beberniss and D. A. Ehrhardt, “High-speed 3D digital image correlation vibration measurement: Recent advancements and noted limitations,” Mech. Syst. Signal. Process. 86, 35–48 (2017). [CrossRef]
22. M. N. Helfrick, C. Niezrecki, P. Avitabile, and T. Schmidt, “3D digital image correlation methods for full-field vibration measurement,” Mech. Syst. Signal. Process. 25(3), 917–927 (2011). [CrossRef]
23. T. Durand-Texte, E. Simonetto, S. Durand, M. Melon, and M. H. Moulet, “Vibration measurement using a pseudo-stereo system, target tracking and vision methods,” Mech. Syst. Signal. Process. 118, 30–40 (2019). [CrossRef]
24. L. Yu and B. Pan, “Single-camera high-speed stereo-digital image correlation for full-field vibration measurement,” Mech. Syst. Signal. Process. 94, 374–383 (2017). [CrossRef]
25. T. Durand-Texte, M. Melon, E Simonetto, S. Durand, and M. H. Moulet, “Single-camera single-axis vision method applied to measure vibrations,” J. Sound. Vib. 465, 115012 (2020). [CrossRef]
26. H. Schreier, J. J. Orteu, and M. A. Sutton, Image Correlation for Shape, Motion and Deformation Measurements (Springer, 2009). [CrossRef]
27. E.M.C. Jones and P.L. Reu, “Distortion of digital image correlation (DIC) displacements and strains from heat waves,” Exp. Mech. 58(7), 1133–1156 (2018). [CrossRef]
28. S. Barone, P. Neri, A. Paoli, and A. V. Razionale, “Low-frame-rate single camera system for 3D full-field high-frequency vibration measurements,” Mech. Syst. Signal. Process. 123, 143–152 (2019). [CrossRef]
29. A. Brandt, Noise and vibration analysis: signal analysis and experimental procedures (Wiley, 2011), Chap. 6.
30. Y. Gao, T. Cheng, Y. Su, X. Xu, Y. Zhang, and Q. Zhang, “High-efficiency and high-accuracy digital image correlation for three-dimensional measurement,” Opt. Lasers Eng. 65, 73–80 (2015). [CrossRef]
31. Y. Su, Z. Gao, Z. Fang, Y. Liu, Y. Wang, Q. Zhang, and S. Wu, “Theoretical analysis on performance of digital speckle pattern: uniqueness, accuracy, precision, and spatial resolution,” Opt. Express 27(16), 22439–22474 (2019). [CrossRef]
32. M. A. Sutton, J. J. Orteu, and H. Schreier, Image correlation for shape, motion and deformation measurements: basic concepts, Theory and Applications (Springer Science & Business Media, 2009), Chap. 5.
33. S. Zhang and P S. Huang, “Novel method for structured light system calibration,” Opt. Eng. 45(8), 083601 (2006). [CrossRef]
34. M. Zannon, “Free vibration of thin film cantilever beam,” Int. J. Eng. Tech. Res. 2, 304–314 (2014).
35. C. Yang, Y. Fu, J. Yuan, M. Guo, K. Yan, H. Liu, H. Miao, and C Zhu, “Damage Identification by using a self-synchronizing multipoint Laser Doppler vibrometer,” Shock. Vib. 2015, 476054 (2015). [CrossRef]
