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REVIEW OF SCIENTIFIC INSTRUMENTS 84, 121702 (2013)

A multi-point laser Doppler vibrometer with fiber-based configuration C. Yang,1,2,3,a) M. Guo,1 H. Liu,1 K. Yan,1 Y. J. Xu,1 H. Miao,2 and Y. Fu1,b) 1 Temasek Laboratories and School of Mechanical and Aerospace Engineering, Nanyang Technological University, 50 Nanyang Drive, Singapore 637553 2 Department of Modern Mechanics, University of Science and Technology of China, Hefei 230027, China 3 Institute of Systems Engineering, China Academy of Engineering Physics, Mianyang 621900, China

(Received 28 May 2013; accepted 17 July 2013; published online 18 December 2013) Laser Doppler vibrometer (LDV) is a non-contact optical interferometric system to measure vibrations of structures and machines with a high precision. Normal LDV can only offer a single-point measurement. Scanning LDV is usually impractical to do measurement on transient events. In this paper, a fiber-based self-synchronized multi-point LDV is proposed. The multiple laser beams with different frequency shifts are generated from one laser source. The beams are projected onto a vibrating object, reflected and interfered with a common reference beam. The signal including vibration information of multiple spatial points is captured by one single-pixel photodetector. The optical system is mainly integrated by fiber components for flexibility in measurement. Two experiments are conducted to measure a steady-state simple harmonic vibration of a cantilever beam and a transient vibration of a beam clamped at both ends. In the first measurement, a numerical interpolation is applied to reconstruct the mode shape with increased number of data points. The vibration mode obtained is compared with that from FEM simulation. In transient vibration measurement, the first five resonant frequencies are obtained. The results show the new-reported fiber-based multipoint LDV can offer a vibration measurement on various spatial points simultaneously. With the flexibility of fiber configuration, it becomes more practical for dynamic structural evaluation in industrial areas. © 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4845335] I. INTRODUCTION

Vibration is a mechanical phenomenon whereby the oscillation or repetitive motion occurs about an equilibrium position. It exists in almost every machining progress in today’s industry and highly affects the performance of machines and structures. Hence, it becomes one of the fundamental physical quantities which needs to be measured in many engineering fields. Transducers, such as velocity pickups and accelerometers, are generally used for vibration measurements and analyses. However, measurement by a transducer is a contact point-wise method and is difficult to apply on a small object as the mass of the transducer can also change the vibration behaviour. Laser Doppler vibrometer1 (LDV) is a non-contact optical interferometric system2 for vibration measurement. The principle of LDV is based on measuring the frequency shift of a laser light when it scatters from a moving surface. The instantaneous velocity of the surface is linear to the change of frequency which can be extracted by the interference of object and reference beam. Normally LDV can only offer a single point measurement. Efforts are made to increase the number of measurement points, thus a scanning LDV or a multi-channel LDV3 has been developed as a multipoint sensor. However, the measurement conditions are assumed to be invariant during scanning process. Unfortunately a lot of unsteady vibrations, such as transient vibration and a) This research was performed while C. Yang was at Temasek Labs, Nanyang

Technological University, Singapore.

b) Author to whom correspondence should be addressed. Electronic mail:

[email protected] 0034-6748/2013/84(12)/121702/6/$30.00

random vibration cannot satisfy the assumption. The multichannel system4 is usually an assembly of several singlepoint vibrometers,5 and multiple detectors or detector array are used.6–8 Hence, synchronization among different channels is necessary. Recently, some new techniques are proposed to provide a simultaneous multi-point measurement9–11 in inplane velocity measurement. In these techniques, several spatially separated laser beams with various frequency shifts are generated by one laser source and acousto-optic devices. The interference signal is captured by a single photodetector and resolved in frequency domain. However, the results presented are limited in the measurement on two or three points, where cross-talk region can be easily separated in the spectrum. The same approach suffers when the measurement points are increased, as all object beams will interfere with a common reference beam, and the object beams will also interfere with each other. In our previous work,12, 13 a new approach was proposed to achieve a self-synchronized vibration measurement on 20 points in a form of 5 × 4 beam array. The beam array with various frequency shifts was generated by a 1550 nm laser source and four acousto-optic devices. The interferometric signal including the useful information of all sampling points was collected using a single photodetector. After bypassing the cross-talks in experiment and signal processing stages, the vibration signals on 20 measurement points can be retrieved from the one-dimensional interference signal. However, the beam array generated by acousto-optic modulators (AOMs) is a regular 1D or 2D pattern, which limits the flexibility of the measurement in practical use. In this paper, an optical design using fiber components is adopted to build a prototype

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of 4-point self-synchronized LDV. This fiber-based version is more flexible to satisfy the requirements of the real applications since it can do measurement on any spatial points of various surfaces. In this paper, a steady-state vibration of a cantilever beam and a transient vibration of a beam clamped at both ends are measured by the proposed multi-point LDV. In steadystate vibration measurement, the spatial vibratory response is obtained by an unequal-spaced sampling, and the numeral interpolation can be used to increase the number of data points. The experimental results are compared with the simulation results from the Finite Element Method (FEM). In the measurement of a transient vibration excited by a shock, the displacement distributions on several points are recorded simultaneously and the first five resonant frequencies are retrieved from spectrum. The results show the proposed prototype performs well in simultaneous vibration measurement on any points of an object. II. FIBER-BASED MULTI-POINT LDV SYSTEM A. Doppler shift and heterodyne interferometry

When a laser light with wavelength of λ is projected on a vibrating object, the motion of the object adds a Doppler shift to the reflected beam. The relationship between the Doppler shift fD (t) and the velocity of the object V(t) can be expressed as fD (t) = 2V (t) · cos α/λ,

(1)

where α is the angle between the laser beam and the velocity vector. In a LDV system, the heterodyne interferometer is normally used to solve the ambiguity problem involved in the interferometric signal. This method introduces a frequency shift into one arm of the interferometer by an AOM to introduce a virtual offset in velocity. The interference intensity can be expressed as Eq. (2):14 I = IDC + IRO cos (2π (fD + fAOM ) t + ϕ) ,

FIG. 1. Schematic layout of fiber-based system to split one input laser into four beams with different frequency shifts.

combination of different types of AOM is proposed to generate beam array cost-effectively. In this 4-point LDV prototype, a four-point beam array with a frequency interval of 20 MHz is generated. A laser beam from a single-frequency 1550 nm laser system is split into a reference beam and an object beam. The object beam is then connected to a pigtailed AOMs with a 50 MHz frequency shift (Brimrose, AMF-501550-2FP). The output laser beam with a 50 MHz frequency shift is then passed through a frequency shifter in the RamanNath regime (Brimrose, AMF-20-1550, separation angle = 12 mrad, RF power tunable, aperture 2 mm). This frequency shifter generates a beam array of five diffraction orders (−2, −1, 0, +1, +2) with a frequency shift of 20 MHz in between. When the incident angle of laser beam is finely adjusted, four beams with quite uniform intensities are observed. Hence, in this application, only four diffraction orders with −20 MHz, 0 MHz, +20 MHz, and +40 MHz frequency shifts are selected. Figure 1 shows the schematic layout of fiber-based system to split one input laser into four beams with frequency shift of +30 MHz, +50 MHz, +70 MHz, and +90 MHz. These four laser beams are projected to an object, reflected and interfered with a reference beam. The interference signal detected by the photodetector can be expressed as I = IDC +

B. Generation of multiple laser beams with different frequency shifts

There are two diffraction types in acousto-optic modulator, Bragg diffraction and Raman-Nath diffraction.15 Only one diffracted beam can be produced in the Bragg regime and several diffracted beams can be produced in the RamanNath regime. In the proposed multi-point laser Doppler vibrometer, multiple laser beams with different frequency shifts are split from one laser source. Obviously, it is not reasonable to use one AOM in Bragg region for each channel. A

IM(i) cos(2π (fD(i) + fAOM(i) )t + φ(i) )

i=1

(2)

where fD and fAOM are Doppler shift and AOM induced frequency shift, respectively. ϕ is the phase difference between the reference beam and object beam. IRO is the modulation factor, determined by the product√of the square root of object and reference beam intensities IR IO . Photodetector converts the intensity fluctuation to a current or voltage signal for later analog or digital decoding.

4

+

3 4

Imn cos(2π [(fD(m) − fD(n) )

m=1 n>m

+ (fAOM(m) − fAOM(n) )]t + φmn ),

(3)

where i = 1, 2, 3, 4; m and n are integers; fAOM (i) are the central frequencies of object beams. The second term is the interference signal between the object beams and reference beam, from which the useful vibration information of four points can be extracted. The third term is the sum of the cross talk between any two object beams, which can be bypassed when the interference signal is processed.13 C. Fiber-based optical system

The beam array generated by AOMs is a regular 1D or 2D pattern, thus a fiber-based optical design is selected to enhance the flexibility of the system in practical applications. Figure 2 shows the schematic layout of the proposed 4-point LDV system. Four object beams with different

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FIG. 2. Schematic layout of four-point laser Doppler vibrometer.

frequency shifts are projected by sensing heads and the reflected beams are collected by the same sensing heads and delivered to photodetector by four fiber circulators. These four object beams are combined and interfered with a common reference beam. The wavelength of laser for measurement is 1550 nm. As it is invisible, a 642 nm pilot laser is used as aiming light. Hence, the sensing head should be achromatic for both wavelengths. Considering the safety of human eyes, the power of measurement laser (1550 nm) emitted from each sensing head is set as 3 mW, and the power of red pilot laser (642 nm) is around 0.2 mW. A 4-channel demodulation system will be connected to the optical system for real-time decoding. In this LDV system, only one photodetector is used. Hence, the system is self-synchronized. Figures 3(a) and 3(b) show the pictures of 4-point LDV system and four pigtailed sensing heads.

III. SPATIAL RESPONSE RECONSTRUCTION

In steady-state vibration measurement, vibratory response is necessary to be measured. In a scanning LDV, it is reconstructed by a scanning measurement procedure, thus more spatial measurement points can be obtained with the cost of longer scanning time. In the proposed 4-point LDV system, the number of measurement points may not be enough to generate the response on a complex structure. However, when the Nyquist sampling theorem is satisfied in spatial domain, it is still possible to increase the number of data points via numerical interpolation. Interpolation is a method of constructing new data points within the range of a discrete set of known data points which are called nodes. After the investigation on several commonly used interpolation algorithms, polynomial interpolation

FIG. 3. (a) Picture of optical and demodulation system; (b) picture of sensing heads.

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FIG. 4. (a) Theoretical normalized 3rd mode shape of an Euler-Bernoulli cantilever beam and six sample points; (b) reconstructed mode shape by polynomial interpolation algorithm.

algorithm is adopted to reconstruct the response on a cantilever beam in this study. It is well known that a function g(x) which is infinitely continuously differentiable can be expanded in the form of a Taylor series expansion as shown in Eq. (4): gn (x) = a0 + a1 x + · · · + an x n .

(4)

Therefore, a signal can be approximated as a polynomial of degree at most n − 1 going through all the n nodes. The polynomial coefficients can be computed from the equations created from the nodes and then the interpolation points can be generated on the polynomial curve. The interpolation is infinitely differential and no data extension or additional condition is acquired. The main disadvantage of this algorithm is that a high degree polynomial may cause oscillatory at the edge points, which is called the Runge’s phenomenon.16 However, since the amount of sampling points in this study is only four, the Runge’s phenomenon can be disregarded. Generally, there are two types of the polynomial interpolation, Lagrange and Newton interpolation algorithms. Normally, they lead to the same results and accuracy. In this application, the Lagrange algorithm is selected for interpolation. The position of sampling points is another consideration that may influence reconstruction precision. In general, sampling points should be more in the location where spatial frequency is high to satisfy the Nyquist sampling theorem. A simulation is conducted to study the effect of sampling position. Figure 4(a) shows the theoretical normalized 3rd mode shape of an Euler-Bernoulli cantilever beam. Six unequally spaced sampling points are selected (marked by cross symbol). High spatial sampling rate is adopted at the edge. Eight points are interpolated between any two adjacent sampling points. The reconstructed mode shape is plotted in Figure 4(b) and shows that the polynomial interpolation performs well. The relative difference between the original theoretical values and the interpolated values is less than 3.5%. The simulation shows once the number of measurement points is increased, the proposed algorithm can be used to reconstruct the vibratory mode shape of various structures.

by a shaker at the position of 10 mm to the free end. The length, width, and thickness of the beam are 270 mm, 20 mm, and 3.1 mm, respectively. The frequency of the exciting force is around 210 Hz, which is close to the second natural frequency of the beam (the 2nd natural frequency is 216.3 Hz, determined by a shock test). The experimental setup is shown in Figure 5. Four unequal-spaced sample points A, B, C, and D on the central-line of the beam are measured. The distances of these four points to the clamping end are 30 mm, 90 mm, 165 mm, and 255 mm, from A to D point, respectively. The stand-off distance of sensing head is around 1.5 m. Figure 6(a) shows the out-of-plane displacement variations of sample points in the first 0.032 s. Figure 6(b) plots the 4-points vibratory response of the beam at t = 0.0012 s. Polynomial interpolation is applied to insert new data points into each of the intervals and reconstruct a 75-points vibratory response curve. These reconstructed points, including the original sampling points and inserted points, are equal-spaced. To verify the accuracy of measurement and interpolated reconstruction, the reconstructed vibratory response is compared with the simulation by FEM. The commercial software ANSYS is used to build the simulative cantilever beam model. Material properties of the actual beam, such as the Young’s modulus, Poisson ratio, and density are tested beforehand and applied to the model. The plane82 8-node quadrilateral element type is chosen, and the model is divided into totally 540

IV. RESULTS AND DISCUSSIONS A. Measurement of steady-state vibration

The experiment is conducted on an aluminium cantilever beam that is subjected to a sinusoidal wave exciting force

FIG. 5. Experimental layout for the steady-state vibration measurement. Reprinted with permission from AIP Conf. Proc. 1457, 219–226 (2012). Copyright 2012 American Institute of Physics.17

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FIG. 7. The aluminium beam under shock test.

the limited spatial sampling. The error can be reduced tremendously with the increase of sampling points. At the moment, a fiber-based 16-point LDV is being developed. With the increase of sampling points, the higher measurement precision can be expected and more complex spatial responses can be studied. B. Measurement of transient vibration

FIG. 6. (a) The out-of-plane displacement distributions of four sample points; (b) the vibratory response of the beam at t = 0.0012 s; and (c) the response curves generated using FEM (dotted line) and reconstructed from the experimental data via interpolation and filtering (solid line).

elements. Both the two response curves obtained by ANSYS (dotted line) and experiment (solid line) are normalized and shown in Figure 6(c). Two curves coincide very well. The difference of simulation and measurement mainly comes from

The transient vibration of an aluminium beam clamped at both ends (shown in Fig. 7) is also measured in this study. The length, width, and thickness of the beam are 400 mm, 19.4 mm, and 4.5 mm, respectively. A pendulum is employed to apply a shock excitation to the beam at the position of 25 mm to one of the clamped ends. Four sample points A, B, C, and D on the central-line of the beam are measured. The laser and the demodulation system are the same as that in Sec. IV A. The demodulation system is triggered when the pendulum almost hit the beam. Figure 8(a) shows the out-of-plane displacement distributions of four measurement points in the first 0.5 s after trigger. Figure 8(b) shows the displacement in the period as indicated in Figure 8(a). Disturbances can be observed in the

FIG. 8. (a) The out-of-plane displacement of four sample points in the first 0.5 s after trigger; (b) the displacement of four points in the period as indicated in Fig. 8(a).

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FIG. 9. Spectrum of the displacement distribution of the points (a) A; (b) B; (c) C; and (d) D.

waveforms, which indicate that higher frequency components are contained in the displacement distribution. As the system is self-synchronized, it is more suitable to measure the transient vibration measurement. In this experiment, even the vibration delay among the sampling points, caused by the propagation of the shock wave along the beam, can be observed in Figure 8(b). Figure 9 shows the spectrum of the displacement of four measurement points. Several resonance peaks can be observed at frequencies of 131.5 Hz, 351.2 Hz, 565.9 Hz, 1253 Hz, and 1813 Hz, which represent the first five resonance frequencies of the beam. The first three resonance frequencies by the ANSYS modal analysis are 129.3 Hz, 344.8 Hz, and 557.9 Hz. The ANSYS model is built with the same material properties to that of the actual beam and divided into 800 elements in the type of plane82 8-node quadrilateral element. The relative difference between experimental and simulative values is within 3%. It is mainly due to the boundary condition in experiment which may not be strictly the same as the simulation. V. CONCLUDING REMARKS

In this paper, a fiber-based four-point LDV is presented for steady-state and transient vibration measurement. Only a single high speed photodetector is used to capture the interferometric signal that includes vibration information of four points. The optical system is mainly integrated by fiber components. The pigtailed fiber sensing heads ensure the flexibility of the measurement. The laser of 1550 nm wavelength is used for measurement, while a laser of 642 nm is applied as a pilot laser for aiming. Compared with a scanning LDV, the proposed system has an advantage in simultaneous measurement of transient events with a low-cost setup. Numerical interpolation is applied to reconstruct the spatial vibratory response to increase the number of data points. Experiments are conducted on two beam-like structures. The steady-state vibration by a simple harmonic excitation and the transient vibration by a shock excitation are measured using the proposed

LDV technique. Vibration parameters are extracted, such as the instantaneous displacement, the spatial vibratory response and the resonance frequencies. The results show the proposed method has potential to be applied in real industrial measurements, especially when the number of measurement points is increased. In Temasek Labs of Nanyang Technological University, Singapore, R&D works are continued to develop a 16-point LDV with fiber-based configuration.

ACKNOWLEDGMENTS

This work is supported by research project TRF10MUPLAD, DRTech, MINDEF, Singapore. 1 P. Castellini, M. Martarelli, and E. P. Tomasini, Mech. Syst. Signal Process.

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