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Vibration and modal analysis of a rotating disc using high-speed 3D digital image correlation
Mechanical Systems and Signal Processing 121 (2019) 201–214
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Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp
Vibration and modal analysis of a rotating disc using high-speed 3D digital image correlation Róbert Hunˇady ⇑, Peter Pavelka, Pavol Lengvarsky´ Technical University of Košice, Faculty of Mechanical Engineering, Letná 9, Košice, Slovakia
a r t i c l e
i n f o
Article history: Received 14 September 2018 Received in revised form 12 November 2018 Accepted 15 November 2018
Keywords: Digital image correlation Rotating disc Rigid body motion elimination Vibration analysis Modal analysis
a b s t r a c t Rotating structures are of interest in many engineering fields, where knowing their dynamic behaviour is important to prevent undesired operating states. However, the experimental measurement of vibrations in rotating structures is still a relatively demanding task that requires the use of special measurement technique or method. The paper deals with the application of high-speed digital image correlation in vibration analysis of rotating structures. The method introduced in this research is based on the elimination of rigid body motion components contained in the primary responses measured by cameras. The process of separation and subsequent elimination of these components is given by numerical post-processing of three-dimensional displacement fields. The basis is to determine rotation matrix and translation vector that optimally describe rigid transformation between two positions of an analysed object. The proposed method is applicable for measuring under both constant and variable speed of rotation. Practical application is presented by three experiments in which vibrations of a flat rotating disc were analysed. The first experiment is focused on determine the operating deflection shapes of the disc rotating at 4700 rpm. The second one is a run-up analysis under variable rotational speed that ranges from 0 to 4700 rpm. The aim of the third measurement is to obtain natural frequencies and modal shapes of the disc rotating at 4800 rpm. The results of the experiments show that the elimination of rigid body motions leads to a higher accuracy of measurement and provides more pronounced frequency response spectra. Ó 2018 Elsevier Ltd. All rights reserved.
1. Introduction Vibration analysis and health monitoring of rotating structures and components, such as impellers, discs, wind turbine blades, etc., is very common in technical practice. It is of paramount importance to periodically monitor the technical condition of these components, in order to avoid resonance or overload problems that may cause undesirable behaviour or even lead to critical failure [1–3]. Various measurement techniques can be employed to capture vibration responses of a rotating object. Conventional contact methods using accelerometers or strain gauges require the installation of wireless device or slip ring to transfer data from the sensors to recording system. This approach has several limitations. The transducers are intrusive because they can influence dynamic properties of the object under investigation due to mass loading effect and local stiffness alteration at the point of transducer attachment. In addition, the measurement is pointwise and limited to only few locations. The use of telemetry is usually costly, time consuming to install, and may introduce electrical noise into the signals [4]. These limitations are largely solved by using contactless methods that have been developed to measure ⇑ Corresponding author. E-mail addresses: [email protected] (R. Hunˇady), [email protected] (P. Pavelka), [email protected] (P. Lengvarsky´). https://doi.org/10.1016/j.ymssp.2018.11.024 0888-3270/Ó 2018 Elsevier Ltd. All rights reserved.
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displacements or deformations. Optical methods suitable for measuring dynamic events include scanning laser Doppler vibrometry (SLDV), electronic speckle pattern interferometry (ESPI), holographic interferometry, digital shearography, photogrammetry, and digital image correlation (DIC). The application of these methods for measuring vibration of a rotating object requires the use of additional devices or special measurement approaches that ensure that the object appears to be in a quasi-steady state. The elimination of rigid body motion components (rotations and translations) is appropriate, regardless of the used measurement technique. The elimination can be realized in several ways. When vibration responses are measured by scanning laser vibrometer, the use of the optical derotator is one of the possible ways. The derotator tracks the motion of the rotating object through a precision controlled optical rotation unit that is synchronised with the rotation frequency. The unit usually contains a Dove prism which rotates at half the speed of the object. It makes the rotating object appear stationary when viewed through the prism. The condition to attain complete optical derotation is that the rotation axis of the object and that of the rotation unit must match [5]. Current optical derotators can be used up to a speed of 24,000 rpm. The disadvantage is their high price. Boedecker et al. [6] described the design of the optical derotator that has been manufactured and subsequently used for experimental vibration analysis of a fan rotating with 3000 rpm. A similar measurement was made by Šároši et al. [7] who used the optical derotator to obtain operating and modal shapes of a rotating disc. They also performed run-up analysis to assess the effect of rotational speed on the shift of natural frequencies. Sever et al. [8] proposed the so-called self-tracking LDV method that is a modification of self-tracking technique proposed by Lomenzo et al. [9]. The method uses two 45-degree mirrors, one of which is fixed and the other (mounted on the drive shaft) rotates with the bladed disc. The modification is given by the use of fixed annular conical mirror instead of the fold mirror. This configuration allows the laser beam to be directed to the blade tips at a right angle. A way that does not require additional equipment is the use of a suitable scanning technique. Generally this is done by utilising the deflection mirrors in the LDV head and by synchronising the signals driving these mirrors with the rotational speed of the object under investigation. The use of scanning techniques can be limited due to inertias of the mirrors that may cause a phase shift of the measured signal [8,10]. The problem can be solved by means of a phase correction algorithm. However, scanning techniques are highly effective in many cases [11–18]. In 2004, Halkon and Rothberg [15] introduced the so-called Synchronised-Scanning Laser Vibrometry method by which the laser beam is capable to track the rotating structure and simultaneously scan the region of interest to provide modal data. They used single-point laser vibrometer with custombuilt scanning system. Maio and Ewins [16] presented how commercially available SLDV system without any hardware modification can be used for synchronous measurement of both stationary and rotating structures. Three different tracking techniques [17] were used in combination with suitable excitation method to analyse vibration of a bladed disc. Gwashavanhu et al. [18] used tracking LDV method for the validation of photogrammetry applied to online turbomachinery blade measurements. A comparison of out-of-plane displacements of a specific point on a rotating blade measured by photogrammetry and displacements captured by laser scanning system showed that there was a good correlation between the methods, both in the time and frequency domains. As mentioned above, another method suitable for vibration analysis is photogrammetry in conjunction with threedimensional point tracking (3DPT). Photogrammetry is a non-contact measurement technique that uses a pair of digital cameras to measure 3D displacements of distinct markers attached to the surface of an object. The use of photogrammetry in vibration analysis of rotating structures is documented by studies [19–23]. Warren et al. [19] used 3D digital stereophotogrammetry to measure vibration responses of a small wind turbine rotating at a speed of approximately 10 Hz. An example of a large-scale application is given by the work [20] in which the authors used 4-camera system to measure the dynamic responses of a wind turbine rotor of 80 m diameter while the turbine was rotating. Photogrammetry has also been used for measurement of helicopter rotor vibrations. Lundstrom et al. in the works [21,22], demonstrated the possibilities of this method when measured the operating deflection shapes of the spinning rotor blades of a Robinson R44 helicopter. In another work by Lundstrom et al. [23], the authors extracted the operating deflection shapes of a wind turbine by applying a harmonic filter to the measured data. If a random speckle pattern is applied on the object surface, the photogrammetry principle can be extended to full-field measurement. This technique is known as digital image correlation (DIC). The use of DIC method in vibration or modal analysis of non-rotating objects is currently common [24–31]. In addition to conventional approaches in 3D DIC method with two cameras, there are also single-camera methods that can also be used to analyse vibrations [32–35]. Yu and Pan [32] proposed a method that uses a single high-speed colour camera in combination with stereo-DIC technique to measure 3D deformations. The separation of recorded colour images into red and blue channel sub-images is the basis of the method. The separated images are then process by regular stereo-DIC algorithm to obtain the desired displacement fields. The method was successfully applied to measure the displacement and velocity of a rotating fan, full-field vibration responses of a rectangular panel, and deformation fields of a balloon during its explosion. In another work [33], Yu and Pan used a singlecamera high-speed stereo-digital image correlation (SCHS stereo-DIC) method with four-mirror adapter for full-field 3D vibration measurement where the mirror system is used to divide the sensor’s image plane into two halves with different views of the object under investigation. Both image areas are subsequently processed as separated images using a conventional stereo DIC procedure to obtain vibration responses. In the study [34], the authors provided a detailed overview and described the strengths and weaknesses of various DIC techniques based on pseudo stereo images recorded with a single camera. The number and results of published works show that DIC measurement techniques and procedures are well established. On the other hand, the works focusing on vibration analysis of rotating structures are really unique. Most available studies
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refer to the measurement of motion or deformation parameters using the triggering method when the sampling rate of the cameras is synchronised with the rotation frequency of an object to be analysed [36–39]. Yashar et al. [40] measured the frequency response functions of a rotating beam using DIC system with one camera that tracked the position of a specific marker through the mirror rotating with the beam. The beam was excited at its base using an electro-dynamic shaker attached to the rotational hub assembly. The disadvantage is that the proposed measurement method requires a special testing rig. In addition, the responses can only be measured at a few points (markers) determined on the structure. Wu et al. [39] measured the in-plane and out-of-plane motion components of a wind turbine blade model using 3D DIC method while the turbine was driven by an electric fan. They investigated the blade displacements of balanced and unbalanced turbine at the different rotational speeds. The measurement was evaluated at ten points spaced along one blade. In the case of the balanced turbine, the out-of-plane displacement frequency spectra showed the presence of the first two harmonic components. When additional weight was added, several sub-harmonic components have also been identified. Since the in-plane displacements corresponded only to the change of the points’ coordinates caused by the rotation of the blade, they did not provide any relevant information related to vibration, neither in time nor in frequency domain. Non-contact measurement methods based on interferometry are also useful in full-field vibration analysis of both nonrotating and rotating structures [41–43]. These methods achieve excellent sensitivity and resolution when measuring displacement and strain, however, their application in industries is relatively difficult. They require complex optical system and vibration isolation to prevent the possibility of phase changes or destroying the fringe patterns [38]. Van der Auweraer et al. [41] proposed a novel high-frequency experimental method to measure modal parameters using an ESPI system. Its basis is time triggered measurements using stroboscopic or pulsed laser illumination. The method has been successfully applied to several industrial case studies. Pérez-López et al. [42] performed an quantitative measurement of dynamic deformations of a rotating compact disc and fan, respectively, using pulsed digital holography and an optical derotator to eliminate fringes that are due to the rotation of the object. In another work by Pérez-López et al. [43], the authors introduced a qualitative method, based on pulsed ESPI, to separate rotation fringes from fringes solely related to vibration. This paper presents a highly efficient method that allows to utilise the full potential of digital image correlation in fullfield vibration and modal analysis of rotating structures. The proposed method is based on the numerical elimination of rigid body motions that are contained in measured data. No additional devices or modifications of correlation system are necessary and measurement can be performed on an object rotating with constant or variable revolutions without having to measure the speed of rotation.
2. Basic principle of 3D digital image correlation Digital image correction method is a full-field contactless method that uses the digital image registration technique to accurately measure the spatial contour as well as the displacements and deformations of an object under load. Its principle is based on the observation of a stochastic speckle pattern that is artificially created on the surface of an object to be analysed. In a conventional 3D digital image correlation technique, the object surface is observed by at least two precise monochromatic cameras from different directions (Fig. 1). Digital images captured by the cameras are divided into smaller square sub-areas called facets in such a way that each of them contains a characteristic part of the speckle pattern. The speckles represent the material points of the object, i.e. they copy surface deformations and move together with the object. It allows to visualize the measured quantities in the whole observed area. An important part of the correlation procedure is the calibration process, the purpose of which is to determine the intrinsic imaging parameters of the both cameras as well as their extrinsic parameters (rotation matrix and translation vector) describing the relative positions and orientations of the cameras with respect to each other. These parameters are needed to calculate the 3D coordinates of the surface points determined by a virtual evaluation grid. The position of each object point in the captured images is identified by a correlation algorithm that tracks grey level of the pixel sets appertaining to corresponding facets. Matching process consists of looking for distributions of correlation coefficients. A target facet is considered to be identical with a reference facet if an extreme is reached at the corresponding correlation coefficient. The position of this extreme then determines the position of the target facet and can be used to calculate the displacement of the facet with respect to its reference position [44].
Fig. 1. Basic principle of 3D digital image correlation method with a stereoscopic configuration of cameras [29].
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Dantec Dynamics correlation systems use a correlation algorithm that is based on pseudo-affine transformations of coordinates between the reference and the target image. The transformation coordinates xT and yT are calculated as follows [45]
xT ¼ a0 þ a1 x þ a2 y þ a3 xy;
ð1Þ
yT ¼ a4 þ a5 x þ a6 y þ a7 xy;
where a0 ; :::; a7 are parameters of affinity transformation that represent translation, shear, stretch and twist of facet (Fig. 2). Within the correlation algorithm the transformation parameters are determined by minimizing the distance between the grey level G2 ðx; yÞ of the pattern in the target image and the grey level G1 ðx; yÞ of the reference pattern
min
a0 ; :::; a7 ; g 0 ; g 1
X
k G1 ðx; yÞ GT ðx; yÞ k;
ð2Þ
x;y
GT ðx; yÞ ¼ g 0 þ g 1 G2 ðxT ðx; yÞ;
yT ðx; yÞÞ;
ð3Þ
where GT ðx; yÞ expresses a change in grey level after transformation of the speckle pattern, and g 0 ; g 1 are photogrammetric corrections, which consider different contrast and intensity levels of the images. 2.1. Practical aspects of high-speed DIC measurements When high-speed cameras are used for measurement, then 3D DIC method can be used to analyse various transient events such as impact, vibration, etc. [10]. Unlike low-speed and static applications, more attention must be paid to light and image conditions. A short exposure time is required to capture sharp images, especially when the object moves/rotates at a high speed. In the case of very short exposure time, it is necessary to use a powerful light source. Equally important is to ensure the same light conditions for both cameras. Measurement of high frequency responses requires the sampling rate of the cameras to be at least two times the upper limit of the desired frequency range. In the case of vibration analysis associated with frequency analysis, it is also important to correctly set the facet size, the width of the evaluation grid, and the smoothing filters, since all these parameters have an impact on the amplitudes of measured responses and vibrational shapes [46]. The influence of other factors on the precision of 3D DIC measurement is described in more detail in [44,45,47,48]. 3. Elimination of rigid body motions When the measurement is performed on a rotating body, the correlation system captures the motion that is composed of a rigid body motion (rotation and translation) and flexible body motion (deflections caused by inertia forces or excitation forces). In that case, it is necessary to separate these two motion components to obtain the responses corresponding only to vibration. The basis is to determine the rotation matrix and translation vector that optimally describe rigid body transformation between two positions of an analysed object. There are several methods that make it possible [49–52]. The method in this study uses singular value decomposition and least-squares estimation to calculate best-fitting rigid transformations. The rotation matrix and translation vector are calculated from a three-dimensional displacement field for each time step according to the procedure published by Sorkine-Hornung and Rabinovich [52]. After the elimination, the remaining data correspond to pure vibration responses and can be subjected to further analysis, such as ODS, OMA, EMA, etc. A rigid body motion from a position P into another position Q can be characterized by a translation vector t and a rotation matrix R. Let P ¼ fp1 ; p2 ; ::: ; pn g and Q ¼ fq1 ; q2 ; ::: ; qn g to be the sets of radius vectors describing two different positions of n corresponding points of the same body in 3D Cartesian coordinate system. A rigid transformation that optimally aligns these two sets is found by the least-square method
Fig. 2. Transformation parameters of potential translation, stretch, shear and distortion [45].
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ðR; t Þ ¼ arg min
n X
wi k ðRpi þ t Þ qi k2
205
ð4Þ
i¼1
where wi > 0 are weights for each point pair. A detailed procedure of the derivation of R and t is given in [52]. Herein, only summary of the steps needed to determine the optimal translation vector and rotation matrix that minimize Eq. (4) is provided: 1. Compute the weighted centroids of both point sets:
Pn wi pi p ¼ Pi¼1 ; n i¼1 wi
Pn wi qi q ¼ Pi¼1 : n i¼1 wi
ð5Þ
2. Compute the centred vectors:
ai ¼ pi p;
bi ¼ qi q;
i ¼ 1; 2; :::; n:
ð6Þ
3. Compute the d d covariance matrix:
C ¼ AWBT ;
ð7Þ
where A and B are d n matrices that have ai and bi as their columns, respectively, and W ¼ diagðw1 ; w2 ; :::; wn Þ. 4. Compute the singular value decomposition:
C ¼ URV T :
ð8Þ
5. Compute the optimal rotation matrix:
0 B B R ¼ VB B @
1
1 ..
C C T CU : C A
. 1
ð9Þ
detðVU T Þ 6. Compute the optimal translation vector:
t ¼ q R p :
ð10Þ
To suppress rigid body motions, we consider vectors pi to correspond to the initial position of the disc, and vectors qi to all subsequent positions in time. Since the highest stiffness of the disc is near the centre of rotation we introduced
wi ¼
1 : k qi k
ð11Þ
After the computation of R and t for each step, the new point coordinates eliminating rigid body motions are determined by inverse transformation
qi ¼ R1 qi t pi :
ð12Þ
The last adjustment is to shift the centre of the disk to origin point and rotate the axis of rotation to match the z-axis of the coordinate system
qi ¼ Sqi r 0 ;
ð13Þ T
where S is a rotation matrix defining the transformation between vector [0 0 1] and average normal vector of the disc in its initial position, r 0 is a position vector of the centre of the disk in its initial position related to the origin point. Fig. 3 shows the disc at the same time step before and after the elimination of rigid body motions. Time variation of displacements of point A is shown in Fig. 4. Supplementary Video 1 shows a sequence corresponding to Fig. 3.
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Video 1.
Fig. 3. Disc at the same time step before (a) and after (b) the elimination of rigid body motions.
The accuracy of the transformation parameters R and t depends on the number of nodal points forming the surface contour of the object. In general, the larger the number, the higher the accuracy is. However, it is necessary to ensure that the number of points in individual time steps is not significantly changed. The loss of data due to uncorrelated facets may introduce inaccuracies into the calculation. For that reason extra attention has to be paid to the realization of the measurement. On the other hand, the losses of facets cannot be completely avoided. These outages can be treated by using a computational mask, common to all steps, that takes into account all uncorrelated facets in the measurement. The mask acts as a spatial data filter that takes value 1 for points in correlated regions and 0 (or NaN) for points in uncorrelated regions.
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Fig. 4. Time variation of displacements of point A before (a) and after (b) the elimination of rigid body motions.
The computational efficiency of the algorithm is also relatively high. The processing of one time step with 8134 points took about 0.17 s. The computations were carried out in Matlab on a computer with CPU IntelÒ CoreTM [email protected] GHz, 16 GB RAM and 64-bit operation system Windows 7.
4. Experimental measurements This chapter presents the results of three experiments that demonstrate the possibilities of the proposed method in vibration analysis of rotating structures. The object of measurement was a plane circular disc made of plastic material PS-1 commonly used in photoelasticimetry. The dimension of the disc and its material properties are shown in Fig. 5a. The disc was attached to the servomotor’s shaft by a screw connection. The servomotor was powered by DC voltage source. A thin preprinted vinyl foil with a stochastic speckle pattern was applied on the front surface of the disc for the purpose of an image correlation process. The responses were measured by the Q-450 Dantec Dynamics system, with two Phantom SpeedSense 9060 high-speed cameras with a resolution of the CMOS sensor of 1280 800 px and the internal memory of 16 GB. The cameras were equipped with ZEISS 50 mm Makro-planar lenses with an aperture of f /8. Due to high rotational speed of the disc, an extremely short shutter time of 20 ns had to be used to capture sharp images. The disc was illuminated by two high-frequency halogen lamps with a total power of 1400 W. The experimental setup can be seen in Fig. 5b. All described measurements have been performed under the same conditions. The region of interest was limited to 824 834 px, as shown in Fig. 6. The facet size was set to 14 px with an overlap of 5 px between the facets. Thus, the surface contour of the disc was approximated by a virtual evaluation grid with 8134 nodal points at which the displacements were measured. The obtained displacement fields were saved as HDF5 files and subsequently imported to Matlab for further processing.
Fig. 5. a) Dimensions and material properties of the disc, b) Experimental setup.
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Fig. 6. Disc in cameras’ views. (Images capture the disc while rotating.)
Fig. 7. Combined frequency spectra of the responses before the elimination of rigid body motions.
Fig. 8. Combined frequency spectra of the responses after the elimination of rigid body motions.
4.1. ODS analysis The aim of the first measurement was to determine operating shapes of vibration of the rotating disc when the disc was excited by inertial forces due to its imbalance. The measurement was performed at constant rotational speed of 4700 rpm. The sampling frequency of the cameras was 4000 fps. The acquisition time was 0.5 s, i.e., 2000 images were captured in total. Image correlation analysis was performed in Istra4D software that is part of the measuring system. The obtained displacement fields were imported to Matlab for post-processing. After the numerical elimination of rigid body motions, the modified displacement fields were processed by DICMAN 3D application to obtain frequency spectra and operating deflection shapes. DICMAN 3D [29] is a software application for spectral and modal analysis that has been specially developed for the correlation system Q-450 Dantec Dynamics.
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Fig. 9. Operating deflection shapes of the disc rotating at 4700 rpm.
Figs. 7 and 8 show the combined frequency spectra of the responses of the rotating disc measured in x, y, z direction, respectively. The spectra have been calculated as complex mode indicator functions. Fig. 7 relates to primary measured displacements, Fig. 8 relates to displacements after the elimination of rigid body motions. From their comparison it is apparent that the higher harmonic and sub-harmonic vibration components are easier identifiable in the frequency response spectrum in which the rigid body motions are suppressed. The operating deflection shapes corresponding to the first nine harmonic frequencies are shown in Fig. 9. It can be seen that the first operating shape does not correspond to the shape to be usually expected. This phenomenon will be detailed explained in chapter 4.3. 4.2. Run-up analysis Run-up analysis is commonly carried out to investigate dynamic behaviour of rotating parts and to identify their failures. The aim of the experiment is to present the operability of the proposed method in the case of variable rotational speed. The measurement was performed under the same conditions, except that the sampling frequency was 1500 fps and the maximum disc speed was about 4800 rpm. The acquisition time was 4 s. The measured data were processed in Matlab. The result
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Fig. 10. Run-up diagram.
Fig. 11. Excitation of the disc.
of the analysis is a run-up diagram shown in Fig. 10. The diagram shows the time-varying frequency spectrum of combined responses measured in z-direction, the magnitudes of which are expressed by colour field plot. The distinctive colour lines that can be seen in the diagram belong to fundamental frequency and its higher harmonics frequencies of the rotating disc.
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Fig. 12. CMIF spectrum of the rotating disc.
Fig. 13. Mode shapes of the disc rotating at 4800 rpm.
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Fig. 14. Comparison of the first mode shape obtained by FEM analysis (left) and DIC measurement (right).
4.3. Operational modal analysis The third experiment presents the measurement of modal parameters of the rotating disc. The disc rotated at a constant rotational speed of 4800 rpm. During the measurement, the disc was continuously excited from an acoustic source of sound acting on the rear side. Fig. 11 shows the location of an excitation nozzle. It should be noted that the exciting point was changing with the rotation. A broadband excitation was achieved by the white noise signal. The excitation signal was not measured, i.e., modal parameters were estimated only from the measured responses. The sampling frequency was set to 3000 fps. The acquisition time was 0.5 s. The measured data were processed in Matlab and subsequently evaluated in DICMAN 3D. Fig. 12 shows the CMIF spectrum of the disc. The spectrum contains peaks corresponding to eigen modes of the disc and peaks corresponding to vibrations caused by the rotation. The Modal Phase Collinearity (MPC) criterion is an effective tool to distinguish them from each other. The criterion expresses the linear functional relationship between the real and the imaginary parts of the unscaled mode shape vector [53]. While the eigen modes are normal, the operating modes are complex. MPC can take values from 0 to 1. The normal modes make MPC value close to 1, the complex modes, close to 0. In the frequency range up to 1500 Hz, nine modes of the disc were identified. Their natural frequencies, MPC values, and mode shapes are shown in Fig. 13. Due to rotation, the multiple modes have also been identified. Similarly as in the case of ODS analysis, the first two mode shapes do not have the appearance that is expected. At first sight, they resemble the higher mode shapes. Šaroši et al. [7], obtained the same first mode shape of a rotating disc when used an optical derotator and SLDV system for measuring. A comparison of the shapes obtained by measurement and FEM analysis has provided an explanation for this phenomenon. As can be seen in Fig. 14, the shape of the mode is correct, but the position of its contour in the evaluation coordinate system is misaligned. The contour is rotated about the axis passing through the plane of the disc. This misalignment resulted in incorrect graphical representation of the shapes in coloured displacement field shown in Fig. 13. All other shapes were displayed correctly. Since the cause of this phenomenon is the process of numerical elimination of rigid body motions, a special attention must be paid to the interpretation of results, especially modal shapes. The problem can be solved by a simple geometric correction of mode shape position that takes into account boundary conditions of the object. 5. Conclusions In the paper, a progressive method of measuring the vibration responses of rotating objects using high-speed digital image correlation has been introduced. The basis of the proposed method is an elimination of rigid body motions, such as rotation and translation, which are primarily contained in time-dependent 3D displacement fields. The elimination is done by numerical post-processing of measured data from which a rotation matrix and a translation vector, describing the homogenous rigid transformation between the initial and current position, are calculated for each time step. In this study, the optimal transformation parameters have been determined using the known method that makes use of the singular value decomposition and least-squares fitting. When the rotation matrix and translation vector are known, the displacement fields corresponding to pure vibration responses of the object are obtained by inverse transformation. Such modified fields are prepared for subsequent vibration and modal analysis. The computational efficiency of the method is very good, even despite the large number of points that are being processed. In addition, the large number of points increases the accuracy of the transformation parameters and thus improves a quality of the elimination process. On the other hand, the number of points should not change significantly, because the lost data can cause inaccuracies into the calculation of rotation matrix and translation vector. Therefore, it is necessary to avoid the loss of facets or to treat the dataset using a computational mask working as a spatial data filter. The method has been successfully applied to measure the operating deflection shapes and mode shapes of a disc rotating with a constant speed. Run-up analysis has shown that the method can also be used for variable speed applications. The results of performed experiments show that the elimination of rigid body motions leads to more
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pronounced (enhanced) response spectra in which individual vibration components are much easier to identify across the entire frequency range. This is particularly useful when vibration is weak. The only deficiency of this method is a position misalignment of the first operating shape as well as mode shape. This misalignment causes their incorrect graphical representation in colour field plot, although the shapes are correct. It can be easily treated by position correction with respect to boundary conditions. Nevertheless, the method is very effective and provides a number of advantages that extend the use of digital image correlation in practice. It is relatively simple and easy to apply, has a wide use and does not require any special devices or modification of measuring system. In addition, there are no special requirements for the position of an object or limitations regarding the speed. The possibilities of the method are largely limited by the possibilities of correlation system. Acknowledgment This research was supported by Scientific Grant Agency of the Ministry of Education of the Slovak Republic and of Slovak Academy of Sciences (Vedecká grantová agentúra MŠVVaŠ SR a SAV) projects VEGA 1/0355/18 and VEGA 1/0731/16. Appendix A. Supplementary data Supplementary data to this article can be found online at https:// doi.org/10.1016/j.ymssp.2018.11.024. References ˇ a, F. Šimcˇák, J. Bocko, P. Trebun ˇ a, Identification of causes of radial fan failure, Eng. Fail. Anal. 16 (7) (2009) 2054–2065. [1] F. Trebun ˇ a, F. Šimcˇák, J. Bocko, R. Hunˇady, M. Pástor, Complex approach to the vibrodiagnostic analysis of excessive vibration of the exhaust fan, Eng. [2] F. Trebun Fail. Anal. 37 (February 2014) 86–95. [3] T. Sedmak, E. Veg, Failure prevention of rotating equipment by vibrodiagnostics, Structu. Integrity Life 12 (2) (2012) 99–104. [4] P. Beauseroy, R. 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Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp
Vibration and modal analysis of a rotating disc using high-speed 3D digital image correlation Róbert Hunˇady ⇑, Peter Pavelka, Pavol Lengvarsky´ Technical University of Košice, Faculty of Mechanical Engineering, Letná 9, Košice, Slovakia
a r t i c l e
i n f o
Article history: Received 14 September 2018 Received in revised form 12 November 2018 Accepted 15 November 2018
Keywords: Digital image correlation Rotating disc Rigid body motion elimination Vibration analysis Modal analysis
a b s t r a c t Rotating structures are of interest in many engineering fields, where knowing their dynamic behaviour is important to prevent undesired operating states. However, the experimental measurement of vibrations in rotating structures is still a relatively demanding task that requires the use of special measurement technique or method. The paper deals with the application of high-speed digital image correlation in vibration analysis of rotating structures. The method introduced in this research is based on the elimination of rigid body motion components contained in the primary responses measured by cameras. The process of separation and subsequent elimination of these components is given by numerical post-processing of three-dimensional displacement fields. The basis is to determine rotation matrix and translation vector that optimally describe rigid transformation between two positions of an analysed object. The proposed method is applicable for measuring under both constant and variable speed of rotation. Practical application is presented by three experiments in which vibrations of a flat rotating disc were analysed. The first experiment is focused on determine the operating deflection shapes of the disc rotating at 4700 rpm. The second one is a run-up analysis under variable rotational speed that ranges from 0 to 4700 rpm. The aim of the third measurement is to obtain natural frequencies and modal shapes of the disc rotating at 4800 rpm. The results of the experiments show that the elimination of rigid body motions leads to a higher accuracy of measurement and provides more pronounced frequency response spectra. Ó 2018 Elsevier Ltd. All rights reserved.
1. Introduction Vibration analysis and health monitoring of rotating structures and components, such as impellers, discs, wind turbine blades, etc., is very common in technical practice. It is of paramount importance to periodically monitor the technical condition of these components, in order to avoid resonance or overload problems that may cause undesirable behaviour or even lead to critical failure [1–3]. Various measurement techniques can be employed to capture vibration responses of a rotating object. Conventional contact methods using accelerometers or strain gauges require the installation of wireless device or slip ring to transfer data from the sensors to recording system. This approach has several limitations. The transducers are intrusive because they can influence dynamic properties of the object under investigation due to mass loading effect and local stiffness alteration at the point of transducer attachment. In addition, the measurement is pointwise and limited to only few locations. The use of telemetry is usually costly, time consuming to install, and may introduce electrical noise into the signals [4]. These limitations are largely solved by using contactless methods that have been developed to measure ⇑ Corresponding author. E-mail addresses: [email protected] (R. Hunˇady), [email protected] (P. Pavelka), [email protected] (P. Lengvarsky´). https://doi.org/10.1016/j.ymssp.2018.11.024 0888-3270/Ó 2018 Elsevier Ltd. All rights reserved.
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displacements or deformations. Optical methods suitable for measuring dynamic events include scanning laser Doppler vibrometry (SLDV), electronic speckle pattern interferometry (ESPI), holographic interferometry, digital shearography, photogrammetry, and digital image correlation (DIC). The application of these methods for measuring vibration of a rotating object requires the use of additional devices or special measurement approaches that ensure that the object appears to be in a quasi-steady state. The elimination of rigid body motion components (rotations and translations) is appropriate, regardless of the used measurement technique. The elimination can be realized in several ways. When vibration responses are measured by scanning laser vibrometer, the use of the optical derotator is one of the possible ways. The derotator tracks the motion of the rotating object through a precision controlled optical rotation unit that is synchronised with the rotation frequency. The unit usually contains a Dove prism which rotates at half the speed of the object. It makes the rotating object appear stationary when viewed through the prism. The condition to attain complete optical derotation is that the rotation axis of the object and that of the rotation unit must match [5]. Current optical derotators can be used up to a speed of 24,000 rpm. The disadvantage is their high price. Boedecker et al. [6] described the design of the optical derotator that has been manufactured and subsequently used for experimental vibration analysis of a fan rotating with 3000 rpm. A similar measurement was made by Šároši et al. [7] who used the optical derotator to obtain operating and modal shapes of a rotating disc. They also performed run-up analysis to assess the effect of rotational speed on the shift of natural frequencies. Sever et al. [8] proposed the so-called self-tracking LDV method that is a modification of self-tracking technique proposed by Lomenzo et al. [9]. The method uses two 45-degree mirrors, one of which is fixed and the other (mounted on the drive shaft) rotates with the bladed disc. The modification is given by the use of fixed annular conical mirror instead of the fold mirror. This configuration allows the laser beam to be directed to the blade tips at a right angle. A way that does not require additional equipment is the use of a suitable scanning technique. Generally this is done by utilising the deflection mirrors in the LDV head and by synchronising the signals driving these mirrors with the rotational speed of the object under investigation. The use of scanning techniques can be limited due to inertias of the mirrors that may cause a phase shift of the measured signal [8,10]. The problem can be solved by means of a phase correction algorithm. However, scanning techniques are highly effective in many cases [11–18]. In 2004, Halkon and Rothberg [15] introduced the so-called Synchronised-Scanning Laser Vibrometry method by which the laser beam is capable to track the rotating structure and simultaneously scan the region of interest to provide modal data. They used single-point laser vibrometer with custombuilt scanning system. Maio and Ewins [16] presented how commercially available SLDV system without any hardware modification can be used for synchronous measurement of both stationary and rotating structures. Three different tracking techniques [17] were used in combination with suitable excitation method to analyse vibration of a bladed disc. Gwashavanhu et al. [18] used tracking LDV method for the validation of photogrammetry applied to online turbomachinery blade measurements. A comparison of out-of-plane displacements of a specific point on a rotating blade measured by photogrammetry and displacements captured by laser scanning system showed that there was a good correlation between the methods, both in the time and frequency domains. As mentioned above, another method suitable for vibration analysis is photogrammetry in conjunction with threedimensional point tracking (3DPT). Photogrammetry is a non-contact measurement technique that uses a pair of digital cameras to measure 3D displacements of distinct markers attached to the surface of an object. The use of photogrammetry in vibration analysis of rotating structures is documented by studies [19–23]. Warren et al. [19] used 3D digital stereophotogrammetry to measure vibration responses of a small wind turbine rotating at a speed of approximately 10 Hz. An example of a large-scale application is given by the work [20] in which the authors used 4-camera system to measure the dynamic responses of a wind turbine rotor of 80 m diameter while the turbine was rotating. Photogrammetry has also been used for measurement of helicopter rotor vibrations. Lundstrom et al. in the works [21,22], demonstrated the possibilities of this method when measured the operating deflection shapes of the spinning rotor blades of a Robinson R44 helicopter. In another work by Lundstrom et al. [23], the authors extracted the operating deflection shapes of a wind turbine by applying a harmonic filter to the measured data. If a random speckle pattern is applied on the object surface, the photogrammetry principle can be extended to full-field measurement. This technique is known as digital image correlation (DIC). The use of DIC method in vibration or modal analysis of non-rotating objects is currently common [24–31]. In addition to conventional approaches in 3D DIC method with two cameras, there are also single-camera methods that can also be used to analyse vibrations [32–35]. Yu and Pan [32] proposed a method that uses a single high-speed colour camera in combination with stereo-DIC technique to measure 3D deformations. The separation of recorded colour images into red and blue channel sub-images is the basis of the method. The separated images are then process by regular stereo-DIC algorithm to obtain the desired displacement fields. The method was successfully applied to measure the displacement and velocity of a rotating fan, full-field vibration responses of a rectangular panel, and deformation fields of a balloon during its explosion. In another work [33], Yu and Pan used a singlecamera high-speed stereo-digital image correlation (SCHS stereo-DIC) method with four-mirror adapter for full-field 3D vibration measurement where the mirror system is used to divide the sensor’s image plane into two halves with different views of the object under investigation. Both image areas are subsequently processed as separated images using a conventional stereo DIC procedure to obtain vibration responses. In the study [34], the authors provided a detailed overview and described the strengths and weaknesses of various DIC techniques based on pseudo stereo images recorded with a single camera. The number and results of published works show that DIC measurement techniques and procedures are well established. On the other hand, the works focusing on vibration analysis of rotating structures are really unique. Most available studies
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refer to the measurement of motion or deformation parameters using the triggering method when the sampling rate of the cameras is synchronised with the rotation frequency of an object to be analysed [36–39]. Yashar et al. [40] measured the frequency response functions of a rotating beam using DIC system with one camera that tracked the position of a specific marker through the mirror rotating with the beam. The beam was excited at its base using an electro-dynamic shaker attached to the rotational hub assembly. The disadvantage is that the proposed measurement method requires a special testing rig. In addition, the responses can only be measured at a few points (markers) determined on the structure. Wu et al. [39] measured the in-plane and out-of-plane motion components of a wind turbine blade model using 3D DIC method while the turbine was driven by an electric fan. They investigated the blade displacements of balanced and unbalanced turbine at the different rotational speeds. The measurement was evaluated at ten points spaced along one blade. In the case of the balanced turbine, the out-of-plane displacement frequency spectra showed the presence of the first two harmonic components. When additional weight was added, several sub-harmonic components have also been identified. Since the in-plane displacements corresponded only to the change of the points’ coordinates caused by the rotation of the blade, they did not provide any relevant information related to vibration, neither in time nor in frequency domain. Non-contact measurement methods based on interferometry are also useful in full-field vibration analysis of both nonrotating and rotating structures [41–43]. These methods achieve excellent sensitivity and resolution when measuring displacement and strain, however, their application in industries is relatively difficult. They require complex optical system and vibration isolation to prevent the possibility of phase changes or destroying the fringe patterns [38]. Van der Auweraer et al. [41] proposed a novel high-frequency experimental method to measure modal parameters using an ESPI system. Its basis is time triggered measurements using stroboscopic or pulsed laser illumination. The method has been successfully applied to several industrial case studies. Pérez-López et al. [42] performed an quantitative measurement of dynamic deformations of a rotating compact disc and fan, respectively, using pulsed digital holography and an optical derotator to eliminate fringes that are due to the rotation of the object. In another work by Pérez-López et al. [43], the authors introduced a qualitative method, based on pulsed ESPI, to separate rotation fringes from fringes solely related to vibration. This paper presents a highly efficient method that allows to utilise the full potential of digital image correlation in fullfield vibration and modal analysis of rotating structures. The proposed method is based on the numerical elimination of rigid body motions that are contained in measured data. No additional devices or modifications of correlation system are necessary and measurement can be performed on an object rotating with constant or variable revolutions without having to measure the speed of rotation.
2. Basic principle of 3D digital image correlation Digital image correction method is a full-field contactless method that uses the digital image registration technique to accurately measure the spatial contour as well as the displacements and deformations of an object under load. Its principle is based on the observation of a stochastic speckle pattern that is artificially created on the surface of an object to be analysed. In a conventional 3D digital image correlation technique, the object surface is observed by at least two precise monochromatic cameras from different directions (Fig. 1). Digital images captured by the cameras are divided into smaller square sub-areas called facets in such a way that each of them contains a characteristic part of the speckle pattern. The speckles represent the material points of the object, i.e. they copy surface deformations and move together with the object. It allows to visualize the measured quantities in the whole observed area. An important part of the correlation procedure is the calibration process, the purpose of which is to determine the intrinsic imaging parameters of the both cameras as well as their extrinsic parameters (rotation matrix and translation vector) describing the relative positions and orientations of the cameras with respect to each other. These parameters are needed to calculate the 3D coordinates of the surface points determined by a virtual evaluation grid. The position of each object point in the captured images is identified by a correlation algorithm that tracks grey level of the pixel sets appertaining to corresponding facets. Matching process consists of looking for distributions of correlation coefficients. A target facet is considered to be identical with a reference facet if an extreme is reached at the corresponding correlation coefficient. The position of this extreme then determines the position of the target facet and can be used to calculate the displacement of the facet with respect to its reference position [44].
Fig. 1. Basic principle of 3D digital image correlation method with a stereoscopic configuration of cameras [29].
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Dantec Dynamics correlation systems use a correlation algorithm that is based on pseudo-affine transformations of coordinates between the reference and the target image. The transformation coordinates xT and yT are calculated as follows [45]
xT ¼ a0 þ a1 x þ a2 y þ a3 xy;
ð1Þ
yT ¼ a4 þ a5 x þ a6 y þ a7 xy;
where a0 ; :::; a7 are parameters of affinity transformation that represent translation, shear, stretch and twist of facet (Fig. 2). Within the correlation algorithm the transformation parameters are determined by minimizing the distance between the grey level G2 ðx; yÞ of the pattern in the target image and the grey level G1 ðx; yÞ of the reference pattern
min
a0 ; :::; a7 ; g 0 ; g 1
X
k G1 ðx; yÞ GT ðx; yÞ k;
ð2Þ
x;y
GT ðx; yÞ ¼ g 0 þ g 1 G2 ðxT ðx; yÞ;
yT ðx; yÞÞ;
ð3Þ
where GT ðx; yÞ expresses a change in grey level after transformation of the speckle pattern, and g 0 ; g 1 are photogrammetric corrections, which consider different contrast and intensity levels of the images. 2.1. Practical aspects of high-speed DIC measurements When high-speed cameras are used for measurement, then 3D DIC method can be used to analyse various transient events such as impact, vibration, etc. [10]. Unlike low-speed and static applications, more attention must be paid to light and image conditions. A short exposure time is required to capture sharp images, especially when the object moves/rotates at a high speed. In the case of very short exposure time, it is necessary to use a powerful light source. Equally important is to ensure the same light conditions for both cameras. Measurement of high frequency responses requires the sampling rate of the cameras to be at least two times the upper limit of the desired frequency range. In the case of vibration analysis associated with frequency analysis, it is also important to correctly set the facet size, the width of the evaluation grid, and the smoothing filters, since all these parameters have an impact on the amplitudes of measured responses and vibrational shapes [46]. The influence of other factors on the precision of 3D DIC measurement is described in more detail in [44,45,47,48]. 3. Elimination of rigid body motions When the measurement is performed on a rotating body, the correlation system captures the motion that is composed of a rigid body motion (rotation and translation) and flexible body motion (deflections caused by inertia forces or excitation forces). In that case, it is necessary to separate these two motion components to obtain the responses corresponding only to vibration. The basis is to determine the rotation matrix and translation vector that optimally describe rigid body transformation between two positions of an analysed object. There are several methods that make it possible [49–52]. The method in this study uses singular value decomposition and least-squares estimation to calculate best-fitting rigid transformations. The rotation matrix and translation vector are calculated from a three-dimensional displacement field for each time step according to the procedure published by Sorkine-Hornung and Rabinovich [52]. After the elimination, the remaining data correspond to pure vibration responses and can be subjected to further analysis, such as ODS, OMA, EMA, etc. A rigid body motion from a position P into another position Q can be characterized by a translation vector t and a rotation matrix R. Let P ¼ fp1 ; p2 ; ::: ; pn g and Q ¼ fq1 ; q2 ; ::: ; qn g to be the sets of radius vectors describing two different positions of n corresponding points of the same body in 3D Cartesian coordinate system. A rigid transformation that optimally aligns these two sets is found by the least-square method
Fig. 2. Transformation parameters of potential translation, stretch, shear and distortion [45].
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ðR; t Þ ¼ arg min
n X
wi k ðRpi þ t Þ qi k2
205
ð4Þ
i¼1
where wi > 0 are weights for each point pair. A detailed procedure of the derivation of R and t is given in [52]. Herein, only summary of the steps needed to determine the optimal translation vector and rotation matrix that minimize Eq. (4) is provided: 1. Compute the weighted centroids of both point sets:
Pn wi pi p ¼ Pi¼1 ; n i¼1 wi
Pn wi qi q ¼ Pi¼1 : n i¼1 wi
ð5Þ
2. Compute the centred vectors:
ai ¼ pi p;
bi ¼ qi q;
i ¼ 1; 2; :::; n:
ð6Þ
3. Compute the d d covariance matrix:
C ¼ AWBT ;
ð7Þ
where A and B are d n matrices that have ai and bi as their columns, respectively, and W ¼ diagðw1 ; w2 ; :::; wn Þ. 4. Compute the singular value decomposition:
C ¼ URV T :
ð8Þ
5. Compute the optimal rotation matrix:
0 B B R ¼ VB B @
1
1 ..
C C T CU : C A
. 1
ð9Þ
detðVU T Þ 6. Compute the optimal translation vector:
t ¼ q R p :
ð10Þ
To suppress rigid body motions, we consider vectors pi to correspond to the initial position of the disc, and vectors qi to all subsequent positions in time. Since the highest stiffness of the disc is near the centre of rotation we introduced
wi ¼
1 : k qi k
ð11Þ
After the computation of R and t for each step, the new point coordinates eliminating rigid body motions are determined by inverse transformation
qi ¼ R1 qi t pi :
ð12Þ
The last adjustment is to shift the centre of the disk to origin point and rotate the axis of rotation to match the z-axis of the coordinate system
qi ¼ Sqi r 0 ;
ð13Þ T
where S is a rotation matrix defining the transformation between vector [0 0 1] and average normal vector of the disc in its initial position, r 0 is a position vector of the centre of the disk in its initial position related to the origin point. Fig. 3 shows the disc at the same time step before and after the elimination of rigid body motions. Time variation of displacements of point A is shown in Fig. 4. Supplementary Video 1 shows a sequence corresponding to Fig. 3.
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Video 1.
Fig. 3. Disc at the same time step before (a) and after (b) the elimination of rigid body motions.
The accuracy of the transformation parameters R and t depends on the number of nodal points forming the surface contour of the object. In general, the larger the number, the higher the accuracy is. However, it is necessary to ensure that the number of points in individual time steps is not significantly changed. The loss of data due to uncorrelated facets may introduce inaccuracies into the calculation. For that reason extra attention has to be paid to the realization of the measurement. On the other hand, the losses of facets cannot be completely avoided. These outages can be treated by using a computational mask, common to all steps, that takes into account all uncorrelated facets in the measurement. The mask acts as a spatial data filter that takes value 1 for points in correlated regions and 0 (or NaN) for points in uncorrelated regions.
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Fig. 4. Time variation of displacements of point A before (a) and after (b) the elimination of rigid body motions.
The computational efficiency of the algorithm is also relatively high. The processing of one time step with 8134 points took about 0.17 s. The computations were carried out in Matlab on a computer with CPU IntelÒ CoreTM [email protected] GHz, 16 GB RAM and 64-bit operation system Windows 7.
4. Experimental measurements This chapter presents the results of three experiments that demonstrate the possibilities of the proposed method in vibration analysis of rotating structures. The object of measurement was a plane circular disc made of plastic material PS-1 commonly used in photoelasticimetry. The dimension of the disc and its material properties are shown in Fig. 5a. The disc was attached to the servomotor’s shaft by a screw connection. The servomotor was powered by DC voltage source. A thin preprinted vinyl foil with a stochastic speckle pattern was applied on the front surface of the disc for the purpose of an image correlation process. The responses were measured by the Q-450 Dantec Dynamics system, with two Phantom SpeedSense 9060 high-speed cameras with a resolution of the CMOS sensor of 1280 800 px and the internal memory of 16 GB. The cameras were equipped with ZEISS 50 mm Makro-planar lenses with an aperture of f /8. Due to high rotational speed of the disc, an extremely short shutter time of 20 ns had to be used to capture sharp images. The disc was illuminated by two high-frequency halogen lamps with a total power of 1400 W. The experimental setup can be seen in Fig. 5b. All described measurements have been performed under the same conditions. The region of interest was limited to 824 834 px, as shown in Fig. 6. The facet size was set to 14 px with an overlap of 5 px between the facets. Thus, the surface contour of the disc was approximated by a virtual evaluation grid with 8134 nodal points at which the displacements were measured. The obtained displacement fields were saved as HDF5 files and subsequently imported to Matlab for further processing.
Fig. 5. a) Dimensions and material properties of the disc, b) Experimental setup.
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Fig. 6. Disc in cameras’ views. (Images capture the disc while rotating.)
Fig. 7. Combined frequency spectra of the responses before the elimination of rigid body motions.
Fig. 8. Combined frequency spectra of the responses after the elimination of rigid body motions.
4.1. ODS analysis The aim of the first measurement was to determine operating shapes of vibration of the rotating disc when the disc was excited by inertial forces due to its imbalance. The measurement was performed at constant rotational speed of 4700 rpm. The sampling frequency of the cameras was 4000 fps. The acquisition time was 0.5 s, i.e., 2000 images were captured in total. Image correlation analysis was performed in Istra4D software that is part of the measuring system. The obtained displacement fields were imported to Matlab for post-processing. After the numerical elimination of rigid body motions, the modified displacement fields were processed by DICMAN 3D application to obtain frequency spectra and operating deflection shapes. DICMAN 3D [29] is a software application for spectral and modal analysis that has been specially developed for the correlation system Q-450 Dantec Dynamics.
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Fig. 9. Operating deflection shapes of the disc rotating at 4700 rpm.
Figs. 7 and 8 show the combined frequency spectra of the responses of the rotating disc measured in x, y, z direction, respectively. The spectra have been calculated as complex mode indicator functions. Fig. 7 relates to primary measured displacements, Fig. 8 relates to displacements after the elimination of rigid body motions. From their comparison it is apparent that the higher harmonic and sub-harmonic vibration components are easier identifiable in the frequency response spectrum in which the rigid body motions are suppressed. The operating deflection shapes corresponding to the first nine harmonic frequencies are shown in Fig. 9. It can be seen that the first operating shape does not correspond to the shape to be usually expected. This phenomenon will be detailed explained in chapter 4.3. 4.2. Run-up analysis Run-up analysis is commonly carried out to investigate dynamic behaviour of rotating parts and to identify their failures. The aim of the experiment is to present the operability of the proposed method in the case of variable rotational speed. The measurement was performed under the same conditions, except that the sampling frequency was 1500 fps and the maximum disc speed was about 4800 rpm. The acquisition time was 4 s. The measured data were processed in Matlab. The result
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Fig. 10. Run-up diagram.
Fig. 11. Excitation of the disc.
of the analysis is a run-up diagram shown in Fig. 10. The diagram shows the time-varying frequency spectrum of combined responses measured in z-direction, the magnitudes of which are expressed by colour field plot. The distinctive colour lines that can be seen in the diagram belong to fundamental frequency and its higher harmonics frequencies of the rotating disc.
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Fig. 12. CMIF spectrum of the rotating disc.
Fig. 13. Mode shapes of the disc rotating at 4800 rpm.
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Fig. 14. Comparison of the first mode shape obtained by FEM analysis (left) and DIC measurement (right).
4.3. Operational modal analysis The third experiment presents the measurement of modal parameters of the rotating disc. The disc rotated at a constant rotational speed of 4800 rpm. During the measurement, the disc was continuously excited from an acoustic source of sound acting on the rear side. Fig. 11 shows the location of an excitation nozzle. It should be noted that the exciting point was changing with the rotation. A broadband excitation was achieved by the white noise signal. The excitation signal was not measured, i.e., modal parameters were estimated only from the measured responses. The sampling frequency was set to 3000 fps. The acquisition time was 0.5 s. The measured data were processed in Matlab and subsequently evaluated in DICMAN 3D. Fig. 12 shows the CMIF spectrum of the disc. The spectrum contains peaks corresponding to eigen modes of the disc and peaks corresponding to vibrations caused by the rotation. The Modal Phase Collinearity (MPC) criterion is an effective tool to distinguish them from each other. The criterion expresses the linear functional relationship between the real and the imaginary parts of the unscaled mode shape vector [53]. While the eigen modes are normal, the operating modes are complex. MPC can take values from 0 to 1. The normal modes make MPC value close to 1, the complex modes, close to 0. In the frequency range up to 1500 Hz, nine modes of the disc were identified. Their natural frequencies, MPC values, and mode shapes are shown in Fig. 13. Due to rotation, the multiple modes have also been identified. Similarly as in the case of ODS analysis, the first two mode shapes do not have the appearance that is expected. At first sight, they resemble the higher mode shapes. Šaroši et al. [7], obtained the same first mode shape of a rotating disc when used an optical derotator and SLDV system for measuring. A comparison of the shapes obtained by measurement and FEM analysis has provided an explanation for this phenomenon. As can be seen in Fig. 14, the shape of the mode is correct, but the position of its contour in the evaluation coordinate system is misaligned. The contour is rotated about the axis passing through the plane of the disc. This misalignment resulted in incorrect graphical representation of the shapes in coloured displacement field shown in Fig. 13. All other shapes were displayed correctly. Since the cause of this phenomenon is the process of numerical elimination of rigid body motions, a special attention must be paid to the interpretation of results, especially modal shapes. The problem can be solved by a simple geometric correction of mode shape position that takes into account boundary conditions of the object. 5. Conclusions In the paper, a progressive method of measuring the vibration responses of rotating objects using high-speed digital image correlation has been introduced. The basis of the proposed method is an elimination of rigid body motions, such as rotation and translation, which are primarily contained in time-dependent 3D displacement fields. The elimination is done by numerical post-processing of measured data from which a rotation matrix and a translation vector, describing the homogenous rigid transformation between the initial and current position, are calculated for each time step. In this study, the optimal transformation parameters have been determined using the known method that makes use of the singular value decomposition and least-squares fitting. When the rotation matrix and translation vector are known, the displacement fields corresponding to pure vibration responses of the object are obtained by inverse transformation. Such modified fields are prepared for subsequent vibration and modal analysis. The computational efficiency of the method is very good, even despite the large number of points that are being processed. In addition, the large number of points increases the accuracy of the transformation parameters and thus improves a quality of the elimination process. On the other hand, the number of points should not change significantly, because the lost data can cause inaccuracies into the calculation of rotation matrix and translation vector. Therefore, it is necessary to avoid the loss of facets or to treat the dataset using a computational mask working as a spatial data filter. The method has been successfully applied to measure the operating deflection shapes and mode shapes of a disc rotating with a constant speed. Run-up analysis has shown that the method can also be used for variable speed applications. The results of performed experiments show that the elimination of rigid body motions leads to more
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pronounced (enhanced) response spectra in which individual vibration components are much easier to identify across the entire frequency range. This is particularly useful when vibration is weak. The only deficiency of this method is a position misalignment of the first operating shape as well as mode shape. This misalignment causes their incorrect graphical representation in colour field plot, although the shapes are correct. It can be easily treated by position correction with respect to boundary conditions. Nevertheless, the method is very effective and provides a number of advantages that extend the use of digital image correlation in practice. It is relatively simple and easy to apply, has a wide use and does not require any special devices or modification of measuring system. In addition, there are no special requirements for the position of an object or limitations regarding the speed. 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