sem.org imac xxvii conf s15p002 vibration

7/28/2019 Sem.org IMAC XXVII Conf s15p002 Vibration

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Vibration Mode Shape Recognition Using the Zernike Moment Descriptor

Weizhuo Wang, J ohn E Mottershead and Cristinel Mares*

Department of Engineering,The University of Liverpool, Liverpool, United Kingdom, L69 7ZF

* School of Engineering and Design,Brunel University, Uxbridge, Middlesex, United Kingdom, UB8 3PH

ABSTRACT

Vibration mode-shape comparison between numerical models and experimental data is an essential step in thestudy of structural finite element model (FEM) updating. The Modal Assurance Criterion (MAC) is the mostpopular method for such comparison at the moment, which works perfectly well for small and medium sizedstructures. MAC provides a measure of closeness between the predicted and measured eigenvectors butcontains no explicit information on shape features. This is especially significant for large and complicatedindustrial systems where the MAC index can rapidly degenerate over the course of just a few modes so that avery detailed finite element model, which surely represents the physical system with good accuracy, cannot bereliably compared to measurements. New techniques, based upon the well-developed philosophies of imageprocessing (IP) and pattern recognition (PR) are considered in this paper. The Zernike moment descriptor(ZMD)is a region-based shape descriptor having outstanding properties in IP including rotational invariance, expression

and computing efficiency, ease of reconstruction and robustness to noise. In this paper the ZMD is applied to theproblem of mode shape recognition for a circular plate. Result shows that the ZMD has considerable advantagesover the traditional MAC index when identifying the cyclically symmetric mode shapes that occur in axisymmetricstructures with repeated eigenvalues.

NOMENCLATURE

,, Geometric Moments ( Central ), Zernike Moments,, , ,, Zernike Polynomials, Radial Polynomials

, ,, Image or Shape Pattern

,,, Cartesian and Polar Coordinates,, Orders of Moments Repetition of Zernike Moments, Kronecker Delta, Real and Imaginary Part of Complex Number

Proceedings of the IMAC-XXVIIFebruary 9-12, 2009 Orlando, Florida USA

2009 Society for Experimental Mechanics Inc.


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Correlation Coefficient Angle of Rotation

1. INTRODUCTION

Experimental validation is a crucial step in obtaining a reliable numerical model (e.g. a finite element model).Vibration mode-shapes are important properties in structural dynamics. The current method for comparing the

predicted and measured mode shapes is the modal assurance criterion (MAC) of Allemang and Brown [1]. TheMAC has been widely and successfully adopted in finite element model updating [2-5] for the last two decadesbecause it provides a simple tool to assess the quality of numerical models. However, all the information onsimilarity between the finite-element prediction and test data is encapsulated in a single number. This makes theappreciation of subtle changes in shape, either locally or globally unrealistic by MAC. This becomes critical in thecase of large and complicated structures with mode shapes that can be fully measured by modern scanning lasertechniques and accurately predicted by detailed finite element models. A new technique is presented in this paperthat allows a more detailed understanding of mode shapes to be obtained in the form of shape features.

Image processing (IP) and pattern recognition (PR) provide many methods for dealing with the shape descriptionand comparison of shapes [6-9]. Within such procedures, a series of shape features with good discriminativecapability should be extracted to form a feature vector shape descriptor (SD). Now the similarity/dissimilarity ofthe shapes can be revealed by the distance of their corresponding SDs in the shape feature space according toappropriate criteria.

The moment descriptor is one of the most popular shape descriptors in image processing and pattern recognitionof two dimensional images and shape patterns [10, 11]. A set of moment invariants constructed by non-linearcombinations of geometric moments was first introduced by Hu [12]. These geometric moment invariants have theproperties of being invariant under image translation, scaling and rotation. However, the basis of geometricmoments is not orthogonal, the moment sequence includes redundant information to a high degree and the high-order moments are very sensitive to noise. Furthermore, the reconstruction of the image from these moments isvery difficult.

The idea of orthogonal moments to recapture the image from the moments based on the theory of orthogonalpolynomials was suggested by Teague [13]. The Zernike moment (ZM) is based on a complete set of orthogonalpolynomials over a circle of unit radius Zernike polynomials.It was found later that the Zernike moment is one ofthe most important region-based shape descriptors because of its outstanding properties resulting from the

orthogonality of the Zernike polynomials. Firstly, expressing an image as a set of mutually independentdescriptors has the effect of reducing the information redundancy of the ZM to a minimum. Secondly, thecontribution of each order of moment to the image reconstruction can be separated, so that the process ofregaining the original image is much easier than by geometric moment descriptors [14]. Rotational invariance [15,16] is another important property of ZM, meaning that rotating an image does not change the magnitudes of itsZernike moments. Also, the ZM is robust to noise [14] and effective, meaning that a small number of Zernikemoments are usually sufficient for shape reconstruction.

Therefore, the Zernike moment descriptor (ZMD) is widely adopted in the image processing and patternrecognition community. It is especially useful for, but not limited to, the study of circular or spherical symmetry, forexample in optometry and ophthalmology [17-19] it is suitable for characterizing the optical aberrations of thecornea. In this paper we include the problem of mode-shape recognition in axisymmetric structures.

The definition of the Zernike moment and its properties are presented in Section 2. The application of mode-

shape recognition by the Zernike moment descriptor to simple plate structures such as circular and rectangularplates are demonstrated in Section 3. A complete description of the ZMD in mode-shape recognition and moreapplications including finite element model updating, based on the sensitivities of the Zernike moments to smallperturbations in system parameters, are described by the present authors [20]. Also, details of further applicationsof other shape descriptors and classification techniques for vibration mode shape recognition may be found in thefirst authors PhD thesis [21].


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2. ZERNIKE MOMENTS

Geometric moment is one of the simplest shape descriptors, defined as,

, , dd , 0,1,2, (1)where , is the image (or shape) to be described. The central moment is defined as,

, , dd , 0,1,2, (2)where

, (3), (4)

Hus seven moment invariants, including rotation, scaling, translation and skew orthogonal invariants, werederived based on equations (1) and (2). However, the kernel functions or are notorthogonal bases, the resulted geometric moment descriptors generally contain redundant information, are verysensitive to noise and shape reconstruction is difficult [13]. Therefore, moment descriptors derived fromorthogonal kernel functions are considered. The Zernike moment descriptor is one of the best choices.

2.1 Definition

The Zernike moment of an image , can be defined as,, 1 , , ,dd (5)

or expressed in a polar coordinate system as,

, 1 , , , dd

(6)

where * denotes the complex conjugate and

,, or

,, is the complex Zernike polynomial introduced

by Zernike [19] and can be expressed as,

,, ,, , (7)where

i 1 Non-negative integer, representing the order of the radial polynomial; Positive and negative integers subject to constraints || even, || representing therepetition of the azimuthal angle; Length of vector from the origin to , ; The azimuthal angle between vector and the -axis in counter-clockwise direction and

,Radial polynomial is defined as

, 1 !! ||2 2 ! ||2 2 !

||

(8)

Complex Zernike polynomials satisfy the orthogonality condition [22],


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,, , ,dd

1 ,, (9)where and , and , are Kronecker deltas.For a digital image, the Zernike moments may be calculated by replacing the integral in equation (5) with thesummation,

, 1 , , , 1 (10)

2.2 Reconstruction

According to the completeness and orthogonality of the Zernike polynomials [23], the original image , mayeasily be reconstructed as,

, ,,,

(11)

where is constrained as in equation (7). However, approximation of the reconstructed image by truncating thehigher orders of Zernike moments must be carried out to reduce the computational demand in real applications.Therefore, the finitely reconstructed image , , which is obtained by keeping the moments from order 0to and discarding the remaining higher order Zernike polynomials, may be written as,

, ,,,

(12)

Furthermore, the right hand side of equation (12) can be expressed with the real-valued functions [16] as,

, , cos, sin, ,

2,

(13)

where

, 2, 2 2 , , cos dd

(14)

and

, 2, 2 2 , , sin dd

(15)

Therefore, the reconstructed image , calculated by equation (13) is guaranteed to be real-valued whateverthe number

is selected. The sufficient large number of

can be determined by a predefined threshold

when comparing the similarity between the original image, and the approximation , . Correlation is oneof the methods that can be applied for such comparison, defined as

, d

d d

(16)

where denotes the internal domain of the unit disc, d is the infinitesimal area, and


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Table 1:Finite Element Model Description of the Disc

Element Type 4-Node Quadrilateral

Element Property Shell

Material Steel

Boundary conditions FreeSolution type Normal Modes

Freq. Range 0 630 Hz

Number of Modes 20Figure 1: Finite Element Model of the Disc

Table 2: Natural Frequencies of the Free-Free Disc

ModeNatural

Frequency(Hz) Mode

NaturalFrequency

(Hz) ModeNatural

Frequency(Hz) Mode

NaturalFrequency

(Hz)1 52.72 6 207.24 11 331.1 16 469.372 52.74 7 207.69 12 356.62 17 535.643 91.12 8 215.27 13 357.73 18 537.384 122.53 9 215.56 14 396.95 19 627.695 122.63 10 330.67 15 468.38 20 628.8

Firstly, the necessary and sufficient order of the ZMD, , needs to be determined to distinguish differentshapes efficiently and to retrieve the original shapes accurately. 8 is adequate for the first 20 modes wasdetermined by equation (16) and the results are shown in Figure 6. The amplitudes of the Zernike moments areshown in Figure 4. It is clear that just a few significant Zernike moments are sufficient to describe each mode.Especially for the first 18 modes, only one or two Zernike moments are sufficient to represent each separatemode shape with very good precision.

According to equation (25), the amplitudes of the Zernike moments of rotated images remain unchanged. The

correlation coefficients of vectors having terms given by the Zernike-moment amplitudes , are shown inFigure 5.

, , , , ,, ,,

, , , ,,

(26)

where and denote the mode number, denotes the Zernike moment and denotes the maximum numberof Zernike moment considered, and , denotes the mean Zernike-moment amplitudes of all modes. It isobvious that the same shape features of the double mode 1 and 2; 4 and 5; etc. are now show 100% correlationaccording to the Zernike moment descriptors. Besides, the Zernike moments of the double mode have zerocorrelation with any other mode shapes.


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Figure 2: Mode Shape Patterns of the Free-Free Disc Figure 3: The AutoMAC of the free-free Disc

Figure 4: Amplitudes of the ZMDs (Free-Free Disc) Figure 5: Correlation Coefficients of theZernike Moment Amplitude (Free-Free Disc)

Furthermore, as discussed in Section 2 and 3, the rotational difference between the double modes can berevealed by equation (25). Figure 8 demonstrates an example where (a) shows shape patter of mode 2 is thesame as mode 1 but rotated through 45, (b) plots the amplitudes of their corresponding Zernike moments. It isclear that only two Zernike moments are significant - , is the largest and , is the second largest which areshown in Figure 7. For the two , in (a), their phase difference is 90.1 and the repetition 2. Then the angle 90.1/2 45.1which is almost exactly the difference between these two modes. An angle of 43.5 isobtained by comparing the angles of the two Zernike moments , in (b).The error 45 43.5 is due to thenumerical error during the discretization of the Zernike polynomials. Similar results are obtained for mode pair 4and 5 ( 30), 6 and 7 ( 90.1), etc.


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Figure 6:Reconstruc

omparisonted Mode Sh

Figu

etween Origapes (Free-F

re 8: Mode S

inal andree Disc)

hape Patter

Figure

and ZMD of

: The , an

(F

Mode 1 and

d , ZMD o

ree-Free)

2 (Free-Free

f Mode 1 an

)

2


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3.2 Rectangular Plate

Since the Zernike moments are defined over a unit circle, mapping to a unit circle should be carried out beforeapplying the ZMD for other structures, e.g. a rectangular plate, in order to preserve the complete shape featuresof the original structure. For example, a polygon shown in Figure 9 may be mapped into a circle by the followingdefinitions,

1. The centroid of the polygon is mapped to the centre of the circle;

2. The triangles formed by the centroid and two neighbouring vertices in the polygon are mapped to the sectorsin the circle;3. The angle of each sector is determined by the area percentage of the corresponding triangle in the polygon

as shown in Figure 10.

According to the definitions above, the polygon may be mapped onto a circle by the following equations [20],

(27)

(28)

A 20mm30mm2mm rectangular plate was modelled with 600 four-node quadrilateral elements and 651 nodes.The first 12 mode shapes are considered. The coordinate mappings were performed before applying the ZMD asshown in Figure 11. The ZMD based on the circular mode-shape patterns are shown in Figure 12 and 13 up toorder 8 which is sufficient for describing the first 12 modes according to the correlation between the original andreconstructed mode shape patterns as show in Figure 14.

Figure 9: Mapping Polygon to Circle Figure 10: Mapping Triangle to Sector

Furthermore, the symmetries of the mode shapes may also be indicated by the Zernike moments [20]. e.g. If theimage is x-symmetric, the imaginary part of the Zernike moment is zero and vice versa. i.e.

, , , 0, (29)If the image is x-antisymmetric, the real part of the Zernike moment is zero and vice versa. i.e.

, , , 0, (30)Therefore, the symmetry of the mode-shape pattern in Figure 11 may clearly be indicated by the Zernikemoments in the Figure 12 which are aligned exactly on the orthogonal x-y axes corresponding to the real andimaginary components of the Zernike moments.


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Figure 11: Mode Shapes (Rectangular to Circular) Figure 12: Zernike Moments of Rectangular Plate

Figure 13: Rectangular Plate - Amplitude of ZernikeMoments

Figure 14: Rectangular Plate - Correlation of Originaland Reconstructed Shapes

4. CONCLUSION

Vibration mode-shape recognition based on image processing and pattern recognition has been considered in this

paper. In particular, the Zernike moment descriptor was investigated and applied to example problems includingthe modes of circular and rectangular plates. Results demonstrate that the ZMD is especially powerful in thediscrimination of circular and spherical shapes. It may be a good supplement or even replacement of theconventional MAC for such structures because the ZMD not only provides the closeness between the predictedand measured mode shapes but also reveals the subtle difference between them. More general andcomprehensive shape recognition techniques are presently being developed and may be used together with finiteelement model updating especially for large and complicated structures.


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ACKNOWLEDGEMENTS

W. Wang wishes to acknowledge the support of an Overseas Research Studentship (ORS), a University ofLiverpool Studentship and an award from the University of Liverpool Graduate Association (Hong Kong).

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