IOMAC'5 6 th International Operational Modal Analysis Conference 5 May-4 Gijón - Spain COMPARISON OF MODE SHAPE VECTORS IN OPERATIONAL MODAL ANALYSIS DEALING WITH CLOSELY SPACED MODES. Olsen P., and Brincker R. Ph.D. student, Aarhus University, pto@eng.au.dk. Professor, Aarhus University, rub@eng.au.dk. ABSTRACT When dealing with operational modal analysis (OMA) the most difficult scenario to encounter is closely spaced modes. In the case of repeated roots the mode shapes often becomes inseparable because the two coupled modes rotate in the plane they span. When evaluating the correlation of mode shape vectors compared to e.g. analysis mode shape vectors the modal assurance criterion (MAC) is often used. The MAC value can be a crude measure when dealing with closely spaced modes. In this paper it is suggested to evaluate the coupled mode shape vectors as a pair and consider the subspace they span as the basis to compare. The angle between the two subspaces spanned by the two set of coupled mode shape vectors can be used as a measure for the correlation between the two set of mode shape vectors. The use of the different measures of correlation between mode shape vectors are evaluated through a numerical case study using simulated time data on systems with closely spaced modes. In the case study the OMA time domain technique Eigen Realization Algorithm (ERA) has been used to estimate the modal parameters. Keywords: Operational, Modal, Analysis, Subspace angle, Model validation. INTRODUCTION Dealing with estimation of modal parameters for systems with closely spaced modes represents a great challenge when estimating the mode shape vectors using OMA. Closely spaced modes are found in a lot of structures. The most general case is axi-symmetric structures where the modes appear in pairs of two. An example could be a cantilever beam where the bending modes will appear in sets one for each axis. It needs not to be axi-symmetric structures it can also be that two modes a bending mode and a torsion mode that are closely spaced. When estimating the modal parameters from experimental test the identification of a set of closely spaced modes results in estimates where the mode shape vectors are rotated relative to the mode shape vectors of e.g. analysis modes which makes the mode shape vectors difficult to separate.[] It shows that an estimated mode shape vector can be expressed as a combination of several analysis mode shape vectors.[] Instead of combining several analysis modes it is possible to use the subspace two closely spaced mode shape vectors span and
compare this to the subspace of two corresponding analysis mode shape vectors. When comparing mode shapes vectors the most common used measure has been the MAC value. The MAC value indicates how well two mode shape vectors correlate but the MAC does not handle the challenge of closely spaced modes very well.[3] The MAC-value often shows the two closely spaced modes to be correlated as a linear combination of the two analysis mode shapes. Therefore when dealing with closely spaced modes it is suggested to consider the subspace that the two modes shape vectors span and compare this to the subspace that two corresponding analysis mode shape vectors span. The angle between the two subspaces will provide information of correlation between the two set of mode shape vectors.. THEORY.. Correlation between mode shapes The modal assurance criterion (MAC) has been used for decades as a measure for the correlation between two set of mode shape vectors.[4] The MAC-value is unity for full correlation and zero for uncorrelated vectors. The definition of MAC is given in () where a i and b j are mode shape vectors. MAC(a i, b j ) = a i H b j a i H a i b j H b j The MAC value is a good indicator of the correlation between mode shape vectors. MAC-values above.8-.9 indicates that the mode shapes are correlated and below as uncorrelated. But the MAC-value is a coarse measure when considering the deviation between to mode shapes vectors especially considering high MAC values. Considering the angle α ij between two mode shapes compared to the MAC value increase the resolution of the deviation. An expression for the angle between two mode shape vectors is given in () α ij = cos ( MAC(a i, b j )) () The angle between the two mode shapes vectors α ij provides a measure for the deviation as long as the modes are well separated. When dealing with closely spaced modes considering the angle between two mode shape vectors by the use of () can results in large deviations caused by the mode shapes vectors rotate in the subspace spanned by the two coupled modes. In the case of closely spaced modes it is suggested that the modes are considered in coupled pairs. Instead of considering the angle between two mode shape vectors it is suggested to consider the angle between two subspaces spanned by the coupled set of mode shape vectors. Considering the angle between two subspaces it is important to distinguish between two angles one angle θ describing the rotation between the two set of vectors in the plane and one φ describing the deviation between the subspaces spanned by the two set of mode shape vectors see Figure. An expression for the rotation angle has been proposed for moderately closely spaced modes and for repeated roots. The expression uses perturbation theory based on the local correspondence principle. For moderately closely spaced modes an approximate solution is found. For repeated roots an eigenvalue problem is set up where the eigenvectors are transformation matrices describing the rotation angle.[] When considering the deviation between the two set of mode shape vectors the rotation angle θ is not of interest in this case it is the subspace angle φ that holds information. ()
a a b θ b b θ b φ a a Figure. The subspace angle φ between two subspaces spanned by two set of mode shape vectors can be used as a measure for the deviation of two set of closely spaced mode shapes. It is important to distinguish between the rotation angle θ and the subspace angle φ. The rotation angle can be used to align the mode shape vectors. The subspace angle φ is determined by establishing the orthonormal bases of the two subspaces and from these determine the subspace angle. The orthonormal bases are established by QR factorizations of the mode shapes matrices A and B. A = Q A R A B = Q B R B (3) Applying singular value decomposition to Q A and Q B results in. Y T Q A T Q B Z = diag(cos(φ k )) (4) Where Q A Y and Q B Z are the orthogonal matrices containing the principal vectors and φ k are the principal angles between the two subspaces spanned by respectively A and B. The dimension of the subspaces A and B must be the same and there are as many principal angles as the common dimension of the subspaces. For the deviation and more information see [5]. In OMA the measurement data is obtained in operational conditions from the response measurements the correlation functions are estimated. Physical information of the modes is extracted by interpreting the correlation functions as free decays. The basic idea in time domain techniques is to fit a model of the system to the correlation functions and solving this as a regression problem.[6] The difference in the time domain techniques is in the approach of solving this. In this paper the time domain identification technique ERA has been applied to estimate the modal parameters. The ERA technique was proposed by Pappa et.al.[7] and based on a state space formulation where an eigenvalue problem is formulated by building a Hankel matrix from the estimated correlation functions. The system matrix is estimated by singular value decomposition. An eigenvalue decomposition of the system matrix is done to extract the modal parameters. 3. NUMERICAL SIMULATION AND RESULTS The numerical simulations were performed as a study of a system with three degrees of freedom (DOF). In the study the frequencies for two of the modes were held constant at. Hz and 3. Hz whereas the frequency for the last mode was varied from.8 Hz to 3. Hz all three modes had a damping ratio of %. New mode shapes for all three modes was generated for each shift in frequency of the varying mode. Mode shapes vectors were created geometrically orthogonal and with random shape. To estimate the modal parameters the time domain technique ERA was used. The numerical
simulations were conducted by simulating responses by the use of the system modal parameters. The excitation of the system were white noise loading. The time series was added artificial noise as white noise at and of the rms value of the simulated signal. To reduce the effect of noise the first five points in the time window of the correlation function was removed and the time window was cut off at % of the maximum amplitude of the correlation function hereby reducing the noise tail. The time step for the simulated responses was set to s and the length of the time series was s. For each set of modal parameters the simulations were repeated times. The mean μ and the standard deviation σ of the absolute percentage error of the estimated frequencies are given in table for noise level, and. Table. The percentage error of the estimated frequencies from the numerical simulations. Where the mean and the standard deviation of the error is given. Mode The relative error of the estimated frequencies System Estimated ERA No noise Noise. Noise μ σ μ σ μ σ [%] [%] [%] [%] [%] [%] [.8 3.].8.4.8.4.9.4 3 3...... The correlation between the estimated mode shape vectors and the mode shapes vectors of the system were determined as the angle α ij given in () hereby building a matrix containing all combinations of the modes. In figure the angles are illustrated in a typical MAC-plot having the estimated modes in one axis and the analysis modes in the other axis plotting all values in the angle matrix in a bar plot. The values of the angle α ij are indicated by colors where α ij = indicates full correlation and α ij = π indicates no correlation between the mode shape vectors. In figure three cases are illustrated the case with well separated modes, closely spaced modes and repeated roots. 3 Mode : : 3:..9 3. 3 Mode : : 3:..99 3. 3 Mode : : 3:.. 3. 3 3 3 Well separated modes Closely spaced modes Repeated roots Figure. The angle between two mode shapes vectors α ij is plotted where the colors indicate the size of the angle. Dark blue (full correlation) α ij = and dark red (uncorrelated) α ij = π. In the three plots the estimated modes are on the y-axis and the system modes are in the x-axis. The values of α ij in the diagonals of the plots indicate the correlation between the mode shape vectors. Figure illustrates three set of modal parameters to illustrate the whole frequency range of the varying mode the diagonal of the matrix holds information of the correlation of the mode shape vectors. These are shown in figure 3 as three curves each representing the correlation of the mode shape vectors for the different modes. In figure 4 the angle φ between the subspace spanned by the mode shape vectors of mode (. Hz) and mode (varying frequency) has been plotted for three noise levels. In figure 5 the subspace angle between the subspaces spanned by the mode shape vectors of mode (varying frequency) and mode 3 (3. Hz) are shown.
.9.8.7 Mode : (Constant) f =. Hz.9.8.7 Mode 3: (Constant) f 3 = 3. Hz.6.6 α [rad].5 α 33 [rad].5 α [rad].9.8.7.6.5 Mode : (Varying) =.5 to 3.5Hz Figure 3. The diagonal terms of the matrix containing the mode shape angle α ii between two mode shapes vectors is plotted for all values of the second mode. Where the three graphs shows each entry in the diagonal corresponding to each mode.. Mode (Red),. Mode (Blue) and 3. Mode (green)..9.8 Noise.9.8 Noise..9.8 Noise.7.7.7.6.5.6.5.6.5 Figure 4. The maximum subspace angle φ between the subspaces spanned by the mode shape vectors of mode (. Hz) and (varying frequency) of the estimated modes and the system modes..9.8 Noise.9.8 Noise..9.8 Noise.7.7.7.6.5.6.5.6.5 Figure 5. The maximum subspace angle φ between the subspaces spanned by the mode shape vectors of mode (varying frequency) and mode 3 (3. Hz) of the estimated modes and the system modes.
4. DISCUSSION In this paper a numerical case study has been made to illustrate how subspace angles can be used as a measure of the correlation between mode shape vectors. The case study illustrates that when modes are closely spaced or have repeated roots the angle α between two mode shape vectors is not a useable measure of the correlation between the mode shape vectors. This can be seen from figure where the mode shape vectors of the system and the estimated modes in the case with well separated modes show good correlation. The systems with closely spaced modes or repeated roots shows the estimated mode shape vectors couples in pairs of the two modes resulting in a low correlation between the mode shape vectors. In figure 3 it can be seen that the angle between the mode shape vectors increase when the varying mode approaches the other two modes at. Hz and at 3. Hz but in the regions where modes are well separated the angle is close to. This increase in the angle is caused by the mode shape vectors rotate in the plane of the two closely spaced mode shape vectors. The rotation can be described by the rotation angle θ but this angle holds no information about the correlation of the two set of mode shape vectors. To describe the correlation between the mode shape vectors of the closely spaced modes it is suggested to consider the subspace spanned by the two closely spaced mode shape vectors. Considering the angle φ between the subspace spanned by the estimated mode shape vectors and the subspace of the mode shape vectors of the system gives a measure for the correlation between the two set of mode shape vectors. In figure 4 the subspace spanned by the mode shape vectors of mode (. Hz) and mode (varying frequency) has been consider and the angle φ between the subspace of the estimated modes and the subspace of the system modes has been plotted. The plots show that when the two modes are closely spaced the subspace angle is close to indicating that the set of modes are correlated. When mode (varying) and mode 3 (3. Hz) are closely spaced using the subspace of mode and mode results in a large angle between the subspaces indicating that the mode shape vector of mode has been rotated in the plane of mode and mode 3. The same behavior as in figure 4 is confirmed by figure 5 where the subspace angle between the subspaces of mode and mode 3 of the estimated mode shape vectors and the system mode shape vectors are shown. To conclude using the subspace angle between two subspaces spanned by two sets of corresponding mode shape vectors is a good indicator of the correlation between the mode shape vectors when dealing with systems that have closely spaced modes or repeated roots. But the measure should only be used in the region of the mode shape vectors that span the subspace. REFERENCES [] Brincker, R.,& Lopez-Aenelle, M. (5). Mode shape sensitivity of two closely spaced eigenvalues. Journal of Sound and Vibration, 334, p. 377-387. [] Walther H., et.al. (4) Model Correlation with Closely Spaced Modes. Procedings of the nd International Modal Analysis Conference, Dearborn, Michigan [3] D Ambrogio, W.,& Fregolent, A. (3). Higher-order MAC for the correlation of close and multiple modes. Mechanical Systems and Signal Processing, 7(3), p. 599 6 [4] Allemang, R. J.,& Brown, D.L. (98). A Correlation Coefficient for Modal Vector Analysis. Procedings of the st International Modal Analysis Conference, (pp.-6), Orlando, Florida [5] Golub G.,& Van Loan C.(996) Matrix Computations 3 rd edition., Section.4, John Hopkins University Press [6] Brincker R.,& Ventura C. (5) Introduction to Operational Modal Analysis., Wiley. [7] Juang J.N.,& Pappa R.S. (985) An eigen system realization algorithm for modal parameter identification and modal reduction. J. Guidance, V. 8, No. 5, p. 6-67.