Journal of Vibration and Control, 8 (1) 19-40, 2002.

Enhanced Proper Orthogonal Decomposition for the Modal Analysis of Homogeneous Structures

S. HANDivision of Mechanical Engineering, Kyungnam University, 449 Wallyoung-dong, Masan, 631-701, Korea

B.F. FEENYDepartment of Mechanical Engineering, Michigan State University, 2555 Engineering Building, East Lansing, MI 48824 U.S.A.

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Abstract: Proper orthogonal decomposition (POD) is studied in an effort to increase its applicability as a modal analysis tool. A modification is proposed to make better use of spatial resolution, and to accommodate arbitrary spacing in the discretization. The theory for this modification is rooted in the discrete approximation of the integral orthogonality condition for continuous normal modes. The modified POD is applied to a finite element beam and an experimental beam sensed with accelerometers, and the resulting proper orthogonal modes (POMs) are compared to the theoretical modes of the beam. The POMs are used as a basis for decomposing the signal ensemble into proper modal coordinates. The proper modal coordinates are used to evaluate the POMs, and to match modes with modal frequencies and damping.

Key Words: Proper orthogonal decomposition, proper orthogonal modes, linear normal modes, modal analysis, Karhunen-Loeve decomposition

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1. INTRODUCTION

Mode shapes, modal frequencies, and modal damping are important quantities in structural vibrations. Experimental modal analysis has widely been used when such information is needed. The method is simple yet powerful in extracting various vibration properties of the structure, such as natural frequencies, modal damping values, and mode shapes of the structure. Recently, Feeny and Kappagantu (1998) showed that proper orthogonal decomposition (POD) can be a supplement for modal analysis of homogeneous structures when only the response measurements are available. This paper reports on further developments of POD as a modal analysis tool.

The development of proper orthogonal decomposition was traced back by Lumley (1970) to independent investigations by Kosambi in 1943, Loève in 1945, Karhunen in 1946, Pougachev in 1953, and Obukhov in 1954. It is primarily a statistical formulation, although it facilitates modal projections of partial differential equations into reduced-order deterministic models. The discrete formulation resembles singular value decompostion. A more thorough history of POD in this context is presented by Ravindra (1999).

POD was applied to turbulence by Lumley (1967), and recently has received considerable attention from structural dynamicists. A common application of POD to structures typically involves the sensed displacements, x1(t), x2(t), …, xM(t), at M locations on the structure, and the construction of a correlation matrix [R]. (The method is also amenable to sensed velocities (Georgiou and Schwartz, 1996; Kreuzer and Kust, 1996).) The eigenvectors of [R] are the proper orthogonal modes (POMs), and the eigenvalues are the proper orthogonal values (POVs). In the analysis of turbulence, the POMs have been shown to represent the optimal distributions of kinetic energy or power, and the proper orthogonal values (POVs) indicate the power associated with these principal distributions (Berkooz et al., 1993; Cusumano et al., 1993). Alternatively, the POMs represent the principal axes of inertia of the data in the measurement space, while the proper orthogonal values indicate the mean squared values of the data in each axis (Feeny and Kappagantu, 1998).

POD has been useful in uncovering spatial coherence in turbulence (Lumley, 1970; Lumley, 1967; Berkooz et al., 1993) and structures (Cusumano and Bai, 1993;

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Cusumano et al., 1993), determining the number of active state variables in a system (Berkooz et al., 1993; Cusumano and Bai, 1993; Cusumano et al., 1993), and in uncovering modal interactions (Davies and Moon, 1997; Georgiou et al., 1999). Proper orthogonal modes have been treated as empirical modes for discretizing nonlinear partial differential equations by means of Galerkin projections in turbulence (Berkooz et al., 1993) and in structural dynamics (FitzSimons and Rui, 1993; Ma and Vakakis, 1999; Ma et al., 2000; Kappagantu and Feeny, 2000), and also for system identification (Yasuda and Kamiya, 1997; Lenaerts et al., 2000).

Cusumano and Bai (1993), Davies and Moon (1997), and Kust (1997) observed that the POMs in particular nonlinear structures resembled the normal modes of the linearized system. A recent analysis has shown the the POMs may indeed converge to linear normal modes in multi-modal free responses of symmetric lumped-mass linear systems, but only if the mass matrix is proportional to the identity matrix (which can be achieved by a coordinate transformation if the mass distribution is known) and if the system is not too strongly damped (Feeny and Kappagantu, 1998). This provides a fundamental tie between the statistically derived POMs and the geometrically based linear normal modes in certain discrete systems. This relationship has been extended to one-dimensional self-adjoint continuous systems. If the mass distribution is known, and the discretization is uniformly spaced, then the POMs converge to an approximation of the discretized normal modes (Feeny, 1997). The approximation is rooted in the discrete approximation of the integral orthogonality condition for the normal modes. The theory was tested on an experimental beam sensed with strain gages (Riaz and Feeny, 1999). The work left a question as to the effect of basis functions, used to convert strains to displacements, on the POD results.

Thus, the door is open for POD to be developed as an alternative to traditional modal analysis. The advantages of POD are that no force sensors are needed. A single free response test suffices. The disadvantages are that the mass distribution must be known, the discretization must be uniform, and a method for associating the modes with frequency and damping information is absent. These issues must be tackled for POD to fully realize its potential as a modal analysis tool.

In this paper we address some of these issues. Our focus continues with homogeneous structures. We extend the application of POD to non-uniform discretizations. The purpose is not to advocate discretizing nonuniformly by choice, rather to adapt the POD

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method to overcome a nonuniform discretization if there is no choice. We test the theory on finite element simulations, and also on an experimental beam sensed with accelerometers. We examine impulse responses, and consider the responses to have many modes contributing. Some auxiliary issues are faced along the way, such as the correlation between mode shapes and frequencies, and the effects of accelerometers in experiments.

The next section contains a brief comparison between the traditional modal analysis and POD formulations.

2. MODAL ANALYSIS AND PROPER ORTHOGONAL DECOMPOSITION

The starting point of the conventional modal analysis is based on the fact that the harmonic response of one of the coordinates of the structure, , caused by a single harmonic force applied at a different coordinate, , can be measured, and that the measured frequency response functions (FRFs) can be expressed in terms of modal properties of the structure as follows (Ewins, 1984):

(1)

where is the FRF representing the harmonic response at point due to input force at point on the structure, is the eigenvalue of the th mode with natural frequency and damping factor incorporated, is the th element of the th eigenvector (i.e., the relative displacement at that point during vibration in the th mode), and is the number of degrees of freedom. Manipulations of the Hjk() can be performed to obtain the mode shapes and associate them with the modal frequencies and damping factors.

On the other hand, proper orthogonal decomposition (POD) is a procedure for simply extracting a basis for a modal decomposition from an ensemble of signals measured from the vibrating structure (Berkooz, et al., 1993). If many modes are excited, it is conceivable that these modes will contribute to this extracted basis. In structures, it is common, but not necessary, to base the POD on displacement measurements ui(tj) at location i, sampled at times tj. POD is performed on the correlation matrix [R], defined as

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(2)

where the ensemble matrix is constructed as

(3)

Thus, each column of the ensemble matrix represents the time history of

displacement (or velocity or acceleration) measured at some point of the structure. Here, represents the number of measurement positions and represents the number of samples of the measured signal. Each element in R in equation (2) represents the expected value of ui(t)uj(t), which is a the cross correlation with no delay. If the means were subtracted from the signals (which we did not do), then the matrix might better be divided by N-1, the statistical degree of freedom, such that each element of R represents a covariance. Indeed, the theory relating POMs to LNMs was derived without subtracting the means, but converges as conditions that are associated with zero mean responses are approached. From a structural point of view, the interpretion the POMs starts with the expansion theorem. The response of a structure can be approximated as a truncated linear sum of normal modes of that structure as

where is the th normal modal coordinate, is the th normal mode, and is the number of modes to be considered, chosen in this case to be the same as the number of displacement measurement signals. As such, the ensemble matrix can be written in the form

(4)where is the modal matrix, and [Q] is an matrix whose column

vectors consist of time history data of the principal coordinates measured at times .

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Since is real and symmetric, its eigenvectors form an orthogonal basis, such that

(5)

On the other hand, the normal modes of the structure have the orthogonality defined as

(6)

In other words, each normal mode of the structure and the corresponding POM differ depending on the mass distribution of the structure, and the POMs might be considered as the normal modes of the structure only when the mass distribution of the structure is homogeneous. This difference between the orthogonality conditions of the normal modes and the POMs is the main obstacle in utilizing the simplicity of the POD in conjunction with the structural vibration analysis. This restriction of the POD can be overcome only with knowledge of the mass distribution of the structure under consideration (Feeny and Kappagantu, 1998; Feeny, 1997).

3. POMS AND THE LINEAR NORMAL MODES

In this section we review the theory that shows that, under certain conditions, the POMs derived from discrete measurements on one-dimensional continuous systems converge to an approximation of the linear normal modes. We then modify the decomposition technique to overcome limitations on the discretization.

3.1. Uniform Self-Adjoint Systems

A continuous one-dimensional self-adjoint system of the form m(x)2y/t2 + Loy = 0 has modal functions which satisfy an orthogonality condition given as

(7)The system can be written as 2u/t2 + Lu = 0 under the coordinate transformation u(x,t) = m(x)1/2y(x,t), where L = m(x)-1/2Lom(x)-1/2 is self adjoint. Thus, if the mass distribution is known, an equivalent system can be modeled with a uniform mass distribution. The orthogonality condition for normalized modal functions for the system with uniform mass has the form

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(8)When we discretize the modal functions with several mesh points along the structure, it is intended to approximate each mode shape function, , with modal vector, , defined as

(9)

If the resolution of discretization is fine enough to truly represent the continuous mode shape functions with discretized vectors, then each modal vector should approximately satisfy the following orthogonality condition which is the discretized version of the orthogonality condition given in equation (8). As such,

(10)

where is the total number of mesh points and is the spatial interval between each mesh point for the approximate integration of equation (7), and is given as following

diagonal matrix:

(11)

Obviously, the error involved in the discretized orthogonality condition in equation (10) can be minimized by increasing the number of mesh points. But instrument availability usually limits the number of mesh points which is the number of response measurement positions in the actual test situation. For the case of uniform spacing, the error associated with the rectangular integration representation of the underlying orthogonality integral is on the order of kx3 where k is proportional to a characteristic curvature in the integrand (Forsythe et al., 1977). In general, the locations of response measurement positions along the structure depend on the choice of the mesh points of the structural elements. Figure 1 shows three different cases of the locations of response measurement positions with respect to the mesh points of one-dimensional structural elements.

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< Fig. 1> Figure 1 (a) is the case where the response measurement positions are located at the mid-point of evenly spaced intervals. In this case, the number of response measurement positions is same as the number of the discretized structural elements and less than the number of mesh points by one. When the rectangular rule is used for the approximate integration in Eq. (10), the orthogonality condition of the modal vectors simply becomes

(12)

with appropriate normalization. Therefore, modal vectors of homogeneous structure can be considered to be orthogonal to each other only when the response measurement positions are evenly spaced at the mid-point of the structural elements.

3.2. Incorporating Enhanced Quadrature When the response measurement positions are located at the midpoints of structural elements of equal length as in Figure 1(a), the midpoint rectangular integration scheme can be used, for which the weighting matrix is [h]=x[I], and Eq. (10) becomes proportional to Eq. (12).

When the response measurement positions are located at the mesh points of the structural elements of equal length as in Figure 1 (b), the approximation of the orthogonality condition given in Eq. (10) can have a different form depending on the integration scheme used. When the rectangular rule is used, either one of the endmost mesh points should be neglected in descretizing the mode shape function depending on whether backward or forward rectangular integration is used. When the discretized points are evenly spaced, the approximation of the orthogonality condition given in Eq. (10) might be improved by applying a better integration scheme such as Simpson’s rule. With Simpson’s rule, the weighting matrix [h] is diagonal. For the case of figure 1(b), the diagonal terms of [h] have the form h11 = x/3, h22 = 4x/3, h33 = 2x/3, h44 = 4x/3, h55 = x/3.

If the number of the mesh points is even or the mesh points at which the response measurement positions are located are not evenly spaced as in Figure 1(c), then we

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might choose the trapezoidal rule as an integration scheme. The weighting matrix [h] is diagonal. For the discretization shown in Figure 1(c), the diagonal elements of [h] are h11 = x1/2, h22=(x1 + x2)/2, h33=(x2 + x3)/2, h44=(x3 + x4)/2, h55 = x4/2. Higher-order quadrature is also possible. When Simpson’s rule or the trapezoidal rule is used for the approximate integration, the matrix in Eq. (10) is not proportional to the identity matrix, and the approximated modal vectors are now orthogonal with respect to the matrix instead of being

orthogonal to each other. In contrast to the lumped mass model of the structure in which the effect of the spacing between mesh points are reflected in the magnitude of the lumped masses, the orthogonality condition of discretized mode shape functions depends on the spacing between the mesh points along the structure. This implies that the effect of the spacing of mesh points should be considered in discretizing the mode shape functions of the structure in the vibration analysis of homogeneous structures.

3.3. Proper Decomposition for Arbitrary Integration Schemes POD, in its raw form, has been shown to generate POMs which approximate linear normal modes in uniform systems with evenly spaced discretizations (Feeny, 1997). It effectively incorporates a rectangular integration scheme with uniform spacing (and thus [h] = x[I]). Here, we modify the decomposition procedure to accommodate other integration schemes. The modification will allow us to overcome a situation in which the discretization is unevenly spaced, and it will provide hope that higher-order integrations may improve the effects of finite resolution.

When the elements of the matrix are chosen such that they reflect the uneven spacing

of the mesh points or the approximation of integration using the trapezoidal rule or Simpson’s rule, the POD should be performed on the nonsymmetric “weighted correlation matrix” instead of on the correlation matrix . Letting [U] = [Q]

[]T, where [Q] is a discrete approximation to the normal modal coordinate ensemble and []T is the matrix of discretized normal modes, we have

(13)

Eq. (10) can be rewritten in the matrix form as follows: (14)The error in this approximation is due to discretization and is O[(x/l)3], where x is a

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characteristic interval size. Eq. (13) is pre-multiplied by and post-multiplied by to give

(15)If the natural frequencies of the structure are well separated to ensure that the correlation between the principal coordinates converges to zero during the measured time duration, then the right hand side of Eq. (15) becomes a diagonal matrix whose elements are the mean squared values of corresponding principal coordinates. Therefore,

(16)

where qii is the mean squared value of the modal coordinate qi(t). If there is strong modal damping, diagonal terms may diminish as N gets large. Eq. (16) indicates that the eigenvalues of the adjusted correlation matrix are the mean squared values of

the principal coordinates and the eigenvectors are the modal vectors of the structure in the approximate sense. These eigenvalues, or POVs, can be used to rank the modes according to modal activities rather than frequencies. Since the adjusted correlation matrix is not symmetric, its eigenvectors are not directly orthogonal to each other.

However, they are orthogonal with respect to matrix [h] as in equation (14), and hence will still be referred to as “proper orthogonal modes,” although “orthogonal” is slightly generalized here. We note that the linear normal modal vectors, obtained by discretizing the modal functions according to an arbitrary spacing, are also nearly orthogonal with respect to [h].

3.4. Proper Modal Coordinates Once the POMs are extracted, the measured response signals can be decomposed into the proper modal coordinates using the POMs as follows: (17)Here, is the modal matrix composed of extracted POMs, and the columns of [Z] are

the sampled proper modal coordinates. The approximation in equation (17) is rooted in the estimation of [] by [], and involves discretization errors of O[(x/l)3] and finite N convergence errors.

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If the POMs were identical to the modal vectors (discretized modal functions), which can currently be approximated for homogeneous structures, then each decomposed proper modal coordinate would be of pure modal content, and would thus contain a single modal frequency (and damping factor if the damping is light and modal). Conceivably, the decomposition of the ensemble into the proper modal coordinates might then be used to obtain frequency and damping information.

Since, in reality, the POMs are not exactly the normal modes of the structure, these decomposed proper modal coordinates will have several other frequency components. But the relative order of the frequency components can be used as a guide to determine which modal component is dominant in the structural responses, and therefore identify POMs with modal frequencies.

Suppose the POMs differ from the LNMs, such that i = i + e i . Then u(t) = [z(t) = [ 1 + e 1, 2 + e 2, …, M + e M] z(t), where z(t) is a vector of proper modal coordinates. By orthogonality of the POMs, i.e. equation (14), z(t) = [T[h]u(t), whence zi(t) = i

T[h]u(t) + e iTu(t) = q i(t) + e i

Tu(t).

The error vector ei can be written in terms of linear normal modes as e i = d1 1 + … + dM M. Hence, zi(t) = qi(t) + d1q1(t) + …+ dM qM(t). Thus the proper orthogonal modal coordinate zi is polluted by other normal modal coordinates depending on the extent of deviation of the POM from the LNM, represented by dj, as well as the relative strengths of linear normal modal components qj(t). Small deviations dj lead to small impurities in zi(t), while large modal coordinate dynamics qj(t) increase the impurity of zi(t).

4. NUMERICAL SIMULATIONS AND EXPERIMENTS

We arranged numerical simulations and experiments for a simple beam, since theoretical modes are then available for evaluating the decomposition results. The boundary conditions of the beam were approximately free-free. Ewins (1984) recommends nearly free-free structures, when possible, for motility measurements.

4.1. Finite-Element Model

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A finite element model was constructed to match the experimental steel (E = 2.0e11 Pa) beam, described later, and to address the practical aspects of the POD method. The finite element model of the test beam was constructed with 20 identical one-dimensional Euler beam elements having 21 mesh nodes with 21 displacement coordinates and 21 rotational coordinates. The effects of the locations of the measurement positions along the structure on the extracted POMs can be found by measuring the responses of selectively chosen node points of this model. To simulate a nearly free-free beam, and at the same time, to prevent the stiffness matrix from being singular, soft linear springs were attached at both sides of the free end. The stiffness of these springs was k = K11/1000, based on the stiffness matrix [K] of the finite-element model. This replaces the rigid body mode of the beam by very low frequency flexural modes with little effect on the natural frequencies and the mode shapes of the actual flexural modes of interest. Natural frequencies obtained from the eigensolution of the finite-element beam, and those extracted from the proper modal coordinates of an impulse response, are shown in Table 1. (The table omits the very-low frequencies that emulate rigid body modes.) For comparison, the first six theoretical non-rigid-body natural frequencies of the free-free Euler beam are also shown. < Table 1 > The beam model had relatively light viscous damping elements to prevent the modes from decaying out quickly. The damping factors shown in Table 1 were achieved by applying a damping matrix in the form [C] = 3.47e-7[K]. The finite element beam model was then subjected to a half-sine type impulse at one of the free ends (node #1). The duration of the impulse controls the frequency contents of the responses and the number of modes to be activated. Only the responses of the displacement coordinates were calculated using 4th order Runge-Kutta direct integration method. The numerical integration step was set to equal to the sampling interval of 1/2048 second which is the same as in the experimental case, and the resulting Nyquist frequency accommodates the first 7 modes of the beam. The responses of the finite element model were then high-pass filtered to eliminate unwanted rigid body modes.

4.2. Experimental Set-up

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A 12.7 12.7 1500 mm uniform steel beam was prepared to extract POMs from the experimental response data. To achieve nearly free-free boundary conditions, the beam was hung horizontally with two nylon strings attached at the ceiling, such that the flexural displacements were horizontal, and the vibration responses in the horizontal direction were measured under an impact in the same direction. The strings were 1 m long and had a thickness of 1 mm. Six equally spaced accelerometers were attached along the length of the beam to measure the acceleration response of the beam. The accelerometers were from two different manufacturers. Each accelerometer was carefully calibrated using a standard back to back method to find its sensitivity (Harris and Crede, 1996). Two of the accelerometers were a little bit heavier than the others. The ratio of the heaviest accelerometer to the mass of the beam was less than 3%. To minimize the mass effect of these relatively heavier accelerometers on the response of the beam, these two heavier accelerometers were located at the mid portion of the beam corresponding to node numbers 3 and 4. The small deviation of weights and time constants of accelerometers were considered as an actual error source that can exist in practice. The dimensions, model numbers, and manufactures of the accelerometers used in the experiment along with the calibration results are given in Table 2. < Table 2 > Beam responses were measured by striking the beam with an impact hammer at one free end, presumably activating all the modes of the beam within the range of frequencies of interest. The first six flexural modal frequencies of the beam are listed in Table 1. The number of modes of the beam to be detected can be controlled by either adjusting the duration of the impact or adjusting the Nyquist frequency of the measurement. All accelerometer signals were simultaneously fed into the B&K 2035 FFT signal analyzer with 8-channel input module. Therefore, each signal represented the response of the beam subjected to the same external disturbance. B&K 2035 FFT signal analyzer captured the time domain signals with 2048 sampling points and converts them to various time and frequency domain functions with 801 discretized points. Since the number of measurement points was six, only six POMs were extracted. But, the preset Nyquist frequency value of the analyzer captured up to the 7th mode of the beam in the frequency domain. The experimental set-up and measurement positions along the beam

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are sketched in Figure 2. < Fig. 2 >

5. RESULTS AND DISCUSSIONS

5.1 Finite-Element Results

The first six POMs extracted from numerically generated responses of 10 node points of the finite element beam model are compared with the theoretical normal modes of the free-free Euler beam in Figure 3. The response measurement positions are chosen to represent the mid-point of 10 identical beam elements as depicted in Figure 1 (a), which correspond to the even numbered nodes among 21 node points. Since the rectangular rule is used in the integration of discretized orthogonality condition, the POD is directly performed on the correlation matrix. The extracted POMs nicely represent the theoretical normal modes of the beam even for the highly fluctuating higher modes. The results of this figure suggest that POD, which is much simpler to implement than the conventional modal analysis, can be an effective tool for the vibration analysis of homogeneous structures. In structural vibration analysis, it is usually convenient to choose the mesh points of the structural elements as the response measurement points to compare the experimental results with the theoretical calculations. When the response measurement positions are located at the mesh points of the structure, the POD should be performed on the adjusted correlation matrix in which the discretized integration scheme such as the trapezoidal rule or Simpson’s rule is reflected. Responses of only the odd numbered node points of the finite element beam model were used to construct the correlation matrix to find out the effect of the discrete integration scheme on the extracted POMs. POMs in Figures 4 and 5 are extracted from the adjusted correlation matrix each of whose matrices is based on the trapezoidal rule and the Simpson’s rule,

respectively. < Fig. 3 > < Fig. 4 > < Fig. 5 > There is little difference among these two cases as long as the lowest two modes are concerned. However, for the next four modes, the quality of the extracted POMs with

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respect to the theoretical normal modes of the beam becomes different depending on the type of the matrix on which the POD is performed. Comparing these two cases of POMs reveals that using a better integration scheme for the approximation of the orthogonality condition given in Eq. (10), which is Simpson’s rule in this case, provides POMS that are closer to the theoretical normal modes of the structure. The norms of the vector differences between the theoretical normal modes and the extracted POMs corresponding to Figures 3, 4 and 5 are compared in Table 3. The results of the table as well as the results of Figures 3, 4 and 5 indicate that locating the response measurement positions at the mid-point of evenly spaced mesh points gives overall more satisfactory results than locating them at the mesh points of the elements. < Table 3 > The quality of the extracted POMs with respect to the normal modes of the structure is also examined by decomposing the response signals into the proper orthogonal modal coordinates. Figures 6, 7 and 8 represent the time histories and their power spectral density functions of decomposed modal coordinates up to the sixth mode using the POMs corresponding to those of Figures 3, 4 and 5, respectively. The trend of deterioration of higher modes with respect to the true normal modes of the beam is clearly indicated by the additional modal components appeared on the power spectral density functions of decomposed modal signals for all of the three cases. But the monotonic purity of the decomposed modal coordinates in the lower modes follows the order of the quality of the approximation of the orthogonality condition, as is the case for the mode shapes. The quality of the first two modal coordinates in Figure 8 is impressive considering the fact that the modal matrix composed of the extracted POMs is only the approximation of the true modal matrix of the beam. The natural frequencies as well as the modal damping values of the beam can be estimated with additional processing of these decomposed modal coordinates. < Fig. 6 > < Fig. 7 > < Fig. 8 > The effect of non-uniform spacing of the response measurement positions on the extracted POMs is shown in Figure 9. The POMs in this figure are extracted based on the ensemble matrix built with only 9 responses of the finite element beam model by ignoring the responses of even numbered node points as well as those of node number 3

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and 5. In this case, the trapezoidal rule is the best choice for the approximation of integration in Eq. (10) because of the uneven spacing of the response measurement positions. The close match in lower modes and deviation in higher modes from the normal modes of the beam is almost same as the cases of the uniformly spaced node points. Close examination of these POMs indicates that the discrepancies between the shape of the POMs and the theoretical normal modes for the higher modes grows bigger toward the left side of the beam where the spatial resolution is worse than the other side of the beam. < Fig. 9 > The main reason for the discrepancies between the POMs and the normal modes of the beam for higher modes is considered to be the poor spatial resolution. Therefore, it is expected that if the number of measurements along the beam is increased, then the POMs will converge to the true normal modes of the beam. Figure 10 represents the POMs extracted from the responses of all 21 node points of the beam using Simpson’s rule as an approximate integration. This figure, in comparison with Figure 5 (the case of eleven node points), illustrates the benefit of increased resolution. In this case, even the higher POMs match nicely with the normal modes of the beam. This indicates that the POD can accurately extract numerous modes of the structure contained in the response signals as long as the number of measurement positions is sufficient. Unfortunately, it is not easy to measure many responses simultaneously in a structure except in some very special cases. It might be possible, however, to conduct sets of experiments and separately measure two-point correlations with two transducers and thus build the correlation matrix, as done by Cusumano et al. (1993). In this case, however, special care should be provided to ensure that each test is carried out with identical input disturbances. < Fig. 10 > If the number of the response measurement positions is reduced to six, as will be the case for the experimental measurement, it turns out that the number of response measurement positions is not enough to adequately approximate the higher modes. In this case, only the lowest two modes can be considered as the true normal modes of the beam as shown in Figure 11. The fifth and sixth POMs of this figure appear to be totally irrelevant to the normal modes of the beam. This indicates that the POD with poor spatial resolution cannot extract all the modal components of the structure from the response signals due to the deterioration of the discretized orthogonality conditions

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given in Eq. (10). Therefore, it is practical to extract only a minimum number of modes of the structure using the POD scheme when a limited number of response measurement positions on the structure are available. < Fig. 11 > 5.2 Experiments Because the experiment was sensed by only six accelerometers, we used the POD modified according to Simpson’s rule in effort to reduce the effects of limited spatial resolution.

Since the eigenvalues of the correlation matrix are simply the mean squared values of the ensemble data (which is response dependent) in the direction of the individual modes extracted, the order of the POMs and the order of the normal modes of the structure can be quite different. Usually, in free structural vibration, the higher frequency modes damp out more quickly than the lower frequency modes. So we might expect the mean squared modal amplitudes to decrease as the modal frequency increases. As we used an acceleration ensemble, our concern is mean squared modal accelerations, which are altered from mean squared displacements by a factor of . This can change the order of the POMs from the otherwise expected order. (This effect was also suspected by Benedettini, 1999, in his work). Measurements based on lasers, probes, or strain gauges would not cause the same frequency scaling effect on the ordering of the POMs as do the accelerometers. However, accelerometers are simple and inexpensive, and so it may be worthwhile confronting this issue. Strain gauges bring forth uncertainties associated with converting a sampled strain field to displacements (Riaz and Feeny, 1999).

Examining the peaks in the power spectral density functions of six acceleration responses of the beam, it was found that the order of the power of each mode was reversed from the usual order expected from displacement signals. In this case, the power of the 7th mode was the largest among the modes contained in the acceleration response (the seventh flexural modal frequency was 713 Hz). The result of linking the POMs with the normal modes of the beam in the reverse order from the 7th mode is shown in Figure 12. As can be seen in this figure, the first two POMs nicely match the 7th and 6th modes of the beam. The poor agreement between the POMs and the other normal modes of the beam, except the first two modes ranked by the proper orthogonal

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values, is probably due to the propagation of error caused by the lack of spatial resolution, as explained with the results of the finite-element beam model. The result of this experiment suggests that the POD scheme can extract even a very high mode of the structure as long as the modal contribution of that mode to the response is significant. But, even though higher modes of the structure can be extracted using POD, it is impractical to express modes of such spatial fluctuation with limited number of node points. < Fig. 12 >

Considering the fact that lower modes of the structure are usually required in the preliminary stage of structural vibration analysis, it is necessary to extract lower modes of the beam effectively from the experimental data. Since the change in the order is due to the term in the acceleration signal, one possible choice is to integrate the acceleration signals into displacement signals, thus changing the modal contribution of each mode. Integration is not trivial and accompanies drift if done in the time domain, and leakage if done in the frequency domain. Another effective way to selectively extract the lower modes of the beam is being tested and will be reported soon.

6. CONCLUSIONS

When vibration responses of a continuous homogeneous structure are simultaneously measured at different locations, POD can be applied to extract the mode shapes of the structure without measuring the full series of the FRFs. The required conditions are that the response signals contain the normal modes to be extracted, the mass distribution be known, and the number of measurement positions provide adequate spatial resolution. For POD to be realized as a modal analysis tool, the two latter limitations must be overcome. Previously, an evenly spaced discretization was also needed (which may be preferred but not always available). We have proposed a modification to POD to overcome this spacing issue. Through this modification, improvements in the modal approximations are enabled by effectively enhancing the numerical integration of the modal orthogonality condition, which underlies the theory associated with the interpretation of the POMs.

We have applied the modified POD to a finite-element model of a beam, and also an experimental beam, with nearly free-free boundary conditions, and compared the results

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to the theoretical free-free modes. Examples of the effects of incorporating Simpson’s rule and the trapezoidal rule of integration into the modified POD were presented. Incorporating Simpson’s rule improved results when the discretization was coarse. Furthermore, we showed examples of the recovery of modes in the presence of an uneven discretization by incorporating a trapezoidal integration. The experimental beam, sensed with six accelerometers, was able to yield two modes accurately, and two qualitatively, with the application of POD based on Simpson’s rule.

Performing POD on acceleration signals can produce an unexpected ordering of modes due to the multiplication of frequencies squared to the modal components of the signals. The strongest modal signals produce the best approximations of mode shapes, regardless of the modal order in terms of frequency. The experimenter may not necessarily obtain the best approximations for the lowest modes, as is often desired.

Using the extracted POMs, proper modal coordinates of the structural responses can be decomposed to evaluate the quality of the POMs. Better modal approximations should produce modal coordinates with stronger single-frequency dominance. The dominant frequency in the proper modal coordinate allows a POM to be associated with the appropriate modal frequency. It is possible that the modal damping values can also be estimated from these decomposed proper modal coordinates.

While this work has been aimed at reducing the current limitations on POD as a modal analysis tool, the limitation of a homogeneous structure (or equivalently a structure with a known mass distribution) remains. So, although POD is applicable to a class of structures, it is currently not applicable to an arbitrary structure. Future studies should address this issue.

Acknowledgement: S. Han is grateful of support from Kyungnam University. B. Feeny is grateful of support from the National Science Foundation (CMS-9624347).

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REFERENCES

Benedettino, F., 1999, personal communication, at the ASME Design Engineering Technical Conferences, September.

Berkooz, G., Holmes, P., and Lumley, J. L., 1993, “The proper orthogonal decomposition in the analysis of turbulent flows,” Annual Review of Fluid Mechanics 25, 539-575.

Cusumano, J. P. and Bai, B. Y., 1993, “Period-infinity periodic motions, chaos and spatial coherence in a 10 degree of freedom impact oscillator,” Chaos, Solitons and Fractals 3(5), 515-535.

Cusumano, J. P., Sharkady, M. T., and Kimble, B. W., 1993, “Spatial coherence measurements of a chaotic flexible-beam impact oscillator,” in Aerospace Structures: Nonlinear Dynamics and System Response, ASME AD-Vol. 33, pp. 13-22.

Davies, M. A. and Moon, F. C., 1997, “Solitons, chaos, and modal interactions in periodic structures,” in Nonlinear Dynamics: The Richard Rand 50th Anniversary Volume, A. Guran, ed., World Scientific, Singapore, pp. 119-143.

Ewins, D. J., 1984, Modal Testing: Theory and Practice, Research Studies Press, Letchworth, Hertfordshire, England.

Feeny, B. F., 1997, “Interpreting proper orthogonal modes in vibrations,” in Proceedings of the ASME DETC, Sacramento, CA, CD-ROM.

Feeny, B. F. and Kappagantu, R., 1998, “On the physical interpretation of proper orthogonal modes in vibrations,” Journal of Sound and Vibration 211(4), 607-616.

Fitzsimons, P. and Rui, C., 1993, “Determining low-dimensional models of distributed systems,” in Advances in Robust Nonlinear Control Systems, ASME DSC Vol. 53, pp. 9-15.

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Forsythe, G. E., Malcolm, M. A., and Moler, C. B., 1977, Computer Methods for Mathematical Computations, Prentice Hall, Englewood Cliffs, NJ.

Georgiou, I. T. and Schwartz, I. B., 1996, “A proper orthogonal decomposition approach to coupled mechanical systems,” in Nonlinear Dynamics and Controls, ASME DE-Vol. 91, pp. 7-12.

Georgiou, I. T., Schwartz, I. B., Emaci, E., and Vakakis, A. F., 1999, “Interaction between slow and fast oscillations in an infinite degree-of-freedom linear system coupled to a nonlinear system: theory and experiment,” ASME Journal of Applied Mechanics 66(2), 448-459.

Harris, C. M. and Crede, C. E., 1996, Shock and Vibration Handbook. McGraw-Hill, New York.

Kappagantu, R. V. and Feeny, B. F., 2000, “Part 2: Proper orthogonal modal modeling of a frictionally excited beam,” Nonlinear Dynamics 23(1), 1-11.

Kreuzer, E. and Kust, O., 1996, “Proper Orthogonal Decomposition--an Efficient Means of Controlling Self-Excited Vibrations of Long Torsional Strings,” in Nonlinear Dynamics and Controls, ASME DE-Vol. 91, pp. 105-110.

Kust, O., 1997, “Modal analysis of long torsional strings through proper orthogonal decomposition,” ZAMM 77(S1), S183-S184.

Lenaerts, V., Kerschen, G., and Golinval, J. C., 2000, “Parameter identification of nonlinear mechanical systems using proper orthogonal decomposition,” in Proceedings of the IMAC XVIII, San Antonio, TX.

Lumley, J. L., 1967, “The structure of inhomogeneous turbulent flow,” in Atmospheric Turbulence and Radio Wave Propagation, A. M Yaglom and V. I. Tatarski, eds., Nauka, Moscow, pp. 166-178.

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Ma, X. and Vakakis, A. F., 1999, “Nonlinear transient localization and low dimensional

22

models of a flexible system with a clearance,” in Proceedings of the ASME Design Engineering Technical Conferences, Las Vegas, NV, CD-ROM.

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Riaz, M. S. and Feeny, B. F., 1999, “Proper orthogonal decomposition of an experimental cantilever beam,” in Proceedings of the ASME DETC, Las Vegas, CD-ROM.

Yasuda, K., and Kamiya, K., 1997, “Experimental identification technique of nonlinear beams in time domain,” in Proceedings of the ASME Design Engineering Technical Conferences, Sacramento, CA, CD-ROM.

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LIST OF FIGURES

Fig. 1. Relationship between the node points of the structural elements and the response measurement positions.

Fig. 2. Experimental set-up for the measurement of acceleration responses of the beam

Fig. 3. POMs extracted from the responses of midpoints of 10 identical beam elements from the finite element beam model. The POD is performed on the correlation matrix .

Fig. 4. POMs extracted from the odd numbered 11 node points of the finite element beam model. The POD is performed on the matrix where is based on

the trapezoidal rule.

Fig. 5. POMs extracted from the odd numbered 11 node points of the finite element beam model. The POD is performed on the matrix where is based on

Simpson’s rule.

Fig. 6. Time histories and power spectral density functions of decomposed modal coordinates of the finite element beam model. POMs in Fig. 3 were used for the modal decomposition.

Fig. 7. Time histories and power spectral density functions of decomposed modal coordinates of the finite element beam model. POMs in Fig. 4 were used for the modal decomposition.

Fig. 8. Time histories and power spectral density functions of decomposed modal coordinates of the finite element beam model. POMs in Fig. 5 were used for the modal decomposition.

Fig. 9. POMs extracted from the responses of unevenly spaced 9 node points of the finite element beam model. The POD is performed on the matrix where is based on the trapezoidal rule.

24

Fig. 10. POMs extracted from the responses of 21 node points of the finite element beam model. The POD is performed on the matrix where is based on

the Simpson’s rule.

Fig. 11. POMs extracted from the responses of odd numbered 6 node points of the finite element beam model. The POD is performed on the matrix where is based on the trapezoidal rule.

Fig. 12. POMs extracted from the time history data of the test beam using POD based on Simpson’s rule. The POMs are compared with the normal mode of the beam in the reverse order starting from the 7th mode.

LIST OF TABLES

Table 1. Modal frequencies obtained from the theoretical Euler beam, obtained from the eigensolution of the finite element model, and extracted from the proper modal coordinates of the finite element impulse response, as shown later in Figure 8. The modal frequencies of the experimental beam are also shown. Modal damping values are from the finite element model.

Table 2. Physical dimensions and sensitivities of the accelerometers used in the experiment

Table 3. Values of

25

Table 1. Modal frequencies obtained from the theoretical Euler beam, obtained from the eigensolution of the finite element model, and extracted from the proper modal coordinates of the finite element impulse response, as shown later in Figure 8. The modal frequencies of the experimental beam are also shown. Modal damping values are from the finite element model.

Modes 1st 2nd 3rd 4th 5th 6th

Modal Frequencies

(Hz)

Theoretical 29.323 80.831 158.46 261.94 391.30 546.53

FEM 29.6 80.9 158.7 262.6 393.3 551.6

Extracted 29 80 158 262 393 551

Experiment 29 80 157 255 390 535

Modal Damping

(%)

FEM 8.59e-3 2.92e-2 6.45e-2 0.11 0.18 0.26

Extracted 9.0e-3 3.0e-2 7.0e-2 0.11 0.19 0.27

Table 2. Physical dimensions and sensitivities of the accelerometers used in the experiment

Manufacturer Model Number

Dimension (mm)Base Dia. X height

Weight (gr)

Sensitivity(mV/g)

PCB 352B22 9.14 X 3.6 0.5 9.97 PCB 338B01 6.35 X 12.5 2.2 10.06 PCB 302A07 12.7 X 33.0 25.0 10.0 PCB 351B11 7.9 X 10.9 2.0 5.35 PCB 303A03 7.14 X 12.2 1.9 11.1 IMV VP4200 12.7 X 25.4 32.6 9.51

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Table 3. Values of

POMsModes

POMs in Fig.(3)

POMs in Fig. (4)

POMs In Fig.(5)

1 0.0657 0.0512 0.0499 2 0.0143 0.0218 0.0093 3 0.0758 0.1123 0.0203 4 0.1209 0.1547 0.0819 5 0.1645 0.3346 0.4990 6 0.2560 0.4134 0.3918

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U1 U2 U3 U4

Xx/2 x x x x/2

(a)

U1 U2 U3 U4 U5

Xx x x x

(b)

U1 U2 U3 U4 U5

X

x1 x2 x3 x4

28

(c)

Fig. 1. Relationship between the node points of the structural elements and the response measurement positions.

29

Fig. 2. Experimental set-up for the measurement of acceleration responses of the beam

30

Fig. 3. POMs extracted from the responses of midpoints of 10 identical beam elements from the finite element beam model. The POD is performed on the correlation matrix .

: Theoretical mode shapes of the free-free Euler beam * : POMs

31

0 0.5 1 1.5-1

0

1

0 0.5 1 1.5-1

0

1

0 0.5 1 1.5-1

0

1

0 0.5 1 1.5-1

0

1

0 0.5 1 1.5-1

0

1

0 0.5 1 1.5-1

0

1

Fig. 4. POMs extracted from the odd numbered 11 node points of the finite element beam model. The POD is performed on the matrix where is based on

the trapezoidal rule. : Theoretical mode shapes of the beam * : POMs

32

0 0.5 1 1.5-1

0

1

0 0.5 1 1.5-1

0

1

0 0.5 1 1.5-1

0

1

0 0.5 1 1.5-1

0

1

0 0.5 1 1.5-1

0

1

0 0.5 1 1.5-1

0

1

Fig. 5. POMs extracted from the odd numbered 11 node points of the finite element beam model. The POD is performed on the matrix where is based on

the Simpson’s rule. : Theoretical mode shapes of the beam * : POMs

33

0 0.5 1-20

0

20

g

0 200 400 600 800

105

g2 /Hz

0 0.5 1-10

0

10

g

0 200 400 600 800

105

g2 /Hz

0 0.5 1-10

0

10

g

0 200 400 600 800

102

104

g2 /Hz

0 0.5 1-5

0

5

g

0 200 400 600 800

102

104

g2 /Hz

0 0.5 1-2

0

2

g

0 200 400 600 80010

010

210

4

g2 /Hz

0 0.5 1-1

0

1

TIME (SEC.)

g

0 200 400 600 80010

010

210

4

FREQUENCY (Hz)

g2 /Hz

Fig. 6. Time histories and power spectral density functions of decomposed modal coordinates of the finite element beam model. POMs in Fig. 3 were used for the modal decomposition.

34

0 0.5 1-20

0

20

g

0 200 400 600 800

105

g2 /Hz

0 0.5 1-10

0

10

g

0 200 400 600 800

105

g2 /Hz

0 0.5 1-10

0

10

g

0 200 400 600 800

102

104

g2 /Hz

0 0.5 1-5

0

5

g

0 200 400 600 800

102

104

g2 /Hz

0 0.5 1-2

0

2

g

0 200 400 600 80010

010

210

4

g2 /Hz

0 0.5 1-1

0

1

TIME (SEC.)

g

0 200 400 600 80010

010

210

4

FREQUENCY (Hz)

g2 /Hz

Fig. 7. Time histories and power spectral density functions of decomposed modal coordinates of the finite element beam model. POMs in Fig. 4 were used for the modal decomposition.

35

0 0.5 1-20

0

20

g

0 200 400 600 800

105

g2 /Hz

0 0.5 1-10

0

10

g

0 200 400 600 800

105

g2 /Hz

0 0.5 1-10

0

10

g

0 200 400 600 800

102

104

g2 /Hz

0 0.5 1-5

0

5

g

0 200 400 600 800

102

104

g2 /Hz

0 0.5 1-2

0

2

g

0 200 400 600 80010

010

210

4

g2 /Hz

0 0.5 1-1

0

1

TIME (SEC.)

g

0 200 400 600 80010

010

210

4

FREQUENCY (Hz)

g2 /Hz

Fig. 8. Time histories and power spectral density functions of decomposed modal coordinates of the finite element beam model. POMs in Fig. 5 were used for the modal decomposition.

36

0 0.5 1 1.5-1

0

1

0 0.5 1 1.5-1

0

1

0 0.5 1 1.5-1

0

1

0 0.5 1 1.5-1

0

1

0 0.5 1 1.5-1

0

1

0 0.5 1 1.5-1

0

1

Fig. 9. POMs extracted from the responses of unevenly spaced 9 node points of the finite element beam model. The POD is performed on the matrix where is based on the trapezoidal rule.

: Theoretical mode shapes of the beam * : POMs

37

0 0.5 1 1.5-1

0

1

0 0.5 1 1.5-1

0

1

0 0.5 1 1.5-1

0

1

0 0.5 1 1.5-1

0

1

0 0.5 1 1.5-1

0

1

0 0.5 1 1.5-1

0

1

Fig. 10. POMs extracted from the responses of 21 node points of the finite element beam model. The POD is performed on the matrix where is based on

the Simpson’s rule. : Theoretical mode shapes of the beam * : POMs

38

0 0.5 1 1.5-1

0

1

0 0.5 1 1.5-1

0

1

0 0.5 1 1.5-1

0

1

0 0.5 1 1.5-1

0

1

0 0.5 1 1.5-1

0

1

0 0.5 1 1.5-1

0

1

Fig. 11. POMs extracted from the responses of odd numbered 6 node points of the Finite element beam model. The POD is performed on the matrix where is based on the trapezoidal rule.

: Theoretical mode shapes of the beam * : POMs

39

0 0.5 1 1.5-1

0

1

0 0.5 1 1.5-1

0

1

0 0.5 1 1.5-1

0

1

0 0.5 1 1.5-1

0

1

0 0.5 1 1.5-1

0

1

0 0.5 1 1.5-1

0

1

Fig. 12. POMs extracted from the time history data of the test beam. The POMs are compared with the normal mode of the beam in the reverse order starting from the 7th mode. : Theoretical mode shapes of the beam * : POMs

40