On the Proper Orthogonal Modes and Normal Modes of Continuous Vibration Systems

Abstract

Contributed by the Technical Committee on Vibration and Sound for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received Feb. 1999; revised Aug. 2001. Associate Editor: A. F. Vakakis.Proper orthogonal decomposition (POD) is a useful experimental tool in dynamics and vibration. A common application of POD to structures involves sensed displacements, x1t,x2t,…,xMt, at M locations on the structure. When the displacements are sampled N times at a fixed sampling rate, we can form displacement-history arrays, such that xi=(xit1,xit2,…,xitN)T, for i=1,…,M. The mean values are often subtracted from the displacement histories. These displacement histories are used to form an N×M ensemble matrix, X=[x1,x2,…,xM].The M×M correlation matrix is R=1/NXTX. Since R is real and symmetric, its eigenvectors form an orthogonal basis. The eigenvectors and eigenvalues of R are the proper orthogonal modes (POMs) and values (POVs). The POMs in certain nonlinear structures have resembled the normal modes of the linearized system 123. The POMs may indeed converge to linear normal modes in multimodal free responses of symmetric lightly damped lumped-mass linear systems, but only if the mass matrix has the form mI, which can be achieved by a coordinate transformation if the mass distribution is known 4. This provides a fundamental tie between the statistically derived POMs and the geometrically based linear normal modes in certain discrete systems. In this note, this relationship is extended to discretized continuous systems. The following analysis relates the POMs to the normal modes in continuous systems with discrete measurements and known mass. We consider a one-dimensional self-adjoint distributed-parameter system of length l: (1)mx ∂2y∂t2+L1y=0,with boundary conditions, where yx,t is a displacement. Letting u=m1/2xy, the system can be rewritten as (2)∂2u∂t2+L2u=0.L2=m−1/2xL1m−1/2x is self-adjoint. Separation of variables leads to normalized eigenvalues and eigenfunctions ϕix that satisfy ∫0Lϕixϕjxdx=δij.Sampling the displacement ux,t at coordinates x1,…,xM, leads to a set of measurements u=[ux1,t…uxM,t]T. The displacement is approximated as a truncated series of linear normal modes, such that ux,t≈∑i=1M^qitϕix=ϕTq, where ϕ=[ϕ1x…ϕM^x]T is a vector of modal functions, and qt=[q1t…qM^t]T is the vector of modal coordinates. We will take M^=M. We define a matrix Φ=[v1…vM]=[ϕx1…ϕxM]T. Thus, the vectors vi=[ϕix1…ϕixM]T are spatial discretizations of the mode shapes ϕix. Then u=Φqtrelates the discrete displacements of the beam to the discretizations of the mode shapes. The displacements are sampled at times ti=iΔt,i=1,…,N, where Δt is the sampling rate. We construct an N×M ensemble matrix U=[ut1…utN]T=[Φqt1…ΦqtN]T, or U=ΦQT,where Q=[qt1…qtN] is an M×N matrix. The correlation matrix is thus R=1/NUTU=1/NΦQQTΦT.We check whether vj is an eigenvector of R by examining Rvj=1/NΦQQTΦTvj. The quantity ΦTvj has elements viTvj. If the spatial discretization is evenly spaced, then, viTvj=∑k=1Mϕixkϕjxk≈1/h∫0Lϕixϕjxdx by the rectangular rule, where h is the spacing of the spatial discretization. Thus we approximate viTvj≈1/hδij. If this approximation is reasonable, then the quantity ΦTvj≈[0…0,1/h,0…0]T=hj has elements of approximately zero, except the jth element which is approximately 1/h. The error associated with the rectangular integration representation of the underlying orthogonality integral is on the order of kh2, where k is proportional to a characteristic curvature in the integrand 5. Then Rvj≈1/NΦQQThj. The ijth elements of QQT are ∑k=1Nqitkqjtk. If the frequencies of oscillation of qit and qjt are distinct, the sampling rate is Δt is fixed, and the time record gets arbitrarily large, then limN→∞1N ∑k=1Nqitkqjtk=0,i≠j.Thus, limN→∞1N QQT=Dwhich is diagonal with elements dii=∑k=1Nqitk2/N, which are the mean squared values of qit.In such case, Rvj→1/NΦDΦTvj≈ΦDhj=Φhjdjj=vjdjj/h. So, for increasing N, with a fixed sampling rate, evenly spaced sensors and distinct modal frequencies, the POMs converge approximately to vj, which are the discretized linear modes. (The POMs converge to vj+ej where ej is an error vector.) Furthermore, the POVs converge to djj/h, which is proportional to the mean squared modal coordinate. Thus, we have an analysis which ties the statistically formulated POMs to the discretization of the nonlinear normal modes for multi-modal free responses of undamped systems with known mass distributions. The role of the mass distribution is critical. The modes of Eq. (1) are orthogonal with respect to the mass distribution (and the linear operator), and are not otherwise perpendicular to each other. Discretized modes are therefore not perpendicular. The POMs, however, are orthogonal (i.e. perpendicular), since RT=R. Thus, for general mass distributions, the POMs cannot represent the discretized linear normal modes. Formulating with respect to the mass, as in Eq. (2), allows us to make a connection between POMs and normal modes in multi-modal responses. The limitation is that the mass distribution must be uniform or known. We apply these ideas to a hinged-hinged beam, for which theoretical modes are readily available for comparison. For each numerical simulation, we choose a uniform mass per unit length of mx=1, a stiffness of EI=1, and a length of L=1. The clamp is at x=0.In putting ten “sensors” on the beam away from the endpoints, the spacing was h=1/11. Here, the modal functions are ϕix=siniπx. The inner product between the discretized modal vectors was, to at least four decimal points, viTvj=δij/h. The modal frequencies, ωi=i2π2, are distinct and widely spaced. Vibrations were induced through the modal variables; q0=[2,1,0.5,0.25,0.12,0.1,0.05,0.05,0.05,0.05]T and q˙(0)=0 were the initial conditions. The vibrations were sampled through four fundamental periods at an interval of Δt=0.0179 (400 samples). Figure 1 shows the comparison between the first two sets of modes. The higher modes visually compared as well as those shown. The norms of the errors between these first two sets of modes are 0.0037, 0.0049, 0.0052 and 0.0055. The mean norm of the error between the ten computed modes is 0.0863. A cantilevered beam model was similarly tested. This case is more sensitive to the spatial discretization effects 6. The discretized modes are not orthogonal, with errors of about 20 percent among the first three modes. Figure 2 compares the first two sets of modes. Since the principal axes optimize the distribution of data from the axes, the dominant POM can be considered as an optimal fit of a “synchronous” nonlinear normal mode during a single-mode response 4 (“synchronous” meaning that the displacement coordinates reach their extrema simultaneously). As the amplitude of the response changes, the path of the synchronous normal mode changes, as does its best fit. This interpretation may extend to multi-modal nonlinear modal responses in some cases 7. However, the relationship between the POMs and the “best fit” of the nonlinear normal modes is generally obscured if more than one mode is active. The equation of motion of a hinged beam with a discrete cubic spring at its midpoint 8 is (3)mu¨+EIu″″+u3δx−l=0,with ux,t=u″x,t=0 at x=0 and x=L=2l, where ux,t is the deflection of the beam and δ is the Dirac delta function. The parameters are m=EI=L=1. Equation (3) can be discretized using the assumed-modes method 9. If u is expanded in a truncated modal series as μx,t=∑i=1Mqitϕix, where ϕix=siniπx/L, the resulting discretized equations of motion are q¨+Λq+fq=0, where Λ is a diagonal matrix of natural frequencies squared, and f(q) has elements fi=ul3 siniπ/2.The numerical solution (fifth-order Runge-Kutta) was based at initial conditions q0=[4.0000,0,0.2244,0,−0.2291,0,0.1023,0,−0.0561,0]T and q˙(0)=0, very nearly exciting only the first nonlinear normal mode. Three periods of motion in this nonlinear mode were sampled at a step size of Δt=0.0255N=180. Figure 3 shows the animated modal vibration and its shape variation with phase. The displacements along the beam were obtained from the truncated modal expansion, and evaluated at M+1 evenly spaced locations along the beam. The redundant “sensor” was added so that the evenly spaced discretization included the midpoint of the beam. The three largest POVs were 46.9601, 0.1227, and 0.0003. Thus the dominant POM comprised about 99.8 percent of the mean signal power. The two dominant POMs are plotted in Fig. 4. The dominant POM of the discrete measurements qualitively fits the animated synchronous motion. Figure 5 plots the displacement of the midpoint of the beam against the displacement at location x=l/6. Superposed on this plot is a projection of the dominant POM onto this coordinate space. The POM is aligned with the principal axis of minimum moment of inertia of the data. Figure 6 plots two modal coordinates against each other. This plot shows how the relative participation of each linear mode changed during the first nonlinear modal vibration. Superposed on the plot is the projection of the first POM computed from an ensemble of modal displacements. The dominant POM is a best fit of the synchronous normal mode, and not its linearization. The POMs approximate the discretized linear normal modes for free multi-modal motions of distributed systems. The approximation depends on the sensors’ spatial resolution. The problem must be formulated in displacement coordinates defined such that the associated mass distribution is uniform. The dominant POM produces a best fit of a single active “synchronous” discretized nonlinear normal mode. The results relate the statistically derived POMs and the geometry of normal modes, for multi-modal responses of a class of linear continuous systems, and also for single mode nonlinear responses. This work was supported by the National Science Foundation (CMS-9624347).