The (CMIF) as a Parameter Estimation Method Allyn W. Phillips, PhD Research Assistant Professor Randall J. Allemang, PhD Professor William A. Fladung Research Assistant Structural Dynamics Research Laboratory Mechanical, Industrial and Nuclear Engineering University of Cincinnati Cincinnati, OH 45221-0072 USA ABSTRACT The (CMIF) has evolved from its original usage as simply an indicator of repeated root order to a full fledged parameter estimation algorithm in its own right. When combined with a properly scaled Enhanced Frequency Response Function (efrf), this spatial domain modal parameter estimation approach provides an effective and efficient technique for the estimation of modal parameters. This paper reviews and presents the development of CMIF and efrf. The paper presents the basic structure of the modal parameter estimation procedure which first finds the modal vector from the CMIF and then finds the modal frequency, damping and scaling. The paper particularly discusses the issues associated with scaling the efrf properly and the variations in indicator function that can be used to improve its sensitivity in estimating mode shape information. Results are presented for two example structural systems. Nomenclature N = Number of modes. Ni = Number of inputs. = Number of outputs. N 0 N, = Number of effective modes. [H1 = Frequency response function matrix. [ t = Hermitian (conjugate transpose) of matrix. [A j = Eigenvalue matrix (diagonal). [V 1 = Eigenvector matrix matrix. j L J = Singular value matrix (diagonal). [U 1 = Left singular vector matrix (unitary). [V 1 = Right singular vector matrix (unitary). {u} = Left singular vector (unitary). {v j = Right singular vector (unitary). Q, = Modal scaling for mode r. If pr = Modal coefficient for DOF p, mode r. {If},= Modal vector for mode r. [M 1 = Reduced mass matrix. M, = Modal mass for mode r. 1. Introduction Modal parameter estimation has evolved and developed extensively over the last twenty-five years due to an increased use and availability of numerical techniques involving least squares estimation and eigenvalue/singular value decomposition methods ll- 3 1. Often times, the development of modal parameter estimation methods appear to be quite complicated and difficult to understand. Much effort has gone into simplifying the theoretical development of these methods to focus on the common characteristics. One common characteristic of most modal parameter estimation 141 methods is the two stage development of the algorithm Traditional time/frequency domain modal parameter estimation algorithms estimate temporal information (modal frequencies or poles) in the first stage and spatial information (modal vectors and modal scaling) in the second stage. The development presented in this paper is an example of a spatial domain method based upon the complex mode indicator function and the enhanced frequency response function. Spatial domain algorithms, like this one, estimate the spatial information (modal vectors) in the first stage and the temporal information (modal frequencies and modal scaling) in the second stage. It is important to understand the fundamental difference between time/frequency (temporal) based algorithms_ and spa~ial based algorithms 141_ Temporal based algonthms ut1hze the magnitude/phase relationships in the temporal information (time/frequency response) to determine the number and characteristics of the modes that have been excited. Spatial based algorithms utilize the magnitude/phase relationships in the spatial information (physical input/output DOFs) to determine the number and characteristics of the modes that have been excited. Most modem algorithms, such as Polyreference Time Domain (PTD), Eigensystem Realization Alforithm (ERA) or Polyreference Frequency Domain (PFD) I4-S, are actually hybrid algorithms that take advantage of both temporal and spatial information. The CMIF algorithm, discussed in the following sections, is an example of a zero order UMPA model that is fundamentally a spatial domain algorithm. 705
2. Theory The development of the complex mode indicator function into a modal parameter estimation method involves a two step process. In contrast to traditional modal parameter estimation methods, modal vectors are estimated in the first phase and modal frequencies and scaling factors are estimated in the second phase. In the first phase, the complex mode indicator function is used to identify the number of modes of vibration that are active and an estimate of the modal vector for each mode of vibration. In the second phase, the mode shape is used to develop an enhanced frequency response function in order to formulate a virtual frequency response function that has limited number of modes present. Normally, there is only one mode of vibration present and single degree-of-freedom modal parameter estimation algorithms can be used to estimate the associated modal frequency and scaling for the modal vector estimated from the CMIF. The theories behind the CMIF and efrf are briefly reviewed in the following sections. 2.1 Complex Mode Indication Function (CMIF) A simple algorithm based on singular value decomposition (SVD) methods applied to multiple reference FRF measurements, identified as the complex mode indicator function, was first developed for traditional FRF data in order to identify the proper number of modal frequencies, particularly when there are closely spaced or repeated modal frequencies 14-71 Unlike the Multivariate Mode Indication Function (MvMIF), which indicates the existence of real normal modes, CMIF indicates the existence of real normal or complex modes and the relative magnitude of each mode. Furthermore, MvMIF yields a set of force patterns that can best excite the real normal mode, while CMIF yields the corresponding mode shape and modal participation vector. The CMIF was originally defined as the eigenvalues, solved from the normal matrix formed from FRF matrix, at each spectral line. The normal matrix is obtained by post-multiplying the FRF matrix by its Hermitian matrix as [H(w)) [H({/))]H [HLv,x.v, [HJZ,x.v, = [V l.v,x.v, l A J.v,x.v, [V lz,xn, The CMIF is the plot of these eigenvalues on a magnitude, or log magnitude, scale as a function of frequency. The peaks detected in the CMIF plot indicate the existence of modes, and the corresponding frequencies of these peaks give the damped natural frequencies for each mode to the nearest spectral line. The CMIF is more commonly formulated by plotting the singular values of the economical SVD of the frequency response function matrix [H] on a frequency by frequency basis. By taking the singular value decomposition of the FRF matrix at each spectral line, a similar expression to the above equation is obtained. Most often, the number of input points (reference points), N;, is less than the number of response points, N"" If the number of effective modes is less than or equal to the smaller dimension of the FRF matrix, ie. N, ~ N;, the singular value decomposition (I) (2) leads to approximate mode shapes (left singular vectors) and approximate modal participation factors (right singular vectors). The singular value is then proportional to the the scaling factor Q, divided by the difference between the discrete frequency and the modal frequency jro- A,. For a given mode, since the scaling factor is a constant, the closer the modal frequency is to the discrete (measured) frequency, the larger the singular value will be. Therefore, the damped natural frequency is near the frequency at which the maximum magnitude of the singular value occurs. Additionally, if different modes are compared, the stronger the mode contribution (larger residue value), the larger the singular value will be. It must be noted that not all peaks in CMIF indicate modes. Errors such as noise, leakage, nonlinearity and a cross eigenvalue (singular value) effect can also generate an apparent peak. The cross eigenvalue effect is a computational characteristic that develops due to the way the CMIF is plotted. In a CMIF plot, the eigenvalue curves are plotted as a function of magnitude - the largest eigenvalue is plotted first at each frequency followed by subsequently smaller eigenvalues. Since the contributions from different modes vary along the frequency axis, at a specific frequency, the contribution of two modes can be approximately equal. At this frequency, these two eigenvalue curves cross each other. Because of the limited frequency resolution and the way that the CMIF is plotted, the lower eigenvalue curve appears to peak, while the higher eigenvalue curve appears to dip. Therefore, the peak in this case is not due to a system pole but is caused by an equal contribution from two modes. This characteristic is identifiable since the peak occurs in the lower eigenvalue curve at the same frequency as a dip in the higher eigenvalue curve. The most common way to avoid this problem is to check the corresponding vectors [U({/))) or [V({/))] associated with each CMIF curve. Since the vector shape is associated with a specific eigenvalue (singular value), the peaks that occur due to this cross eigenvalue effect can easily be identified by comparing the vectors associated with each eigenvalue on a spectral line to spectral line basis. The modal assurance criterion (MAC) 181 is used to evaluate this characteristic by comparing the MAC value between vectors associated with adjacent spectral lines. This process is referred to as tracking and eliminates the cross eigenvalue problem. Since the mode shapes that contribute to each peak do not change much around each peak, several adjacent spectral lines from the FRF matrix can be used simultaneously for a better estimation of mode shapes. By including several spectral lines of data in the singular value decomposition calculation, the effect of the leakage error can be minimized. If only the quadrature (imaginary) part of the FRF matrix is used in the CMIF calculation, more sharply defined peaks will be found (along with real modes.) 2.2 Enhanced Frequency Response Functions (efrf) A virtual measurement, known as the enhanced frequency response function (efrf), is used to identify the modal frequencies and scaling of a single degree-of-freedom characteristic that is associated with each peak in the CMIF 1 7-91. The efrf is developed based upon the concept of physical to modal coordinate transformation and is used to manipulate frequency response 706
functions so as to enhance a particular mode of vibration. The left singular values, associated with the peaks in the CMIF, are used as an estimate of the modal filter which accomplishes this. Starting with: Redefining: ~ Q,IJiqr Hpq(w) = "-' IJipr -.--).- r=l JW- r The enhanced frequency response function (efrf) can now be defined: efrf,(w) = _Q,IJfqr JW- A, The efrf,(w) is defined as the Enhanced Frequency Response Function for mode r. The efrf has only to do with the input location and is constant for a given column of the FRF matrix. The efrf can be formulated from measured frequency response function data in the following manner: 2N Q {H(w)) = L {IJI), ~ r=l JW-.1., From the orthogonality conditions: efrf (w) = M Q,IJiqr.r s jw-j., (3) (4) (5) (6) (7) (9) (10) The last equation indicates that an estimate of the reduced mass matrix is needed to develop the enhanced frequency response function. In many situations, when the mass distribution is adequately represented by the measured degrees-of-freedom (good spatial representation), the modal vector can be used directly. Essentially, for the efrf to yield only one degree-of-freedom, this vector must represent a modal filter for that mode [ 91. In reality, the goal of the efrf is to allow a simple modal parameter estimation algorithm to be used to estimate modal frequency and scaling. If other modes are still observable in the efrf, these modes can be handled with residuals in the modal parameter estimation. In the technique utilized in the following examples, the left singular vectors, associated with the singular values at the peaks in the CMIF, are used as this modal filter. and damping of the associated modal frequency. In order for the efrf to also be used to estimate the modal scaling (modal mass and/or modal A), the correct scaling (correct magnitude and phase) of the efrf must be accounted for. Since the right and left singular vectors in the singular value decomposition are unitary and scaled consistently as a set, the arbitrary scaling of the left singular vector must be accounted for in the efrf computation. For this case, where the modal vector used in the efrf is estimated from the left singular vector associated with a peak in the CMIF plot, the efrf is scaled by utilizing the values of the left and right singular vectors, associated with the significant singular value, at the driving point location as follows: where: efrf,(w) =SF,eFRF * {u,} ixn)h(w)]n,xn; { v,} N,xl (12) SF,eFRF is the scaling factor that corrects the magnitude/phase of the efrf for mode r. 3. Examples Two example tests are presented to demonstrate the method presented in this paper. The first is a seven reference test on a circular plate laboratory test structure which has repeated modes. The second is a fifteen reference test from a set of data taken on a bridge as part of a damage detection research program. 3.1 Circular Plate Example The data presented in Figures I through 5 is representative of the seven reference circular plate test object. Figure I shows the basic CMIF plot. Several cross eigenvalue effects are noticeable (for example, peaks in the 3'd and 4'h CMIF curves between 1250 and 1300 Hz.) Figure 2 has mode tracking enabled. This provides a clearer indication of the evolution of the shape as a function of frequency. Expanding the region around 1300 Hz, Figure 3, the tracked CMIF curves show clearly four distinct modes in the region. Finally, a more sharply defined CMIF plot using only the imaginary part of the FRF is shown in Figure 4. The enhanced FRF, Figure 5, is shown for the 760 Hz peak identified in Figure 2. This plot shows both the efrf and the estimated/synthesized SDOF fit for the region. Note that most single degree-of-freedom modal parameter estimation methods would work well on this data. The efrf is typically used, together with single degree-of-freedom modal parameter estimation methods, to estimate the frequency 707
1o-'.-----,r------,------,----,.---,-----,---,.---1... - ""' 10~... 10 ~0L0----~4~00~--~s=oo~---s=o=o--~1=o~oo~--~12~o~o--~14~o~o~~1~6o~o~-,1~aoo Figure 1. CMIF - Untracked 1050 1100 1150 1200 1250 1300 1350 1400 1450 1500 1550 Figure 3. CMIF - Tracked 1 o-' r------r-----.-------.------r------.------r-------,----~ 10-5,------,---,.-_:_:.,.------,---,.-----,--l ----------~.. -...... -..., 10~oLo 4~o-o solo a~o~o--~1o~o~o-~1=2~oo~-~14~o=o~~1~6~0~0-~1aoo Figure 2. CMIF - Tracked Figure 4. Imaginary CMIF - Tracked 708
~tbo :!:'... 0 -., "' \ 764.028 Hz, 0.00340 zeta f10-5 --+- to-' 0 10-7 0 10_, ii ii /\ I \ J \ '0 " I \ :g "' 10-7 0> / "- ::1 "' ~to_. ::!; to--' to- to-''--------''---------'----'-----'----'-----'------'----' 200 400 600 BOO 1000 t200 1400 1600 1800 Figure 5. Enhanced Frequency Response Function 3.2 Bridge Example The data presented in Figures 6 through 9 is representative of data taken with an impact testing method on a highway bridge. Fifteen reference accelerometers were utilized for this testing situation. It is very difficult to utilize this data in conventional modal parameter estimation algorithms due to the level of damping and number of modes excited. Figure 6 shows an overlay of some typical frequency response functions. Figure 7 shows the basic CMIF plot with mode tracking enabled. This provides a clear indication of the number of active shapes in each frequency region. Two of the ten enhanced FRFs, Figures 8 and 9, are shown for the 8.2 Hz and 12.7 Hz peaks identified in Figure 7. These plots show both the efrf and the estimated/synthesized SDOF fit for the region. to'.----.----..----.----.----.----. g; 180 '0 i 0 to t2 t4 t6 tb Figure 7. CMIF - Tracked 8.244 Hz, 0.02449 zeta., ~0~ ~---~--~---~---~----+----] 1o- 6L... L..,1L-O...J12:c--------'14:c----!-.t6:-----!18 Figure 8. Enhanced Frequency Response Function c; tbo.. :!:'. 0 5l., ~0~ t2.737 Hz, 0.0279t zeta ---...,,. 10-6'-----aL--------'1o'------------'12,_'--"'>= :-L.J...:...::~ 18 Figure 6. Typical FRF Data - Highway Bridge to-'.. '0 ~ 10-1 go ::1 to- /' '-... 10-10 6 8 10 12 14 16 18 Figure 9. Enhanced Frequency Response Function 709
4. Acknowledgments The authors would like to thank the University of Cincinnati, Cincinnati Infrastructure Institute for use of the bridge data. [9] Shelley, S.J., "Investigation of Discrete Modal Filters for Structural Dynamic Applications", Doctoral Dissertation, Department of Mechanical Engineering, University of Cincinnati, 269 pp., 1991. 5. Summary - Conclusions Using CMIF with the efrf produces an effective, efficient modal parameter estimation technique. It has been used successfully on many different types of structures and on data that was not amenable to more sophisticated techniques. The fifteen reference MRIT test on a bridge is a good example. The use of CMIF/eFRF produced surprisingly accurate results from the relatively poor quality data, whereas none of PTD, ERA, or PFD were able to process the data reasonably. Since CMIF/eFRF is a shape based technique, it is less sensitive to shifts in modal frequency. The mode shape remains relatively unchanged and is extracted first. This is then used to enforce a single modal frequency on the data set for the extracted shape. Because the technique is shape based, it is important to have good spatial definition with many references. 6. References [I) Strang, G., Linear Algebra and Its Applications, Third Edition, Harcourt Brace Jovanovich Publishers, San Diego, 1988, 505 pp. [2) Lawson, C.L., Hanson, R.J., Solving Least Squares Problems, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1974, 340 pp. [3] Jolliffe, I.T., Principal Component Analysis Springer Verlag New York, Inc., 1986, 271 pp. [4] Allemang, R.J., Brown, D.L., Fladung, W., "Modal Parameter Estimation: A Unified Matrix Polynomial Approach", Proceedings, International Modal Analysis Conference, pp. 501-514, 1994. [5] Allemang, R.J., Brown, D.L., "Modal Parameter Estimation" Experimental Modal Analysis and Dynamic Component Synthesis, USAF Technical Report, Contract No. F33615-83-C-3218, AFWAL-TR-87-3069, Vol. 3, 130 pp., 1987. [6] Shih, C.Y., Tsuei, Y.G., Allemang, R.J., Brown, D.L., "Complex Mode Indication Function and Its Application to Spatial Domain Parameter Estimation", Mechanical System and Signal Processing, Vol. 2, No.4, pp. 367-377, 1988. [7] Fladung, W.D., Phillips, A.W., Brown, D.L., "Specialized Parameter Estimation Algorithms for Multiple Reference Testing", Proceedings, International Modal Analysis Conference, pp. I 078-1087, 1997. [8] Allemang, R. J., "Investigation of Some Multiple Input/Output Frequency Response Function Experimental Modal Analysis Techniques", Doctoral Dissertation, University of Cincinnati, 1980, 358 pp. 710