AN ITERATIVE APPROACH TO VISCOUS DAMPING MATRIX IDENTIFICATION

Transcription

1 AN ITERATIVE APPROACH TO VISCOUS DAMPING MATRIX IDENTIFICATION Deborah F. Pilkey & Daniel J. Inman Department of Engineering Science and Mechanics Virginia Tech Blacksburg, VA ABSTRACT INTRODUCTION A method has been designed to calculate the damping matrix of a structural system given only limited information about the structure. The method presented is an iterative procedure that is valid for all damping matrices in which the eigenvalues occur in com.uler conjugate pairs for small damping. Th,e results ax accurate and can be used as a good estimate men for slightly no%sy data. Existing methods are limited and provide only a rough estimate of the damping matrix. The proposed technique is intended to we experimentally obtained eigenvalues and eigenvectors as well a.3 an analytical ma,yn matrix to solve for the damping matriz. To illustrate the procedure, an erample is presented where the exact solution is known in advance and convergence to this solution using the iteration procedure can be seen. The results are n useful tool in areas such as damage detection and diagnostics of structures. NOMENCLATURE M : Mass matrix C : Damping matrix K : Stiffness matrix : Eigcnvalue ; : Eigenvector x : Displacement a : Eigenvector matrix A : Eigenvalue matrix c : Damping ratio w : Frequency The synthesis of damping in structural systems and machines is extremely important if the model is to be used in predicting transient responses, transmissibility, decay times or other characteristics in design and analysis that are dominated by cncrgy dissipation. Methods for determining t,he mass and stiffness matrices of a system are more straight, forward t,han those for determining the damping matrix a t,hey represent quantities which can be measured and evaluate d by static tests. Damping, on the other hand must he determined by dynamic testing. This makes the process of modeling and experimental verification difficult,. It is assumed here that acceptable models of the mass and stiffness are available and that it is desired to use rigenvalue and eigenvector information to construct a damping matrix. This is known as an inverse eigenvalue problem. While the eigenvalues and eigenvrct~ors arc dcpcndent on measured information. these issues arc not, discussed here. One applicat,ion of t,he inverse eigenvalue problem is diagnost,ics. The idea is to test for changes in a structure s proper& by looking at changes in measurable values such as mnodt: slopes or frequencies. Here the underlying assumption is that charlges in t,he damping values correspond to some sort, of change in the structure s health. The problem being investigated assumes a structural systcm consist,ing of mass (MI), damping (C), and stiffness (K) matrices such t,hat t,hr response z(t) satisfies where 3: is an n x vector varying with time, representing the displscemcnts of t,he masses in a lumped mass system. The vectors j: and 52 represent the velocity and acceleration respectively of the lumped masses. For an n dimensional non-gyroscopic system, M, C, and K 52

2 are all symmetric, positive definite n x n matrices. The quadratic pencil for the system is written as: (MA? + cxi + K)$i = 0 ( 4 where 4; is a onzero vector of constants known as the eigenvector or mode shape, and Xi is the corresponding eigenvalue. The idea behind the inverse eigenvalue problem is to find the physical parameters of a system (mass, damp ing and stiffness) from the eigenvalues and eigenvectors. In this case, we are given the eigenvalues, eigenvectors, and mass of the system, the goal is to find the damping matrix. A comparison will be made with several other methods that also attempt to find a damping matrix given some other attainable data. It will be see that often these methods are limited and provide only a rough estimate of the damping matrix. The method presented is a iterative procedure that is valid for all damping matrices in which the eigensystem occurs in complex conjugate pairs for small damping. Damping and stiffness matrices are calculated simultaneously. The results are accurate and can be used as a good estimate eve for slightly noisy data. BACKGROUND Matrix identification can be done by several methods, many of which are described below. Several approaches make limiting assumptions such as diagonal or proportional damping matrices, while others assume the cxperimental data is incomplete, and use only partial eigensystems to solve for damping. Minas and Inman [5] developed a method to find a unique identification of the damping matrix of a struture that has complex mode shapes. This method is capable of sing incomplete modal testing results by forcing the mode shapes to be in a certain part of the complex space to, among other things, make the matrix symmetric. Linear or weighted least squares or pseudoinverse approach is used to solve for the damping matrix. This assumes that only a partial set of eigenvalues and eigenvectors can be determined experimentally. Minas and Inman set up a matrix whose elements are the real and imaginary parts of the known eigenvectors, which is multiplied by a vector containing the elements of the damping matrix (unknown) and set equal t,o a vector containing real and imaginary parts of -(idma + Km- ) (3) where 9 and A represent the collection of eigenvectors and eigenvalues that arc available. Clough and Penzien [2] propose a method for constructing the damping matrices of proportional viscous systems. Rayleigh dampirlg ( C = a M + 0 K ) is essential for the formulatiorl of the equations, but the concept is extended in a wa,y t,hat allows for a number of combinations in series eq& to the number of frequencies (u;) and damping ratios (Ci) available. The accuracy of the method is highly dependent on the number of frequencies available. If an odd number of terms is used, the systenl becomes negatively damped, and the results are not valid. In a alternative formulation. Clough and Pen&n calculate the proportionally damped matrix by starting with the mass matrix, eigenwlues, and eigenvectors. Initially, C and w can be calculated via wj = -Tt3d(xi)/~~ (4) The mass is diagonalized with the matrix of eigenvectors: M = WM@ so that C: = 2ciw;M; (c is also a diagonal matrix whos elements are CT). The damping matrix is then found by c - c**-l To reduce the large computational effort required to find the inverse of ip, the equation can be rewritten as c = (Mml*- )C M) Hasselman s method [3] ses phase differences between coincident and quadrature components of the acceleration response to construct the off diagonal terms of the dampirlg matrix. This car, be done only if pure modes are obtainable. Hasselnlan uses the method of normal modes for analysis of structures. First the equations of motion are written im terms of modal coordinates. In most cases, damping is introduced after this initial step, and it is assumed to be diagonal. Based on the eigenvect,ors of t,he damped system as well as the damped system, this lethod firlds off diagonal terms of linear viscorls damping matrices. The damped eigenval- es are considered a small perturbation. Since higher order tarns are neglected in this formulation, the real part of the damped eigenvcctors are disregarded in the final equations. Thus, this method becornes more accurate as the eigenvalues of the undamped system approach the real part (5) V-3) 53

3 of the eigenvalues of the damped system. There is an underlying assumption that the damped eigenvalues rep resent only a small perturbation with respect to the undamped system. The method becomes similar to the Clough and Pen&n formulation if the difference between the undamped and real damped eigenvectors becomes negligible. Roemer and Mook [6] propose a method to find mass, stiffness and damping matrices from noisy measurements. This is done using a combination of several methods that result in a time domain technique. Beliveau [] uses a modified Newton-Raphson technique to modify parameter estimates. This work assumes that phase angles, mode shapes, damping constants, and natural frequencies are known a priori. An eigenvector and eigenvalue perturbation method is used, where the maa and stiffness matrices don t need to be symmetric or positive definite. Proportional damping is not assumed in this case. A variance on the parameters is achieved, but the actual model can not be evaluated using this method. Lancaster (96) [4] developed an inverse method to calculate the coefficient matrices in a quadratic pencil of the form given in equation 2. Lancaster s formulation is intended to compute mass, damping, and stiffness matrices of a system given only the eigenvalues and eigenvectors. The input data must be normalized in a very specific way for the method to work. In particular, his formulation requires that the rrz.s and damping matrices be used to normalize the eigenvectors, which are used to calculate M, C and K. Thus, as it stands, his formulation can not be used directly. This method, as stated, is appropriate if eigenvalues and properly normalized eigenvectors are provided. The problem comes in starting with information so difficult, if not impossible to obtain. As Lancaster states, the method will become practical for use in the future when this information becomes readily attainable. Currently, though, to find M, C, and K it is necessary to start with at least M and C to properly normalize the eigenvectom. The only piece of information returned that was previously unknown is the stiffness matrix, K. ALGORITHM Using equations 7 0, an algorithm has been formed to solve a different type of problem. The problem of interest is to identify the damping matrix. The goal is to construct a damping and stiffness matrix given all the other necessary components of the quadratic pencil. Vibration response measurements of the structure can provide data such as natural frequencies, and complex mode shapes. The mass matrix is also considered known a priori and can be found experimentally. Given thii information, an iteration of the above formulation can be used to solve for the unknown components: damping and stiffness. The iterative procedure is pictured in figure I. The Lancaster method is very specific to systems with only viscous damping where M, C, and K are symmetric. All of the zeros of the quadratic pencil must also arise in complex conjugate pairs. That is, the system must be underdamped. The method is summarized here. Note that the accent mark represents the transpose of the matrix, not the conjugate transpose. Let A = diagonal matrix of eigenvalues Xi and let Q = matrix whose columns are $; (free modes of vibration, which are complex valued). If the eigenvectors are normalized such that ++ (2MXi + C)& = (7) Then the solution to the inverse problem becomes: ---, M=(@A@ ++A+)- (8) K = (-CA@ - GJ A s )- (9) C = M(+A% + 5 x2 $)A4 00) Figure : Iterative procedure for estimation of damping matrix 54

4 Starting with calculated or experimental values of the mass and the eigensystem, the first step in the procedure involves guessing an initial damping matrix. For an nth order system, this can be any reasonable n dimensional matrix, such as the identity matrix. Next, the eigenvectors must be normalized using 4; (2M& + Co)& = () M, C, and K are then solved for using equations 8, 9, and 0. Since the initial guess for C is not going to match the new value of C, it is necessary to iterate. In the next iteration, the eigenvectors are again normalized, this time using the initial mass matrix and the updated C matrix: & (2MoX; + C&i+ = (2) The mass, stiffness, and damping are again calculated using equations 8 to 0. The iterative procedure continues using an updated damping matrix each time to normalize the eigenvectors until the error between successive damping matrices is small enough to declare convergence. Most structural systems can be solved using this method with only a few exceptions. The system should be underdamped (in other words, eigenvectors and eigenvalues must occur in complex conjugate pairs). The only case where the iterative procedure diverges occurs when the difference between the damping and the mass matrices is small. If the values of the damping matrix are too close to the values of the stiffness matrix, then the iterative procedure will produce a damping matrix that oscillates between two solutions, both near the expected value. EXAMPLE Consider the multi-degree-of-freedom lumped mass system shown in figure 2. Figure 2: lumped mass system r C= (5) For this example we assume the mass of the system is known. The eigenvalues and eigenvectors can be obtained experimentally, although in this case, we use exact values obtained analytically. The complex conjugates of the matrix of cigcnvalues and eigenvectors are discarded. This reduces the mat,ricrs to four by four from eight by eight XI = i x2 = i X3 = % Xq = i i i i Oi i i Oi i Begin by choosing an initial value for C. In this case, the identity matrix is the initial guess. G= / (6) After one iteration, the values for M, C, and K are: r M= I I (3) M = /

5 c = [ K = After live iterations: M = i PO.0072 c = K = After sixteen iterations: M = I c = I K = I This is the exact solution to the problem. As more iterations are performed, the values tend toward the exact solution with higher precision. Convergence to the desired solution has been achieved. In a case such as the above example where the model is known, the damping values at specific locations on the structure can be determined. The example presented will have a damping matrix of the form: cl + c2 -c2 0 0 c= -; c2 + c3 -c3 0 -c3 c3 + c4 -c4 0 0 PC4 c4 (7) It becomes evident that the values for ci are for all i. This is a valuable tool in diagnostics of structures. If a real system were involved, the test would be run again at a later time to determine any differences in the resulting damping matrix. Since the location of excess energy dissipation can be determined by comparing damping matrices, the damage to the structure can be identified. COMPUTATIONAL ISSUES Since structures tend to be represented by models with many degrees of freedom, it is necessary to determine the usefulness of an algorithm for large models. For example, if a structure were represented by a 00 degree of freedom model, what would be the computation time for an inverse method to find a damping matrix? If it is too slow, on the order of hours or days, then the algorithm has very little practical use. If, on the other hand, the routine is acceptable for real time computation, then the question becomes one of optimizing speed. What methods are there to decrease computation time? How much faster can the algorithm go? The iterative algorithm in this paper can be coded in Matlab and run on a SUN Spar&&ion 2 at a speed of 773,935 microseconds for a 00 degree of freedom model. If the algorithm is coded in High Performance Fortran (HPF), a parallel language, then significant speedup can be seen with an increase in the number of processors used. There is an optimal number of processors when programming in parallel. This is occurs when the communication time between processors and the computation time together are minimized. Running the code in serial (with only one processor) on an IBM SP-2 requires 04,97 microseconds. Using HPF (block, block), (block, *), and (*, block) storage structures for the data, computation time can be reduced. 56

6 Processors Block Distr. (block, block) (block, block) Run Time(ps) 70,884 70,43 67,85 54,54 54,465 94,30 07,400 35,448 information. Journal of Vibration and Accoustics, 33:29-224, April 99. [6] M. J. Roemer and D. J. Mook. Mass, stiffness, and damping matrix identification: An integrated ap preach. ASME Journal of Vibration and Accoustics, 4: , July 992. CONCLUSION A method has been suggested to synthesize the damping matrices for mechanical systems of structures. This paper presents an iterative method of performing the inverse eigenvalue problem. By iterating on a group of equations that require updated normalized eigenvectors a viscous damping matrix can be developed. Real time computation was shown possible with the use of multiple processors. The results are a useful tool in areas such as damage detection and diagnostics of structures. ACKNOWLEDGMENTS The first author wishes to acknowledge the support of the Virginia Space Grant Consortium and the Virginia/ICASE/LaRC program in High Performance Computing. The second author acknowledges the Army Research Office through grant number DAAL G-080 supervised by Dr. Gary Anderson. References [I] Jean-Guy Beliveau. Identification of viscous damping in structures from modal information. ASME Jourml of Applied Mechanics, 43: , June 976. [2] Ray W. Clough and Joseph Penzien. Dynamics of Structures. McGraw Hill, second edition, 993. [3] T.K. Hasselman. A method of constructing a full modal damping matrix from experimental measurements. AIAA Journal, 0: , 972. [4] P. Lancaster. Expression for damping matrices in linear vibration problems. Journal of the Aerospace Sciences, page 256, March 96. [5] C. Minas and D. J. Inman. Identification of a nonproportional damping matrix from incomplete modal 57