AIAA JOURNAL Vol. 40, No. 4, April 2002 Estimating Natural Frequencies and Mode Shapes from Forced Response Calculations J. Gregory McDaniel, ¤ Ferry Widjaja, † Paul E. Barbone, ‡ and Allan D. Pierce § Boston University, Boston, Massachusetts 02215 A method is proposed for condensing a structural eigenvalue problem by using more than one forced response vector in a condensation. The method approximates an eigenvector by a linear combination of forced response vectors, resulting in a signicantly smaller eigenvalue problem that retains the eigenpairs that were most excited by the force vectors. The condensation is extremely economical in cases where the forced response vectors have been computed for other purposes. In cases where some eigenpairs are approximately known, the method produces accurate eigenvalue estimates by choosing the force vectors to be parallel to the approximate eigenvectors. This idea forms the basis of an iterative procedure that nds all of the eigenvalues in a specied frequency range. Theoretical error bounds on the estimated eigenvalues are also presented. Numerical examples illustrate unique features of the condensation in its noniterative and iterative forms. Nomenclature A = system matrix, K ¡ ¸M C n = participationfactor of the nth mode d = vector of amplitudes in eigenvector approximation K = stiffness matrix M = mass matrix P = matrix of primary amplitudes R = Rayleigh’s quotient r = residual error in eigenvalue problem S = matrix of secondary amplitudes X = matrix holding forced response vectors x = forced response vector ² = scaling factor for error in eigenvector estimate · = condensed stiffness matrix K P = diagonal matrix of primary eigenvalues K S = diagonal matrix of secondary eigenvalues ¸ = frequency parameter, ! 2 ¸ n = n th eigenvalue Q ¸ n = approximationof the nth eigenvalue ¹ = condensed mass matrix U = matrix holding eigenvectors Q U = matrix holding approximate eigenvectors Á n = n th eigenvector Q Á n = approximationof the nth eigenvector à .0/ = shape of error in eigenvector estimate ! = radian frequency Subscripts max = maximum value min = minimum value m = forced response index n = eigenpair index Received 12 April 2001; revision received 20 September 2001; accepted for publication 22 September 2001. Copyright c ° 2002 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. Copies of this paper may be made for personal or internal use, on condition that the copier pay the $10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; include the code 0001-1452/02 $10.00 in correspondence with the CCC. ¤ Assistant Professor, Department of Aerospace and Mechanical Engineer- ing; jgm@bu.edu. Member AIAA. † Research Assistant, Department of Aerospace and Mechanical Engineer- ing. ‡ Associate Professor, Department of Aerospace and Mechanical Engi- neering. § Professor, Department of Aerospace and Mechanical Engineering. Senior Member AIAA. Superscript . p/ = iterate Introduction M ANY analysts are concerned with the free and forced vibra- tionspredictedby structuralmodels over specied frequency ranges. To estimate a structure’s frequency response, for example, an engineer may compute the forced response of a structure at a number of frequencies in a given frequency range. Given this re- sponse, the engineer is then faced with the question, “Which of the myriad of vibrational modes present in the structure are primarily responsiblefor the observedvibrationpattern?”This paperpresents an efcient and straightforwardpostprocessingmethod that answers this question directly. The method can also be used as the kernel of an iterativeschemeto nd allvibrationalmodesin a givenfrequency band. We present both applications here. The present method belongs to a class of methods in which an eigenvector is approximated by a linear combination of ba- sis vectors. Use of this approximation in Hamilton’s principle is often referred to as the Rayleigh– Ritz method, and the vectors are known as Ritz vectors or assumed modes. When inserted into the full eigenvalue problem, the approximation produces a much smaller eigenvalue problem that retains approximations of the de- sired eigenpairs. The distinguishing characteristic among methods is the choice of basis vectors. Thorough and interestingdiscussions of the Rayleigh– Ritz method in eigenvalue problems are found in the texts by Hughes 1 and Bathe. 2 One of the earliest expressions of the idea is found in Crandall’s monograph, 3 whichwas publishedin 1956.In describingatruncated Lanczosmethod,he proposeda procedurefor estimatingeigenpairs by expanding an eigenvector in a series of orthogonal vectors. Re- quiring stationarity of Rayleigh’s quotient with respect to the coef- cients in the expansioncreated a reduced eigenvalueproblem that Crandall referred to as an “::: eigenvalueproblem within an eigen- value problem.” In the same year, Turner et al. 4 presenteda method for condensing structural matrices by expanding displacements in polynomialbasis functions.This expansionin kineticand strain en- ergyexpressionsyieldedreducedmatricesof thesame form as those in Crandall’s reduced eigenvalue problem. 3 Since that time, a variety of methods have been developed based on alternate choices of basis vectors. Bhat 5 chose basis vectors that satised the geometric and natural boundary conditions on the structure. Matta 6 proposed a rationale for selecting other ba- sis vectors based on the diagonal terms of the mass and stiffness matrices. Other researchers prescribed basis vectors that identied and removed unimportant coordinates. At the time, such methods were not recognized as Rayleigh– Ritz methods. For example, the mass condensation method developed by Irons 7;8 and Guyan 9 in 758 Downloaded by BOSTON UNIVERSITY on February 18, 2023 | http://arc.aiaa.org | DOI: 10.2514/2.1710