An accurate method for transcendental eigenproblems with a new criterion for eigenfrequencies

Abstract A new computational method for the transcendental eigensolution of structural dynamics is proposed. It is based on a new criterion for detecting the eigenfrequency. Instead of trying to find the roots of the determinant of the dynamic stiffness matrix, this method seeks the frequency that makes the last energy norm vanish, so matching the physical definition of eigenfrequency except for those eigenfrequencies for which the last degree of freedom is nodal in the mode, for which the method is suitably modified. By using the property of derivatives of energy norms, the eigenproblem is transformed safely into a specific initial value problem of an ordinary differential equation. Among many available methods to solve the resulting ordinary differential equation, the one-step Runge–Kutta method is proved to be a simple and efficient way to obtain eigensolutions, as confirmed by a numerical example.

[1]  A. Simpson On the solution of S(ω)x=0 by a Newtonian procedure , 1984 .

[2]  A. R. Collar,et al.  Matrices and Engineering Dynamics , 1987 .

[3]  Leonard Meirovitch,et al.  Computational Methods in Structural Dynamics , 1980 .

[4]  W. H. Wittrick,et al.  An automatic computational procedure for calculating natural frequencies of skeletal structures , 1970 .

[5]  F. W. Williams,et al.  An Algorithm for Computing Critical Buckling Loads of Elastic Structures , 1973 .

[6]  Inclusion of elastically connected members in exact buckling and frequency calculations , 1986 .

[7]  F. Williams,et al.  Reliable use of determinants to solve non-linear structural eigenvalue problems efficiently , 1988 .

[8]  J. S. Fleming,et al.  Dynamics in engineering structures , 1973 .

[9]  Jianqiao Ye,et al.  A successive bounding method to find the exact eigenvalues of transcendental stiffness matrix formulations , 1995 .

[10]  J. R. Banerjee,et al.  Concise equations and program for exact eigensolutions of plane frames including member shear , 1983 .

[11]  David Kennedy,et al.  More efficient use of determinants to solve transcendental structural eigenvalue problems reliably , 1991 .

[12]  S. BŁaszkowiak,et al.  Iterative methods in structural analysis , 1966 .

[13]  Romualdas Bausys,et al.  Adaptive h-version eigenfrequency analysis , 1999 .

[14]  F. W. Williams,et al.  Mode Finding in Nonlinear Structural Eigenvalue Calculations , 1977 .

[15]  F. W. Williams,et al.  A GENERAL ALGORITHM FOR COMPUTING NATURAL FREQUENCIES OF ELASTIC STRUCTURES , 1971 .

[16]  P. Hager,et al.  Adaptive eigenfrequency analysis by superconvergent patch recovery , 1999 .

[17]  W. P. Howson,et al.  A compact method for computing the eigenvalues and eigenvectors of plane frames , 1979 .

[18]  K. Bathe Finite Element Procedures , 1995 .

[19]  H. Saunders Book Reviews : NUMERICAL METHODS IN FINITE ELEMENT ANALYSIS K.-J. Bathe and E.L. Wilson Prentice-Hall, Inc, Englewood Cliffs, NJ , 1978 .

[20]  L. Mirsky,et al.  Introduction to Linear Algebra , 1965, The Mathematical Gazette.

[21]  Frederic W. Williams,et al.  Exact Buckling and Frequency Calculations Surveyed , 1983 .

[22]  Hamid R. Ronagh,et al.  Calculation of Eigenvectors with Uniform Accuracy , 1995 .

[23]  F. Williams,et al.  CALCULATING THE MODES FROM DYNAMIC STIFFNESS MATRIX ANALYSIS OF PIECEWISE CONTINUOUS STRUCTURES , 1999 .