Mass condensation and simultaneous iteration for vibration problems

“…(1) this approximate value for the second eigenvalue followed by application of Eqs. (2)(3)(4)(5)(6)(7) to obtain now a virtually exact second eigenvalue and an approximation to the third eigenvalue. The continuation of this process of solution will result in the virtually exact eigensolution for the lowest p eigenvalues and corresponding p eigenvectors of the original system.…”

Section: Algorithm For Dynamic Condensationmentioning

confidence: 99%

“…(1) may be written as (2) in which [7] is the transformation matrix defined in the following relation (4) where [K] and [M\ are, respectively, the stiffness and mass matrices of the reduced system. The reduced mass matrix is calculated from and [D] the dynamic matrix which satisfies the relation where (5) (6) while the reduced stiffness matrix is determined using Eq. (4).…”

Section: Algorithm For Dynamic Condensationmentioning

confidence: 99%

“…2 ' 6 The majority of previous investigations on this subject is limited to the case with the environment temperature at absolute zero. More recently, Faghri and Sparrow 7 applied a numerical scheme developed earlier by Patankar and Spalding to solve the problem of laminar flow in a horizontal pipe subjected to external natural convection and radiation.…”

Section: Introductionmentioning

confidence: 99%

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Dynamic Condensation

Paz

1984

AIAA Journal

141

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The relationship between this perturbation method and the direct differential methods now will be clarified.The variable e may be equated to an increment dv k in the Ath design variable V k . Consequently the matrix Q becomes equal to the variation in (\A + E) due to a unit change in V k .We may then compare the power series (4) and (5) with the corresponding Taylor series. (17) giving ax, dV k Wtr (18) (19)close eigenvalues, then the second eigenvalue sensitivity will be large.The second-order eigenvector sensitivity involves /5 /y from Eq. (16). In this case, the frequency difference is represented by (X, -X y ) 2 in the denominator. Hence, whatever condition (good or bad) is present in the second eigenvalue sensitivity will be exaggerated in the second eigenvector sensitivity. ConclusionsThe extrapolation of first-order sensitivities for gross changes is dependent on the size of the second-order sensitivity (which must be negligible).The second-order sensitivity of the eigenvalue may be computed from knowledge of the first-order sensitivity of eigenvectors. A necessary (although not sufficient) condition for this second-order eigenvalue sensitivity to be negligible is the sufficient separation of the chosen eigenvalues from all other eigenvalues.the results which were to be established here. Physical Significance of Higher-Order SensitivitiesThe authors previously cited have remarked at the intuitive appeal of the first-order sensitivities. The sensitivity of frequency (or X) can be computed solely on the basis of knowledge of the associated eigenvector. The eigenvector sensitivity can be computed similarly using data which are natural products of previous analyses. It should be noted, however, that the term (X, -X 7 ) in the denominator of Eq. (14) shows that numerical difficulties may be encountered for relative sensitivities of eigenvectors with close natural frequencies. The equation will degenerate for X, = X y and Eq. (14) cannot be used. The paper by Lancaster provides analyses which may be useful if this occurs.Turning now to the second-order sensitivities and discussing only the frequency sensitivity, it can be seen that the second-order sensitivity of the eigenvalue involves the first-order sensitivity of the eigenvector. Thus, if the first eigenvector sensitivity is large (poorly conditioned) due to

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“…The decoupled vectors W then follow from the matrix multiplication W = VT* ( 8 ) (The off-diagonal elements of T* are skew-symmetric.) In the present algorithm V is initially copied into the store for W and then, for each coefficient t$(j < i), t$vj is added to wi and tijvi is subtracted from wj as soon as t; has been formed.…”

Section: Simultaneous Iterationmentioning

confidence: 99%

A simultaneous iteration algorithm for symmetric eigenvalue problems

Corr

Jennings

1976

Numerical Meth Engineering

149

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SUMMARYA FORTRAN IV algorithm is presented for determining sets of dominant eigenvalues and corresponding eigenvectors of symmetric matrices. It is also extended to the solution of the equations of natural vibration of a structure for which symmetric stiffness and mass matrices are available. The matrices are stored and processed in variable bandwidth form, thus enabling advantage to be gained from sparseness in the equations. Some of the procedures may also be used to solve symmetric positive definite equations such as those arising from the static analysis of structures loaded within the elastic range. SIMULTANEOUS ITERATIONSimultaneous iteration is an extension of the power method which avoids the deflation process otherwise involved in obtaining the subdominant eigenvalues. Although it was first adopted by Bauer in 1958 it was not until fast convergence rates were obtained by Jennings' in 1967 and Rutishauser2 in 1969 that the method became of much practical use. The present algorithm is based on Jennings' method and has been tested on structural problems over a period of several years in almost its present form.3 A version of the method which includes the use of backing store has been used to solve problems of several thousand variables on ICL 1907 and 1906s computers. The following is a brief description of the main operations involved in the partial eigensolution of a symmetric matrix. Consider a symmetric matrix A having eigenvalues Izi such that lL1l 2

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Mass condensation and simultaneous iteration for vibration problems

Cited by 14 publications

References 14 publications

The Time Dimension: Semi-Discretization of Field and Dynamic Problems

The Time Dimension: Semi-Discretization of Field and Dynamic Problems

Dynamic Condensation

A simultaneous iteration algorithm for symmetric eigenvalue problems

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