Approximate Vibration Reanalysis of Structures

Abstract

Thecombinedapproximationsmethodisdevelopedforvibrationreanalysisofstructures.Inthesolution process, the terms of local (series) approximations are used as basis vectors in global (reduced basis ) approximations. The solution steps are straightforward, and the method can be readily used with a general e nite element system. The computational effort needed by the method is usually much smaller than the effort needed for complete vibration analysis. The efe ciency of the calculations and the accuracy of the results can be controlled by the amount of information considered. Efe cient solutions are obtained by low-order approximations and accurate results can be achievedbyconsideringhigher-orderterms.Numericalexamplesshowthataccurateresultsareachievedefe ciently for very large changes in the design. Nomenclature B = matrix dee ned by Eq. (21) C = matrix dee ned by Eq. (16) E = modulus of elasticity I = moment of inertia I = identity matrix K = stiffness matrix KR = reduced stiffness matrix L = length M = mass matrix MR = reduced mass matrix R = right-hand-side vector r = displacement vector rB = matrix of basis vectors s = number of basis vectors U = upper triangular matrix y = vector of coefe cients ® = coefe cient dee ned by Eq. (35) ¯ = angle between vectors 1K = change in the stiffness matrix 1M = change in the mass matrix ± = Kronecker delta ¸ = eigenvalue ! = circular frequency