Nonlinear inverse perturbation method in dynamic analysis

Abstract

An analytical method is presented for the automated redesign of the modal characteristics of undamped mechanical systems. The method is based on a perturbation of the eigensystem, and all nonlinear terms are carried so that large changes can be considered. The user requests frequency and mode shape changes. The structural changes are expressed through design parameters a,-representing conventional thicknesses, moments of inertia, etc. A penalty function method is used in which the original objective function is a minimum weight condition and the penalty term is a properly normalized set of residual nodal force errors. Solutions converge well for typical example problems using five cycles of the conjugate gradient method. The method is superior to linear methods, which have shown error when significant mode shape changes are required. It is believed to be computationally superior to series solutions, such as Taylor's series, when only one or two frequencies and mode shapes are of interest. In addition, the method is not subject to a limited radius of convergence in the design variables as is encountered in series solutions. h [*],[*'] [Am], [Am, L M N P( ) [R] x [T] V [AX] [*],[«'] Nomenclature = coefficient matrices in Eq. (20) = modified objective function in penalty function method = original objective function = thickness of cantilever beam = assembled stiffness, original and perturbed = generalized stiffness, original and perturbed = element coefficient matrices in Eq. (20) = changes in stiffness in system and elements = assembled mass, original and perturbed = generalized mass, original and perturbed = element coefficient matrices in Eq. (20)= changes in mass, in system and elements = number of elements to be perturbed = number of modes used in N dimensional system, (<Af) = dimension of dynamic system = penalty term = residual error in force equilibrium = vector of variables in minimization problem = starting point and kih iteration point = design variables of element e in stiffness and mass changes = Ah e /h e fractional thickness change in element e -dimensionless frequency design variable, AX/X = normalization matrix, NxN -gradient operator = diagonal matrix of eigenvalues, original and perturbed = eigenvalue perturbation = penalty parameter = modal matrix, original and perturbed = specified and unspecified mode shape changes

Redesign of marine structures by perturbation

“…[mk] = X [mke]= --X [ke]ae (7) e=l e=l where several [k~]a~ terms may refer to the same element but different properties like bending, torsion, stretching, etc. For example, for a beam element we may have four change properties; that is, bending in two directions, torsion and axial tension.…”

“…These may be related directly to changes to displacements or indirectly to reduction of stress levels. Selecting out of eqn (8) only those degrees of freedom for which constraints on displacement changes are imposed we get a system of simultaneous nonlinear implicit equations and/or inequalities in %. Due to their complexity they must first be modified using the algorithm developed in Section 3 before they can be used to produce the values of aes by employing the solvers presented in Section 4.…”

“…For larger scale structures they used dynamic condensation which introduces an additional source of numerical error but reduces the prohibitively high cost of the standard methods. 8,9 In previous work the authors have developed algorithms for solution of the nonlinear perturbation redesign problem for only static 16 Perturbation-based redesign does not produce globally optimal design. Actually redesign problems may or may not be formulated as optimization problems depending on the number of constraints and allowable changes (see Section 4).…”

Redesign of plates by large admissible perturbations

“…The meaning of linear here is that differences in redesign variables and response between the initial and the improved structural states are small. Nonlinear perturbation methods 4 " 17 are used to find the objective state for large differences-of the order of 100%-between the two structural states. Perturbation methods calculate the objective state without trial and error or repeated finite element analyses (FEAs) by postprocessing the FEA results of the initial design.…”

Uncorrelated Modes Driven Inverse Eigensensitivity Method for Finite Element Model Updating

“…The use of eigendata sensitivity for analytical model updating in an iterative framework was first proposed by Collins et al [8], Kim et al [9] studied the effect of including secondorder sensitivities in this framework. Lin et al [ 10] employed both the analytical and the experimental modal data for evaluating sensitivity coefficients to frame updating equations.…”