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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, (

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