Altering structures to move vibration frequencies using acceptable changes

Structures vibrate with discrete natural frequencies, which can be moved by tuning the structural stiffness, or the mass, or both. Changes might clear any unwanted frequencies from a given range, but they might also alter the structure so that it no longer serves its purpose, and clearly be unacceptable. This paper starts by considering a basic problem of moving a single frequency from a band, together with a constraint on the acceptability of any changes. The frequency clearance problem is always solvable, but may not be with the added constraint. This is shown by constructing an unsolvable problem, settling the question of universal solvability in a very straightforward way, but revealing little about the relationship between the unconstrained and constrained problems, and under what circumstances the constrained problem has solutions. Changes producing the desired frequency clearance, and those that are acceptable form sets in the vector space of the structure freedoms, and descriptions of these sets are developed. Solutions of the constrained problem, which are intersections of these sets, are then considered. These are developed in eigenspaces, where they are simply written, but are computationally intensive. The paper concludes by showing how relevant calculations, formulated in an eigenspace, can be evaluated while knowing very little about this space that defines them.