A method of design against vibration resonance of beams and shafts

We consider problems of designing vibrating beams or rotating shafts to obtain as large gaps as possible in the spectra of natural frequencies or critical speeds in order to avoid resonance or whirling instability for a wide range of external excitation frequencies or service speeds. Governing equations for such design problems are derived by variational analysis of a bound formulation, and a numerical solution to an example problem is discussed.