Tuning of vibration absorbers and Helmholtz resonators based on modal density/overlap parameters of distributed mechanical and acoustic systems

This paper presents a study on the tuning of vibration absorbers and Helmholtz resonators used to control at target resonance frequencies respectively the flexural response of lightly damped distributed structures and the acoustic response of lightly damped cavities subject to broadband stochastic excitations. The vibration and acoustic control effects of these two devices are assessed considering the broadband flexural response and acoustic response of typical structures and enclosures encountered in practical applications, that is: beam, thin plate and thin walled cylinder structures and one-dimensional duct, two-dimensional slender volume, three-dimensional volume enclosures. The herein proposed multi-mode tuning approaches are based on the H₂ cost functions of the total flexural kinetic energy of the three structures and of the total acoustic potential energy of the three enclosures. The H₂ cost functions are defined within finite frequency bands centred at the target structural or acoustic resonance frequencies respectively. A comprehensive overview of the modal density and modal overlap functions for the flexural response of the three structures and for the acoustic response of the three enclosures is also presented to provide physical interpretations and practical guidelines on the performance and applicability of classical single-mode and the proposed multi-mode tuning approaches. The study shows that classical tuning laws can be straightforwardly employed to control the response at low resonance frequencies such that the modal overlap is not greater than one, whereas to control the response at higher resonance frequencies, where the modal overlap is significantly greater than one, the proposed multi-mode tuning approach should be adopted.