Dynamic stiffness of hollowed cylindrical rubber vibration isolators — The wave-guide solution

The dynamic stiffness of hollowed cylindrical vibration isolators using a wave-guide modelling approach is given. The isolators consist of rubber and metal elements in series. The boundary conditions at the lateral and radial surfaces of each rubber component are locally non-mixed and simultaneously satisfied by using the modes corresponding to the dispersion relation for axial waves in infinite hollow cylinders while the metal plates are assumed rigid, following Newton’s second law. The modes of the rubber elements exactly satisfy the stress free boundary conditions at the curved radial boundaries, while the displacement conditions on the flat cylinder ends are satisfied in mean by a mode matching approach. The rubber is modelled as nearly incompressible with deviatoric visco–elasticity based on a fractional derivative, standard linear solid. The stiffness is found to depend strongly on frequency, displaying resonances and anti-resonances. Further, results indicate a great potential for optimising vibration isolators of the studied kind. By choosing appropriate combinations of metal plates and rubber elements, the dynamic transfer stiffness in higher frequency bands can be suppressed to be magnitudes lower than the static stiffness, a highly sought feature in vibration isolators. Finally, simplified models including mass-spring systems and the long rod theory are implemented and compared to the derived formulation.