ON THE WAVEGUIDE MODELLING OF DYNAMIC STIFFNESS OF CYLINDRICAL VIBRATION ISOLATORS. PART II: THE DISPERSION RELATION SOLUTION, CONVERGENCE ANALYSIS AND COMPARISON WITH SIMPLE MODELS

Abstract

Based on a waveguide model presented in a companion paper (L. Kari 2001 Journal of Sound and Vibration244, 211–233 [1]), the influences of higher order modes and structure-borne sound dispersion on the axial dynamic stiffness for cylindrical vibration isolators are investigated. On the whole, a moderate mode number results in an accurate stiffness prediction while an accurate stress point value prediction requires more modes. The dispersion relation is solved by a modified Newton–Raphson method with initial values given by an asymptotic expansion or a winding integral method. The integral technique is based on the argument principle; but, as the square root operators in the dispersion relation yield branch points, some modifications are needed. To create single-valued functions conforming to the argument principle, the winding integral search domain is split into branch cut absent subdomains, containing adaptively defined square root operators. The subregion method used for the fulfilment of the boundary conditions at the lateral surfaces is shown to converge faster than for the point-matching method. However, the latter reveals a similar convergence rate as the former at overdetermination. Comparisons with simple stiffness models are made. These models, known as the long rod, the Love, the Bishop, the Kynch, the Mindlin and Herrmann and the Mindlin and McNiven theories are shown to diverge substantially from the presented “exact” theory. To a great extent, the pertinent stress and displacement fields, derived from the presented waveguide model, explain the discrepancies reported for the approximate theories.