Acoustic scattering in circular cylindrical shells: a modal approach based on a generalised orthogonality relation

Abstract

During the past 60 years fluid-structure interaction in a wide range of three dimensional circular cylinder problems have been studied. Initial problems considered a rigid wall structure which were solved using impedance model comparisons. Soon after, further solution techniques were used, such as computer simulation, transfer matrix methods and finite element techniques. However such problems were only valid for low frequencies when compared with experiments, this was because that did not include higher order modes. The importance of higher order modes was then established and studies have since included these modes. More recently, mode matching methods have been used to find the amplitudes of waves in structures comprising two or more ducts. This has been done with using an orthogonality relation to find integrals which occur from the application this method. This methodology is demonstrated in as background information and is applied to prototype problems formed of rigid ducts. The rigid duct theory led to the consideration of elastic shells, of which several shell modelling equations were available from the vibration theory. In this thesis, the Donnell-Mustari equations of motion are used to model thin, elastic, fluid-loaded shells of circular cross-section. It is demonstrated that generalised orthogonality relations exist for such shells. Two such relations are found: one for shells subject to axisymmetric motion and one for shells subject to non-axisymmetric motion. These generalised orthogonality relations are new to the field of acoustics and are specific to shells modelled with the Donnell-Mustari equations of motion. The mode matching method is used to find the amplitudes of waves propagating in prototype problems and the generalised orthogonality relations are used to find integrals which occur through this method. Expressions for energy for all considered structure types are used to find the resulting energy for each prototype problem and results for equivalent problems are compared. In addition, verification of the resulting amplitudes is done by ensuring that the matching conditions are suitably satisfied. It is anticipated that the method will have application to the understanding and control of the vibration of cylindrical casings such as those enclosing turbo-machinery. Another application of the method would be the tuning of cylindrical casings, such as those featured on car exhaust systems or HVAC (heating, ventilation and air conditioning) systems.