Nonlinear statics and dynamics of a simply supported nonuniform tube conveying an incompressible inviscid fluid

Nonlinear static and dynamic behaviour of a simply supported fluid-conveying tube, which has a constant inner diameter and a variable thickness is analysed analytically and numerically. Nonlinear static bending is considered in two loading cases: (i) a tube subjected to supercritical axial compressive forces acting at its edges or (ii) a tube loaded by concentrated bending moments, which provide a symmetrical (with respect to the mid-span) shape of a tube. The nonlinear governing equations of motions are derived by using Hamilton's principle. The elementary plug flow theory of an incompressible inviscid fluid is adopted for modelling a fluid–structure interaction. The flow velocity is taken as the sum of a principal constant ‘mean’ velocity component and a fairly small pulsating component. Firstly, eigenfrequencies and eigenmodes of a deformed tube are found from linearised equations of motions. Then resonant nonlinear oscillations of a tube about its deformed static equilibrium position in a plane of static bending are considered. A multiple scales method is used and a weak resonant excitation by the flow pulsation is considered in a single-mode regime and in a bi-modal regime (in the case of an internal parametric resonance) and the stability of each of them is examined. The brief parametric study of these regimes of motions is carried out.