Fluid-conveying damped Rayleigh-Timoshenko beams in transverse vibration analyzed by use of an exact finite element part I: Theory

Abstract

Free and forced nonsynchronous harmonic linear vibrations of piecewise uniform straight beam structures supported by a Winkler-type ambient medium and conveying a piecewise constant-speed plug flow of material along the deflected beam axis are studied by means of finite elements. Hysteretic and viscous damping in the beam material and ambient medium are considered. Large static axial loads may act on the beam and fluid calling for use of a second-order theory. In a companion paper, exact complex-valued nonsymmetric transcendentally frequency-dependent member stiffness matrices have been established for uniform Rayleigh-Timoshenko beams, uniform Euler-Bernoulli beams (with infinite shear stiffness), and also for uniform strings (with zero bending stiffness). In the present paper, the corresponding approximate member stiffness matrices based on standard polynomial shape functions are given. Fluid-conveying rigid bodies are treated. Six numerical examples are presented for beam structures. Comparisons between results found by use of exact and approximate finite elements are performed. Forced and free harmonic vibrations are studied including the calculation of critical fluid velocities and mechanical power flows.