Damped second‐order Rayleigh‐Timoshenko beam vibration in space—an exact complex dynamic member stiffness matrix

Abstract

SUMMARYA uniform linear beam in a uniform linear ambient medium is studied. The beam performs stationary harmonic damped nonsynchronous space vibration in simultaneous tension, torsion, bending and shear in the presence of a large static axial load. Hysteretic and viscous dampings of the beam material and ambient medium are considered. Generalized complex KolouSek functions are derived. A 12 x 12 complex symmetric stiffness matrix is established for a supported beam member excited at its ends by prescribed harmonic translations and rotations which have the same frequency but may be out of phase. This matrix allows for an exact analysis of nonproportionally damped built-up beam structures, thus avoiding assumed mode shapes and lumped or consistent masses. A general notation is suggested. Numerical examples are given, including applications of the computer program SFVIBAT-DAMP. OBJECTS AND RESULTSThe main object of this study is to establish, for the damped uniform beam member in Figure 1 supported by a damped uniform ambient medium (a distributed Winkler-type foundation), the 12 x 12 complex member stiffness matrix F* = F*(w) of the relation (* = complex quantity)

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