Damped second-order Rayleigh-Timoshenko semi-infinite beam vibration—an exact complex dynamic member stiffness matrix

A uniform linear semi-infinite beam in a uniform linear ambient medium is studied. The beam performs stationary harmonic damped non-synchronous space vibration in simultaneous tension, torsion, bending and shear. Hysteretic and viscous dampings of the beam material and ambient medium are considered. Four new generalized complex Kolousek functions are derived. A 6 × 6 complex symmetric stiffness matrix is established for a semi-infinite beam member excited at its end by prescribed harmonic translations and rotations which have the same frequency but may be out of phase. This matrix extends the range of application of the so called ‘exact analysis’ of non-proportionally damped built-up beam structures as described in a previous paper by Lundén and Åkesson.¹ Numerical examples are given, including applications of the computer program SFVIBAT-DAMP. Power flows are studied.