Combined Linear Aeroelastic and Aero-viscoelastic Effects in da Vinci--Euler--Bernoulli and Timoshenko Beams (Spars) with Random Properties, Loads and Physical Starting Transients, and with Moving Shear Centers and Neutral Axes. Part I: Theoretical

Abstract

Following the unified approach to viscoelastic Timoshenko beams (spars) formulated in \bodc{hh18mesa}  similar inclusive analyses are formulated  for elastic and viscoelastic combined unsymmetrical bending-torsion during level flight and for vehicle rolling motion. The overall bending degrees of freedom considered are plunging, in plane and chord-wise motions.  Bending-torsion effects on and changes in angles of attack due the rolling velocity as well as the influence of moving shear centers and neutral axes and of material failures are considered during simultaneous occurrences. The final goal is to establish conditions for bending and torsional flutter, torsional divergence, control effectiveness and ultimate survival time of the wing due to material failures and structural instabilities (buckling) with future extensions to the entire vehicle under the rubric of  system of systems (SoS) approach, leading to a single pair of critical velocities and frequencies including material failure effects. A new stress invariant stochastic generalization  to the original Shanley-Ryder stress ratio failure criterion is derived and utilized. The latter has the advantage of having an unlimited number of arbitrary coefficients to be used to in fitting analytical expressions to stochastic experimental data. The multi-D numerical example results are displayed as a single figure of multiple 2--D parallel coordinates (\parc1 \hspace{-4.65pt}), as opposed to numerous simultaneous, but separate, 2--D traces of  a multi--D aeroelastic/aero--viscoelastic combined stability, buckling and material failure surface. In the present analyses, the use of \parc1 clearly graphically demonstrates the individual and collective influences of many parameters on critical velocities without recourse to a multidimensional critical surface representation.The critical velocities are also displayed as separate 2--D traces for each of the divers parameters, as well as in  parc1 renderings. A small sample of randomly chosen subsonic wing parametric variation calculations show that compared to the free standing bending-torsion configuration,  combinations involving plunging and in-plane bending, control effectiveness and reversal, with or without positive or negative roll velocities, and absent or present Timoshenko effects, produce substantially altered flutter velocities and their paired frequencies.