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Combined Linear Aeroelastic and Aero-viscoelastic Effects in da Vinci--Euler--Bernoulli and Timoshenko Spars (Beams) with Random Properties, Loads and Physical Starting Transients, and with Moving Shear Centers and Neutral Axes. Part III: Nonlinear Cons
Harry H. Hilton, David Du, Eric Brzeski, Andy Song

Last modified: 2023-06-29


The present paper is a continuation and expansion of  previous Parts I  and II  to emphasize nonlinear material constitutive, failure criteria, second order strain displacement relations, nonlinear deformations, and nonlinear aerodynamic behaviors  in conjunction with aero - elastic/viscoelastic phenomena. It is the last of the trilogy on the combined plunging/in plane bending and twisting of wings under level flight as well as pitching, rolling and yawing maneuvers about three normal axes.

Absent any existence or uniqueness theorem, any and all solutions to nonlinear PDEs and IPDEs, including those presented here, must be viewed with reserve and extreme caution. At this time, such solutions can only verified and authenticated by extensive multi-dimensional experimental data, which unfortunately is missing even for material properties  due to the unavailability of proper multi-D testing equipment.

Nonlinear constitutive and failure relations are developed ready to receive deterministic and stochastic experimental data to match through least square protocols the now open analytical parameters and complete the material behavior descriptions. Additionally, nonlinear lift and drag coefficients are included to allow for realistic high angle of attack representations, including stall and post-stall phenomena.

The analytical protocols of Poincare's and Poincare-Lighthill-Kuo's successive approximations and the numerical Runge-Kutta shooting procedure for boundary conditions are also developed.

Examples are provided of a wing with nonlinear elastic and/or viscoelastic spars under nonlinear bending and torsion including Timoshenko effects, and moving shear center and neutral axis. The analyses demonstrate the behavior of isotropic spars may depend on 28 variables, while a full anisotropic one calls for 168 variables. Under either conditions, graphical representations of more than three axes (variables) are not practical unless one turns to Inselberg's parallel coordinates as was done with some of the results produced as part of this project.