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Stabilization of periodic flap-lag dynamics in rotorcraft. (English) Zbl 0981.70021
Moon, Francis C. (ed.), IUTAM symposium on new applications of nonlinear and chaotic dynamics in mechanics. Proceedings of the IUTAM symposium held in Ithaca, NY, USA, July 27-August 1, 1997. Dordrecht: Kluwer Academic Publishers. Solid Mech. Appl. 63, 493-502 (1999).
Summary: An approach developed to stabilize periodic orbits in chaotic attractors is generalized and applied to nonchaotic flap-lag instability in helicopter rotor blades. The coupled flap-lag motion is described by the nonlinear system $$\ddot\beta+ \sin\beta \cos\beta (1+\dot \zeta)^2 +\omega^2_\beta \beta= \int^1_0 F_\beta rdr$$, $$\cos^2\beta \ddot\zeta-2\sin\beta \cos\beta(1+ \dot\zeta)\dot \beta+\omega^2_\zeta \zeta=\cos \beta \int^1_0 F_\zeta rdr$$ $$(\dot z\equiv dz/d\psi$$, $$\psi=\Omega t)$$, where $$\Omega$$ is angular velocity, $$\beta$$ is flap angle, $$\zeta$$ is lag angle, and $$F_\beta$$ and $$F_\zeta$$ are dimensionless airloads on the blade in flap and lag directions. The basic procedure of E. Ott, C. Grebogi and J. A. Yorke [Phys. Rev. Lett. 64, No. 11, 1196-1199 (1990); Erratum, No. 23, 2837 (1990; Zbl 0964.37501)] is extended in following ways: 1) the control law is optimized; 2) unstable orbits possessing two complex unstable eigenvalues are controlled; 3) the control is implemented without knowledge of fixed point location; 4) constraints on the performance in steady state are satisfied, and 5) control strategies are used to cope with lack of knowledge of state variables and also with delay in the implementation of control.
For the entire collection see [Zbl 0930.00076].
##### MSC:
 70Q05 Control of mechanical systems 70K55 Transition to stochasticity (chaotic behavior) for nonlinear problems in mechanics 70K20 Stability for nonlinear problems in mechanics 74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)