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Research Papers

# Application of the First Order Generalized-$α$ Method to the Solution of an Intrinsic Geometrically Exact Model of Rotor Blade Systems

[+] Author and Article Information
F. Khouli

Department of Mechanical and Aerospace Engineering, Carleton University, 1125 Colonel By Drive, Ottawa, ON K1S 5B6, Canadafkhouli@connect.carleton.ca

F. F. Afagh, R. G. Langlois

Department of Mechanical and Aerospace Engineering, Carleton University, 1125 Colonel By Drive, Ottawa, ON K1S 5B6, Canada

J. Comput. Nonlinear Dynam 4(1), 011006 (Nov 12, 2008) (12 pages) doi:10.1115/1.3007972 History: Received August 19, 2007; Revised May 08, 2008; Published November 12, 2008

## Abstract

An energy decaying integration scheme for an intrinsic, geometrically exact, multibody dynamics model with composite, dimensionally reducible, active beamlike structures is proposed. The scheme is based on the first order generalized-$α$ method that was proposed and successfully applied to various nonlinear dynamics models. The similarities and the differences between the mathematical structure of the nonlinear intrinsic model and a parallel nonlinear mixed model of chains are highlighted to demonstrate the effect of the form of the governing equation on the stability of the integration scheme. Simple $C°$ shape functions are used in the spatial discretization of the state variables owing to the weak form of the model. Numerical solution of the transient behavior of multibody systems, representative of various rotor blade system configurations, is presented to highlight the advantages and the drawbacks of the integration scheme. Simulation predictions are compared against experimental results whenever the latter is available to verify the implementation. The suitability and the robustness of the proposed integration scheme are then established based on satisfying two conservational laws derived from the intrinsic model, which demonstrate the retained energy decaying characteristic of the scheme and its unconditional stability when applied to the intrinsic nonlinear problem, and the dependance of its success on the form of the governing equations.

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## Figures

Figure 1

Model of an articulated rotor blade configuration. All joints are assumed to be simple revolute joints and articulation links to be rigid.

Figure 2

Time histories of (a) the blade normalized total mechanical energy and its difference after the loading is ceased and (b) the second intrinsic conservation law [lb in./sec] of the blade after the loading is ceased of case 2(a)

Figure 3

Time histories of (a) the axial force F1 and (b) the shear flapping force of the blade F3 of the 25th beam element after the loading is ceased of case 2(a)

Figure 4

Time histories of (a) the normalized total mechanical energy of the blade during the initial steps of the simulation and (b) the normalized total mechanical energy of the blade during the final steps of the simulation of cases 2(b), 2(c), and 2(d)

Figure 5

Time history of the second intrinsic conservation law of the blade after the loading is ceased of cases 2(b), 2(c), and 2(d)

Figure 6

Time history of the total mechanical energy of the fully articulated rotor system of case 3

Figure 7

Time histories of (a) the shear flapping force F3, (b) the tip flap displacement u3, and (c) the flap hinge angle ϕflap of case 3

Figure 8

Normalized relative error based on the signal rms of (a) Cases 4 using the linear flap velocity V3, the tip displacement u3, and the shear flapping force F3 of the 25th beam element, and (b) the second intrinsic conservation law

Figure 9

(a) Normalized FFT of the tip displacement u3 of cases 5(a), 5(b) and 5(c); (b) time history of the twisting moment of the second beam element of cases 5(a) and 5(c)

Figure 10

Time histories of (a) The experimental/theoretical flap hinge angle and (b) the experimental/theoretical strain at 40% blade length station of the drop test/droop stop impact from 9.7deg

Figure 11

Size of the time step used by the solver during the initial stages of the simulation of case 2(c)

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