A recursive formulation for the dynamic analysis of open-loop deformable multibody systems

A recursive nonlinear lagrangian formulation for the spatial kinematic and dynamic analysis of open chain mechanical systems containing interconnected deformable bodies, each of which may undergo large translational and rotational displacements, is developed. While approximation techniques such as the finite element, Rayleigh-Ritz methods, or experimentally identified modal parameters can be used to introduce the elastic coordinates that describe the deformation of the bodies with respect to selected body references, the large relative motion between the neighboring deformable bodies are described by the large relative translational and rotational displacements between a set of intermediate coordinate systems using a minimum set of relative translational and rotational coordinates. The nonlinear terms that represent the dynamic coupling between the large relative motion and the small elastic deformations are identified and presented in terms of a set of time-invariant quantities that depend on the assumed displacement field. These sets of invariant matrices provide a systematic approach to study the spatial dynamics of open loop kinematic chains. The system differential equations are developed in terms of these invariants using Lagrange's equation of motion.; In order to demonstrate the potential use of the nonlinear recursive formulation proposed in this thesis, a computer algorithm is developed and used to determine the nonlinear vibration response of a coupled rotor-fuselage system of a helicopter model. This displacement of the deformable rotor is expressed in terms of a set of mode shapes obtained using the finite element method. The differential equations of motion are expressed in terms of a coupled set of relative joint variables and modal elastic coordinates. These equations are integrated forward in time using a direct numerical integration method. Two cases are considered. In Case A, the rotor lies in a plane normal to the rotor axis of rotation while in Case B, it is assumed that the rotor has built in coining angle at its inner attachment point. It is shown that the elastic rotors are in general stiffer in the in-place direction (Case A) than in out-plane direction and the natural frequencies are correspondingly higher. By comparing the two cases, it is also concluded that decreasing the coining angle of the blade as well as design of the rotor blades so that their mass is confined to the plane of rotation leads to a more stable solution. It is also shown that by adding damping, a more stable coupled rotor-fuselage system can be achieved.