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In order to study dynamics of flapping wing moving at low Reynolds number in ambient fluid, we adopt geometric modelling approach of fully coupled wing-fluid system, incorporating Boundary Integral Method for calculating added masses, and Lie group rigid body integrator. Our aim is to explore numerical advantages of such an approach in comparison to the standard procedures that comprise volume discretization of a fluid domain.

By using principles of geometric mechanics - under assumption of its Hamiltonian structure - it is possible to treat solid-fluid system in a monolithic way without separating its fluid and solid part, as it is standard procedure in the contemporary computational practice. This strategy involves symplectic and Lie-Poisson reduction of the airfoil-fluid Hamiltonian system and allows for significantly improved numerical efficiency, since full discretization of the fluid domain becomes non-obligatory. However, in order to take into account viscous effects of fluid flow, numerical imposing of Kutta condition on the airfoil trailing edge is mandatory. To this end, a new hybrid approach - combining analytical and numerical modeling techniques as well as model reduction procedures - for dynamics simulation of airfoil motion in fluid flow with vortices will be described.

Unit quaternion representation is widely used in flight simulation to overcome the limitations of the standard numerical ordinary-differential- equations (ODEs) based on three-parameters rotation variables (such as Euler angels), as they may impose kinematic singularities. However, these benefits do not come without a price, since the classical way of integrating rotational quaternions includes solving of differential- algebraic equations (DAEs) that requires numerical stabilization of the additional algebraic constraint enforcing the quaternion unitary norm. This can pose a problem in the case of longer flight simulations since improper numerical treatment of the quaternion- normalisation constraint may induce numerical drift into the simulation results. As a remedy, the proposed novel algorithm circumvents DAE problem of quaternion integration by shifting update-integration-process from configuration manifold to the local tangential level of the incremental rotations. This can be done due to the isomorphism of the Lie algebras of the SO(3) group and the unit quaternion Sp(1) group. Besides avoiding DAE formulation by reducing integration process to standard three ODEs problem (by using incremental rotation vector as integration variable at Lie algebra level), the proposed algorithm may also exhibit numerical advantages. Depending on the numerical case at hand - as it is in the presented flight example with 'slower' dynamics and steady rotation component - the method allows for utilization of longer integration steps and overall better numerical accuracy. This is because numerical integration of three non-linear ODEs in terms of incremental rotation vector within the proposed novel algorithm follows a non-linear kinematical rotation update more accurately than integration of a four (linear!) ODEs (as it appears in the standard algorithm), resulting in more accurate simulation results. The effect of better numerical accuracy will be more accented if overall flight vehicle dynamics allows for longer integration steps and aircraft motion pattern involves steady rotational component within its 3D motion.

The paper surveys geometric mathematical framework for computational modeling of multibody system dynamics. Starting with the configuration space of rigid body motion and analysis of it’s Lie group structure, the elements of respective Lie algebra are addressed and basic relations pertinent to geometrical formulations of multibody system dynamics are surveyed. Dynamical model of multibody system on manifold introduced, along with the outline of geometric characteristics of holonomic and non-holonomic kinematical constraints.