A Logarithmic Complexity Divide-and-conquer Algorithm Formulti-flexible Articulated Body Dynamics

This paper presents an efficient algorithm for the parallel implementation of dynamics simulation and analysis of multi-flexible-body systems. This algorithm formulates and solves the nonlinear equations of motion for mechanical systems with interconnected flexible bodies subject to the limitations of modal superposition, and body substructuring, with arbitrarily large rotations and translations. The large rotations or translations are modelled as rigid body degrees of freedom associated with the interconnecting kinematic joint degrees of freedom. The elastic deformation of the component bodies is modelled through the use of modal coordinates and associated admissible shape functions. Apart from the approximation associated with the elastic deformations, this algorithm is exact, non-iterative and applicable to generalized multiflexible chain and tree topologies. In its basic form, the algorithm is both time and processor optimal in its treatment of the nb joint variables, providing O(log(nb)) turn around time per temporal integration step, achieved with O(nb) processors. The actual cost associated with the parallel treatment of the nf flexible degrees of freedom depends on the specific parallel method chosen for dealing with the inversion of the individual coefficient matrices which are associated locally with each flexible body. 1 To appear in ASME Journal of Computational and Nonlinear Dynamics Rudranarayan M. Mukherjee, Kurt S. Anderson r Position vector V Spatial velocity, a 6 × 1 column matrix A Spatial acceleration, a 6 × 1 column matrix ω Angular velocity v Translational velocity of a body reference point α Angular acceleration a Translational acceleration of a body reference point φ Admissible shape function for translational components of deformation ψ Admissible shape function for rotational components of deformation Φ Spatial matrix containing the shape functions [ ψ φ ] qi k i-th Modal Coordinate of a flexible body k q̇ k i Time derivative of i-th modal coordinate of a flexible body k q̈ k i Second time derivative of i-th modal coordinate of a flexible body k ρ Mass density ζ Inertia coupling terms for individual body or subassembly Υ Inertia coupling terms for assembly nb Number of bodies in the system ν Number of assumed mode shapes for a flexible body k H Joint motion subspace map of joint connecting body k+1 to body k D Orthogonal complement of H u Generalized relative speed u̇ Time derivative of generalized relative speed N ωr k+ r-th partial angular velocity of body k w.r.t frame N N vr k+ r-th partial velocity of body k w.r.t frame N P k r r-th spatial partial velocity of body k w.r.t frame N S Shift Matrix between joint k and k+1 [ U r× 0 U ] b× 3×3 skew symmetric matrix for cross product of any vector b C Transpose of any arbitrary matrix C Fc i Spatial constraint force at joint i τc i Constraint torque at joint i fc i Constraint force at joint i τ̃c i Measure numbers of constraint torque at joint i f̃c i Measure numbers of constraint force at joint i f Body force at point p k̂i Unit vector in direction i U 3× 3 Identity matrix 0 3× 3 Zero matrix φ|P φ evaluated at point P Table 1: The Nomenclature 2 To appear in ASME Journal of Computational and Nonlinear Dynamics