Hybrid state-space time integration of rotating beams

Steen Krenk, Martin Bjerre Nielsen

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An efficient time integration algorithm for the dynamic equations of flexible beams in a rotating frame of reference is presented. The equations of motion are formulated in a hybrid state-space format in terms of local displacements and local components of the absolute velocity. With inspiration from Hamiltonian mechanics, where displacement and momentum have similar roles, both sets of state-space variables are interpolated between nodes by the same shape functions, leading to a general format where all inertia effects are represented via the classic constant mass matrix, while effects of the system rotation enter via global operations with the angular velocity vector. The algorithm is based on an integrated form of the equations of motion with energy and momentum conserving properties, if a kinematically consistent non-linear formulation is used. A consistent monotonic scheme for algorithmic energy dissipation in terms of local displacements and velocities, typical of structural vibrations, is developed and implemented in the form of forward weighting of appropriate mean value terms in the algorithm. The algorithm is implemented for a beam theory with consistent quadratic non-linear kinematics, valid for moderate finite rotations. The equations of this non-linear beam theory are generated in explicit form by extension of the constitutive stiffness and the geometric stiffness of a linear beam theory at the element level. The performance of the algorithm is illustrated by numerical examples.
Original languageEnglish
JournalComputer Methods in Applied Mechanics and Engineering
Pages (from-to)243-254
Publication statusPublished - 2012


  • Conservative time integration
  • Beam elements
  • Structural dynamics
  • Algorithmic dissipation
  • Dynamics in rotating frame

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