INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2011; 87:1301–1324 Published online 7 March 2011 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/nme.3169 Hybrid state-space time integration in a rotating frame of reference Steen Krenk ∗, † and Martin B. Nielsen Department of Mechanical Engineering, Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark SUMMARY A time integration algorithm is developed for the equations of motion of a flexible body in a rotating frame of reference. The equations are formulated in a hybrid state-space, formed by the local displacement components and the global velocity components. In the spatial discretization the local displacements and the global velocities are represented by the same shape functions. This leads to a simple generalization of the corresponding equations of motion in a stationary frame in which all inertial effects are represented via the classic global mass matrix. The formulation introduces two gyroscopic terms, while the centrifugal forces are represented implicitly via the hybrid state-space format. An angular momentum and energy conserving algorithm is developed, in which the angular velocity of the frame is represented by its mean value. A consistent algorithmic damping scheme is identified by applying the conservative algorithm to a decaying response, which is rendered stationary by an increasing exponential factor that compensates the decay. The algorithmic damping is implemented by introducing forward weighting of the mean values appearing in the algorithm. Numerical examples illustrate the simplicity and accuracy of the algorithm. Copyright  2011 John Wiley & Sons, Ltd. Received 18 October 2010; Revised 20 January 2011; Accepted 20 January 2011 KEY WORDS: conservative time integration; dynamics in rotating frame; energy conservation; algorithmic energy dissipation; structural dynamics 1. INTRODUCTION Conservative time-integration algorithms for the equations of dynamics have received an increasing interest over the last two decades. The basic idea is to develop the algorithm to reproduce the integrated equations of motion in such a way that exact conservation properties are obtained for the invariants of the problem, such as momentum and energy [1]. The discretization of time in the equations leads to lack of ability to resolve phenomena at frequencies exceeding the Nyquist frequency, and the conservative algorithms are, therefore, often extended to include algorithmic dissipation terms, aiming at attenuating the part of the response at the higher frequencies. The basic idea of energy conserving algorithms for kinematically non-linear solids and structures was introduced by Simo et al. [2–4]. A central idea is that a representative value over a time increment should integrate to the correct momentum or energy increment over the interval. This can be formalized in the notion of a ‘finite time derivative’ [5]. In particular, linear elastic models in which the energy is quadratic in the strains, which are quadratic in the displacements, lead to a simple hierarchy of mean values, where the representative internal force can be expressed in explicit form as a standard mean plus a correction in terms of the global geometric stiffness [6]. Conservative ∗ Correspondence to: Steen Krenk, Department of Mechanical Engineering, Technical University of Denmark, DK- 2800 Kgs. Lyngby, Denmark. † E-mail: sk@mek.dtu.dk Copyright  2011 John Wiley & Sons, Ltd.