Hybrid state‐space time integration in a rotating frame of reference

Abstract

SUMMARYA 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.

Frames of reference in multibody dynamics

“…We will stick to the spin tensor in our forthcoming discussions. j There is full symmetry between the frames f and f à which means that the relation (8) is equivalent to the following Proof: Equations (14) and (15) are immediate consequences of equations (11) and (12) and the observa-…”

“…where L rel;pa c is the angular momentum relative to principal axis frame. An alternative hybrid-coordinate formulation has been developed in Krenk and Nielsen [11] for dynamic analysis of structures in a rotating frame of reference. The hybrid state-space is formed by the local displacements and the local components of the global velocities.…”

Flexible body dynamics in a local frame with explicitly predicted motion

“…Simo and Wong [16], Bauchau and Bottasso [17], Hairer et al [18], and Krenk [19] for formulations involving rotations. This section gives a brief summary of a momentum based integration scheme in a moving frame of reference, adapted from Krenk [20].…”

“…In order to satisfy energy conservation, the constant stiffness matrix K should be replaced by an appropriately rotated modification as discussed in [20]. The solution proceeds in the standard way by eliminating the absolute velocity…”