How Mode Veering and Mode Crossing affects Global Modal Parametrization and solutions to overcome these problems
Gert H. K. Heirman, Frank Naets, Wim Desmet · Lirias · 2010
The presence of both differential and algebraic equations in the model equations, as well as the number of degrees of freedom needed to accurately represent flexibility, prohibit fast simulation of flexible multibody systems (e.g. real-time). In this research the capabilities for fast simulation, of the model reduction technique for multibody systems Global Modal Parameterization (GMP) [1] are investigated. The reduction of the model is achieved by projection on a judiciously chosen curvilinear subspace: the motion of the multibody system is expressed in terms of its dominant, system-level, configuration-dependent eigenmodes and static deformation patterns. The system is thus represented by a curvilinear coordinate system that maps the subspace spanning the dominant dynamic phenomena. The coordinate axes of this curvilinear coordinate system are defined by the eigenmodes and static deformation patterns incorporated in the GMP-mode set. As opposed to a fixed vector space, using a curvilinear reduction subspace requires significantly less degrees of freedom to represent the system’s dynamics with the same level of accuracy [2] [3], which is a significant step towards faster simulation of flexible multibody systems. When eigenfrequencies converge in the configuration space, Mode Veering or Mode Crossing occurs. These phenomena cause strong gradients or, in the case of Mode Crossing, even discontinuities in the eigenmodes throughout the configuration space. This results in strongly curved or even discontinuous coordinate axes for the reduced model, such that difficulties are to be expected. The numerical experiments in this work indeed show that the strong gradients of the eigenmodes linked to Mode Veering imposes the use of a fine discretization space, limiting the advantages of GMP. For cases exhibiting Mode Crossing, the GMP-description fails altogether. This paper discusses opportunities to overcome this problem. The vector space spanned by veering or crossing modes is continuous throughout the configuration space as opposed to the individual vectors themselves. This enables to define continuously varying vectors to be used as the coordinate axes of the curvilinear coordinate system instead of the veering/crossing eigenmodes themselves. As this curvilinear coordinate system maps the same reduction subspace, the reduced model is still able to capture the same dynamical phenomena, while being relieved from its strongly curved of even discontinuous coordinate axes, making simulation of the reduced model significantly cheaper or, in the case of Mode Crossing, removing the singularity and thus enabling the use of the model reduction technique.