A Null Space Projection Approach for Modally Reduced Flexible Multibody Systems

Abstract

Light-weight structures and high-performance mechanical systems lead to an increasing amount of vibrations. In order to comply with standards in noise and vibration limits, the simulation of flexible multibody systems is inevitable. Due to the size of the finite element models of real-life mechanical systems, a model order reduction is necessary for the efficient simulation of such large scale flexible multibody systems. Currently, the most widely used technique for modelling and simulation of large scale flexible multibody systems is based on the Floating Frame of Reference Formulation (FFRF) of the modally reduced bodies. Recently, alternatives to the FFRF have been proposed, e.g. the Generalized Component Mode Synthesis (GCMS) which uses an absolute or inertial description of the modes. GCMS leads to a concise form of the equations of motion and a constant mass matrix. Within the context of the GCMS method, the rigid body motion is described with twelve coordinates while the deformation of the body is represented with nine coordinates for each flexible mode. The main drawback of the GCMS method is that the number of flexible coordinates is nine times higher as compared to the classic FFRF and therefore when more modes are needed the efficiency of the method can be impaired. Therefore, the objective of the present paper is the further reduction of the new flexible coordinates by means of a null space projection method. Null space methods have been extensively used in order to develop efficient integration algorithms for rigid bodies, flexible beams and shells; however their applicability to modally reduced flexible multibody systems has not been studied intensively. In the paper herein, we develop a new formulation for modally reduced flexible multibody systems which involves a projection onto the null space of properly defined (internal) constraint conditions imposed to the flexible coordinates. It is important to note that focus is put on the description of the projection in the continuous case rather than the discrete which will be addressed in later developments. The proposed formulation is derived in great detail and it is shown that the simple form of the equations of motion of the GCMS method is almost retained. Finally, the applicability and performance of the method is assessed by means of a numerical example.