Abstract

A simple approach is developed to approximate the full mass matrix in the rotational-coordinate-based beam formulation, which can significantly improve the efficiency of Jacobian calculations in the corresponding dynamic analyses. The rotational-coordinate-based beam formulation adopts only rotational coordinates as generalized coordinates, and the position vectors are derived by integrations from the slope vectors, which are expressed by rotational coordinates. While the rotational-coordinate-based beam formulation can reduce numbers of elements and generalized coordinates, its mass matrix is a full matrix, such that the Jacobian matrix is also full, which is time-consuming to solve. Two approximations are adopted in this work: (1) a double integral is approximated by an integral; and (2) a full matrix is approximated by a sum of a series of rank-one matrices. Through this way, the full mass matrix is approximately decomposed as a band-diagonal sparse matrix and multiplication of low-rank matrices, and its inverse can be efficiently calculated using Sherman-Woodbury formula. Several numerical examples are presented to demonstrate the performance of the current approach, and accuracy and efficiency of the current approach are presented and discussed.

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