Superconvergent isogeometric free vibration analysis of Euler–Bernoulli beams and Kirchhoff plates with new higher order mass matrices

Abstract

A superconvergent isogeometric free vibration analysis is presented for Euler–Bernoulli beams and Kirchhoff plates. This method is featured by new higher order mass matrices. For the 1D Euler–Bernoulli beam problem, it is shown that a new higher order mass matrix can be directly established by optimizing a reduced bandwidth mass matrix. The reduced bandwidth mass matrix is designed based upon the consistent mass matrix and it contains adjustable parameters to be determined via maximizing the order of accuracy of the vibration frequency. As a result, 4th and 6th orders of accuracy are observed for the proposed quadratic and cubic higher order mass matrices, while their corresponding consistent mass matrices are 2nd and 4th order accurate, respectively. Thus the higher order beam mass matrices have more superior frequency accuracy simultaneously with smaller bandwidth compared with their corresponding consistent mass matrices. While for the 2D Kirchhoff plate problem, in order to compute arbitrary frequency in a superconvergent fashion, a mixed mass matrix is formulated through a linear combination of the consistent mass matrix and the reduced bandwidth mass matrix. Then by introducing the wave propagation angle, the higher order plate mass matrix can be rationally derived from the mixed mass matrix. It is proved that the optimal combination parameter for higher order mass matrix depends on the wave propagation angle. Consequently a particular higher order mass matrix can always be set up for a superconvergent computation of arbitrary vibration frequency. It is shown that the quadratic and cubic higher order mass matrices possess 4th and 6th orders of accuracy, which are two orders higher than those of the consistent plate mass matrices. The construction of higher order mass matrices and the analytical results for vibration frequencies are systematically demonstrated by a set of numerical examples.