A review of mass matrices for eigenproblems

A novel integration scheme for solution of consistent mass matrix in free and forced vibration analysis

“…It can be observed in Table 1 that the results obtained by present method are the same as those of SFEM, and it indicates that the novel integration scheme of consistent mass matrix, which no coordinate mapping is required, possesses the same accuracy as the traditional consistent mass matrix. Because in the calculation of FEM, the lumped mass matrix overestimates the mass effect and the consistent mass matrix underestimates the mass effect [17], the exact natural frequencies should be located in the interval of results obtained by FEM with lumped mass matrix and those with consistent mass matrix. Hence, it can be confirmed that the results given by present method are close to the exact solutions, see from Table 1.…”

“…The lumped mass matrix, which is obtained by concentrated mass at the degree of freedom (DOF) of nodes, is diagonal and independent of shape function. Generally, the lumped mass matrices overestimate the mass effect and hence give lower natural frequencies than the exact ones [17]. On the other hand, the underestimation of consistent mass matrices causes higher natural frequencies.…”

Accelerated Approach for the Band Structures Calculation of Phononic Crystals by Finite Element Method

“…We think FEM is a potential method for solving the above-mentioned problems. The well-known supporting evidence for the finite element theory are: the consistent mass matrix underestimates the mass effects and gives higher Eigen frequencies (upper limit) than the exact ones, while the lumped mass matrix overestimates the mass effects and gives lower Eigen frequencies (lower limit) with similar mathematical calibrations [20]. Therefore, it is reasonable to assume that the true solution of the Eigen frequencies should be located between the upper and lower limits after applying the Bloch's theorem.…”

An Inverse Eigenvalue Formulation for Optimizing the Dynamic Behaviour of Pin-Jointed Structures

“…This external forced response is formulated in terms of the modi"cation parameters, thus creating a modi"ed eigenvalue problem. More recently, Zhang and Kim [9] investigated the use of mass matrix modi"cation to achieve the desired natural frequencies. McMillan and Keane [10] investigated a method of shifting eigenfrequencies of a rectangular plate by adding concentrated mass elements.…”