Element Mass

The mass matrix for an element can be derived as described above. This is known as a consistent mass matrix. In many situations it is convenient to simplify the mass matrix.

One way of doing this is to diagonalize the mass matrix. In this case all the terms relating to rotations are zeroed (except for explicit analysis) and then the translational terms are accumulated on the diagonal of the matrix.

The original diagonalization used a row summation where the diagonal is the sum of the terms in the row.

$$m_{ii} = \sum_{j}m_{ij}$$

This has the effect of lumping all the mass on the diagonal but has the disadvantage that for higher order elements some of the mass terms are negative. This method has been superseded by the Hinton-Rock-Zienkiewicz$^8$ (HRZ) method. In this the diagonal terms of the consistent mass matrix are summed for each direction, $n$

$$m_{sum} = \sum_i m_{ii}\qquad i \in n$$

This is then used to scale the diagonal terms

$$m_{ij} = \begin{cases} \frac{m_{elem}}{m_{sum}}m_{ij} & i = j\\ 0 & i \neq j \end{cases}$$

The other, less used, simplification that is used is to ignore the mass of all elements except lumped masses.

With both these simplifications it may be possible to diagonalize the structure mass matrix, however this is not possible if the inertias of nodal masses are non-zero or if there are rigid elements in the structure. (Rigid elements generate off diagonal terms when the masses are replaced by inertias at the primary.)

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$^8$ E. Hinton, T. Rock, O.C. Zienkiewicz, A note on mass lumping and related processes in the finite element method, Earthquake Engineering and Structures Dynamics 4 (1976) 245-249