Entities: Consistent vs Lumped Mass

Description

Inertia effects in dynamic analysis are captured by considering the mass of the structure. To represent the mass distribution of a structure in a finite element model, the mass of an element is proportionally applied to its nodes. In the general case, six components of mass (three translational and three rotational) are possible at each node within the element; rotational terms appear when the element includes rotational degrees of freedom (i.e., beams and plates).

Two options are available to convert the element mass into a matrix of nodal translational and rotational mass values: the consistent and the lumped mass options.

Consistent Mass Matrix

The calculation of the consistent mass matrix is based the following integral:

where

= Material mass density,

= Element shape function matrix, and

= Element volume domain.

This integral is evaluated analytically or numerically (using an approximate Gauss integration scheme). The term "consistent" refers to the fact that the shape functions, , used to calculate the element mass matrix are the same as those used to calculate the element stiffness matrix.

As an example, the consistent mass matrix for a two-dimensional truss with two degrees of freedom at each end is shown below:

where is the total mass of the element.

The consistent mass matrix provides a more accurate representation of the mass distribution in a continuum. It includes the effect of mass coupling between different degrees of freedom and different nodes, including the effects of the rotational inertia for elements with rotational degrees of freedom.

Lumped Mass Matrix

Lumped mass approximation is a simpler and often more efficient approach for the representation of the distributed mass within a structure. In this intuitive approach, the mass within each element is assumed to be lumped equally onto the nodes such that the sum of the nodal masses associated with the translational degrees of freedom for each global direction equals the total mass of the element. Usually, only the translational mass is included, even for elements with rotational degrees of freedom, and all off-diagonal terms are excluded (i.e., there is no mass coupling between the different degrees of freedom). Without rotational mass terms, overall rotational inertia comes from the relative translation of the nodes.

For a two-dimensional truss, with two degrees of freedom at each end, a lumped mass matrix like the one below is produced:

where is the total mass of the element.

Intuitively, one can generate this mass matrix by simply lumping half of the element mass onto each of its nodes.

Although the consistent mass matrix is a better approximation of the element mass distribution, the lumped mass matrix is often preferred for efficiency reasons, particularly for reasonably well refined meshes. As the lumped mass matrix is diagonal, the storage requirement is much lower and the amount of numerical effort to solve the model can be greatly reduced.

Notes

The lumped mass matrix will theoretically produce a lower level of accuracy than the consistent mass matrix. But in practice, the accuracy of the lumped mass approximation is comparable to the accuracy of the consistent mass when the mesh is sufficiently refined. This is particularly true for the lower vibration modes of a structure. Of course there are situations that cannot be modelled with a lumped diagonal mass matrix that considers only translational terms. An example is the torsional vibration of a beam modelled with beam elements. Unless a rotational mass term corresponding to the twist rotation of the beam is included in the mass matrix, the torsional vibration modes cannot be calculated.
Lumped mass matrices usually yield natural frequencies that are lower than the exact values, particularly for coarse meshes.
Most simple hand calculation methods use the lumped mass approach. Thus despite the fact that the consistent mass approximation will provide more accurate results in most situations, better agreement with simple benchmark tests is often noted with the lumped mass approach because the theory uses the same simplifying assumptions.
To fully account for the effects of beam and plate offsets, nodal rotational mass in a UCS, beam and plate rotational mass for elements not aligned with the global axis system, and point masses assigned to links, the consistent mass option is needed. In Strand7, if a dynamic analysis is requested using the lumped mass matrix, but the solution requires consistent mass (due to the presence any of these features) the consistent mass is automatically used by the solvers for the relevant degrees of freedom.