Lumped and consistent mass matrices interpretation (bar FEM)

Discusses the physical interpretation of the consistent and lumped mass matrices of the FEM for dynamic problems w/ the linear bar element.

At Craig Jr. and Kurdila (2006) the stiffness matrix and the (consistent) mass matrix for a linear bar element are presented as

where h is the element lenght, E is the elasticity modulus, A is the element cross section area and ρ is the bar material density,

Most engineers are familiar to the interpretation of the stiffness matrix elements as the force at the node i caused by an unitary displacement at the node j. In this physical interpretation there is a kinematic hypothesis involved (about how the displacement is distributed between the two nodes) but, as the solution of static elasticity problems are ubiquitous it looks as something natural (that matches the exact solution found in many practical situations).

The mass matrix elements analogously represent the apparent (inertia) force acting over node i caused by an unitary acceleration of the node j. To make a calculation of this force consistent with the kinematic hypothesis used in obtaining the stiffness matrix one needs to take into account that the bar inertia and the acceleration associated with each node are also linearly distributed. Using a local coordinate ξ to map each element ( at node 1 and at node 2) and applying Newton's second law to each differential lenght along the element, one gets

Although this is the consistent mass matrix, one can alternatively use the lumped mass matrix, that is obtained lumping half the mass of each element at each node and dismissing the apparent force that would be caused at one node by the acceleration of the other. Doing so

A carpenter could say that the lumped mass matrix is like dividing each element by a cross cut at its middle (disregarding the possibility of one half to pull the other when it is accelerated). The consistent mass matrix would be like cutting each element through a diagonal, obtaining two wedges, but gluing them in such a way that the acceleration of a wedge would drag part of the other.

The script presented here can be used to verify that the superior accuracy of the consistent mass matrix becomes really evident only for the highest natural frequencies estimated using coarse grids (e.g. the 4th. natural frequency obtained using 5 elements). For finer grids and lower natural frequencies both matrices provide equally accurate results.