chap92 - Mass Matrices A mass matrix pre-multiplies the...

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1 Mass Matrices ± A mass matrix pre-multiplies the acceleration vector in the equation of motion. where are nodal accelerations (translational or rotational) 4 3 2 1 41 31 22 21 14 13 12 11 ... ... d d d d m m m m m m m m & & & & & & & & j d & & , 0 other all while 1 say, If, 1 = = j d d & & & & then the result of the above multiplication is the first column. ± Thus, the j th column is the vector of nodal loads that must be applied to induce a unit acceleration in the jth dof with zero acceleration in the others. = 41 31 21 11 1 41 31 21 11 ) ( m m m m d m m m m & &

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2 Mass Lumping ± One way to account for inertia effects is to lump the mass of a structure at the nodes of a FE model. L A, ρ v 1 v 2 •• ρ AL/2 ρ AL/2 v 2 v 1 ± To find the mass matrix for this finite element, we apply a unit acceleration to node 1 and none to node 2. ± The forces required for that are F 1 = m 11 = ρ AL/2, F 2 = m 21 = 0 F 1 = ( ρ AL/2)(1) F 2 =0 1 1 = v & & 0 2 = v & &
3 Mass Matrix of a Lumped System ± A similar procedure for the second node gives m 12 = 0, m 22 = ρ AL/2 ± Collecting these in a matrix: ± Thus, the mass matrix for a lumped system is diagonal . ± Note that sum of elements is equal to total mass of element

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4 Acceleration Field In a Lumped System ± Mass lumping corresponds to a discontinuous acceleration field in which the two halves of the element accelerate separately.
5 A Continuous Acceleration Field ± In reality, mass and, therefore, inertia force “ma” is distributed throughout a body. That is, acceleration is continuous. ± Recall : The displacement field in a bar element with a single dof at each node is linear .

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chap92 - Mass Matrices A mass matrix pre-multiplies the...

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