u m = PAL u {d} = u m = PAL 102 {d}= V m 1 [m] = [m] a) = 20 m 61 01 156 22L 54 -13L : Lumped mass matrix m 22L 4 13L -31 Consistent [m] = 420 : Consistent mass matrix 2 54 13L 156 -13L-3 -22L 4 -22L mass matrix b) Figure 1 Figure la shows a uniform 2-node bar element having length L, cross-sectional area A, and mass density p. Total element mass is therefore m = pAL. Using lumped mass matrix places half of the total mass to each node for the bar element. The consistent mass matrix converts the mass to the nodal degrees of freedom utilizing the same shape functions as used to obtain the stiffness matrix, i.e.: L [m] = [[N]" [N]pAdx. 0 Figure 1b shows the consistent mass matrix for a uniform 2-node beam element having length L, cross-sectional area A, and mass density p. Consider the kinetic energies of the 2-node elements shown in Figure 1 in the following three rigid- body plane motions: lateral translation, rotation about the mass center, and rotation about the left end. Check whether the mass matrices provide the correct kinetic energies or not. a) What are the correct kinetic energies for the three rigid body motions? b) Consider the 2-node bar element with the lumped mass matrix shown in Figure 1a. c) Consider the 2-node bar element with the consistent mass matrix shown in Figure 1a.