By Albeverio S., et al.

ISBN-10: 0821836242

ISBN-13: 9780821836248

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17) The acceleration of node 1 is given in this model problem, since node 1 is a prescribed displacement node. The acceleration of the prescribed displacement node can be obtained from the prescribed nodal displacement by differentiating twice in time. Obviously, the prescribed displacement must be sufficiently smooth so that the second derivative can be taken; this requires it to be a C1 function of time. If the mass matrix is not diagonal, then the acceleration on the prescribed displacement node, node 1, will contribute to the Eq.

The mass matrix which results from a consistent derivation from the weak form is called a consistent mass matrix. In many applications, it is advantageous to use a diagonal mass matrix called a lumped mass matrix. Procedures for diagonalizing the mass matrix are often quite ad hoc, and there is little theory underlying these procedures. 26) J where the sum is over the entire row of the matrix, MIJC is the consistent mass matrix and MIJD is the diagonal or lumped, mass matrix. 27) where we have used the fact that the sum of the shape functions must equal one; this is a reproducing condition discussed in Chapter 8.

3. Three-node quadratic displacement element. A 3-node element of length L0 and cross-sectional area A0 is shown in Fig. 4. Node 2 is placed between nodes 1 and 3; although in this analysis we do not assume it to be midway between the nodes, it is recommended that it be placed midway between the nodes in most models. 36) where N(ξ) is the matrix of Lagrange interpolants, or shape functions, and ξ is the element coordinate. 38) We have used the fact that in one dimension, ξ ,x = X,−1 ξ . 39) The internal nodal forces are given by Eq.

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