Comparison of some single-step methods for the numerical solution of the structural dynamic equation

✍ Scribed by S. N. Penry; W. L. Wood

• Publisher: John Wiley and Sons
• Year: 1985
• Tongue: English
• Weight: 613 KB
• Volume: 21
• Category: Article
• ISSN: 0029-5981
• DOI: 10.1002/nme.1620211102

✦ Synopsis

This paper compares the performance of the SS22 and SS23 (References 1 and 2) single step algorithms for the numerical solution of the second-order structural dynamic equation and a related new algorithm SS32B applied to the equivalent first-order system, with sine and step forcing functions. Various aspects of stability relevant to these equations are discussed.

We consider the structural dynamic system given by the N equations

arising from the finite element discretization of a structure. M, C and K are the mass, damping and stiffness matrices, respectively. x is the vector of displacements and f(t) is the forcing function. We make the usual assumption of Rayleigh damping (i.e. the matrix C is some linear combination of the matrices M and K), then, since the matrices M and K are symmetric and positive definite due to their finite element origin, we can use the theory of Wilkinson3 to show that M, C and K effectively have a common complete set of eigenvectors u,, r = 1,2,. . . , N . Hence we can make a modal decomposition and show that the exact solution of the system of equations ( 1) is N x = C a,exp(;l,t)u, r = l where A, satisfies m , & ! + prAr + k, = 0, r = 1,2,. . . , N (3) Mu, = m,u,, Cur = pu,, Ku, = k,u,

and Another approach to the solution of the system of equations ( 1) is to reduce is to a first-order system as in, for example, References 4 and 5, by putting v = x and rewriting the equations as

(5) where