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A weighted residual development of a time‐stepping algorithm for structural dynamics using two general weight functions
Author(s) -
Golley B. W.
Publication year - 1998
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/(sici)1097-0207(19980515)42:1<93::aid-nme353>3.0.co;2-w
Subject(s) - mathematics , residual , matching (statistics) , stability (learning theory) , algebraic equation , displacement (psychology) , weight function , series (stratigraphy) , function (biology) , method of mean weighted residuals , limiting , degrees of freedom (physics and chemistry) , taylor series , algorithm , mathematical analysis , finite element method , computer science , nonlinear system , psychotherapist , engineering , biology , psychology , paleontology , quantum mechanics , machine learning , evolutionary biology , galerkin method , thermodynamics , mechanical engineering , statistics , physics
An unconditionally stable single‐step implicit algorithm for the integration of the equations of motion arising in structural dynamics is presented. Within a time step, the displacement for a single degree of freedom system is approximated by a function which is cubic in time. The four coefficients of the cubic are chosen to satisfy the two initial conditions and two weighted integral equations. By considering general weight functions, six additional coefficients arise. In a series of steps, these coefficients are selected to (i) maximize algebraic accuracy by matching terms of Taylor's expansions of exact and approximate solutions, (ii) ensure unconditional stability and (iii) optimize numerical conditioning of the equations in a limiting case. Equations required to implement the procedure are presented. The method as presented has no algorithmic damping of higher modes, although it is indicated how this may be achieved. The error in period elongation obtained using the proposed method is shown to be far less than using alternative procedures. © 1998 John Wiley & Sons, Ltd.