A weighted residual development of a time-stepping algorithm for structural dynamics using two general weight functions

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 coe cients of the cubic are chosen to satisfy the two initial conditions and two weighted integral equations. By considering general weight functions, six additional coe cients arise. In a series of steps, these coe cients are selected to (i) maximize algebraic accuracy by matching terms of Taylorʹs 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