SUMMARYThis paper discusses the Bossak-Newmark algorithm, which is an extension of the well-known Newmark algorithm' for the numerical integration of the equations of discretized structural dynamics problems. The extra parameter introduced here enables the method (when used on the test equation i = -0 2 x ) to be simultaneously second order, unconditionally stable and with positive artificial damping.Com arisons are made with another modification of Newmark introduced by Hilber, Hughes andTaylor. PIn many structural dynamics applications the equations of motion for the discretized system have the formwhere M, C, K are the mass, damping and stiffness matrices, respectively; x, x, f are the displacement, velocity and acceleration vectors, respectively; and F is the external force vector.The well-known Newmark algorithm' for the numerical integration of equation ( where the Newmark parameters are distinguished by the subscript 'N' to avoid confusion with other parameters in this paper. The idea of introducing an additional parameter for controlling the damping properties of Newmark's algorithm was proposed in 1977 by Hilber, Hughes and Taylor.' Hilber, Hughes and Taylor introduce a parameter, called here aH to avoid confusion, which they apply to the equation without natural damping and with which equation (4) is replaced by A particular version of this algorithm has also been used by A b~u d i .~

SUMMARYA general algorithm for a single step time marching scheme for use in dynamic or diffusion equations is presented. This algorithm is easily programmed in its universal form for all orders of approximation and covers most of the currently used schemes as well as presenting many new possibilities. In many cases it presents a computationally advantageous form over conventional procedures-this is particularly so when compared with the Newmark algorighm and its variants.

SUMMARYThis paper investigates the phenomenon of 'noise' which is common in most time-dependent problems. The emphasis is on the achievable accuracy that is obtained with various time-stepping algorithms and how this can be improved if noise is artificially damped to an acceptable level. A series of experiments are made where the space domain is discretized using the finite element method and the variation with time is approximated by several finite difference methods. The conclusion is reached that the Crank-Nicolson scheme with a simple averaging process is superior to the other methods investigated.

SUMMARYPart 1 of this paper' presents a very general single step algorithm SSpj for the numerical integration of first and second order time dependent differential equations. In Part 2 we present and discuss results of the accuracy and stability analysis for SS11, SS21, SS31, SS22 and SS32. There is also a detailed comparison of SS22 and the Newmark algorithm.

This paper describes numerical experiments made to investigate the stability of some time-stepping algorithms applied to the equation ii + P(u) = 0 representing a nonliner elastic spring. These algorithms would be unconditionally stable when applied to linear problems, but here they may be only conditionally stable. Ways of improving the stability are demonstrated; the effect of linearization is also investigated.