A TIME-STEPPING PROCEDURE FOR STRUCTURAL DYNAMICS USING GAUSS POINT COLLOCATION

When a cubic function is interpolated between the prescribed initial displacement and velocity and the exact displacement and velocity at the end of a time step for a single degree of freedom system, the error, or residual, in the governing equation is zero at a number of times. It is shown that for a general undamped system, in the limit as the time step approaches zero these times correspond to Gauss points. This observation is verified by considering a general collocation procedure in which the displacement in any time step is approximated as a cubic function of time, with two coefficients chosen to satisfy the displacement and velocity at the beginning of the time step with the other two coefficients being chosen to satisfy the governing differential equation at any two times. It is shown that optimum accuracy is obtained if these points are the Gauss points. Detailed expressions are then presented for this particular case, and stability of the algorithm is investigated showing that the procedure is conditionally stable. For time steps which are a small proportion of the least period of vibration of the structure, the algorithm is considered to be the most accurate possible procedure based on cubic approximation of the displacement