ABSTRACT
The closed form solutions for single-degree-of-freedom problems are not possible if the applied external excitations are complex loading time histories. As a result, most arbitrary complex loading conditions require some form of numerical time stepping for integration of the differential equations of motion. The numerical techniques deemed most appropriate for solving such problems are introduced in this chapter. The algorithms used can be categorized into two main groups, depending on their approach to satisfying the differential equation of motion. The first group includes Duhamel's integral solved in a piecewise-constant and piecewise-linear fashion and the second group includes the finite difference method, fourth-order Runge–Kutta method, Newmark method, Wilson method, and the Hilber, Hughes, and Taylor alpha (HHT-) method. The initial method presented in the finite difference method include the Euler, modified Euler (or Heun), Runge–Kutta, and central difference methods. Next, the constant, average, and linear acceleration Newmark methods are described. The Wilson-theta method is then presented and the chapter culminates with the presentation of the HHT-method.
