Appendix 11A Newmark Method

Equations of motion of a linear multi-DOF dynamic system can be written as

where [M], [C], and [K] are the mass, damping, and stiffness matrices: {f(t)] is the vector of externally applied loads, is the displacement vector and the ‘ dot ' is the time derivative.

Direct Integration Methods:

In direct integration equations in (11A.1) are integrated using a numerical step-by-step procedure. In this method instead of trying to satisfy equation (11A.1) at any time, t , it is aimed to satisfy only at discrete time intervals, .This means that, basically, equilibrium, which includes the effect of inertia, stiffness, damping and external forces, is sought at discrete time points within the interval of solution. Therefore, it appears that all solution techniques employed in the static analysis can probably also be used effectively in the direct integration, and variation of displacements, velocities, and accelerations within each time interval is assumed.

Assumptions : If the equilibrium relation in (11A.1) is regarded as a system of ordinary differential equations with constant coefficients. It follows that any convenient finite difference expressions to approximate the accelerations and velocities in terms of displacements can be used. The following assumptions are used in the Newmark method:

and