The Newmark-beta method is a method of numerical integration used to solve certain differential equations. It is widely used in numerical evaluation of the dynamic response of structures and solids such as in finite element analysis to model dynamic systems. The method is named after Nathan M. Newmark, former Professor of Civil Engineering at the University of Illinois at Urbana–Champaign, who developed it in 1959 for use in structural dynamics. The semi-discretized structural equation is a second order ordinary differential equation system,

is the mass matrix is the damping matrix are internal force per unit displacement and external forces, respectively.

Using the extended mean value theorem method states that the first time derivative (velocity in the equation of motion) can be solved as,

where

therefore

Because acceleration also varies with time, however, the extended mean value theorem must also be extended to the second time derivative to obtain the correct displacement. Thus,

where again

The discretized structural equation becomes

Explicit central difference scheme

Average constant acceleration (Middle point rule)

Stability Analysis

. Assume the time-integration scheme is

is the spectral radius.

For the linear structural equation

, and

of the structural system, which are solved by the generalized eigenvalue problem

For each eigenmode, the update matrix becomes

The characteristic equation of the update matrix is

As for the stability, we have

Explicit central difference scheme.

Average constant acceleration (Middle point rule)) is unconditionally stable.

References