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Concept
Newmark-beta method
Summary
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,
here is the mass matrix, is the damping matrix, and are internal force per unit displacement and external forces, respectively.
Using the extended mean value theorem, the Newmark- 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 is obtained by setting and
Average constant acceleration (Middle point rule) is obtained by setting and
A time-integration scheme is said to be stable if there exists an integration time-step so that for any , a finite variation of the state vector at time induces only a non-increasing variation of the state-vector calculated at a subsequent time . Assume the time-integration scheme is
The linear stability is equivalent to , here is the spectral radius of the update matrix .
For the linear structural equation
here is the stiffness matrix. Let , the update matrix is , and
For undamped case (), the update matrix can be decoupled by introducing the eigenmodes 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 ( and ) is stable when .
Official source
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