The Newmark-beta method is a
method
Method ( grc, μέθοδος, methodos) literally means a pursuit of knowledge, investigation, mode of prosecuting such inquiry, or system. In recent centuries it more often means a prescribed process for completing a task. It may refer to:
*Scien ...
of
numerical integration
In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical solution of differential equations ...
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
Nathan Mortimore Newmark (September 22, 1910 – January 25, 1981) was an American structural engineer and academic, who is widely considered one of the founding fathers of earthquake engineering. He was awarded the National Medal of Science fo ...
,
former Professor of Civil Engineering at the
University of Illinois at Urbana–Champaign
The University of Illinois Urbana-Champaign (U of I, Illinois, University of Illinois, or UIUC) is a public land-grant research university in Illinois in the twin cities of Champaign and Urbana. It is the flagship institution of the Univer ...
, who developed it in 1959 for use in
structural dynamics
Structural dynamics is a type of structural analysis which covers the behavior of a structure subjected to dynamic (actions having high acceleration) loading. Dynamic loads include people, wind, waves, traffic, earthquakes, and blasts. Any structur ...
. 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
In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time.''Encyclopaedia of Physics'' (second Edition), R.G. Lerner, G.L. Trigg, VHC Publishers, 1991, ISBN (Ver ...
) 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
Stability Analysis
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
In mathematics, the spectral radius of a square matrix is the maximum of the absolute values of its eigenvalues. More generally, the spectral radius of a bounded linear operator is the supremum of the absolute values of the elements of its spectru ...
of the update matrix
.
For the linear structural equation
here
is the stiffness matrix. Let