The Newmark-beta method is a method of numerical integration used to solve 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, who developed it in 1959 for use in Structural dynamics.
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
Newmark showed that a reasonable value of is 0.5, therefore the update rules are,
Setting β to various values between 0 and 1 can give a wide range of results. Typically β = 1/4, which yields the constant average acceleration method, is used.
Famous quotes containing the word method:
“If all feeling for grace and beauty were not extinguished in the mass of mankind at the actual moment, such a method of locomotion as cycling could never have found acceptance; no man or woman with the slightest aesthetic sense could assume the ludicrous position necessary for it.”
—Ouida [Marie Louise De La Ramée] (1839–1908)