GeneralizedAlpha
The generalized- method provides second order accuracy with controllable algorithmic damping on high frequency response.
Syntax
Two forms are available.
integrator GeneralisedAlpha (1) [2]
integrator GeneralizedAlpha (1) [2]
# (1) int, unique tag
# [2] double, spectral radius at infinite frequency, default: 0.5
integrator GeneralisedAlpha (1) (2) (3)
integrator GeneralizedAlpha (1) (2) (3)
# (1) int, unique tag
# (2) double, \alpha_f
# (3) double, \alpha_m
Governing Equation
The generalized alpha method assumes that the displacement and the velocity are integrated as such,
The equation of motion is expressed at somewhere between and .
which can also be explicitly shown as
where and are two additional parameters.
Default Parameters
To obtain an unconditionally stable algorithm, the following conditions shall be satisfied.
Only one parameter is required, the spectral radius that ranges from zero to one.
The following expressions are used to determine the values of all constants used.
So that the resulting algorithm is unconditionally stable and has a second order accuracy. The target numerical damping for high frequencies is achieved while that of low frequencies is minimized.
The recommended values of the spectral radius range from to .
Some special parameters can be chosen.
method
Newmark
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HHT-
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WBZ-
Accuracy Analysis
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