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RayleighNewmark

Newmark Algorithm With Rayleigh Damping Model

The RayleighNewmark integrator is simply a wrapper of the Newmark integrator with Rayleigh model implemented on global level.

For the moment, MPC cannot be considered in all global damping models.

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# valid starting from version 2.x
integrator RayleighNewmark (1) (2) (3) (4) (5) (6) (7)
# (1) int, unique tag
# (2) double, alpha (beta in some references) in Newmark method, normally 0.25
# (3) double, beta (gamma in some references) in Newmark method, normally 0.5
# (4) double, alpha
# (5) double, beta
# (6) double, theta
# (7) double, eta

Theory

By default, the Rayleigh type damping model is used. No matter what other damping modifiers are defined, this integrator always set the element damping matrix to be

\[ C=\alpha{}M+\beta{}K_C+\theta{}K_I+\eta{}K_T \]

by the Rayleigh modifier in addition to the existing damping matrix defined in the element. In the above formulation, \(K_C\) is the current converged stiffness, \(K_I\) is the initial stiffness of the element, \(K_T\) is the trial stiffness of current iteration. Please note if \(K_T\) is involved, the damping force is then a function of trial displacement. The quadratic convergence rate is lost then.