Newmarkbeta
The Newmark-beta method is a family of time-integration schemes used to solve second-order differential equations of motion in structural dynamics, typically written as M x¨ + C x˙ + K x = F(t). Developed by Nathan M. Newmark in 1959 for finite element analysis, the method advances displacement and velocity in time through predictor–corrector formulas controlled by two parameters, gamma (γ) and beta (β).
At each time step, given x_n, v_n, and a_n, the method forms x_{n+1} and v_{n+1} with:
x_{n+1} = x_n + v_n Δt + [ (1/2) − β ] a_n Δt^2 + β a_{n+1} Δt^2
v_{n+1} = v_n + (1 − γ) a_n Δt + γ a_{n+1} Δt
The acceleration a_{n+1} is obtained by enforcing dynamic equilibrium at time t_{n+1}:
M a_{n+1} + C v_{n+1} + K x_{n+1} = F_{n+1}
Substituting x_{n+1} and v_{n+1} yields a linear system for a_{n+1}, which is solved to update x_{n+1} and
Choice of (γ, β) determines stability and numerical damping. The canonical average-acceleration case (γ = 1/2, β = 1/4) is unconditionally
Applications of the Newmark-beta method span civil and mechanical engineering analyses, including earthquake simulations and dynamic