Newarks Method

Newmark's method is a one step integration method. The state of the system at a given time ${t}_{n+1}={t}_{n}+h$ is computed using Taylor's formula:(1)
$f\left({t}_{n}+h\right)=f\left({t}_{n}\right)+h{f}^{\prime }\left({t}_{n}\right)+\frac{{h}^{2}}{2}{f}^{\left(2\right)}\left({t}_{n}\right)+...+\frac{{h}^{s}}{s!}{f}^{\left(s\right)}\left({t}_{n}\right)+{R}_{s}$
(2)
${R}_{s}=\frac{1}{s!}\underset{{t}_{n}}{\overset{{t}_{n}+h}{\int }}{f}^{\left(s+1\right)}\left(\tau \right){\left[{t}_{n}+h-\tau \right]}^{s}d\tau$
The preceding formula allows the computation of displacements and velocities of the system at time ${t}_{n+1}$ :(3)
${\stackrel{˙}{u}}_{n+1}={\stackrel{˙}{u}}_{n}+\underset{{t}_{n}}{\overset{{t}_{n+1}}{\int }}\stackrel{¨}{u}\left(\tau \right)d\tau$
(4)
${u}_{n+1}={u}_{n}+h{\stackrel{˙}{u}}_{n}+\underset{{t}_{n}}{\overset{{t}_{n+1}}{\int }}\left({t}_{n+1}-\tau \right)\stackrel{¨}{u}\left(\tau \right)d\tau$
The approximation consists in computing the integrals for acceleration in 式 3 and in 式 4 by numerical quadrature:(5)
$\underset{{t}_{n}}{\overset{{t}_{n+1}}{\int }}\stackrel{¨}{u}\left(\tau \right)d\tau =\left(1-\gamma \right)h{\stackrel{¨}{u}}_{n}+\gamma h{\stackrel{¨}{u}}_{n+1}+{r}_{n}$
(6)
$\underset{{t}_{n}}{\overset{{t}_{n+1}}{\int }}\left({t}_{n+1}-\tau \right)\stackrel{¨}{u}\left(\tau \right)d\tau =\left(\frac{1}{2}-\beta \right){h}^{2}{\stackrel{¨}{u}}_{n}+\beta {h}^{2}{\stackrel{¨}{u}}_{n+1}+{{r}^{\prime }}_{n}$
By replacing 式 3 and 式 4, you have:(7)
${\stackrel{˙}{u}}_{n+1}={\stackrel{˙}{u}}_{n}+\left(1-\gamma \right)h{\stackrel{¨}{u}}_{n}+\gamma h{\stackrel{¨}{u}}_{n+1}$
(8)
${u}_{n+1}={u}_{n}+h{\stackrel{˙}{u}}_{n}+\left(\frac{1}{2}-\beta \right){h}^{2}{\stackrel{¨}{u}}_{n}+\beta {h}^{2}{\stackrel{¨}{u}}_{n+1}+{{r}^{\prime }}_{n}$
According to the values of $\gamma$ and $\beta$ , different algorithms can be derived:
• $\gamma =0,\beta =0$ : pure explicit algorithm. It can be shown that it is always unstable. An integration scheme is stable if a critical time step exists so that, for a value of the time step lower or equal to this critical value, a finite perturbation at a given time does not lead to a growing modification at future time steps.
• $\gamma =1/2,\beta =0$ : central difference algorithm. It can be shown that it is conditionally stable.
• $\gamma =1/2,\beta =1/2$ : Fox & Goodwin algorithm.
• $\gamma =1/2,\beta =1/6$ : linear acceleration.
• $\gamma =1/2,\beta =1/4$ : mean acceleration. This integration scheme is the unconditionally stable algorithm of maximum accuracy.