# 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:

$$f\left({t}_{n}+h\right)=f\left({t}_{n}\right)+h{f}^{\prime}\left({t}_{n}\right)+\frac{{h}^{2}}{2}{f}^{\left(2\right)}\left({t}_{n}\right)+\mathrm{...}+\frac{{h}^{s}}{s!}{f}^{\left(s\right)}\left({t}_{n}\right)+{R}_{s}$$

$${R}_{s}=\frac{1}{s!}{\displaystyle \underset{{t}_{n}}{\overset{{t}_{n}+h}{\int}}{f}^{\left(s+1\right)}\left(\tau \right){\left[{t}_{n}+h-\tau \right]}^{s}d\tau}$$

The preceding formula allows the computation of displacements and velocities of the system at time ${t}_{n+1}$ :

$${\dot{u}}_{n+1}={\dot{u}}_{n}+{\displaystyle \underset{{t}_{n}}{\overset{{t}_{n+1}}{\int}}\ddot{u}\left(\tau \right)d\tau}$$

$${u}_{n+1}={u}_{n}+h{\dot{u}}_{n}+{\displaystyle \underset{{t}_{n}}{\overset{{t}_{n+1}}{\int}}\left({t}_{n+1}-\tau \right)\ddot{u}\left(\tau \right)d\tau}$$

The approximation consists in computing the integrals for acceleration in Equation 3 and in Equation 4 by numerical quadrature:

$$\underset{{t}_{n}}{\overset{{t}_{n+1}}{\int}}\ddot{u}\left(\tau \right)d\tau}=\left(1-\gamma \right)h{\ddot{u}}_{n}+\gamma h{\ddot{u}}_{n+1}+{r}_{n$$

$$\underset{{t}_{n}}{\overset{{t}_{n+1}}{\int}}\left({t}_{n+1}-\tau \right)\ddot{u}\left(\tau \right)d\tau}=\left(\frac{1}{2}-\beta \right){h}^{2}{\ddot{u}}_{n}+\beta {h}^{2}{\ddot{u}}_{n+1}+{{r}^{\prime}}_{n$$

By replacing Equation 3 and Equation 4, you have:

$${\dot{u}}_{n+1}={\dot{u}}_{n}+\left(1-\gamma \right)h{\ddot{u}}_{n}+\gamma h{\ddot{u}}_{n+1}$$

$${u}_{n+1}={u}_{n}+h{\dot{u}}_{n}+\left(\frac{1}{2}-\beta \right){h}^{2}{\ddot{u}}_{n}+\beta {h}^{2}{\ddot{u}}_{n+1}+{{r}^{\prime}}_{n}$$

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.