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)

$$f\left(t_n+h\right)=f\left(t_n\right)+h{f}^{\prime }\left(t_n\right)+\frac{h^2}{2}{f}^{\left(2\right)}\left(t_n\right)+\mathrm{...}+\frac{h^s}{s!}{f}^{\left(s\right)}\left(t_n\right)+R_s$$

(2)

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

The preceding formula allows the computation of displacements and velocities of the system at time $t_{n+1}$ :(3)

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

(4)

$$u_{n+1}=u_n+h{\dot{u}}_n+{\displaystyle \underset{t_n}{\overset{t_{n+1}}{\int }}\left(t_{n+1}-\tau \right)\ddot{u}\left(\tau \right)d\tau }$$

The approximation consists in computing the integrals for acceleration in Equation 3 and in Equation 4 by numerical quadrature:(5)

$${\displaystyle \underset{t_n}{\overset{t_{n+1}}{\int }}\ddot{u}\left(\tau \right)d\tau }=\left(1-\gamma \right)h{\ddot{u}}_n+\gamma h{\ddot{u}}_{n+1}+r_n$$

(6)

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

By replacing Equation 3 and Equation 4, you have:(7)

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

(8)

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

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.