====== Appendix 10: Dynamics parameters ======
===== Choice of Newmark parameters  =====
The conditional or non conditional stability (limited time step) of the Newmark's algorithm depends on the value of the parameters $\beta$ and $\gamma$. To stay in the non conditional stability zone, two parameters $\alpha$ and $k$ which scan this zone must be introduced: \[\beta=\frac{(1+\alpha)^2}{4}\] \[\gamma=\frac{1}{2}+\alpha k\] With $\alpha>0$ and $0 \leq k \leq 1$ in the non conditionally stable zone. \\
{{:appendices:a10_a.png|}} \\
Moreover, a numerical damping appears if $\gamma \neq 0.5$ and it is all the more important as $\gamma$ is different from 0.5 and as the time step is great relating to the fundamental period. \\
If $\beta$ is greater than 0.25 and $\gamma$ greater than 0.5, a frequency distortion appears, all the more great as the time step is great. \\
That's why, one advises to take: $0 \leq \alpha \leq 1$ and $0 \leq k \leq 0.5$ \\
This is valid in the linear case. In the non linear case, some modifications may appear.
{{:appendices:a10_2.png?325|}}{{:appendices:a10_3.png?350|}}

===== Choice of INITV =====
Before any iteration, there are various ways to estimate the geometry at the end of the step $X_B$. \\
Considering: $X_B=X_A+\Delta X$ \\
with $X_A$ = geometry at the beginning of the step \\
$X_B$ = geometry at the end of the step \\
$\Delta X$ = increment of displacements

The following choices are available: \\
^INITV^Increment of displacements^Method^
| \[= 0\] | \[\Delta X = 0\] |Most effective choice|
| \[= 1\] | \[\Delta X = \dot{V}_0 \Delta t + \ddot{V}_0 \frac{\Delta t^2}{2}\] |Central difference method|
| \[= 2\] | \[\Delta X = \dot{V}_0 \Delta t + \frac{1}{2} \ddot{V}_0 \Delta t^2 \left(1-\frac{2 \beta}{\gamma}\right)\] |Newmark's algorithm with $ \dot{V}_1=\dot{V}_0$|
| \[= 3\] | \[\Delta X = \dot{V}_0 \Delta t \left(1-\frac{\beta}{\gamma}\right) + \frac{1}{2} \ddot{V}_0 \Delta t^2 \left(1-\frac{2 \beta}{\gamma}\right)\] |Newmark's algorithm with $ \dot{V}_1=0$|
| \[= 3\] | \[\Delta X = \dot{V}_0 \Delta t + \frac{1}{2} \ddot{V}_0 \Delta t^2 \left(1-2 \beta\right)\] |Newmark's algorithm with $ \ddot{V}_1=0$|

With: 
  * $\dot{V}_0$ = speeds at the beginning of the step
  * $\ddot{V}_0$ = accelerations at the beginning of the step
  * $\dot{V}_1$ = speeds at the end of the step
  * $\ddot{V}_1$ = accelerations at the end of the step