appendices:a10
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| ====== Appendix 10: Dynamics parameters ====== | ====== 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. \\ | 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|}} \\ | {{:appendices:a10_a.png|}} \\ | ||
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| That's why, one advises to take: $0 \leq \alpha \leq 1$ and $0 \leq k \leq 0.5$ \\ | 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. | 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 | ||
appendices/a10.1561113388.txt.gz · Last modified: 2020/08/25 15:33 (external edit)
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