Linearization of the equations of motion
========================================

Assume calculated the state of the structure at iteration :math:`n` of time :math:`i`, :math:`n=1` corresponds to the prediction phase. The weak form of the equilibrium equations is, at this iteration [:ref:`éq 8.1-1 <éq 8.1-1>`]:

 :math:`W({\phi }_{i}^{n},{\dot{\phi }}_{i}^{n},{\ddot{\phi }}_{i}^{n};\delta \phi )={W}_{\text{int}}({\phi }_{i}^{n};\delta \phi )-{W}_{\text{iner}}({\phi }_{i}^{n},{\dot{\phi }}_{i}^{n},{\ddot{\phi }}_{i}^{n};\delta \phi )-{W}_{\text{ext}}({\phi }_{i}^{n};\delta \phi )` **eq 9-1**


* If this quantity is quite small, in the sense of the stopping criterion [:ref:`bib10 <bib10>`], we consider that this :math:`n` -th iteration gives the state of the structure at time :math:`i`.


*Otherwise,**corrections** for displacement :math:`\Delta {\phi }_{i}^{n+1}` are calculated such as:

 :math:`\begin{array}{}L\{W\left[({\phi }_{i}^{n}+\Delta {\phi }_{i}^{n+1}),{({\phi }_{i}^{n}+\Delta {\phi }_{i}^{n+1})}^{\text{.}},{({\phi }_{i}^{n}+\Delta {\phi }_{i}^{n+1})}^{\text{.}\text{.}};\delta \phi \right]=\\ W({\phi }_{i}^{n},{\dot{\phi }}_{i}^{n},{\ddot{\phi }}_{i}^{n};\delta \phi )+\text{DW}({\phi }_{i}^{n},{\dot{\phi }}_{i}^{n},{\ddot{\phi }}_{i}^{n};\delta \phi )\text{.}\Delta {\phi }_{i}^{n+1}=0\text{.}\end{array}` **eq 9-2**

:math:`\text{DW}({\phi }_{i}^{n},{\dot{\phi }}_{i}^{n},{\ddot{\phi }}_{i}^{n};\delta \phi )\text{.}\Delta {\phi }_{i}^{n+1}` is the Fréchet differential of :math:`W({\phi }_{i}^{n},{\dot{\phi }}_{i}^{n},{\ddot{\phi }}_{i}^{n};\delta \phi )` in the :math:`\Delta {\phi }_{i}^{n+1}` [:ref:`An4 <An4>`] direction.




Stiffness matrices
--------------------

They result from Fréchet's differentiation from :math:`{W}_{\text{int}}(\phi ;\delta \phi )` in the :math:`\Delta \phi` direction. According to equations [:ref:`éq 6.3-4 <éq 6.3-4>`] and [:ref:`éq 6.4-1 <éq 6.4-1>`]:

:math:`{W}_{\text{int}}(\phi ;\delta \phi )={\int }_{{s}_{1}}^{{s}_{2}}\left\{B\delta \phi \right\}\text{.}\Pi \left\{\begin{array}{c}F\\ M\end{array}\right\}\text{ds}\text{.}`

Either:

 :math:`\begin{array}{}{\mathrm{DW}}_{\text{int}}\text{.}\Delta \phi ={\int }_{{s}_{1}}^{{s}_{2}}\left[D\left\{B\delta \phi \right\}\text{.}\Delta \phi \right]\text{.}\Pi \left\{\begin{array}{c}F\\ M\end{array}\right\}\text{ds}\\ +{\int }_{{s}_{1}}^{{s}_{2}}\left\{B\delta \phi \right\}\text{.}\left[(D\Pi \text{.}\Delta \phi )\left\{\begin{array}{c}F\\ M\end{array}\right\}\right]\text{ds}+{\int }_{{s}_{1}}^{{s}_{2}}\left\{B\delta \phi \right\}\text{.}\left[\Pi (D\left\{\begin{array}{c}F\\ M\end{array}\right\}\text{.}\Delta \phi )\right]\text{ds}\text{.}\end{array}` **eq 9.1-1**

Now, according to equation [:ref:`éq 6.3-5 <éq 6.3-5>`]:

:math:`\left\{B\delta \phi \right\}=\left\{\begin{array}{c}\delta {x}_{o}\text{'}+{x}_{o}\text{'}\wedge \delta \Psi \\ \delta \Psi \text{'}\end{array}\right\}\text{.}`

So [:ref:`éq A4-2 <éq A4-2>`]:

:math:`D\left\{B\delta \phi \right\}\text{.}\Delta \phi =\left\{\begin{array}{c}\Delta {x}_{o}\text{'}\wedge \delta \Psi \\ 0\end{array}\right\}\text{.}`

On the other hand [:ref:`éq A4-5 <éq A4-5>`]:

:math:`D\Pi \text{.}\Delta \phi =\left[\begin{array}{ccc}\stackrel{ˆ}{\Delta \Psi }& {R}_{\text{tot}}& 0\\ 0& \stackrel{ˆ}{\Delta \Psi }& {R}_{\text{tot}}\end{array}\right]\text{.}`

Finally, according to [:ref:`§6.4 <§6.4>`]:

:math:`\begin{array}{c}D\left\{\begin{array}{c}F\\ M\end{array}\right\}\Delta \phi =CD({\Pi }^{T}\left\{\begin{array}{c}\varepsilon \\ \chi \end{array}\right\})\text{.}\Delta \phi \\ ={C\Pi }^{T}B\Delta \phi -C{\Pi }^{T}\left[\begin{array}{cc}\stackrel{ˆ}{\Delta \Psi }& 0\\ 0& \stackrel{ˆ}{\Delta \Psi }\end{array}\right]\left\{\begin{array}{c}\varepsilon \\ \chi \end{array}\right\},\end{array}`

because [:ref:`éq 6.3-2 <éq 6.3-2>`] and [:ref:`éq 6.3-5 <éq 6.3-5>`]:

:math:`D\left\{\begin{array}{c}\varepsilon \\ \chi \end{array}\right\}\text{.}\Delta \phi =B\Delta \phi ,`

and [:ref:`éq A4.6 <éq A4.6>`]:

:math:`D{R}_{\text{tot}}^{T}\text{.}\Delta \phi =-{R}_{\text{tot}}^{T}\stackrel{ˆ}{\Delta \Psi }\text{.}`

We show [:ref:`A5 <A5>`] that:


* the sum of the first two integrals of the second member of [:ref:`éq 9.1-1 <éq 9.1-1>`] can be in the form:

:math:`{\int }_{{s}_{1}}^{{s}_{2}}\delta {\phi }^{T}{Y}^{T}\mathrm{EY}\Delta \phi \text{ds};`


* the third integral can be written as:

:math:`{\int }_{{s}_{1}}^{{s}_{2}}\delta {\phi }^{T}{B}^{T}{\Pi C\Pi }^{T}B\Delta \phi \mathrm{ds}+{\int }_{{s}_{1}}^{{s}_{2}}\delta {\phi }^{T}{B}^{T}Z\Delta \phi \text{ds},`

with:

 :math:`Y=\left[\begin{array}{cc}\frac{d}{\text{ds}}1& 0\\ 0& \frac{d}{\text{ds}}1\\ 0& 1\end{array}\right];`

 :math:`E=(\begin{array}{ccc}0& 0& -f\wedge \\ 0& 0& -m\wedge \\ f\wedge & 0& {\stackrel{ˆ}{x}}_{o}\text{'}\stackrel{ˆ}{f}\end{array});` **eq 9.1-2**

 :math:`Z=\left[\begin{array}{cc}0& {R}_{\text{tot}}{C}_{1}{R}_{\text{tot}}^{T}{({R}_{\text{tot}}{C}_{1}^{-1}{R}_{\text{tot}}^{T}f)}^{v}\\ 0& {R}_{\text{tot}}{C}_{2}{R}_{\text{tot}}^{T}{({R}_{\text{tot}}{C}_{2}^{-1}{R}_{\text{tot}}^{T}m)}^{v}\end{array}\right]` **eq 9.1-3**

:math:`{C}_{1}` and :math:`{C}_{2}` being two sub-matrices of :math:`C`:

:math:`\begin{array}{}{C}_{1}=\text{DIAG}\left[\text{EA},{\text{GA}}_{2},{\text{GA}}_{3}\right],\\ {C}_{2}=\text{DIAG}\left[{\text{GI}}_{1},{\text{EI}}_{2},{\text{EI}}_{3}\right]\text{.}\end{array}`

 :math:`{Y}^{T}\mathrm{EY}` is called the geometric rigidity matrix **eq 9.1-4**

 :math:`{B}^{T}{\Pi C\Pi }^{T}` is called material stiffness matrix **eq 9.1-5**

Finally, during the differentiation, a matrix appears that is not in [:ref:`bib2 <bib2>`]:

 :math:`{B}^{T}Z` that we call complementary matrix **eq 9.1-6**




Inertia matrices
------------------

They result from the Fréchet differentiation of :math:`{W}_{\mathrm{iner}}` [:ref:`éq 7-4 <éq 7-4>`] in the :math:`\Delta \phi` direction. More precisely, we place ourselves in the configuration of the :math:`n` -th iteration of the instant :math:`i` and we differentiate in the direction :math:`\Delta {\phi }_{i}^{n+1}`.




Differentiation of translational inertia :math:`\rho A\ddot{{x}_{0}}`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

We immediately have, according to the equations [:ref:`éq An4-4 <éq An4-4>`] on the one hand and [:ref:`éq An3.1-4 <éq An3.1-4>`] on the other hand:

 :math:`D(\rho A{\ddot{x}}_{o,i}^{n})\mathrm{.}\Delta {x}_{o,i}^{n+1}=\rho A{\ddot{x}}_{o,i}^{n+1}=\frac{\rho A}{\beta \Delta {t}^{2}}\Delta {x}_{o,i}^{n+1}` **eq 9.2.1-1**




Differentiation of rotational inertia :math:`{I}_{\rho }\dot{\omega }+\omega \wedge {I}_{\rho }\omega`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~

From equation [:ref:`éq 7-3 <éq 7-3>`]:

:math:`{I}_{\rho }=R{J}_{\rho }{R}^{T}`

:math:`{J}_{\rho }` being a constant tensor.




Terms derived from the differentiation of :math:`{I}_{\rho }`
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

According to equations [:ref:`éq A4-3 <éq A4-3>`] and [:ref:`éq A4-4 <éq A4-4>`], these terms are:

 :math:`{\left[-{({I}_{\rho }\dot{\omega })}^{\text{^}}+{I}_{\rho }\dot{\stackrel{ˆ}{\omega }}-\stackrel{ˆ}{\omega }{({I}_{\omega }\omega )}^{\text{^}}+\stackrel{ˆ}{\omega }{I}_{\rho }\stackrel{ˆ}{\omega }\right]}_{i}^{n}\Delta {\Psi }_{i}^{n+1}` **eq 9.2.2.1-1**

all the quantities in the bracket being taken at iteration :math:`n` from time :math:`i`.




Terms derived from the differentiation of :math:`\omega` and :math:`\dot{\omega }`
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

According to the expressions [:ref:`éq A3.2-3 <éq A3.2-3>`] and [:ref:`éq A3.2-4 <éq A3.2-4>`], the angular speed and acceleration at the iteration of :math:`n` at time :math:`i` are:

:math:`\begin{array}{}{\omega }_{i}^{n}={R}_{i}^{n}{R}_{i}^{n-\mathrm{1,}T}{\omega }_{i}^{n-1}+\frac{\gamma }{\beta \Delta t}{R}_{i}^{n}{R}_{i-1}^{T}({\Psi }_{i-\mathrm{1,}i}^{n}-{\Psi }_{i-\mathrm{1,}i}^{n-1})\\ {\dot{\omega }}_{i}^{n}={R}_{i}^{n}{R}_{i}^{n-\mathrm{1,}T}{\dot{\omega }}_{i}^{n-1}+\frac{1}{\beta \Delta {t}^{2}}{R}_{i}^{n}{R}_{i-1}^{T}({\Psi }_{i-\mathrm{1,}i}^{n}-{\Psi }_{i-\mathrm{1,}i}^{n-1})\text{.}\end{array}`

A variation :math:`\Delta \Psi` of the rotation vector can only affect the quantities relating to this iteration :math:`n` of the instant :math:`i`, since, in the two previous relationships, the other quantities are fixed. In other words, the only difference is :math:`{R}_{i}^{n}\text{et}{\Psi }_{i-\mathrm{1,}i}^{n}`, the vector-rotation increment from the moment to :math:`i-1`, the iteration :math:`n` and the instant :math:`i`.

Gold [:ref:`éq A4-5 <éq A4-5>`]:

:math:`D{R}_{i}^{n}\text{.}\Delta {\Psi }_{i}^{n+1}=\stackrel{ˆ}{{\Delta \Psi }_{i}^{n+1}}{R}_{i}^{n}\text{.}`

And, according to [:ref:`bib3 <bib3>`]:

:math:`D{\Psi }_{i-\mathrm{1,}i}^{n}\text{.}\Delta {\Psi }_{i}^{n+1}=T({y}_{i-\mathrm{1,}i}^{n})\Delta {\Psi }_{i}^{n+1},`

with:

:math:`T(\Psi )=\frac{1}{{\parallel \Psi \parallel }^{2}}\Psi {\Psi }^{T}+\frac{\text{tan}()}{}\left[1-\frac{1}{{\parallel \Psi \parallel }^{2}}\Psi {\Psi }^{T}\right]-\frac{\stackrel{ˆ}{\Psi }}{2}\text{.}`

The terms resulting from the differentiation of :math:`\omega` and :math:`\dot{\omega }` are therefore:

 :math:`\begin{array}{}-{I}_{\rho ,i}^{n}{({R}_{i}^{n}{R}_{i}^{n-\mathrm{1,}T}{\dot{{\omega }_{i}}}^{n-1})}^{\text{^}}-\left[{\stackrel{ˆ}{\omega }}_{i}^{n}{I}_{\rho ,i}^{n}-{({I}_{\rho ,i}^{n}{\stackrel{ˆ}{\omega }}_{i}^{n})}^{\text{^}}\right]{({R}_{i}^{n}{R}_{i}^{n-\mathrm{1,}T}{\omega }_{i}^{n-1})}^{\text{^}}\\ \frac{+1}{\beta \Delta {t}^{2}}\left\{{I}_{\rho ,i}^{n}+\gamma \Delta t\left[{\stackrel{ˆ}{\omega }}_{i}^{n}{I}_{\rho ,i}^{n}-{({I}_{\rho ,i}^{n}{\omega }_{i}^{n})}^{\text{^}}\right]\right\}\\ \left\{{\left[-{R}_{i}^{n}{R}_{i-1}^{T}({\Psi }_{i-\mathrm{1,}i}^{n}-{\Psi }_{i-\mathrm{1,}i}^{n-1})\right]}^{\text{^}}+{R}_{i}^{n}{R}_{i-1}^{T}T({\Psi }_{i-\mathrm{1,}i}^{n})\right\}\end{array}` **eq 9.2.2.2-1**

the combination of the three preceding matrices being multiplied by :math:`\Delta {\Psi }_{i}^{n+1}`.