9. Linearization of the equations of motion

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

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

• If this quantity is quite small, in the sense of the stopping criterion [bib10], we consider that this \(n\) -th iteration gives the state of the structure at time \(i\).

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

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

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

9.1. Stiffness matrices

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

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

Either:

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

Now, according to equation [éq 6.3-5]:

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

So [éq A4-2]:

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

On the other hand [éq A4-5]:

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

Finally, according to [§6.4]:

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

because [éq 6.3-2] and [éq 6.3-5]:

\(D\left\{\begin{array}{c}\varepsilon \\ \chi \end{array}\right\}\text{.}\Delta \phi =B\Delta \phi ,\)

and [éq A4.6]:

\(D{R}_{\text{tot}}^{T}\text{.}\Delta \phi =-{R}_{\text{tot}}^{T}\stackrel{ˆ}{\Delta \Psi }\text{.}\)

We show [A5] that:

• the sum of the first two integrals of the second member of [éq 9.1-1] can be in the form:

\({\int }_{{s}_{1}}^{{s}_{2}}\delta {\phi }^{T}{Y}^{T}\mathrm{EY}\Delta \phi \text{ds};\)

• the third integral can be written as:

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

with:

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

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

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

\({C}_{1}\) and \({C}_{2}\) being two sub-matrices of \(C\):

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

\({Y}^{T}\mathrm{EY}\) is called the geometric rigidity matrix eq 9.1-4

\({B}^{T}{\Pi C\Pi }^{T}\) is called material stiffness matrix eq 9.1-5

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

\({B}^{T}Z\) that we call complementary matrix eq 9.1-6

9.2. Inertia matrices

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

9.2.1. Differentiation of translational inertia \(\rho A\ddot{{x}_{0}}\)

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

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

Differentiation of rotational inertia \({I}_{\rho }\dot{\omega }+\omega \wedge {I}_{\rho }\omega\) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~

From equation [éq 7-3]:

\({I}_{\rho }=R{J}_{\rho }{R}^{T}\)

\({J}_{\rho }\) being a constant tensor.

9.2.1.1. Terms derived from the differentiation of \({I}_{\rho }\)

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

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

all the quantities in the bracket being taken at iteration \(n\) from time \(i\).

9.2.1.2. Terms derived from the differentiation of \(\omega\) and \(\dot{\omega }\)

According to the expressions [éq A3.2-3] and [éq A3.2-4], the angular speed and acceleration at the iteration of \(n\) at time \(i\) are:

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

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

Gold [éq A4-5]:

\(D{R}_{i}^{n}\text{.}\Delta {\Psi }_{i}^{n+1}=\stackrel{ˆ}{{\Delta \Psi }_{i}^{n+1}}{R}_{i}^{n}\text{.}\)

And, according to [bib3]:

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

with:

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

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

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

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