12. Appendix B: second member vector

The expressions considered here relate only to terms related to contact friction. It is therefore a question of discretizing the expression of virtual works given by the equations () to (). Remember that the system to be solved is:

\[\]

: label: eq-366

left [Kright]left{Delta Wright}mathrm {=}mathrm {-}left{Lright}

12.1. Balance terms

We start with the expression for the contact reaction in the equilibrium equation, starting with \({G}_{c}\) (), which we write on the known configuration at time step \(k\), since the start of the Newton process:

(12.1)\[ {G} _ {c}\ mathrm {=} {\ mathrm {\ int}}} _ {{\ Gamma} _ {c}} {\ stackrel {} {S}}} _ {u, k}} ^ {{g}} ^ {{g} _ {n}} {n}} _ {n, k}\ tilde {\ delta} {d}} ^ {g} _ {n}} _ {n} d {\ Gamma} _ {c}\]

After discretization:

(12.2)\[ \ mathrm {-} {G} _ {c}\ stackrel {\ text {Discretization}} {\ to}\ left\ {{L} _ {c} ^ {e}\ right\},\ left\ {e}\ right\},\ left\ {{L}}\ right\},\ left\ {{L}}\ right\}\]

The sign \(–\) comes from the expression for balance in (). Using () and separating the slave and master contributions, we get:

(12.3)\[ \ left\ {{L} _ {c} ^ {e}\ right\}\ right\ right\}\ mathrm {=}\ mathrm {-} {\ omega} _ {c} {c} {\ stackrel {} {S}}}\ right\ right\ right\}\ right\ right\ right\}}\ mathrm {=}} {\ mathrm {-} {\ omega} _ {g} _ {g}} {g}} {c} {c} {\ stackrel {}} {g}} {c} {\ stackrel {}} {g}} {g}} {\ stackrel {}} {g}}} {n, k} _ {n, k}} _ {n, k} {\ left [{N} ^ {e}\ right]}} ^ {T}\ mathrm {\ {} {n} _ {h}\ mathrm {\}} \ left\ {{L} _ {c} ^ {m}\ right\}\ right\}\ right\ right\}\ mathrm {=} + {\ omega} _ {c} {c} {c} {s}}} _ {u, k}}} _ {u, k}} _ {u, k}} _ {u, k}} _ {u, k}} _ {u, k}} _ {u, k}} _ {u, k}} _ {u, k}} _ {u, k}} _ {u, k}} _ {u, k}} _ {u, k}} _ {u, k}} _ {u, k}} _ {u, k}} _ {u, k}} _ {u, k}}} _ {u, k}} ackrel {} {N}}} ^ {m}\ right]} ^ {T}\ mathrm {\ {} {n} _ {h}\ mathrm {\}}\ mathrm {\}}\]

These terms are only active when there is contact (\({S}_{u,k}^{{g}_{n}}\mathrm{=}1\)).

We then consider the friction reaction ():

\[\]

: label: eq-370

{G} _ {f} = {int} _ {{mathrm {Gamma}} _ {c}}mathrm {mu} {widehat {mathrm {lambda}}}} _ {k}} {k} {widehat {S}}}} _ {f} _ {f} _ {f} _ {f}, k} ^ {{h} _ {mathrm {tau}}}}}} {h}} {h} _ {mathrm {tau}, k}cdot {stackrel {~} {mathrm {delta}}} d {mathrm {tau}}, k}cdot {stackrel {~}} {} {} {} {mathrm {delta}} d {mathrm {tau}} d {mathrm {gamma}} _ {mathrm {delta}} d {mathrm {tau}} {Gamma}} _ {c}}}mathrm {mu} {mathrm {mu} {widehat}}}} _ {widehat {S}} _ {u, k} _ {u, k}} ^ {u, k}} ^ {u, k}} ^ {u, k}} ^ {u, k}} ^ {u, k}} ^ {u, k}} ^ {u, k}} ^ {u, k}} ^ {u, k} ^ {u, k}} ^ {u, k} ^ {u, k}} ^ {u, k} ^ {u, k}} ^ {u, k} ^ {u, k}} ^ {u, k}} ^ {u, k} ^ {u, k}} ^ {thrm {tau}}}right) {widehat {widehat {mathrm {tau}}}} _ {k}cdot {stackrel {~} {mathrm {delta}} d} {mathrm {delta}} d {mathrm {delta}}} d {mathrm {gamma}}} _ {c}

Using the projection operator (). After discretization:

\[\]

: label: eq-371

mathrm {-} {G} _ {f}stackrel {text {Discretization}} {to}left{{L} _ {f} ^ {e}right},left{e}right},left{{L}}right},left{{L}}right}

We get:

\[\]

: label: eq-372

left{{L} _ {f} ^ {e}right}rightright}mathrm {=}mathrm {-}mu {stackrel {} {lambda}} _ {k} {k} {stackrel {}} {e}rightrightright}rightrightrightrightrightrightrightrightrightrightrightrightrightrightrightrightrightrightrightrightright}}mathrm {-}}mu {stackrel {}} {lambda}}} _ {k} {stackrel {e} {e} {e}rightrightrightrightrightrightrightrightrightright} {left [{N} ^ {e}right]} ^ {e}right]} ^ {T}left [{P} ^ {tau}right] ({stackrel {} {S}}} _ {f, k} ^ {{h}} ^ {h}} ^ {h}}} _ {tau, k} ^ {{h}} ^ {h}} ^ {h}} ^ {h} _ {f, k} ^ {h} _ {{h} _ {h} _ {{h}} _ {{h}} _ {{h}} _ {{h}} ^ {h} _ {{h}} _ {{h}} _ {tau, k} ^ {h}} ^ {h} _ {{h} _ {h}} mathrm {-} {stackrel {} {S}} {S}}} _ {f, k}} _ {{h} _ {tau}})left{{stackrel {}} {tau}} {tau}}} _ {k}right}) left{{L} _ {f} ^ {m} {m}right}rightright}mathrm {=} +mu {stackrel {}} {stackrel {} {S}}}rightright}}rightright}}rightright}}}mathrm {=} +mu {stackrel {}} {lambda}}} _ {k} {stackrel {} {S} {S}} {S}}} {S}}} {S}}} _ {c} {S}} {S}}} {S}}} _ {c} {left}} _ {c} {left [stackrel {} {N}}} ^ {m}right]}}right]}} ^ {T}left [{P} ^ {tau}right] ({stackrel {} {S}}} _ {f, k}} ^ {m}} ^ {h}}} ^ {h}}} _ {f, k} ^ {h}}} _ {f, k} ^ {h}} ^ {h}} ^ {h}} ^ {h}} _ {f, k} ^ {h}} ^ {h}} ^ {h}} ^ {h}} ^ {h}} ^ {h}} ^ {h}} ^ {h}} ^ {h}} ^ {h}} ^ {h}} ^ {h}} ^ {h}} (1mathrm {-} {stackrel {} {S}} {S}}} _ {f, k} ^ {{h} _ {tau}})left{{stackrel {} {}} {tau}} {tau}} _ {k}right})

An active term is observed in sliding contact (\({\stackrel{ˆ}{S}}_{f,k}^{{h}_{\tau }}\mathrm{=}0\)):

(12.4)\[ \ left\ {{L} _ {f} ^ {e}\ right\}\ right\ right\}\ mathrm {=}\ mathrm {-}\ mu {\ stackrel {} {\ lambda}} _ {k} {k} {\ stackrel {}} {e}\ right\ right\ right}\ right\ right\ right\ right\ right\ right\ right\ right\ right\ right\ right\ right\ right\ right\ right\ right\ right\ right\ right\ right\ right\}}\ mathrm {-}}\ mu {\ stackrel {}} {\ lambda}}} _ {k} {\ stackrel {e} {e} {e}\ right\ right\ right\ right\ right\ right\ right\ right\ right\ right\} {\ left [{N} ^ {e}\ right]} ^ {e}\ right]} ^ {T}\ left [{P} ^ {\ tau}\ right] ({\ stackrel {} {S}}} _ {f, k} ^ {{h}} ^ {h}} ^ {h}}} _ {\ tau, k} ^ {{h}} ^ {h}} ^ {h}} ^ {h} _ {f, k} ^ {h} _ {{h} _ {h} _ {{h}} _ {{h}} _ {{h}} _ {{h}} ^ {h} _ {{h}} _ {{h}} _ {\ tau, k} ^ {h}} ^ {h} _ {{h} _ {h}} mathrm {-} {\ stackrel {} {S}} {S}}} _ {f, k}} _ {{h} _ {\ tau}})\ left\ {{\ stackrel {}} {\ tau}} {\ tau}}} _ {k}\ right\}) {\ left\ {{L} _ {f} ^ {m} {m}\ right\}}} _ {\ text {sliding}}\ mathrm {=} +\ mu {\ stackrel {}} {\ lambda}} {\ lambda}}}} _ {\ omega}} _ {\ omega} _ {c} {J}} _ {c} {J} _ {c}} {c}} {\ left [{\ stackrel {} {N}} {N}} {\ lambda}}} _ {k} {\ lambda}}} _ {k} {\ omega}} _ {c} {J} _ {c} {J} _ {c} {J} _ {c} {J} _ {c}} {\ left [{\ stackrel {]} ^ {T}\ left [{P} ^ {\ tau} ^ {\ tau}\ right]\ left\ {{\ stackrel {} {\ tau}}} _ {k}\ right\}\]

And an active term in adherent contact (\({\stackrel{ˆ}{S}}_{f,k}^{{h}_{\tau }}\mathrm{=}1\)):

\[\]

: label: eq-374

{left{{L} _ {f} ^ {e}right}}} _ {text {member}}mathrm {-}mathrm {-}mu {stackrel {}} {lambda} {lambda}} {lambda}}}} _ {left [{N} ^ {e}lambda}}} {lambda}}} _ {omega}} _ {omega} _ {c} _ {c} {c} {c} {left [{N} ^ {e}right}right]} ^ {e}right]} ^ {e}right]} ^ {T}left [{P} ^ {tau}right]right]left{{stackrel {} {h}}} _ {tau, k}right} {left{{L} _ {f} ^ {m} {m}right}}} _ {text {adherent}}mathrm {=} +mu {stackrel {}} {lambda}}}} _ {k} {lambda}}}} _ {omega}} _ {c} {J}} _ {c} {J} _ {c} {J} _ {c} {J} _ {left [{stackrel {} {N}} {N}}} {lambda}}} _ {k} {lambda}}} _ {omega} _ {c} {J} _ {c} {J} _ {c} {J} _ {c} {J} _ {c}} {left [{]} ^ {T}left [{P} ^ {tau} ^ {tau}right]left{{stackrel {} {h}}} _ {tau, k}right}

12.2. Terms of Signorin’s law

We will now consider the term corresponding to the unknown in contact pressure, starting from ():

(12.5)\[ {\ tilde {G}} _ {c}\ mathrm {=}\ mathrm {=}\ mathrm {-}\ frac {1} {{\ rho}} {\ mathrm {\ int}}} _ {\ Gamma}} _ {\ Gamma} _ {\ int}}} _ {\ int}}} _ {\ int}} _ {\ int}} _ {\ int}} _ {\ int}} _ {\ int}} _ {\ int}} _ {\ int}} _ {\ int}} _ {\ int}} _ {\ int}} _ {\ int}} _ {\ int}} _ {\ int}} _ {\ int}} _ {\ int}}} _ {\ int}} _ {\ int}}} _ {\ stackrel {} {S}} _ {u, k}} ^ {{g}} ^ {{g} _ {n}} {g}} _ {n, k}\ right\}\ right\}\ right\}\ delta\ lambda d {\ Gamma} _ {c}\]

After discretization:

(12.6)\[ {\ tilde {G}} _ {c}\ stackrel {\ text {Discretization}} {\ to}\ left\ {{L} _ {c}} ^ {c} ^ {c}\ right\}\]

We get:

\[\]

: label: eq-377

left{{L} _ {c} ^ {c}right}rightright}mathrm {=}mathrm {-}frac {1} {{rho} _ {n}}} ({stackrel {}}} ({stackrel {}}}} ({stackrel {}}}}} ({stackrel {}}}} ({stackrel {}}}} ({stackrel {}}}} ({stackrel {}}}} ({stackrel {}}}} ({stackrel {}}}} ({stackrel {}}}} ({stackrel {}}}} ({{{g} _ {n}} {stackrel {} {g}} {g}}} _ {n, k}) {omega} _ {c} {J} _ {c}left{left{psiright}

This expression can be divided into two parts. If the contact is inactive (\({S}_{u,k}^{{g}_{n}}\mathrm{=}0\)), we have:

(12.7)\[ {\ left\ {{L} _ {c} ^ {c} {c}\ right\}}} _ {\ text {contactless}}\ mathrm {-}\ frac {1} {{\ rho} {{\ rho} _ {\ rho} _ {\ rho} _ {\ rho} _ {\ rho} _ {n}}}} _ {n}}} {n}}} {n}}} {n}}} {n}}} {n}}} {n}}} {\ stackrel {}} {\ lambda}} _ {\ omega} _ {c} {J} {J} {\ rho} _ {\ rho} _ {c}} _ {c}} {c}}\ left\ {\ psi\ right\}\]

If the contact is active (\({\stackrel{ˆ}{S}}_{u,k}^{{g}_{n}}\mathrm{=}1\)), we have terms that can be simplified between \({\stackrel{ˆ}{\lambda }}_{k}\) and \({\stackrel{ˆ}{g}}_{n,k}\), which gives:

(12.8)\[ {\ left\ {{L} _ {c} ^ {c} {c}\ right\}}} _ {\ text {with contact}}\ mathrm {-} {\ stackrel {} {d}} {d}}} _ {n, k}\ right}}} _ {c} {J} _ {c}\ left\ {\ psi\ right\}\]

12.3. Coulomb’s law terms

Finally, the term corresponds to the unknown in friction pressure, based on the weak expression of Coulomb’s law ():

\[\]

: label: eq-380

begin {array} {cc} {tilde {G}}} _ {f}mathrm {=}} &frac {1} {{stackrel {} {rho}} _ {t}}} {mathrm {int}} {mathrm {int}} {mathrm {int}}} {mathrm {int}}} _ {int}} _ {int}} _ {int}} _ {int}} _ {int}} _ {int}} _ {int}} _ {int}} _ {int}} _ {int}} _ {k} {stackrel {} {S}} _ {u, k} _ {u, k}} ^ {{g} _ {n}} {stackrel {}} {Lambda}} _ {k}mathrm {cdot}\ mathrm {cdot}\cdot}\cdot}\cdot}deltaLambda d {Lambda}deltaLambda d {Lambda}deltaLambda d {Gamma} _ {c}mathrm {-}}\ &frac {1} {{stackrel {cdot}}deltaLambda d {Lambda}}deltaLambda d {Lambda}}rho}} _ {t}} {mathrm {int}}} _ {{Gamma} _ {c}}mu {stackrel {} {lambda}} _ {k} {stackrel {} {S}}} _ {S}}} _ {S}} _ {S}} _ {S}} _ {S}} _ {S}} {f, k} ^ {{h} _ {tau}}} {stackrel {}}} {stackrel {}} {tau, k}mathrm {cdot}deltaLambda d {Gamma} _ {Gamma} _ {Gamma} _ {Gamma} _ {Gamma} _ {Gamma} _ {Gamma} _ {Gamma} _ {c}mathrm {-}}mathrm {-}}mathrm {-}}\ frac {1} {stackrel {}} {rho}} _ {Gamma}} _ {Gamma} _ {Gamma} _ {Gamma} _ {Gamma} _ {Gamma} _ {m {int}} _ {{Gamma} _ {c}}mu {stackrel {}} {stackrel {}} {stackrel {} {S}} _ {u, k} ^ {{g} _ {n} _ {n}}} {n}}} (1mathrm {-}} {lambda}}} _ {stackrel {} {S}}} _ {f, k} ^ {{g} _ {n} _ {n}}} (1mathrm {-}}} (1mathrm {-}} {stackrel {} {S}}} _ {f, k} ^ {{g} _ {n} _ {n}}}} (1mathrm {-}} {tau}}) {stackrel {} {tau}} _ {tau}} _ {k}mathrm {cdot}deltaLambda d {Gamma} _ {c} +\ & {mathrm {int}}}} _ {int}}} _ {int}}} _ {int}}} _ {int}}} _ {Gamma} _ {Gamma} _ {Gamma} _ {Gamma} _ {Gamma} _ {Gamma} _ {Gamma} _ {Gamma} _ {gamma} _ {c}}} (1mathrm {-}} {S}}} _ {u, k}} ^ {{g} _ {n}}) {stackrel {} {} {Lambda}} _ {k}mathrm {cdot}deltaLambda d {Gamma} d {Gamma} _ {Gamma} _ {c}end {array}

After discretization:

\[\]

: label: eq-381

{tilde {G}} _ {f}stackrel {text {Discretization}} {to}left{{L} _ {f}} ^ {f} ^ {f}right}

We get:

\[\]

: label: eq-382

begin {array} {cc}left{{L} _ {f} _ {f} ^ {f}right}mathrm {=} &frac {1} {{stackrel {} {rho}} {rho}}} _ {t}}} _ {rho}}} _ {k}} _ {k} {stackrel {} {S}}} _ {u, k} _ {u, k} ^ {{g} _ {n}} {left [psiright]} {left [Tright]}} ^ {T}}{T}\\\}leftleftleft{left}\ &frac {1} {{stackrel {} {lambda}} {rho}} _ {t}}}mu {omega} _ {c} {stackrel {} {lambda}} {lambda}}} _ {k} {rho}}} _ {stackrel {} {S}}} _ {u, k} {g} _ {n}} {stackrel {}} {lambda}} _ {stackrel {}}} _ {stackrel {}} {lambda}}} _ {stackrel {}} {lambda}}} _ {stackrel {}} S}} _ {f, k} ^ {{h} _ {tau}} {tau}} {left [psiright]} ^ {T} {left [Tright]} ^ {T}left{{stackrel {}} _ {tau} {h}}} _ {tau, k}right]}\ &frac {1} {stackrel {} {stackrel {} {h}}} _ {tau, k}right}right}\ &frac {1} {stackrel {} {stackrel {} {h}}} rel {} {rho}} _ {t}} mu {omega} _ {c} {J} _ {J} _ {c} {stackrel {}} {lambda}} _ {stackrel {} {S}}} _ {u, k}} ^ {{g}} ^ {g} _ {g} _ {g} _ {g} _ {g} _ {g} _ {g} _ {g} _ {g} _ {g} _ {g} _ {g} _ {g} _ {g} _ {g} _ {g} _ {g} _ {g} _ {g} _ {g} _ {g} _ {g} _ {g} _ {g} _ {g} _ {g} _ {g} _ {g} _ {g} _ {g} _ {g} _ {g} _ {tau}}) {left [psiright]}} ^ {T}} {left [Tright]} ^ {T}left{stackrel {} {tau}}} _ {k}right}}} _ {k}\ right}}} _ {k}right}} _ {k}right}} _ {k}right}} _ {k}right}}} _ {k}right}}right}} +\ right}} +\ & (1mathrm {-} {stackrel {} {S}}} _ {u, k} {tau}}} _ {k}tau}}} _ {k}\ rightright}} _ {n}}) {omega} _ {c} {J} _ {J} _ {c} _ {c} {left [psiright]} ^ {T}} ^ {T}left{{stackrel {{stackrel {}} {stackrel {}} {stackrel {}} {stackrel {}} {stackrel {}} {stackrel {}} {stackrel {}} {Lambda}}} _ {k}right}end {array}

This complex expression can be broken down into three parts. If the contact is inactive (\({S}_{u,k}^{{g}_{n}}\mathrm{=}0\)), we have:

(12.9)\[ {\ left\ {{L} _ {f} ^ {f} ^ {f}\ right\}}} _ {\ text {without contact}}\ mathrm {=} {\ omega} _ {J} _ {c} _ {c} {\ c} {\ {c} {\} {\ c} {\} {\ stackrel {} {\ stackrel {\} {\ stackrel {\} {\ stackrel {\} {\ left} {\ stackrel {\} {\ left} {\ lambda} {\ left}} _ {k}\ right\}\]

If we are in sliding contact (\({S}_{u,k}^{{g}_{n}}\mathrm{=}1\) and \({S}_{f,k}^{{h}_{\tau }}\mathrm{=}0\)), we have:

\[\]

: label: eq-384

{left{{L} _ {f} ^ {f} {f}right}}} _ {text {slippery}}mathrm {=}frac {1} {{stackrel {} {rho} {rho}} {rho}}} _ {rho}}} _ {rho}}} _ {t}}} _ {t}}}mu {stackrel {} {rho}}} _ {t}}} _ {t}}}mu {stackrel {} {rho}}} _ {t}}} _ {t}}}mu {stackrel {} {rho}}} _ {t}}} _ {stackrel {{c} {left [psiright]}} ^ {T} {left [Tright]} {left [Tright]} ^ {T} (left{stackrel {}} {Lambda}}} _ {k}rightright}right}right}}right}right})

If we are in adherent contact (\({\stackrel{ˆ}{S}}_{u,k}^{{g}_{n}}\mathrm{=}1\) and \({\stackrel{ˆ}{S}}_{f,k}^{{h}_{\tau }}\mathrm{=}1\)), we have, some terms are simplified between the first and the second line of (), in particular, we have:

\[\]

: label: eq-385

left{{stackrel {}} {Lambda}} {Lambda}} _ {Lambda}}} _ {tau, k}right}} _ {tau, k}right}rightright}}rightright}}mathrm {=}rightright}}rightright}}mathrm {-}mathrm {-}left {stackrel {} {rho}} _ {tau, k}rightright}}rightright}}rightright}}mathrm {-}mathrm {-}left{{mathrm {}stackrel {Ø} {x} {x}mathrm {}} _ {tau, k}right}

With:

\[\]

: label: eq-386

left{{stackrel {}} {Lambda}} {Lambda}} _ {Lambda}}} _ {tau, k}right}} _ {tau, k}right}rightright}}rightright}}mathrm {=}rightright}}rightright}}mathrm {-}mathrm {-}left {stackrel {} {rho}} _ {tau, k}rightright}}rightright}}rightright}}mathrm {-}mathrm {-}left{{mathrm {}stackrel {} {x} {x}mathrm {}} _ {tau, k}right}mathrm {-}mathrm {-} {mathrm {-} {} {stackrel {} {} {mathrm} {} {mathrm {}stackrel {} {x} {x}mathrm {}} _ {k}right}

So:

(12.10)\[\begin{split} {\ left\ {{L} _ {f} ^ {f} {f}\ right\}}} _ {\ text {adherent}}\ mathrm {-}\ mu {\ stackrel {}} {\ lambda} {\ lambda}}} {\ lambda}}}} _ {\ omega}} _ {\ omega} _ {\ omega} _ {c} _ {c} {\ left [\\\ right]}} ^ {T} {\ right]} {\ lambda}} {\ lambda}}} _ {\ omega}} _ {\ omega} _ {\ omega} _ {c} _ {c} {\ left [\\\ right]}} ^ {T} {\ right]} {\ lambda}} {\ lambda}}} _ {\ omega} left [T\ right]} ^ {T}\ left [{P} ^ {P} ^ {\ tau}\ right]\ left\ {{\ mathrm {}\ stackrel {} {x}\ mathrm {}}\ mathrm {}}} _ {k}\ right\}\end{split}\]

In summary:

Contactless	\({\left\{{L}_{c}^{c}\right\}}_{\text{sans contact}}\mathrm{=}\mathrm{-}\frac{1}{{\rho }_{n}}{\stackrel{ˆ}{\lambda }}_{k}{\omega }_{c}{J}_{c}\left\{\psi \right\}\) \({\left\{{L}_{f}^{f}\right\}}_{\text{sans contact}}\mathrm{=}{\omega }_{c}{J}_{c}{\left[\psi \right]}^{T}{\left[T\right]}^{T}\left\{{\stackrel{ˆ}{\Lambda }}_{k}\right\}\)
Sliding contact	\(\left\{{L}_{c}^{e}\right\}\mathrm{=}\mathrm{-}{\left[{N}^{e}\right]}^{T}\mathrm{\{}{n}_{h}\mathrm{\}}{\stackrel{ˆ}{g}}_{n,k}{\omega }_{c}{J}_{c}\) \({\left\{{L}_{c}^{m}\right\}}_{\text{glissant}}\mathrm{=}+{\left[{N}^{m}\right]}^{T}\mathrm{\{}{n}_{h}\mathrm{\}}{\stackrel{ˆ}{g}}_{n,k}{\omega }_{c}{J}_{c}\) \({\left\{{L}_{f}^{e}\right\}}_{\text{glissant}}\mathrm{=}\mathrm{-}\mu {\stackrel{ˆ}{\lambda }}_{k}{\omega }_{c}{J}_{c}{\left[{N}^{e}\right]}^{T}\left[{P}^{\tau }\right]\left\{{\stackrel{ˆ}{\tau }}_{k}\right\}\) \({\left\{{L}_{f}^{m}\right\}}_{\text{glissant}}\mathrm{=}+\mu {\stackrel{ˆ}{\lambda }}_{k}{\omega }_{c}{J}_{c}{\left[{\stackrel{ˉ}{N}}^{m}\right]}^{T}\left[{P}^{\tau }\right]\left\{{\stackrel{ˆ}{\tau }}_{k}\right\}\) \({\left\{{L}_{c}^{c}\right\}}_{\text{glissant}}\mathrm{=}\mathrm{-}{\stackrel{ˆ}{d}}_{n,k}{\omega }_{c}{J}_{c}\left\{\psi \right\}\) \({\left\{{L}_{f}^{f}\right\}}_{\text{glissant}}\mathrm{=}\frac{1}{{\stackrel{ˉ}{\rho }}_{t}}\mu {\stackrel{ˆ}{\lambda }}_{k}{\omega }_{c}{J}_{c}{\left[\psi \right]}^{T}{\left[T\right]}^{T}(\left\{{\stackrel{ˆ}{\Lambda }}_{k}\right\}\mathrm{-}\left\{{\stackrel{ˆ}{\tau }}_{k}\right\})\)
Member contact	\({\left\{{L}_{c}^{e}\right\}}_{\text{adhérent}}\mathrm{=}\mathrm{-}{\left[{N}^{e}\right]}^{T}\mathrm{\{}{n}_{h}\mathrm{\}}{\stackrel{ˆ}{g}}_{n,k}{\omega }_{c}{J}_{c}\) \({\left\{{L}_{c}^{m}\right\}}_{\text{adhérent}}\mathrm{=}+{\left[{N}^{m}\right]}^{T}\mathrm{\{}{n}_{h}\mathrm{\}}{\stackrel{ˆ}{g}}_{n,k}{\omega }_{c}{J}_{c}\) \({\left\{{L}_{f}^{e}\right\}}_{\text{adhérent}}\mathrm{=}\mathrm{-}\mu {\stackrel{ˆ}{\lambda }}_{k}{\omega }_{c}{J}_{c}{\left[{N}^{e}\right]}^{T}\left[{P}^{\tau }\right]\left\{{\stackrel{ˆ}{h}}_{\tau ,k}\right\}\) \({\left\{{L}_{f}^{m}\right\}}_{\text{adhérent}}\mathrm{=}+\mu {\stackrel{ˆ}{\lambda }}_{k}{\omega }_{c}{J}_{c}{\left[{\stackrel{ˉ}{N}}^{m}\right]}^{T}\left[{P}^{\tau }\right]\left\{{\stackrel{ˆ}{h}}_{\tau ,k}\right\}\) \({\left\{{L}_{c}^{c}\right\}}_{\text{adhérent}}\mathrm{=}\mathrm{-}{\stackrel{ˆ}{d}}_{n,k}{\omega }_{c}{J}_{c}\left\{\psi \right\}\) \({\left\{{L}_{f}^{f}\right\}}_{\text{adhérent}}\mathrm{=}\mathrm{-}\mu {\stackrel{ˆ}{\lambda }}_{k}{\omega }_{c}{J}_{c}{\left[\psi \right]}^{T}{\left[T\right]}^{T}\left[{P}^{\tau }\right]\left\{{\mathrm{〚}\stackrel{ˆ}{x}\mathrm{〛}}_{k}\right\}\)