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:

.. math::
: label: eq-366

    \ left [K\ right]\ left\ {\ Delta W\ right\}\ mathrm {=}\ mathrm {-}\ left\ {L\ right\}




Balance terms
------------------

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

.. math::
 :label: eq-367

    {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:

.. math::
 :label: eq-368

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

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

.. math::
 :label: eq-369

    \ 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 (:math:`{S}_{u,k}^{{g}_{n}}\mathrm{=}1`).

We then consider the friction reaction ():

.. math::
: 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:

.. math::
: 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:

.. math::
: label: eq-372

    \ 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\}\ right\ right\}\ mathrm {=} +\ mu {\ stackrel {}} {\ stackrel {} {S}}}\ right\ right\}}\ right\ right\}}\ right\ right\}}}\ 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}} (1\ mathrm {-} {\ stackrel {} {S}} {S}}} _ {f, k} ^ {{h} _ {\ tau}})\ left\ {{\ stackrel {} {}} {\ tau}} {\ tau}} _ {k}\ right\})

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

.. math::
 :label: eq-373

    \ 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 (:math:`{\stackrel{ˆ}{S}}_{f,k}^{{h}_{\tau }}\mathrm{=}1`):

.. math::
: 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\}




Terms of Signorin's law
-----------------------------

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

.. math::
 :label: eq-375

    {\ 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:

.. math::
 :label: eq-376

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

We get:

.. math::
: label: eq-377

    \ left\ {{L} _ {c} ^ {c}\ right\}\ right\ right\}\ 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\ {\ psi\ right\}

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

.. math::
 :label: eq-378

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

.. math::
 :label: eq-379

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




Coulomb's law terms
---------------------------

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

.. math::
: 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}\ delta\ Lambda d {\ Lambda}\ delta\ Lambda d {\ Lambda}\ delta\ Lambda d {\ Gamma} _ {c}\ mathrm {-}}\\ &\ frac {1} {{\ stackrel {\ cdot}}\ delta\ Lambda d {\ Lambda}}\ delta\ Lambda 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}\ delta\ Lambda 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}}} (1\ mathrm {-}} {\ lambda}}} _ {\ stackrel {} {S}}} _ {f, k} ^ {{g} _ {n} _ {n}}} (1\ mathrm {-}}} (1\ mathrm {-}} {\ stackrel {} {S}}} _ {f, k} ^ {{g} _ {n} _ {n}}}} (1\ mathrm {-}} {\ tau}}) {\ stackrel {} {\ tau}} _ {\ tau}} _ {k}\ mathrm {\ cdot}\ delta\ Lambda d {\ Gamma} _ {c} +\\ & {\ mathrm {\ int}}}} _ {\ int}}} _ {\ int}}} _ {\ int}}} _ {\ int}}} _ {\ Gamma} _ {\ Gamma} _ {\ Gamma} _ {\ Gamma} _ {\ Gamma} _ {\ Gamma} _ {\ Gamma} _ {\ Gamma} _ {\ gamma} _ {c}}} (1\ mathrm {-}} {S}}} _ {u, k}} ^ {{g} _ {n}}) {\ stackrel {} {} {\ Lambda}} _ {k}\ mathrm {\ cdot}\ delta\ Lambda d {\ Gamma} d {\ Gamma} _ {Gamma} _ {c}\ end {array}

After discretization:

.. math::
: label: eq-381

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

We get:

.. math::
: 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 [\ psi\ right]} {\ left [T\ right]}} ^ {T}}\ {T}\\\\\\}\ left\ left\ left\ {\ 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 [\ psi\ right]} ^ {T} {\ left [T\ right]} ^ {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 [\ psi\ right]}} ^ {T}} {\ left [T\ right]} ^ {T}\ left\ {\ stackrel {} {\ tau}}} _ {k}\ right\}}} _ {k}\\ right\}}} _ {k}\ right\}} _ {k}\ right\}} _ {k}\ right\}} _ {k}\ right\}}} _ {k}\ right\}}\ right\}} +\\ right\}} +\\ & (1\ mathrm {-} {\ stackrel {} {S}}} _ {u, k} {\ tau}}} _ {k}\ tau}}} _ {k}\\ right\ right\}} _ {n}}) {\ omega} _ {c} {J} _ {J} _ {c} _ {c} {\ left [\ psi\ right]} ^ {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 (:math:`{S}_{u,k}^{{g}_{n}}\mathrm{=}0`), we have:

.. math::
 :label: eq-383

    {\ 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 (:math:`{S}_{u,k}^{{g}_{n}}\mathrm{=}1` and :math:`{S}_{f,k}^{{h}_{\tau }}\mathrm{=}0`), we have:

.. math::
: 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 [\ psi\ right]}} ^ {T} {\ left [T\ right]} {\ left [T\ right]} ^ {T} (\ left\ {\ stackrel {}} {\ Lambda}}} _ {k}\ right\ right\}\ right\}\ right\}}\ right\}\ right\})

If we are in adherent contact (:math:`{\stackrel{ˆ}{S}}_{u,k}^{{g}_{n}}\mathrm{=}1` and :math:`{\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:

.. math::
: label: eq-385

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

With:

.. math::
: label: eq-386

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

So:

.. math::
 :label: eq-387

    {\ 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\}

In summary:

.. csv-table::

    "Contactless", ":math:`{\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\}`
    :math:`{\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", ":math:`\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}`
    :math:`{\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}`
    :math:`{\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\}`
    :math:`{\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\}`
    :math:`{\left\{{L}_{c}^{c}\right\}}_{\text{glissant}}\mathrm{=}\mathrm{-}{\stackrel{ˆ}{d}}_{n,k}{\omega }_{c}{J}_{c}\left\{\psi \right\}`
    :math:`{\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", ":math:`{\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}`
    :math:`{\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}`
    :math:`{\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\}`
    :math:`{\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\}`
    :math:`{\left\{{L}_{c}^{c}\right\}}_{\text{adhérent}}\mathrm{=}\mathrm{-}{\stackrel{ˆ}{d}}_{n,k}{\omega }_{c}{J}_{c}\left\{\psi \right\}`
    :math:`{\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\}`"





.. _refnumpara__15574_1415021697: