Benchmark solution
=====================

Calculation method used for the reference solution
--------------------------------------------------------

For a system without damping, the differential equation to be solved is written as:

.. math::
: label: EQ-None

    \ {\ begin {array} {} m\ ddot {r} +kr=\ mu\ mid {F} _ {n}\ mid\\ r (t=0) = {r} _ {0}\ ge 0\ ddot {r} +kr=\ mu\ mid {r} _ {r} (t=0) = {r} (t=0) = {r} _ {0}\ {0}\ ge 0\ ge 0\\ dot {r} (r)

We show [:ref:`bib1 <bib1>`] that the solution to the differential equation is written as:

:math: `r (t) =\ frac {\ mathrm {\ mu} | {F} _ {n} |} {k} +\ left ({r} _ {0} -\ frac {\ mathrm {\ mu} | {F} {\ mu} | {F} _ {mu}} | {f} _ {n} |} |} {k} |} {k}\ right)\ mathrm {cos} ({\ mathrm {\ omega}}}} _ {0} | {0}} t) `with:math:` {\ mathrm {\ omega}} _ {0} =\ sqrt {\ frac {k} {m}}} `

The amplitude of the extrema, which all :math:`{t}_{n+1}=\frac{n\pi }{{\omega }_{0}}` originate, obeys the following law of recurrence:

:math:`r({t}_{n+1})={(-1)}^{n}\left[{r}_{0}-\frac{\mu \mid {F}_{n}\mid }{k}\right]\mathrm{cos}{\omega }_{0}t`

with :math:`n=\mathrm{1,2},\dots ,N` such as :math:`∣\frac{r({t}_{n+1})}{{r}_{0}}∣<\frac{\mu \mid {F}_{n}\mid }{k{r}_{0}}`

The movement stops when :math:`∣\frac{r({t}_{n+1})}{{r}_{0}}∣<\frac{\mu \mid {F}_{n}\mid }{k{r}_{0}}` is in position :math:`r({t}_{n+1})`.


Benchmark results
----------------------

Values of the movements in the :math:`\mathrm{\theta }=45°` direction for the times of change of sign of speed :math:`r({t}_{1}),r({t}_{2}),r({t}_{3}),\mathrm{...}` established above.


Uncertainty about the solution
----------------------------

Analytical solution.


Bibliographical references
---------------------------

F. AXISA - Analysis methods in nonlinear structural dynamics: contact nonlinearities - Course IPSI from May 28 to 30, 1991