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

Modeling A
--------------

This case models a mass that bounces off an elastic surface (modeling A)


Benchmark results
~~~~~~~~~~~~~~~~~~~~~~~~

The mass is released with an initial speed :math:`{V}_{0}` (configuration a), as long as it has not touched the ground, it only experiences gravity. When it touches the elastic ground, it is subjected, in addition to gravity, to a force of the type :math:`k\mathrm{.}\mathrm{\delta }x` (configuration b). Once the rebound is over, it is only subjected to gravity (configuration a) with an initial speed :math:`{V}_{1}`.


.. image:: images/10000000000006280000044A084203840137B87C.png
    :width: 3.9374in
    :height: 2.7417in


.. _RefImage_10000000000006280000044A084203840137B87C.png:

**Table** 2.1.1-1 **:** Behavioral model

Equation during free fall (configuration a):

.. math::
 :label: eq-1

    m\ mathrm {.} \ ddot {x} =-m\ mathrm {.} g

     And so, by integration:

.. math::
 :label: eq-2

    x=-g\ mathrm {.} \ frac {{t} ^ {2}} {2}} {2} + {V} _ {0}\ mathrm {.} t

     The duration of the fall is obtained by solving :math:`x=h`:

.. math::
 :label: eq-3

    {t} _ {\ mathit {fall}} =\ frac {1} {g}}\ left [{V} _ {0} +\ sqrt {{V} _ {0} ^ {2} -2.g\ mathrm {.} h}\ right]

Equation during rebound (configuration b):

.. math::
: label: eq-4

    m\ mathrm {.} \ ddot {x} =-m\ mathrm {.} g+k\ mathrm {.} (h-z)

     And, therefore:

.. math::
: label: eq-5

    x= {C} _ {1}\ mathrm {sin} (w\ mathrm {.} \ stackrel {~} {t}) + {C} _ {2}\ mathrm {cos} (w\ mathrm {.} \ stackrel {~} {t}) + (h-g/ {w} ^ {2})

     The two constants are valid:

.. math::
 :label: eq-6

    {C} _ {2} =g/ {w} ^ {2}; {C} _ {2}; {C} _ {2}} =\ frac {-1} {w}\ sqrt {{V} _ {O} _ {O} ^ {2} -2.g\ mathrm {.} h}

     The time corresponding to the detachment is obtained by solving :math:`x=h`:

.. math::
 :label: eq-7

    \ mathrm {tan}\ left (w\ mathrm {.} \ frac {{\ stackrel {~} {t}}}} _ {\ mathit {decol}}} {2}\ right) = {w} ^ {2}\ mathrm {.} \ frac {{C} _ {1}}} {g}


B modeling
--------------

This case models a mass that bounces off an absorbent surface.


Benchmark results
~~~~~~~~~~~~~~~~~~~~~~~~

The mass is released with an initial speed :math:`{V}_{0}` (configuration a), as long as it has not touched the surface, it only experiences gravity. When it touches the absorbent surface, in addition to gravity, it is subjected to a force of the :math:`k\mathrm{.}\mathrm{\delta }x+c\mathrm{.}\dot{x}` type (configuration c). Once the rebound is over, it is only subjected to gravity (configuration a) with an initial speed :math:`{V}_{1}`.


.. image:: images/1000000000000628000003FDA7D229F2BC03F93C.png
    :width: 3.9374in
    :height: 2.5492in


.. _RefImage_1000000000000628000003FDA7D229F2BC03F93C.png:

**Table** 2.2.1-1 **:**: ** Behavioral model

Equation during free fall (configuration a)

.. math::
 :label: eq-8

    m\ mathrm {.} \ ddot {x} =-m\ mathrm {.} g

     And so, by integration:

.. math::
 :label: eq-9

    x=-g\ mathrm {.} \ frac {{t} ^ {2}} {2}} {2} + {V} _ {0}\ mathrm {.} t

     The duration of the fall is obtained by solving :math:`x=h`:

.. math::
 :label: eq-10

    {t} _ {\ mathit {fall}} =\ frac {1} {g}}\ left [{V} _ {0} +\ sqrt {{V} _ {0} ^ {2} -2.g\ mathrm {.} h}\ right]

Equation during rebound (configuration c):

.. math::
 :label: eq-11

    m\ mathrm {.} \ ddot {x} =-m\ mathrm {.} g+k\ mathrm {.} (h-z) -c\ mathrm {.} \ dot {x}

     And so:

.. math::
 :label: eq-12

    x=\ mathrm {exp}\ left (\ frac {-c\ mathrm {.} \ stackrel {~} {t}} {2.m}\ right)\ left [{C} _ {1}\ mathrm {sin}\ left (\ frac {\ mathit {Rd}\ mathrm {.} \ stackrel {~} {t}} {2}\ right) + {C} _ {2}\ mathrm {cos}\ left (\ frac {\ mathit {Rd}\ mathrm {Rd}\ mathrm {.} \ stackrel {~} {t}} {2} {2}\ right)\ right] + (h-g/ {w} ^ {2})
    \ textrm {with}
    \ stackrel {~} {t} =t- {t} _ {\ mathit {fall}}

     The constants are equal to:

.. math::
 :label: eq-13

    {C} _ {2} =g/ {w} ^ {2}; {C} _ {1} =\ frac {c\ mathrm {.} g-2.k\ mathrm {.} \ sqrt {{V} _ {O} ^ {2} -2.g\ mathrm {.} h}} {k\ mathrm {.} \ sqrt {4.k\ mathrm {.} m- {c} ^ {2}/{m} ^ {2}}}}

     The time corresponding to the detachment is obtained by solving :math:`x=h` numerically.


Uncertainty about solutions
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

No uncertainty (analytical solution).