2. Benchmark solution#

2.1. Modeling A#

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

2.1.1. Benchmark results#

The mass is released with an initial speed \({V}_{0}\) (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 \(k\mathrm{.}\mathrm{\delta }x\) (configuration b). Once the rebound is over, it is only subjected to gravity (configuration a) with an initial speed \({V}_{1}\).

_images/10000000000006280000044A084203840137B87C.png

Table 2.1.1-1 : Behavioral model

Equation during free fall (configuration a):

(2.1)#\[ \begin{align}\begin{aligned} m\ mathrm {.} \ ddot {x} =-m\ mathrm {.} g\\ And so, by integration:\end{aligned}\end{align} \]
(2.2)#\[ \begin{align}\begin{aligned} x=-g\ mathrm {.} \ frac {{t} ^ {2}} {2}} {2} + {V} _ {0}\ mathrm {.} t\\ The duration of the fall is obtained by solving :math:`x=h`:\end{aligned}\end{align} \]
(2.3)#\[ {t} _ {\ mathit {fall}} =\ frac {1} {g}}\ left [{V} _ {0} +\ sqrt {{V} _ {0} ^ {2} -2.g\ mathrm {.} h}\ right]\]

Equation during rebound (configuration b):

\[\]

: label: eq-4

mmathrm {.} ddot {x} =-mmathrm {.} g+kmathrm {.} (h-z)

And, therefore:

\[\]

: label: eq-5

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

The two constants are valid:

(2.4)#\[ \begin{align}\begin{aligned} {C} _ {2} =g/ {w} ^ {2}; {C} _ {2}; {C} _ {2}} =\ frac {-1} {w}\ sqrt {{V} _ {O} _ {O} ^ {2} -2.g\ mathrm {.} h}\\ The time corresponding to the detachment is obtained by solving :math:`x=h`:\end{aligned}\end{align} \]
(2.5)#\[ \ mathrm {tan}\ left (w\ mathrm {.} \ frac {{\ stackrel {~} {t}}}} _ {\ mathit {decol}}} {2}\ right) = {w} ^ {2}\ mathrm {.} \ frac {{C} _ {1}}} {g}\]

2.2. B modeling#

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

2.2.1. Benchmark results#

The mass is released with an initial speed \({V}_{0}\) (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 \(k\mathrm{.}\mathrm{\delta }x+c\mathrm{.}\dot{x}\) type (configuration c). Once the rebound is over, it is only subjected to gravity (configuration a) with an initial speed \({V}_{1}\).

_images/1000000000000628000003FDA7D229F2BC03F93C.png

Table 2.2.1-1 :: ** Behavioral model

Equation during free fall (configuration a)

(2.6)#\[ \begin{align}\begin{aligned} m\ mathrm {.} \ ddot {x} =-m\ mathrm {.} g\\ And so, by integration:\end{aligned}\end{align} \]
(2.7)#\[ \begin{align}\begin{aligned} x=-g\ mathrm {.} \ frac {{t} ^ {2}} {2}} {2} + {V} _ {0}\ mathrm {.} t\\ The duration of the fall is obtained by solving :math:`x=h`:\end{aligned}\end{align} \]
(2.8)#\[ {t} _ {\ mathit {fall}} =\ frac {1} {g}}\ left [{V} _ {0} +\ sqrt {{V} _ {0} ^ {2} -2.g\ mathrm {.} h}\ right]\]

Equation during rebound (configuration c):

(2.9)#\[ \begin{align}\begin{aligned} m\ mathrm {.} \ ddot {x} =-m\ mathrm {.} g+k\ mathrm {.} (h-z) -c\ mathrm {.} \ dot {x}\\ And so:\end{aligned}\end{align} \]
(2.10)#\[ \begin{align}\begin{aligned} x=\ mathrm {exp}\ left (\ frac {-c\ mathrm {.} \ stackrel {~} {t}} {2.m}\ right)\ left [{C} _ {1}\ mathrm {sin}\ left (\ frac {\ mathit {Rd}\ mathrm {.} \ stackrel {~} {t}} {2}\ right) + {C} _ {2}\ mathrm {cos}\ left (\ frac {\ mathit {Rd}\ mathrm {Rd}\ mathrm {.} \ stackrel {~} {t}} {2} {2}\ right)\ right] + (h-g/ {w} ^ {2}) \ textrm {with} \ stackrel {~} {t} =t- {t} _ {\ mathit {fall}}\\ The constants are equal to:\end{aligned}\end{align} \]
(2.11)#\[ \begin{align}\begin{aligned} {C} _ {2} =g/ {w} ^ {2}; {C} _ {1} =\ frac {c\ mathrm {.} g-2.k\ mathrm {.} \ sqrt {{V} _ {O} ^ {2} -2.g\ mathrm {.} h}} {k\ mathrm {.} \ sqrt {4.k\ mathrm {.} m- {c} ^ {2}/{m} ^ {2}}}}\\ The time corresponding to the detachment is obtained by solving :math:`x=h` numerically.\end{aligned}\end{align} \]

2.2.2. Uncertainty about solutions#

No uncertainty (analytical solution).