ARUP; ISBN: 978-0-9562121-5-3 - CMBBE 2012 - Cardiff University

K
sh
=
2.
3E
sh
2
( ) 2
1
2
1−ν
( 4)
Rsh
sh
The bulk modulus of the fluid has no effect on the contact stiffness of the shell.
3-2: Analytical Model: Implicit equation for Fmax
With the assumption of a conservative system and by writing the principle of Energy
Conservation before the impact and at the point of maximum contraction:
5
1 2 1 2 1 2 2 1 2 1
2
2
mshVsh + meVe = Ksh∆ xsh + KH ∆ xH + ( msh + me ) Vsh−e + msh ( VshSinα )
2 2 2 5 2 2
(5)
∆ is the deformation due to the membrane and bending of shell, Vsk is the system
xsh
velocity of shell-torus object in the maximum contraction point, and ∆xH is the
Hertzian contact deformation in the shell and torus body together. The impact occurs
in "n" direction and the value ( ) 2
1
msh VshSinα is kinetic energy of the shell in "t"
2
direction during the impact. Using the principle of conservation of momentum:
m V + m V = m + m V (6)
h
( )
sh sh−n e e sh e sh−e− n
Since the impact occurs in "n" direction, in the above equation, Vsk-n is the velocity of
shell in "n" direction before impact, and Vsh-e-n is the common velocity of the two
objects during the impact in "n" direction. By substituting Vsk-n in equation 6 and
solving for Vsh-e-n, we obtain:
xsh
V
sh−e m V Cosα + m V
=
m + m
sh sh e e
sh e
∆ and ∆xH could be obtained from the equation of maximum transferred force
(7)
2
⎛ F 3
max ⎞
x = ⎜
⎟
H
(8)
K H
Fmax
∆ xsh
= And ∆
K sh
⎝ ⎠
By replacing equations 7 and 8 into equation 5 and arranging:
Where
2
max
5
4 F 3
max
sh
2
5
K 3
H
F
∆ U = + (9)
K
∆ U = m V Cos α + m V −
( ) 2
m V Cosα + m V
2 2 2 sh sh e e
sh sh e e
msh + me
Equation 9 is an implicit and approximate representation for the maximum transferred
force which contains both Hertzian contact deformations in shell and torus object and
also membrane and bending effect in the shell. It can be solved analytically for Fmax to
obtain the maximum deformation and maximum acceleration of the fluid filled shell.
∆x
2
3
max max
max ⎟ F ⎛ F ⎞
= ∆x
+ ∆ = + ⎜
sh xH
K sh K H
a
max =
F
m
max
sh
⎝
(11)
⎠
(10)