Improved Model of a Vibrating Sieve
397
Fig. 2 The elastic bars of
the system
1
2
n
3
δ i j =
x
2
i
3x j − x i
6E I
(3)
where δ i j represents the deformation occurred in section i when a force equal to
the unit is applied in section j, E denotes the Young modulus and I is the moment
of inertia of the cross-section, with respect to the neutral axis, x i is the abscissa of
section i, while j is the abscissa of section j.
The elastic bars are discretized by replacing the mass m, which is uniformly
distributed, with n concentrated masses equal to
m
n
, arranged equidistantly along the
bar [7, 8] (Fig. 2).
By discretizing the bar in n elements, the mass matrix has the form:
[M] =
⎡
⎢
⎣
m 11 · · · 0
. . .
. . .
. . .
0 · · · m nn
⎤
⎥
⎦,
(4)
where, for an arbitrary section
m ii = 2
m
n
,
(5)
while for the sections where the sieves are hinged,
m ii = 2
m
n
+ m s ,
(6)
where m s is the mass of the sieve.
For the studied system, the damping matrix has the expression:
[C] = χ [K ] = χ [δ]
−1
,
(7)
where χ denotes a viscosity coefficient