Avoidance of brake squeal by a separation of the brake ... - tuprints

3.3 Free vibrations of a rectangular plate with a rectangular or circular hole
a)
b)
y
a
y
l x
y
y m
m l y
b
a
r
b
x m
x
x m
x
Figure 3.3: a) Rectangular plate with a rectangular hole. b) Rectangular plate
with a circular hole.
mass and stiffness matrices of the plate with hole is then conducted using
the approach described in section 3.1.2. Since local shape functions are used
for the formulation of these elements, the entries of the coordinate coupling
matrix S are given by local shape functions Uj loc (ξ i ,η i ,τ i ), Vj loc (ξ i ,η i ,τ i ) and
Wj loc (ξ i ,η i ,τ i ) evaluated at local element coordinates ξ i in x-direction, η i in y-
direction and τ i in z-direction (if a cartesian coordinate system is used). Thus
for each node in the mesh of the modification it is necessary to determine the
element in the mesh of the basis structure where the node is placed and to
determine the local coordinates from the global position x i of the node. In
general, it could be an elaborate procedure to determine the element in which
the modification’s node is placed, e.g. if different kinds of elements were used
in one mesh. However, for the rectangular plate presented here, a mesh is used
which consists only of bricks with edge angles of 90 o shown in Fig. 3.4, a),
which makes it very easy to determine the element belonging to the considered
hole node at position x i . The second important point is to determine
the local coordinates ξ i , η i and τ i from the position x i of the actual node.
Depending on the choice of the element type used to mesh the body (e.g.
plate elements), their shape and the local coordinate system used, this might
require additional calculations. Nevertheless, the utilization of isoparametric
elements, which use (local) shape functions not only to describe displacements
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