ABSTRACT
Chances are that the experimenter has only measured the first set. The node line pattern of both sets is the same, except that those of the second set are shifted by
π/2.
In my terminology, this case is a limiting case of node splitting because both sets occur at the same natural frequency. See Figure 4.2 for the basic mode set for n
=
Let us now take the same shell, but consider an attached small point mass that disturbs the axisymmetry. The result is that the modes now have a preferential direction. Their approximate theoretical expressions are still given by Equations 4.1 and 4.2, but their lineup is now such that in the set described by Equation 4.1 the antinode is located at the location of mass while the set described by Equation 4.2
U x m x L n mnl3 ( , ) sin ( / ) cosθ π θ=
U x m x L n mn3 2( , ) sin( / ) sinθ π θ=
will always have a node at the mass location (see Figure 4.3). That is, the origin of the
θ
coordinate is at the mass. The set described by Equation 4.2 occurs still at the natural frequencies of the shell without mass, namely
(4.3)
= .
