Figure 1
shows the updated acoustic mesh near the footprint region. The geometric
changes associated with the updated mesh are taken into account in the coupled
acoustic-structural analyses.
The eigenvalues of the air cavity, the tire, and the coupled tire-air system
are tabulated in
Table 1.
The resonant frequencies of the uncoupled air cavity are computed using the
original configuration. We obtain two acoustic modes at frequencies of 228.58
Hz and 230.17 Hz. These frequencies correspond to two identical modes rotated
90° with respect to each other, as shown in
Figure 2
and
Figure 3;
the magnitudes of the frequencies are different since we have used a nonuniform
mesh along the circumferential direction. We refer to the two modes as the
fore-aft mode and the vertical mode, respectively. These eigenfrequencies
correspond very closely to our original estimate of 230 Hz. The table shows
that these eigenfrequencies occur at almost the same magnitude in the coupled
system, indicating that the coupling has a very small effect on the acoustic
resonance. The difference between the two vertical modes is larger than the
difference between the fore-aft modes. This can be attributed to the geometry
changes associated with structural loading. The coupling has a much stronger
influence on the structural modes than on the acoustic modes, but we expect the
coupling to decrease as we move away from the 230 Hz range.
Figure 4
to
Figure 7
show the response of the structure to the spindle excitation.
Figure 4
and
Figure 5
compare the response of the coupled tire-air system to the response of a tire
without the air cavity.
Figure 6
and
Figure 7
show the acoustic pressure measured in the crown and side of the air. We draw
the following conclusions from these figures. The frequencies at which
resonance is predicted by the steady-state dynamic analysis correspond closely
to the eigenfrequencies. However, not all the eigenmodes are excited by the
spindle excitations. For example, the fore-aft mode is not excited by vertical
loading. Similarly, the vertical mode is not excited by fore-aft loading. In
addition, only some of the structural modes are excited by the spindle loads,
while others are suppressed by material damping. These figures further show
that the air cavity resonance has a very strong influence on the behavior of
the coupled system and that the structural resonance of the coupled tire-air
system occurs at different frequencies than the resonance of the tire without
air. As expected, this coupling effect decreases as we move further away from
the cavity resonance frequency.
The eigenfrequencies obtained in the substructure analysis are identical to
the eigenfrequencies obtained in the equivalent analysis without substructures.
The reaction force obtained at the road reference node is also compared to the
reaction force at the same node in the equivalent analysis without
substructures. As shown in
Figure 8
the results for the two steady-state dynamics steps in the substructure
analysis are virtually identical, and they compare well, in general, with the
reaction force obtained in the nonsubstructure analysis. The observed small
differences are due to modal truncation and the fact that constant material
properties are used to generate the substructure.
