Dynamic analysis of an air-filled tire with rolling transport
effects
This example examines the effect of steady-state rolling transport
on the acoustic response of a tire and its associated air cavity after it has
been subjected to the inflation pressure and footprint load.
The air cavity resonance in a
tire is often a significant contributor to the vehicle interior noise,
particularly when the resonance of the tire couples with the cavity resonance.
This coupled resonance phenomenon, however, is affected by the rotating motion
in the fluid and the solid. This example extends the analyses of
Coupled acoustic-structural analysis of a tire filled with air
to include rolling transport effects in the tire and air. The acoustic cavity
is modeled as part of an axisymmetric model, which is inflated, revolved,
reflected, and deformed to obtain a footprint in a manner consistent with the
aforementioned example.
A detailed description of the tire model is provided in
Symmetric results transfer for a static tire analysis.
We model the rubber as an incompressible hyperelastic material. Viscoelasticity
in the material is ignored in this example.
The air cavity in the model is defined as the space enclosed between the
interior surface of the tire and a cylindrical surface of the same diameter as
the diameter of the bead. A cross-section of the tire model is shown in
Figure 1.
The values of the bulk modulus and the density of air are taken to be 426 kPa
and 3.6 kg/m3, respectively, and represent the properties of air at
the tire inflation pressure.
The simulation assumes that both the road and rim are rigid. We further
assume that the contact between the road and the tire is frictionless during
the preloading analyses. However, we use a nonzero friction coefficient in the
subsequent coupled acoustic-structural analyses.
To assess the effect of rolling motion on the dynamics of the coupled
tire-air system, we first generate dynamic results for the stationary tire. In
a subsequent analysis the tire and air are set into rolling motion, and
corresponding dynamic results are obtained.
Model definition
We use a tire cross-section that is identical to that used in the simulation described in Symmetric results transfer for a static tire analysis. The air cavity is discretized using linear acoustic
elements and is coupled to the structural mesh using a surface-based tie constraint with the
secondary surface defined on the acoustic domain. We model the rigid rim by applying fixed
boundary conditions to the nodes on the bead of the tire, while the interaction between the
air cavity and rim is modeled by a traction-free surface; i.e., no boundary conditions are
prescribed on the surface.
We first create an axisymmetric mesh of half of the cross-section of the
tire and air, then revolve it into half-symmetry. Symmetric model generation
and symmetric results transfer, together with a static analysis procedure, are
used to generate the preloading solution, which serves as the base state in the
subsequent coupled acoustic-structural analyses.
In the first coupled analysis we reflect the revolved tire and air model
into a full three-dimensional configuration. We then compute the real
eigenvalues of the stationary tire and air cavity system. During frequency
extraction steps, fixed boundary conditions are imposed automatically on the
tire-road interface in the contact normal direction. Fixed boundary conditions
are also applied in the tangential direction for points that are sticking.
Points that are slipping are free to move in the tangential direction. This
analysis is followed by a direct steady-state dynamic analysis in which we
obtain the response of the tire-air system subjected to imposed harmonic motion
of the road.
In the second coupled analysis the reflected model is restarted from the
first step, after the footprint equilibrium configuration had been established.
The steady-state transport procedure is used to obtain the free-rolling state
of the tire at 60 km/h. The corresponding magnitude of the transport velocity
is determined independently in a separate analysis. In this step the acoustic
flow velocity signifies that the acoustic medium is also in rotational motion.
It is assumed that the air inside the tire rotates with the same angular
velocity as the tire. The direct steady-state analysis, with similar parameters
to those used in the stationary case, is repeated.
Additional analyses using a substructure that was generated with the effect
of acoustic flow velocity are also included.
Loading
The loading sequence for computing the footprint solution is identical to
that discussed in
Symmetric results transfer for a static tire analysis.
The simulation starts with an axisymmetric model, which includes the mesh for
the air cavity. Only half the cross-section is modeled. The inflation pressure
is applied to the structure using a static analysis. In this example the
application of pressure does not cause significant changes to the geometry of
the air cavity, so it is not necessary to update the acoustic mesh. However, we
perform adaptive mesh smoothing after the pressure is applied to illustrate
that the updated geometry of the acoustic domain is transferred to the
three-dimensional model when symmetric results transfer is used.
The axisymmetric analysis is followed by a reflection—symmetric
three-dimensional analysis in which the footprint solution is obtained. The
footprint load is established over several load increments. The deformation
during each load increment causes significant changes to the geometry of the
air cavity. We update the acoustic mesh by performing five mesh sweeps after
each converged structural load increment using the adaptive mesh domain. At the
end of this analysis sequence we activate friction between the tire and road
using a change to friction properties. This footprint solution, which includes
the updated acoustic domain, is transferred to a full three-dimensional model.
This model is used to perform the coupled analysis. In the first coupled
analysis we extract the eigenvalues of the undamped system, followed by a
direct-solution steady-state dynamic analysis in which we apply a harmonic
excitation to the reference node of the rigid surface that is used to model the
road.
In the stationary and rolling analyses we compute the response of the
coupled system in the same frequency range used in
Coupled acoustic-structural analysis of a tire filled with air
(200 to 260 Hz). This band includes the natural frequencies of the fore-aft and
vertical acoustic modes, at 225.67 Hz and 230.94 Hz, respectively.
The model is excited by a boundary condition specified at the road reference
node in the direct-solution steady-state dynamic step. A small amount of
stiffness proportional damping is applied to the rubber to avoid computing
unbounded response at the eigenfrequencies.
Results and discussion
The characteristic frequencies of the coupled tire-air system are affected
by the rolling motion. Generally, we expect a mode observed in the stationary
coupled tire-air system to convert to a pair of modes, corresponding to forward
and backward wave travel. This does not always occur in complex systems,
because the stationary modes are not all affected by the rolling motion to a
similar degree. However, the mode split described above can be observed for
several of the structural modes and the fundamental acoustic modes of the
cavity. For an observer in a nonrotating reference frame attached to the axle
of the tire, similar to the reference frame used by the steady-state transport
procedure, these modes appear as waves traveling clockwise and counterclockwise
along the circumference of the tire. The mode corresponding to forward wave
travel increases in frequency, whereas the mode corresponding to backward wave
travel decreases in frequency. The modes of a stationary tire appear as static
vibrations. The resonant frequencies of the stationary case are computed using
the real-valued frequency extraction procedure. The complex frequency procedure
will yield almost identical results for the stationary case, since the damping
used in this example is relatively low. However, for the rolling analysis the
complex frequency procedure must be used to obtain accurate results using all
of the element contributions due to rotation. In
Table 1
an example is shown of a pair of structural and acoustic modes splitting into a
pair of corresponding modes in the rolling case. The structural mode is a
radial mode of circumferential order two. If there is no footprint loading, the
stationary case will predict identical frequencies for these modes; however,
the split would still be observed for a rotating tire. The modes are shown in
Figure 2,
Figure 3,
and
Figure 4.
Similar behavior is observed for radial modes of higher order.
Figure 5,
Figure 6,
and
Figure 7
show the response of the structure to the imposed vertical motion at the
spindle.
Figure 5
compares the acoustic response, at the crown, of the coupled tire-air system
for the stationary case and the rolling case at 60 km/h.
Figure 6
shows the fore-aft reaction force, and
Figure 7
shows the vertical reaction force.
These figures further show that the rolling motion of the solid has a very
strong influence on the behavior of the coupled system and that the rolling
motion of the air exerts a similarly strong effect in the frequency range
observed here. In particular, the resonances affecting the reaction force occur
at different frequencies for the stationary and rolling cases. The resonances
observed in the reaction force frequency response diagram also proliferate as
rolling is introduced since waves traveling against and with the direction of
rolling propagate at different speeds with respect to the observer. For a
stationary tire, a vertical excitation to the road produces negligible fore-aft
reaction forces. However, the fore-aft reaction forces due to a vertical
excitation are significant in the case of a rolling tire.
The same effect can be shown when using a substructure generated with the
acoustic flow velocity. The tire model with a coarser mesh is used following
the same pattern of actions: axisymmetric model followed by the revolution,
reflection, and the steady-state transport analysis for a rolling tire. The
substructure is used within the frequency range of 200–250 Hz using a direct
steady-state dynamic procedure. The same reaction force split resonance can
also be observed for the full finite element model used in place of the
substructure.
Acoustic(fundamental modes of the acoustic cavity)
225.67 Hz, 230.94 Hz
218.48 Hz, 237.31 Hz
Figures
Figure 1. Cross-section of tire and air. Figure 2. Radial mode, stationary case, at 95.46 Hz. Figure 3. Radial mode, rolling case, backward, at 86.7 Hz (value at angle 90°).
Figure 4. Radial mode, rolling case, forward, at 103.8 Hz (value at angle 90°).
Figure 5. Acoustic pressure at crown due to imposed vertical motion. Figure 6. Fore-aft reaction force due to road displacement excitation. Figure 7. Vertical reaction force due to road displacement excitation.
