Editorial Type:
Article Category: Research Article
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Online Publication Date: 01 Jun 2009

Validation of a Belt Model for Prediction of Hub Forces from a Rolling Tire4

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Page Range: 62 – 102
DOI: 10.2346/1.3130984
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Abstract

An integrated model is under development which will be able to predict the interior noise due to the vibrations of a rolling tire structurally transmitted to the hub of a vehicle. Here, the tire belt model used as part of this prediction method is first briefly presented and discussed, and it is then compared to other models available in the literature. This component will be linked to the tread blocks through normal and tangential forces and to the sidewalls through impedance boundary conditions. The tire belt is modeled as an orthotropic cylindrical ring of negligible thickness with rotational effects, internal pressure, and prestresses included. The associated equations of motion are derived by a variational approach and are investigated for both unforced and forced motions. The model supports extensional and bending waves, which are believed to be the important features to correctly predict the hub forces in the midfrequency (50–500 Hz) range of interest. The predicted waves and forced responses of a benchmark structure are compared to the predictions of several alternative analytical models: two three dimensional models that can support multiple isotropic layers, one of these models include curvature and the other one is flat; a one-dimensional beam model which does not consider axial variations; and several shell models. Finally, the effects of internal pressure, prestress, curvature, and tire rotation on free waves are discussed.

Keywords: tire belt model; hub forces; interior tire noise; shell theories; three dimensional cylindrical model; three dimensional plate model; one-dimensional model; rotating tire; prestress; orthotropy
Copyright: The Tire Society
FIG. 1.
FIG. 1.

Geometry of the infinite cylinder.

FIG. 2.
FIG. 2.

Comparison of prediction of resonances by using the three dimensional and the belt model. The resonances predicted from the three dimensional model are the zeros (blue regions) of the color map. The resonances predicted by the belt model are represented by crosses. The four figures vary by the axial order, m=0,…,3.

FIG. 3.
FIG. 3.

Comparison of prediction of resonances by using the three dimensional and the belt model. The resonances predicted from the three dimensional model are the zeros (blue regions) of the color map. The resonances predicted by the belt model are represented by crosses. The four figures vary by the axial order, m=4,…,7.

FIG. 4.
FIG. 4.

Magnitude of the forced responses predicted by the belt model. The same color map is used for each plot: The logarithm of the absolute value goes from −23 (dark blue) to −15 (dark red). The figures vary by the axial order, m=0,…,3 and by the normal (“Norm.”) or tangential (“Tang.”) directions of the force and displacement response. The force and response refer to the “plus” side.

FIG. 5.
FIG. 5.

Magnitude of the forced responses predicted by the three dimensional cylindrical viscoelastic model. The same color map is used for each plot: The logarithm of the absolute value goes from −23 (dark blue) to −15 (dark red). The figures vary by the axial order, m=0,…,3 and by the normal (“Norm.”) or tangential (“Tang.”) directions of the force and displacement response. The force and response refer to the “plus” side.

FIG. 6.
FIG. 6.

Comparison of prediction of resonances by using the belt model and the 1D model for an axial order m=0. The resonances are represented by crosses for the belt model and by circles for the 1D model.

FIG. 7.
FIG. 7.

Comparison of prediction of resonances by using the three dimensional plate model and the belt model. The resonances predicted from the plate model are the zeros (blue regions) of the color map. The resonances predicted by the belt model are represented by crosses. The four figures vary by the axial order, m=0,…,3.

FIG. 8.
FIG. 8.

Comparison of prediction of resonances by using the three dimensional plate model and the belt model. The resonances predicted from the plate model are the zeros (blue regions) of the color map. The resonances predicted by the belt model are represented by crosses. The four figures vary by the axial order, m=4,…,7.

FIG. 9.
FIG. 9.

Magnitude of the forced responses predicted by the three dimensional plate model. The same color map is used for each plot: The logarithm of the absolute value goes from −23 (dark blue) to −15 (dark red). The figures vary by the axial order, m=0,…3 and by the normal (“Norm.”) or tangential (“Tang.”) directions of the force and displacement response. The force and response refer to the ”plus” side.

FIG. 10.
FIG. 10.

Comparison of prediction of resonances by using the three dimensional and the Epstein–Kennard shell model. The resonances predicted from the three dimensional model are the zeros (blue regions) of the color map. The resonances predicted by the shell model are represented by crosses. The four figures vary by the axial order, m=0,…,3.

FIG. 11.
FIG. 11.

Comparison of prediction of resonances by using the three dimensional and the Epstein–Kennard shell model. The resonances predicted from the three dimensional model are the zeros (blue regions) of the color map. The resonances predicted by the shell model are represented by crosses. The four figures vary by the axial order, m=4,…,7.

FIG. 12.
FIG. 12.

Comparison of prediction of resonances by using the three dimensional and the DMSR shell model. The resonances predicted from the three dimensional model are the zeros (blue regions) of the color map. The resonances predicted by the shell model are represented by crosses. The four figures vary by the axial order, m=0,…,3.

FIG. 13.
FIG. 13.

Comparison of prediction of resonances by using the three dimensional and the DMSR shell model. The resonances predicted from the three dimensional model are the zeros (blue regions) of the color map. The resonances predicted by the shell model are represented by crosses. The four figures vary by the axial order, m= 4,…,7.

FIG. 14.
FIG. 14.

Prediction of resonances by using the belt model considering internal cavity air pressure. The axial order is equal to m=0. The resonances are represented by crosses. The six figures vary by the values of prestress in circumferential and axial directions.

FIG. 15.
FIG. 15.

Prediction of resonances by using the belt model considering internal cavity air pressure. The axial order is equal to m=3. The resonances are represented by crosses. The six figures vary by the values of prestress in circumferential and axial directions.

FIG. 16.
FIG. 16.

Magnitude of the forced responses of the multilayer structure predicted by the three dimensional cylindrical model. The same color map is used for each plot: The logarithm of the absolute value goes from −23 (dark blue) to −15 (dark red). The figures vary by the axial order, m=0,…,3 and by the normal (“Norm.”) or tangential (“Tang.”) directions of the force and displacement response. The force and response refer to the “plus” side.

FIG. 17.
FIG. 17.

Magnitude of the forced responses of the multilayer structure predicted by the three dimensional plate model. The same color map is used for each plot: The logarithm of the absolute value goes from −23 (dark blue) to −15 (dark red). The figures vary by the axial order, m=0,…,3 and by the normal (“Norm.”) or tangential (“Tang.”) directions of the force and displacement response. The force and response refer to the “plus” side.

FIG. 18.
FIG. 18.

Prediction of resonances by using the belt model considering the tire rotation. The resonances are represented by crosses or circles. The four figures vary by the axial order, m=0,…,3.

FIG. 19.
FIG. 19.

Comparison of prediction of resonances by belt model for isotropic and anisotropic materials. The resonances are represented by crosses or circles. The four figures vary by the axial order, m=0,…,3.

FIG. 20.
FIG. 20.

Comparison of prediction of resonances by belt model for isotropic and anisotropic materials. The resonances are represented by crosses or circles. The four figures vary by the axial order, m=4,…,7.

FIG. A1.1
FIG. A1.1

Representation of the external forces applied on an infinitesimal surface area of the belt.

FIG. A1.2
FIG. A1.2

Representation of the external forces applied on infinitesimal elements at the boundary of the belt.

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