Vibrations of a rolling tyre

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We investigate vibrations of an unloaded and loaded tyre rolling at constant speed without slipping in the contact area. A previously proposed analytical model of a reinforced tyre is considered. The surface of the tyre is represented by flexible tread, combined with parts of two tori (sidewalls of the tyre). The contact between the wheel and the ground plane occurs by the part of the tread. The natural frequencies (NF) and mode shapes (MS) are determined analytically for unloaded tyre and numerically for loaded tyre. The results were compared with experiments for the non-rotating tyre. In the case of loaded rotating tyre, the increasing of the angular velocity of rotation implies that NF decrease. Moreover, a phenomenon of frequency loci veering is visible here: NF as functions of angular velocity approach each other and then veer away instead of crossing. The MS interact in veering region and, as a result, interchange.

 The analytical model of a wheel with a reinforced tyre

In the case of loaded non-rotating (LNR) and loaded rotating (LR) tyre, in determining the frequency of the vibrations, the length of the contact area was taken as constant, since its variation determines a second order of smallness correction to the frequency, in the model chosen. For LNR tyre, the increasing of the contact area (due to the increasing of the vertical load) implies that NF also increase. In fact, the mass of the vibrating part of the tyre decreases with the increasing of the contact area, as the tyre does not vibrate in the contact area. Each mode of an unloaded non-rotating (UNR) tyre is double and for each NF two identical MS exist. The fixed contact points of the tyre cause a loss of the circular symmetry. Thus, for each NF of an UNR tyre there are two different NF of a LNR tyre. This is explained by the disturbance of free wave motion due to contact. The identical modes split into two not identical ones. The MS subdivides to a symmetric and an anti-symmetric shapes. For symmetric MS the corrections to the longitudinal reactions at the boundary points of the contact area have opposite signs. As for the antisymmetric MS the corrections carry the same sign. The mass centre of the tyre does not move in the longitudinal direction for the symmetric MS, and it moves for the antisymmetric MS. Thus, the antisymmetric MS “sways” from side to side.

 The basic mode shapes of a loaded non-rotating tyre (Input II, Omega=0 rad/s) nu_1=92.15 Hz nu_2=99.35 Hz nu_3=109.72 Hz

The quantities of obtained NF for the LNR tyre were compared with the results of experiment.

 The test rig for measurements of NF of an UNR and LNR tyre (2005, LAMI, ENPC)

In the case of LR tyre, the increasing of the angular velocity implies that NF decrease. The split of NF of an UR tyre caused by rotation disappears under rolling conditions due to the disturbed symmetry.

 The basic mode shapes of a loaded rotating tyre (Input I, Omega=175 rad/s), the rolling occurs in the clockwise direction nu_2=94.93 Hz nu_3=100.46 Hz nu_5=112.78 Hz

In addition, an interesting phenomenon of frequency loci veering is visible here: NF-lines approach each other and suddenly veer away instead of crossing. The MS interact in frequency loci veering region and finally interchange.

 The NF as a function of angular velocity in Eulerian specification (phenomenon of frequency loci veering)

For Omega=3, 100, 130 rad/s the third MS of a LR tyre has, respectively, three, four, five nodes and is similar to the third, fourth, fifth MS of a LNR tyre. Thus, the third MS changes from a three-node to a five-node shape, while the NF decreases from 116.91 Hz to 102.81 Hz.

 The evolution of the third mode shape of a loaded rotating tyre (Input I) Omega=3 rad/s, nu_3=116.91 Hz Omega=100 rad/s, nu_3=107.71 Hz Omega=130 rad/s, nu_3=102.81 Hz

 The movement of a point of tread for the third mode shape of a loaded rotating tyre (Input I, Omega=50 rad/s, nu_3=114 Hz)

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