Abstract <p>We investigate forced vibrations of a loaded tyre rolling at constant speed with a variable contact area (CA) length. A previously proposed analytical model of a reinforced tyre is considered. The surface of the tyre is represented by flexible tread, combined with elastic sidewalls. The contact between the wheel and the ground plane occurs by the part of the tread. The equations of motion and conditions on the unknown in advance boundary of CA are obtained from the Hamilton-Ostrogradsky variational principle. The main feature is that the dynamic boundary conditions determining the solution of the problem do not need to be invented, since they are obtained automatically as equations from the variational principle. The inhomogeneous partial differential equation of fourth order describing the tyre radial in-plane vibrations about steady-state regime of rolling and satisfying the kinematic and dynamic boundary conditions is solved analytically. The functions of time determining the dynamic components of a variable CA boundaries are found. The mode shapes (MS) of loaded rotating tyre are determined analytically in the case of forced vibrations with harmonic input with different driving frequencies in horizontal and vertical directions. The counter-rotating wave, which is the superposition of two MS, is observed.</p>