The nonlinear problem of twisting cylindrical bearing for John’s harmonic material is solved. Application of this model allows us to use the methods of the theory complex functions. The exact and simple analytical solution of the problem is obtained. The dependence of the twisting moment on an angle of rotation of the bearing (rigid characteristic of a bearing) is established. Calculated rigidity of a bearing for different angles of rotation is compared to known experimental results for elastomeric bearing. Formulation of the problem. The problem of a cylindrical hinge torsion. The inner and outer surface of the hinge considered non-deformable, the inner surface stationary, while the outer rotates through an angle ω torque. The challenge is flat and axially symmetric.