Rubber and rubberlike materials (elastomers) are able to absorb input energy much better than other construction materials, therefore elastomers are widely used in manufacturing of the compensating devices, shock-absorbers, vibration isolators, joints, etc. In this paper the rubber - metal vibroisolators with movable nondeformable side surface, operating under axial compression, are considered. Such rubber-metal devices with adjustable stiffness allow to implement nonlinear characteristics "force-displacement" both of rigid and soft type even at small axial deformation, that cannot be obtained in classical forms of rubber-metal products. Based on the Ritz method, using the principle of minimum of total potential energy of deformation, the variational method of obtaining the stiffness characteristics of "force-displacement" is described. The weak compressibility of the rubber components is taking into account. As an example the solid cylindrical rubber - metal device with the movable outer side surface, working under axial compression, is presented. It is shown that under condition of axial compression the weak compressibility of the rubber layer neglecting may lead to significant errors in the calculation of "force-displacement" characteristics of the rubber-metal device.