h tot h p tot p e B 1 V V B 1 V V B B + + = 1 1 (2) where Vp is volume of the pipeline, Vh is volume of the hoses and Vtot total volume. By means of specific displacement (displacement/revolution) and specific torque (torque/pressure difference) of the hydraulic motor, stiffness of the hydraulic system was obtained. Shaft Line The thrust bearing consist of two spherical roller bearings having contact angle about 45 degrees. Thus, the thrust bearing was capable to carry both axial and radial loads. The propeller bearing was a toroidal roller bearing, so called CARB-type bearing. This type of bearing carries only radial load and allows both axial and angular deflections of the rotor shaft. Stiffness of the thrust and propeller bearings at operational loads were also determined by the linearization method. The operational loads used for linearization were static reaction forces during normal operation. Cross coupling terms were neglected. Bearings were modeled as spring elements having stiffness coefficients equivalent to the bearing stiffness, radial and axial for thrust bearing and only radial for propeller bearing. One end of this spring element was connected to the shaft and other end by means of constraint equations to the outer race of the bearing housing. It was found that radial stiffness of the bearings were much higher than bending stiffness of the rotor shaft. Therefore calculation results were not very sensitive to the bearing stiffness values. Electromechanical Interaction Magnetic field around the rotor of electrical machine induces electromagnetic forces between rotor and stator. When the rotor is not concentric with the stator bore, the magnetic forces are unbalanced around the rotor. The total force tends to deflect the rotor in the direction of minimum air gap. This force is called unbalanced magnetic pull (UMP). Because the force is directly proportional to the radial displacement of the rotor, it can be modeled by a spring having negative spring constant. Effect of this force is to reduce the effective spring constant of the shaft and thus reduce the rotor natural frequency. The spring constant can be roughly estimated by following equation [5]: 0 0 2 4 μ δ πr r r e d l B k = (3) where dr is inner diameter of the stator, lr is stator core length, Br is amplitude of magnetic flux density, ȝ0 is permeability of air and į0 is air gap length. In FE-model the unbalanced magnetic pull was modeled by spring elements between stator and rotor. The elements were distributed in five cross sections and there were 8 spring elements in each cross section equally directed around the cross section so that total negative spring constant in the radial direction equals the estimated value. In reality the electromagnetically induced force consists of several harmonic components and has components also in axial and tangential directions. However, in this study these components were not taken into account. Rotordynamics Influence of gyroscopic forces caused by rotation for vibratory behavior of the shaft line was studied by a separate rotor model. As a consequence of the gyroscopic forces the natural frequencies of the rotor are functions of rotation speed and the natural modes split into forward and backward whirling modes. Considering angular momentum balance of spinning disk, it can be shown that linearized equations of motion for spinning disk become [6]: 4
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