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Analysis of forced oscillations is performed to predict the behavior of a structure under external actions which change in accordance with the harmonic law. These actions include force and/or kinematic excitation. In addition to it, the impact of the system damping may be taken into account. The objective of the forced oscillation analysis is to obtain a dependence of the system's response on frequency of compelling actions. As a result of calculations we get amplitudes of displacements, vibration acceleration, and vibration overload at the preset compelling frequency. According to these results, one can obtain, for a frequency range, the dependencies of amplitudes and vibration acceleration on frequency of compelling actions, which is important with the evaluation of vibration stability of the system in this preset frequency range. The rotation of a shaft or spindle in the unbalanced state on elastic supports may serve an example of harmonic compelling force. Kinematic excitation is applied when values of compelling forces are not known, in contrast to amplitudes of oscillations of some structure elements which are known. When considering forced oscillations, it is important to take into account the impact of damping forces. The process of energy dissipation of mechanic oscillations, leading to step-by-step attenuation of lump-sum produced oscillations of the system, is termed damping. Damping forces may have different origin, namely: friction between dry sliding surfaces, friction between lubricated surfaces, internal friction, air or liquid resistance, etc. It is usually presumed that the damping force is proportional to velocity (viscous damping). Resistance forces changing under the voluntary law are replaced for equivalent damping forces, proceeding from the condition that in one cycle, they dissipate the same amount of energy as real forces.