The problem of calculating harmonic oscillations of a linear damped system with inhomogeneous damping is considered. Along with the solution available in literature, which requires the inversion of the matrix that determines the eigenvalues of the undamped system, the authors propose a new solution that does not require the inversion of the above mentioned matrix, but requires the inversion of the system damping matrix. The cases of hysteretic, viscous and mixed damping are considered. It is shown that the known solution gives an error near the resonance. At the resonance point, the result is not defined at all, and near the resonance it may be incorrect. An example of building the amplitude-frequency characteristic of a system with two mass tuned dynamic dampers and three peaks in the amplitude-frequency characteristic is given. The proposed formulas for calculating displacements are convenient for constructing the amplitude-frequency characteristics of damped systems with viscous, hysteretic and mixed types of damping.

Seismic isolation of a road bridge in the seismic region of Uzbekistan with a design seismicity of 9 degrees on the MSK-scale is considered. A special feature of the bridge is the large mass of the spans, which is almost 50 times greater than the reduced pier mass. At a first glance, this rules out tuning the isolation to mass dynamic regime. The rigidity of seismic isolation is determined basing on the condition of limiting the mutual displacement of spans. Even at this limitation, seismic loads were reduced by approximately two times, i.e. by one intensity number. A more effective isolation method has been proposed, that is, when one span is rigidly fixed to the pier and the other is seismically isolated, and then the isolation can be tuned, ensuring the operation of the isolated span as the mass damper. In this case, the load on one of the piers is reduced by more than two times and that on the adjacent pier by more than four times.