Mathematical modeling of seismic-induced vibrations of high-rise structures with liquid reservoir

This study investigates the dynamic behavior of high-rise engineering structures equipped with cylindrical liquid reservoirs under seismic and harmonic excitations. The research focuses on the development of a mathematical model describing the coupled vibration of an elastic vertical structure and a contained incompressible fluid. The governing differential equations of transverse vibrations are formulated considering complex boundary conditions and the interaction between the structure, the fluid, and the supporting soil. Analytical solutions for the natural frequencies and vibration modes of the system are obtained using integral transform techniques and Krylov functions. The influence of structural parameters, fluid mass, and boundary conditions on the vibration characteristics is analyzed. Numerical calculations are performed using computational tools to determine the roots of the frequency equation and evaluate dynamic displacement functions. The results demonstrate that the presence of fluid significantly affects the dynamic response of the structure and may alter its vibration frequencies depending on the filling level of the reservoir. The proposed model can be applied in seismic engineering and vibration analysis of industrial structures such as towers, tanks, and hydro-engineering facilities. The obtained theoretical results provide a basis for improving structural safety and reliability under dynamic loads.