Liquid Vibrations in Cylindrical Tanks with and Without Baffles Under Lateral and Longitudinal Excitations

The paper is devoted to issues of estimating free surface elevations in rigid cylindrical fluid-filled tanks under external loadings. The possibility of baffles installation is provided. The liquid vibrations caused by lateral and longitudinal harmonic loadings are under consideration. Free, forced and parametrical vibrations are examined. Modes of the free liquid vibrations are considered as basic functions for the analysis of forced and parametric vibrations. The modes of the free liquid vibrations in baffled and un-baffled cylindrical tanks are received by using single-domain and multi-domain boundary element methods. Effects of baffle installation are studied. The problems of forced vibrations are reduced to solving the systems of second order ordinary differential equations. For parametric vibrations the system of Mathieu equations is obtained. The numerical simulation of free surface elevations at different loadings and baffle configurations is accomplished. Beat phenomena effects are considered under lateral harmonic excitations. The phenomenon of parametric resonance is examined under longitudinal harmonic excitations.

The Dynamics of One Way Coupling in a System of Nonlinear Mathieu Equations

Background: This paper extends earlier research on the dynamics of two coupled Mathieu equations by introducing nonlinear terms and focusing on the effect of one-way coupling. The studied system of n equations models the motion of a train of n particle bunches in a circular particle accelerator. Objective: The goal is to determine (a) the system parameters which correspond to bounded motion, and (b) the resulting amplitudes of vibration for parameters in (a). Method: We start the investigation by examining two coupled equations and then generalize the results to any number of coupled equations. We use a perturbation method to obtain a slow flow and calculate its nontrivial fixed points to determine steady state oscillations. Results: The perturbation method reveals the existence of an upper bound on the amplitude of steady state oscillations. Conclusion: The model predicts how many bunches may be included in a train before instability occurs.

Bounded solutions of self-adjoint second order linear difference equations with periodic coeffients

In this work we obtain easy characterizations for the boundedness of the solutions of the discrete, self–adjoint, second order and linear unidimensional equations with periodic coefficients, including the analysis of the so-called discrete Mathieu equations as particular cases.

Free Vibration of Thin Shallow Elliptical Shells

This research presents a study of the free vibration of thin, shallow elliptical shells. The equations of motion for the elliptical shell, which are developed from Love's equations, are coupled and nonlinear. In this research, a new approach is introduced to uncouple the transverse motion of the shallow elliptical shell from the surface coordinates. Through the substitution of the strain-compatibility equation into the differential equations of motion in terms of strain, an explicit relationship between the curvilinear surface strains and transverse strain is determined. This latter relationship is then utilized to uncouple the spatial differential equation for transverse motion from that of the surface coordinates. The approach introduced provides a more explicit relationship between the surface and transverse coordinates than could be obtained through use of the Airy stress function. Angular and radial Mathieu equations are used to obtain solutions to the spatial differential equation of motion. Since the recursive relationships that are derived from the Mathieu equations lead to an infinite number of roots, not all of which are physically meaningful, the solution to the eigenvalue problem is used to determine the mode shapes and eigenfrequencies of the shallow elliptical shell. The results of examples demonstrate that the eigenfrequencies of the thin shallow elliptical shell are directly proportional to the curvature of the shell and inversely proportional to the shell's eccentricity.