The comparison of the frequencies of free oscillations of a ship in a seaway based on the coupled and uncoupled differential equations of motion 1

Abstract

The motion of a ship in a seaway may be expressed by two simultaneous linear unhomogeneous second-order differential equations with constant coefficients. The terms in those equations which cause them to be coupled arc usually neglected in engineering practice, and the frequencies of free oscillations are determined from the uncoupled equations of motion.

The problem dealt with by the author of the present paper is as follows: to find out the connection of the frequencies of free oscillations of a ship when determined from the uncoupled viz. coupled equations of motion, i.e., in other words, to estimate the influence of the neglected “coupling” upon the frequencies of free oscillations.

By introducing the principal coordinates, the author has proposed an analysis of the problem as a function of three parameters, i.e.:

a. ${\mathit{\chi }}_1={\mathit{N}}_{\mathit{c}}\mathrm{/}\mathit{i}$ of the ratio of the abscissa of the center of gravity of the flotation surface and of the radius of gyration of the flotation surface for transverse axis.

b. ${\mathit{\chi }}_2={\mathit{i}}_1\mathrm{/}\mathit{i}$ of the ratio of the radius of gyration of the mass of the ship for transverse axis and the radius of gyration of the flotation surface for transverse axis.

c. K of the coefficient of added masses.

The author has shown that in certain cases the frequencies of the free oscillations cannot be determined from uncoupled equations of motion. However, such a treatment, connected with various assumptions is usual in engineering practice, and even widely recommended in technical literature.

The author has introduced the coefficients ${\mathit{\mu }}_1$ and ${\mathit{\mu }}_2$ , in order to be able to estimate this influence. The analytical expressions, derived by the author, were used for calculations of numerical values of those coefficients, the variations of parameters ${\mathit{\chi }}_1$ , ${\mathit{\chi }}_2$ and K being within the limits encountered in practice.

The values of the coefficients ${\mathit{\mu }}_1$ and ${\mathit{\mu }}_2$ are given in tables, whereas the plotted diagrams may be useful for practice.