International Scientific Journals
of Scientific Technical Union of Mechanical Engineering "Industry 4.0"

In the paper, a two-degrees-of-freedom ship model with quadratic coupled pitch and roll modes under sinusoidal harmonic excitation is considered. A straightforward expansion allows for obtaining both the resonant values of external excitation frequency and one internal resonance. The Multiple Scales method yields the first-order expansions for the special resonant case where the excitation frequency is close to the roll frequency. The time series and the frequency – amplitude curves provided by numerical integration are contrasted with those given by the perturbation technique for different combination of system’s parameters. If the parameters are selected within the pre-ordered range, the results of both methods are in excellent or, at least, in pretty good agreement.

In a companion paper, the response of a two-degrees-of-freedom ship model with nonlinear coupled pitch and roll modes under sinusoidal harmonic excitation was studied analytically by means of the Multiple Scales method for the case where the pitch frequency is twice the roll frequency. Five resonant cases were analysed and the governing equations for the transition towards the steady-state solutions, the first-order approximations for these solutions and the frequency-amplitude relationships were derived. The present contribution aimed to verify the accuracy of the analytical results by contrasting them with the numerical results provided by direct integration of the equations of motion. The two sets of results were found to be in excellent or, at least, in decent agreement every time the system parameters were selected without a flagrant violation of the order’s magnitude.

In the paper, a two-degrees-of-freedom ship model with quadratic coupled pitch and roll modes under sinusoidal harmonic excitation is considered. The Multiple Scales perturbation technique is applied to yield the first-order expansions for the special internal resonant case where the pitch frequency is twice the roll frequency. Increasing the wave frequency from zero to infinity, five resonant situations are detected. For each case, the governing equations for the transition towards the steady-state solutions, the first-order approximations for these solutions and the frequency-amplitude relationships are presented. A detailed analysis is performed only for the case where the excitation frequency is half of the roll frequency. The reliability of the analytical results derived in the paper is checked in a companion contribution by comparison with numerical solutions.

The standard map is an apparently simple system that is well suited to explain the transition from regular behaviour to global chaos. Its dynamics depends strongly on a control parameter that influences the degree of chaos. For low or high parameter values the resulting dynamics is entirely regular or chaotic. At intermediate parameter values, however, the map exhibits a complex behaviour characterized by a mixture of chaotic and regular regions in the phase space. It is the purpose of this paper to emphasize this remarkable dynamics. Using phase planes and the Orthogonal Fast Lyapunov Indicator (OFLI) plots we try to determine the control parameter levels at which the main transformations take place and determine how quickly the chaotic orbits replace the regular ones in the phase space. Some comments referring the implementation and the efficiency of the OFLI test are included in the paper.

The plunge galloping is a high-amplitude, low-frequency oscillation of a slender structure, such as iced conductors of a power transmission line or bridge decks, essentially perpendicular to the wind direction. In the paper, an idealized model for the plunge galloping is shortly reviewed and then a slight modification of the variational iteration method, applicable to weakly nonlinear problems, is employed to obtain a system of two amplitude-frequency equations that provide both the transitional and long-term behaviours. The approximate analytical results derived in the paper have been applied to a typical section model and the numerical results are contrasted with those provided by the direct integration of equation of motion.

In the paper, HE’s energy balance method and HE’s Hamiltonian approach are used to derive simple approximate formulas for the dependence of the frequency of a pendulum-like nonlinear oscillator on its amplitude. Such kind of oscillators are frequently encountered in many fields of engineering and three examples from mechanical domain are given and utilized in the numerical simulations. By comparison with the exact solution, it is shown that obtained formulas lead to high accuracy for initial amplitudes lower than 900 (when the relative errors do not exceed 0.2%) and acceptable closeness for amplitudes well beyond the small-angle regime (here the relative errors are about 5-6% for amplitudes as high as 1500). Results furnished by energy balance method and Hamiltonian approach are contrasted with those provided by other techniques which generally need much more sophisticated procedures

The motion of a ship in long beam seas could be described by a second-order non-linear differential equation, having the roll angle as variable and depending on four parameters. With the direct forcing amplitude as bifurcation parameter, the dynamical system exhibits either periodic or chaotic behaviour, the route to chaos being realized by a period doubling sequence of periodic motions. Some accepted indicators, like bifurcation diagrams, phase planes and Poincare sections have been computed and they confirm the transition from order to chaos. In the main part of the paper, the harmonic balance method is used to obtain approximate solutions for the periodic motions and to predict the period doubling bifurcations by a stability analysis.

The non-linear ship capsize equation derived by Thompson et al., that incorporates both direct and parametric excitation, is examined numerically in an attempt to deepen our understanding on the influence of the parameters involved in the final ship’s response.
Because our interest is focused on the binary outcome of capsize-non-capsize, no remark of the steady-state onto which a non-capsize motion may settle is made. The four-dimensional phase-control space includes the non-dimensional damping coefficient, the ratio between wave frequency and ship’s natural frequency, and the direct and parametric forcing amplitudes. All the computed boundaries between capsizing and non-capsizing regions in bi-dimensional projections of control parameter space show fractal features.

The non-linear ship capsize equation derived by Thompson et al., that incorporates both direct and parametric excitation, is examined numerically in an attempt to deepen our understanding on the influence of the parameters involved in the final ship’s response.
Because our interest is focused on the binary outcome of capsize-non-capsize, no remark of the steady-state onto which a non-capsize motion may settle is made. The four-dimensional phase-control space includes the non-dimensional damping coefficient, the ratio between wave frequency and ship’s natural frequency, and the direct and parametric forcing amplitudes. All the computed boundaries between capsizing and non-capsizing regions in bi-dimensional projections of control parameter space show fractal features.