Parametric Resonance in Dynamical Systems

Recent years have shown several incidents with dramatic damage on container vessels caused by parametric resonance. When the resonance starts, the roll oscillation at a sub-harmonic frequency of the wave excitation may be completely unexpected. Timely warning about the onset of the resonance phenomenon could make the navigator to change ship’s speed and heading, and these remedial actions could make the vessel escape the bifurcation. This chapter proposes nonparametric methods to detect the onset of parametric roll resonance. Theoretical conditions for parametric resonance are re-visited and signal-based methods are developed to detect its onset. Hypothesis testing is derived for the particular distribution of the indicators for resonance. Robustness is investigated by analyzing forced roll and disturbances in real weather conditions. The performance of the novel methods is demonstrated on experimental data from towing tank tests and data from a container ship passing an Atlantic storm.

This chapter attempts to provide a guideline for quantifying the magnitude of parametric roll of a ship in irregular seaways. First, in the light of physical model experiments with a scaled ship model in irregular water waves, it is shown that parametric roll of this ship model is a practically nonergodic process. Second, a guideline is developed by randomly sampling time series from the measured records. It is tentatively recommended that we should conduct physical and/or numerical experiments of more than 12-realizations and more than 360 excitation cycles for accurately estimating the magnitude of parametric rolling. The results of parametric roll estimation sufficiently above the threshold should be used for practical stability assessment. Finally, a container ship example is used to compare the numerical results with physical data.

Parametric roll can occur in case of trimaran vessels, in particular when the outriggers have limited draught and/or limited transversal separation. Due to the significant nonlinearities of the restoring moment and due to the often complex shape of the metacentric height in waves for this type of ships, parametric roll clearly shows quite peculiar characteristics. Starting by the description of a simplified 1-DOF nonlinear mathematical model for parametric roll in longitudinal regular waves the paper then describes how the Floquet theory can be directly applied to the linearized model in order to determine the instability regions for the upright position. An example of calculation of stability map for the upright position is provided. Also another example for prediction of rolling amplitude in the nonlinear range to highlight the peculiar aspects relevant to the considered trimaran configuration is discussed. In both cases experimental data are also reported for comparison. Finally the paper describes how the position of the outriggers can be optimally chosen in order to minimize the variations of metacentric height in waves.

The aim of the present chapter is to advocate for a very effective stochastic procedure based on the First-Order Reliability method (FORM) and Monte Carlo simulations (MCS), for prediction of parametric rolling in stationary stochastic seaways. Due to efficient optimization procedures implemented in standard FORM codes and the short duration of the time domain simulations needed, the calculation of the mean out-crossing rates of any given roll response is very fast. Thus complicated nonlinear effects can be included. Furthermore, the FORM analysis also identifies the most probable wave episodes leading to given roll responses. Because of the linearization in the FORM procedure the results are, however, only asymptotically exact and thus MCS often also needs to be performed. Here a scaling property inherent in the FORM procedure is investigated for use in MCS in order to reduce the necessary simulation time. More specifically, the MCS results for the reliability index β for parametric rolling of a ship suggest that the relation \(\beta = a + b/{H}_{\mathrm{s}}\) can provide an accurate scaling of the reliability index with significant wave height Hs. MCS can therefore be performed for sea states higher than the design sea state and thereafter be scaled down. From the reliability index the mean out-crossing rate and the probability of exceedance are then readily obtained.

A new 6-DOF nonlinear mathematical model based on Taylor-series expansions with coupling terms defined up to the third-order is introduced and validatedl for head seas parametric rolling for the case of a fishing vessel and a container vessel. Additionally, a large and deep drafted cylindrical SPAR platform is also simulated. The nonlinear algorithm is systematically simulated for different wave conditions. Parametric amplification domains (PADs) are thus obtained for the three hulls. On a comparative basis some of their main characteristics are then examined: influence of coupling, relevance of third-order coupling terms, impact of roll/roll nonlinearities, influence of wave amplitude, and initial conditions. The main differences in the PADs for the three distinct floating vessels are then interpreted and aspects of interest for the modeling and simulation of different hull forms are discussed.

The chapter considers statistical properties of parametric roll; the source of data is numerical simulation of parametric roll of a C11 class container carrier in head seas. The characteristics considered include estimates of spectral density, autocorrelation function. Special attention is paid to uncertainty of estimates of mean value and variance.

This chapter reviews three recent full-scale events with parametric rolling for Ro–Ro Large Car and Truck Carriers (LCTC). The events represent three principally different modes of parametric rolling: principal parametric resonance where the period of encounter is half of the roll natural period in following seas (I) and in head seas (II), and fundamental parametric resonance where the period of encounter coincides with the roll natural period in following seas (III). Roll motion, course, and speed recorded during the events are presented and analyzed together with the present weather situation based on recordings, forecasts, and re-analysis. Different aspects of on-board operational guidance for assisting crews in avoiding parametric rolling are discussed in relation to the presented events. Involved complexities and considerations that are normally not included in well defined model tests or numerical simulations are exposed.

In this work, we develop a ship model for parametric roll resonance under non-constant speed. Variations in speed changes the encounter frequency, i.e., the Doppler-shifted frequency of the waves as seen from the ship. Commonly, a Mathieu-type equation is used to describe the ship dynamics in parametric roll resonance; however, the derivation of this model assumes constant encounter frequency. In this chapter, we show that Mathieu-type equations are not able to accurately capture the dynamics of the ship if the encounter frequency is subject to changes. We derive a highly accurate six-degree-of-freedom computer model of the ship and use this to verify a simplified one-degree-of-freedom analytical model, valid also for a non-constant encounter frequency.

In this work, a control strategy for ship parametric roll resonance is developed by changing the frequency of the parametric excitation. This is achieved by varying the ship’s forward speed. This changes the perceived frequency of the waves, known as the encounter frequency (and thus the frequency of parametric excitation), via Doppler shift. A novel control philosophy based on a 1-DOF roll model, capable of describing the roll motion when the encounter frequency is non-constant, is proposed. The stability properties of the closed-loop system are mathematically proven and the effectiveness of the control system is demonstrated by simulating the closed-loop system using both a 6-DOF ship model and a simplified 1-DOF model describing the roll motion. The simulations are in agreement with the theoretical analysis.

In this chapter, two strategies to actively control ships experiencing parametric roll resonance are proposed. Both approaches aim at changing the frequency of the parametric excitation by controlling the Doppler shift, which, in recent results, has been shown to be effective to reduce the roll angle significantly. However, exactly how to change the frequency of the parametric excitation to stabilize the parametric oscillations has remained an open research topic. Thus, two optimal control philosophies that alter the ship’s forward speed and heading angle, which in turn changes the encounter frequency and consequently the frequency of the parametric excitation, are presented. This is referred to as frequency detuning. As a first approach, the methodology of extremum seeking (ES) control is adapted for ships in parametric roll resonance to iteratively determine the optimal setpoint of the encounter frequency in order to avoid one of the conditions for parametric roll. The encounter frequency is consequently mapped to the ship’s forward speed and heading angle by a control allocation. This is formulated as a constrained nonlinear optimization problem. Second, the use of a model predictive control (MPC) is proposed. By addressing both states and input constraints explicitly, the MPC formulation is used to change the ship’s forward speed and heading angle in an optimal manner to reduce parametric roll oscillations. The effectiveness of the proposed control approaches to stabilize parametric oscillations, by simultaneously changing the ship’s forward speed and heading angle optimally, is illustrated by computer simulations. This clearly verifies the concept of frequency detuning.

In this work, two main results are presented. (1) A 2-DOF (roll and tank fluid) model for a u-tank-equipped ship is developed. The ship–tank interaction is modeled with Lagrangian (analytical) mechanics, while nonconservative forces and forces due to the surrounding ocean are added to the model based on Newtonian mechanics. The model is suitable for arbitrarily-shaped u-tanks and system responses of any magnitude. (2) Based on a simplification of this model, a controller giving global (uniform) exponential stability of the origin (roll at zero, tank fluid in equilibrium) is developed for ships in parametric roll resonance. The controller is easily implementable and requires very little energy or power. Simulation results with both the simple and the full models confirm the theoretical analysis.

In this chapter, nonlinear resonances in a coupled shaker-beam-top mass system are investigated both numerically and experimentally. The imperfect, vertical beam carries the top mass and is axially excited by the shaker at its base. The weight of the top mass is below the beam’s static buckling load. A semi-analytical model is derived for the coupled system. In this model, Taylor-series approximations are used for the inextensibility constraint and the curvature of the beam. The steady-state behavior of the model is studied using numerical tools. In the model with a single beam mode, parametric and direct resonances are found, which affect the dynamic stability of the structure. The model with two beam modes not only shows an additional second harmonic resonance, but also reveals some extra small resonances in the low-frequency range, some of which can be identified as combination resonances. The experimental steady-state response is obtained by performing a (stepped) frequency sweep-up and sweep-down with respect to the harmonic input voltage of the amplifier-shaker combination. A good correspondence between the numerical and experimental steady-state responses is obtained.

Parametric resonance is a resonant phenomenon which takes place in systems characterized by periodic variations of some parameters. While seen as a threatening condition, whose onset can drive a system into instability, this chapter advocates that parametric resonance may become an advantage if the system undergoing it could transform the large amplitude motion into, for example, energy. Therefore the development of control strategies to induce parametric resonance into a system can be as valuable as those which aim at stabilizing the resonant oscillations. By means of a mechanical equivalent the authors review the conditions for the onset of parametric resonance, and propose a nonlinear control strategy in order to both induce the resonant oscillations and to stabilize the unstable motion. Lagrange’s theory is used to derive the dynamics of the system and input–output feedback linearization is applied to demonstrate the feasibility of the control method.