Random Wave Excitation Research Articles

Lyapunov Function based Criteria for Ship Rolling in Random Beam Seas

Abstract The aim of this study is to present a Lyapunov function which can be used to derive an intact stability criterion for a ship in random beam seas. First, the mathematical model of the rolling motion of ships in random beam seas is introduced. The random wave excitation is described by a spectrum which is depended on the wave energy spectrum and the amplitude of the moment of roll. This spectrum is generated by a second order linear filter. Second, the methodology of creating a Lyapunov function is explained briefly. Then, there is outlined the way by which Lyapunov function can be used as the intact stability criterion for a ship. The proposed criterion is derived by considering the weather criteria for German naval vessels. Finally, the coherence of the boundary of safe basin obtained by Lyapunov function with the numerical results obtained by Euler-Maruyama Method is presented. From the results it can be deduced that the Lyapunov function can be used to define an intact stability criterion.

Investigation of dynamic characteristics of a monopile wind turbine based on sea test

Wind energy plays an important role in the field of renewable energy due to its lower cost and the maturity of its manufacturing technology. During the last few years, there has been an increase in offshore wind turbine (OWT) projects especially in China. Considering the fact that ocean environment where these offshore wind turbines operate usually has a negative impact on their structural health, the demand for investigating the dynamic characteristics of offshore wind turbines based on sea test is increasing. However, current studies are mainly focused on method development, theoretical analysis and numerical simulation, and few attention has been given to the study of the dynamic characteristics of an actual OWT in different operational states based on sea test. To improve the current studies, in September 2016, November 2016 and April 2017 a series of monitoring campaigns of an actual in service OWT and its monopile were conducted at an offshore wind farm in the sea area near Jiangsu Province, China. For the purpose of acquiring high-quality vibration signal, one type of artificial excitation was adopted to excite the OWT vibration. In these monitoring campaigns, with the help of a group of wireless health monitoring sensor nodes developed by Ocean University of China (OUC), we successfully acquired the vibration signal of the wind turbine in parked state and normal operational state and acquired the vibration signal of the monopile in construction, which means no tower was installed. Afterwards, two modal analysis methods, the eigensystem realization algorithm (ERA) and stochastic subspace identification (SSI), were used to obtain the modal parameters of the tested structure using the vibration signal under artificial excitation and random wave excitation, respectively. The results show that the ERA method can successfully identify the first two order modal parameters of the monopile and the first three order modal parameters of the wind turbine in parked or operational state. However, the SSI method failed to identify the third-order modal parameters of the wind turbine, regardless of whether it is in parked state or operational state (only the first two order modal parameters can be identified); meanwhile, SSI failed to identify the modal parameters of the monopile. Concerning the high-level environmental noise in an offshore site, we applied weak-mode identification (WMI) for the first time to obtain the modal parameters of OWT using vibration signal under both artificial excitation and random wave excitation. The results indicated that WMI has achieved the same good performance as the ERA when the vibration signal under artificial excitation is used. When the vibration signal under random wave is used, WMI successfully identified the first order modal parameters of the monopile and the first three order modal parameters of the wind turbine in parked or operational state, which indicated a better ability than that of the SSI method. We can conclude that the modal parameters of the wind turbine can be obtained by the WMI method and ERA method, and the ERA method is the best way to obtain the modal parameters of monopile under artificial excitation. These monitoring campaign of an actual OWT not only provide modal information support for structural maintenance and operation safety assessment, but also provide a real data reference for wind turbine design.

Dynamic analysis of an offshore wind turbine under random wind and wave excitation with soil-structure interaction and blade tower coupling

This study investigates the dynamic response of a 5 MW offshore wind turbine with monopile foundation subjected to wind and wave actions. The work includes dynamic interaction between the monopile and the underlying soil subjected to realistic offshore random wind and wave loading modeled using Von karman spectrum and Pierson Moskowitz spectrum respectively. The study also incorporates the effect of blade tower coupling in the analysis. The offshore wind turbine tower is modeled herein as a multi-degree of freedom system (MDOF) and it comprises of a rotor blade system, a nacelle, and a flexible tower. The mass of the rotor, blade, and nacelle are lumped at the top of the tower for simplicity. Separately, the effect of the rotation of blades has also been incorporated in the work. The rotational effect of the blades is taken into account considering shape filters using von Karman spectrum. The soil-structure interaction effect at the foundation level is modeled using equivalent spring-dashpot model for embedded foundations. The results are studied in time as well as frequency domain for both wind and wave loading. It has been observed that soil structure interaction effect greatly alters the response of the offshore wind turbine structure not only in the parked condition but also in operational conditions when blade tower coupling is also included. The effect of blade tower coupling and SSI on the response of the structure are observed more coherently in the case of wave induced loading.

Dynamic and first passage analysis of ship roll motion with inelastic impacts via path integration method

The inelastic impacts may happen in cold region and affect the stability of ship roll motions, which is worth studying in stochastic cases. This paper is devoted to analyzing the inelastic impacted ship roll motion under random wave excitations by path integration (PI) method. Ivanov non-smooth transformation is applied to combine impact condition and ship roll equation into one continuous equation; then, a further development on Gauss–Legendre path integration (GLPI) method is proposed. Moreover, a first passage type ship roll stability is studied based on path integration results. The stationary probability density function (PDF) is calculated in stable cases, while the first passage probability are carried out in unstable cases. The influences of different parameters on the stationary PDF and the ship roll stability are studied in this paper. The accuracy of our expended GLPI and solutions is verified by Monte Carlo simulation, which indicates that our improved PI method has wide prospect in application.

Time-Domain Hydro-Elastic Analysis of a SFT (Submerged Floating Tunnel) with Mooring Lines under Extreme Wave and Seismic Excitations

Global dynamic analysis of a 700-m-long SFT section considered in the South Sea of Korea is carried out for survival random wave and seismic excitations. To solve the tunnel-mooring coupled hydro-elastic responses, in-house time-domain-simulation computer program is developed. The hydro-elastic equation of motion for the tunnel and mooring is based on rod-theory-based finite element formulation with Galerkin method with fully coupled full matrix. The dummy-connection-mass method is devised to conveniently connect objects and mooring lines with linear and rotational springs. Hydrodynamic forces on a submerged floating tunnel (SFT) are evaluated by the modified Morison equation for a moving object so that the hydrodynamic forces by wave or seismic excitations can be computed at its instantaneous positions at every time step. In the case of seabed earthquake, both the dynamic effect transferred through mooring lines and the seawater-fluctuation-induced seaquake effect are considered. For validation purposes, the hydro-elastic analysis results by the developed numerical simulation code is compared with those by a commercial program, OrcaFlex, which shows excellent agreement between them. For the given design condition, extreme storm waves cause higher hydro-elastic responses and mooring tensions than those of the severe seismic case.

Dynamic response analysis of the equivalent water depth truncated point of the catenary mooring line

The real-time computer-controlled actuators are used to connect the truncated parts of moorings and risers in the active hybrid model testing system. This must be able to work in model-scale real time, based on feedback input from the floater motions. Thus, mooring line dynamics and damping effects are artificially simulated in real time, based on a computer-based model of the problem. In consideration of the nonlinear characteristics of the sea platform catenary mooring line, the equations of the mooring line motion are formulated by using the lumped-mass method and the dynamic response of several points on the mooring line is investigated by the time and frequency domain analysis method. The dynamic response of the representative point on the mooring line is analyzed under the condition of two different corresponding upper endpoint movements namely sine wave excitation and random wave excitation. The corresponding laws of the dynamic response between the equivalent water depth truncated points at different locations and the upper endpoint are obtained, which can provide technical support for further study of the active hybrid model test.

Stochastic roll response for a vessel with nonlinear damping models and steady heeling angles in random beam seas

Loss of ship stability is most frequently associated with extreme roll motion. For the case of random beam sea, a single-degree-of-freedom (SDOF) model is applied in the present study in order to represent the stochastic rolling behavior. The random wave excitation term in the SDOF model is approximated as a filtered white noise process by applying a second order linear filter. Accordingly, the original SDOF model is extended into a four-dimensional (4D) dynamic system. The coupled dynamic system can be viewed as a Markov system whose probabilistic properties are governed by the corresponding Fokker–Planck equation. Based on the convenient Markov property, a host of useful response statistics can be obtained by an efficient path integration (PI) method. Different nonlinear damping models, i.e. the linear-plus-quadratic damping (LPQD) model and the linear-plus-cubic damping (LPCD) model, and their effects on the stochastic roll response are investigated and the influence of the steady heeling angle on the response level associated with ship rolling in random seas is also studied. Furthermore, the accuracy of the response statistics computed by the PI technique is verified by means of the versatile Monte Carlo simulation (MCS) technique.

Extraction of the mode shapes of a segmented ship model with a hydroelastic response

ABSTRACT The mode shapes of a segmented hull model towed in a model basin were predicted using both the Proper Orthogonal Decomposition (POD) and cross random decrement technique. The proper orthogonal decomposition, which is also known as Karhunen-Loeve decomposition, is an emerging technology as a useful signal processing technique in structural dynamics. The technique is based on the fact that the eigenvectors of a spatial coherence matrix become the mode shapes of the system under free and randomly excited forced vibration conditions. Taking advantage of the sim-plicity of POD, efforts have been made to reveal the mode shapes of vibrating flexible hull under random wave ex-citation. First, the segmented hull model of a 400 K ore carrier with 3 flexible connections was towed in a model basin under different sea states and the time histories of the vertical bending moment at three different locations were meas-ured. The measured response time histories were processed using the proper orthogonal decomposition, eventually to obtain both the first and second vertical vibration modes of the flexible hull. A comparison of the obtained mode shapes with those obtained using the cross random decrement technique showed excellent correspondence between the two results.

Stochastic nonlinear ship rolling in random beam seas by the path integration method

Nonlinear ship rolling in random seas is a serious threat to ship stability. In this work, the dynamic stability of the vessel in random seas is evaluated by means of probabilistic methods. Specifically, the Markov diffusion theory is applied in order to study the stochastic aspects of the roll motion driven by random wave loads. The roll motion is modeled as a single-degree-of-freedom (SDOF) model in which the nonlinear damping and restoring terms as well as the random wave excitation are all incorporated. The stationary wave excitation moment in the SDOF model is represented as a filtered white noise by employing a second order linear filter. Therefore, a four-dimensional (4D) Markov dynamic system is established by combing the SDOF model with the linear filter model. Because the probabilistic property of the 4D Markov system is governed by the Fokker–Planck (FP) equation, the response statistics of roll motion can be obtained by solving the FP equation via an efficient 4D path integration (PI) method, which is based on the Markov property of the coupled dynamic system. Furthermore, the random wave excitation is approximated by an equivalent Gaussian white noise, and a two-dimensional (2D) PI technique is applied in order to obtain the response statistics of the dynamic system driven by this Gaussian white noise. The rationality and accuracy of applying the equivalent Gaussian white noise to simulate nonlinear ship rolling in random seas is studied. Moreover, the accuracy of the response statistics computed by the 4D and 2D PI techniques is verified by means of the versatile Monte Carlo simulation technique.

Filter models for prediction of stochastic ship roll response

In this paper, the shaping filter technique is introduced to study the stochastic roll response of a vessel in random beam seas. Specifically, the roll motion is described as a single-degree-of-freedom (SDOF) model in which the stationary random wave excitation term is approximated as the output of a shaping filter, i.e. a filtered white noise process. Therefore, the original SDOF model can be extended into a four-dimensional (4D) and a six-dimensional (6D) dynamic system, respectively, when this second-order nonlinear differential equation couples with the second-order linear filter and the fourth-order linear filter used to model the random wave excitation. Basically, the fourth-order linear filter provides a better approximation, but the subsequent dynamic system would be more complicated to tackle. In this regard, the 4D coupled system can be viewed as a Markov system whose probability properties are governed by the Fokker–Planck (FP) equation. Furthermore, the 4D path integration (PI) method, an efficient approximate technique based on the Markov property of the dynamic system, is applied to solve the 4D FP equation and then the response statistics are derived. In contrast, the statistics of the roll response in the 6D extended system are evaluated directly by a 6D Monte Carlo simulation. The advantages of the 4D PI method as well as the feasibility of simplifying the 6D system by the 4D system in terms of determining the statistics of high-level roll response are demonstrated through various cases corresponding to different sea states. Furthermore, the influence of external excitations on the statistics of high roll response levels is also illustrated by these cases.

Dynamic stability analysis of caisson breakwater in lifetime considering the annual frequency of severe storm

In the dynamic stability analysis of a caisson breakwater, most of current studies pay attention to the motion characteristics of caisson breakwaters under a single periodical breaking wave excitation. And in the lifetime stability analysis of caisson breakwater, it is assumed that the caisson breakwater suffers storm wave excitation once annually in the design lifetime. However, the number of annual severe storm occurrence is a random variable. In this paper, a series of random waves are generated by the Wen Sheng-chang wave spectrum, and the histories of successive and long-term random wave forces are built up by using the improved Goda wave force model. It is assumed that the number of annual severe storm occurrence is in the Poisson distribution over the 50-year design lifetime, and the history of random wave excitation is generated for each storm by the wave spectrum. The response histories of the caisson breakwater to the random waves over 50-year design lifetime are calculated and taken as a set of samples. On the basis of the Monte Carlo simulation technique, a large number of samples can be obtained, and the probability assessment of the safety of the breakwater during the complete design lifetime is obtained by statistical analysis of a large number of samples. Finally, the procedure of probability assessment of the breakwater safety is illustrated by an example.

Ship Nonlinear Rolling and Roll Angle Reconstruction Based on FIR

The paper analyzes the nonlinear damping torque and nonlinear restoring moment and establishes the nonlinear motion equation of ship rolling under random sea. The Runge-Kutta method is used in the time domain to solve the equation, and to simulate the time history of random wave excitation ship rolling and free decay process. This paper used the ship attitude inertial measurement system and its algorithm technology to construct the ship rolling angle from the acceleration measuring signals and to simulate the angle acceleration measurement signal. To study on test conditions of these parameters and the method to improve the accuracy of parameters, the effect of acceleration measurement noise on angle reconstruction accuracy is studied. Under SNR 20, reconstructed angle RMSE is 0.0019 rad. Studies show that the inertial measurement system and algorithm can effectively reconstruct ship rolling angle from the acceleration measurement signal, and the algorithm has the ability to suppress measurement noise.

Robust mixed $$H_2 /H_\infty $$ H 2 / H ∞ active control for offshore steel jacket platform

This paper presents a robust mixed $$H_2 /H_\infty $$ control method for wave-excited offshore jacket platforms. Its objective was to design a controller that minimizes the upper bound of the $$H_2 $$ performance measure on platform dynamics satisfying some $$H_\infty $$ norm bound constraint simultaneously. Based on mixed $$H_2 /H_\infty $$ control theory and linear matrix inequality techniques, a novel approach to stabilize offshore platform vibration with constrained $$H_2 /H_\infty $$ performances is proposed. Uncertainties of the wave excitation are considered in dynamic performance analysis of offshore platforms. A reduced mode offshore platform structure under wave excitation is analyzed, and simulations are used to verify the effectiveness of the proposed approach. Compared with existing $$H_\infty $$ control methods, the proposed approach makes a significant improvement for dynamic performances of offshore platforms under random wave excitation.

Time series prediction of nonlinear ship structural responses in irregular seaways using a third-order Volterra model

To predict the nonlinear structural responses of a ship traveling through irregular waves, a third-order Volterra model was applied based on the given irregular data. A nonlinear wave–body interaction system was identified using the nonlinear autoregressive with exogenous input (NARX) technique, which is one of the most commonly used nonlinear system identification schemes. The harmonic probing method was applied to extract the first-, second- and third-order frequency response functions of the system. To achieve this, a given set of time history data of both the irregular wave excitation and the corresponding midship vertical bending moment for a certain sea state was fed into the three-layer perceptron neural network. The network parameters are determined based on the supervised training. Next, the harmonic probing method was applied to the identified system to extract the frequency response function of each order. While applying the harmonic probing method, the nonlinear activation function (i.e., the hyperbolic tangent function) was expanded into a Taylor series for harmonic component matching. After the frequency response functions were obtained, the structural responses of the ship under an arbitrary random wave excitation were easily calculated with rapidity using a third-order Volterra series. Additionally, the methodology was validated through the in-depth analysis of a nonlinear oscillator model for a weak quadratic and cubic stiffness term, whose analytic solutions are known. It was confirmed that the current method effectively predicts the nonlinear structural response of a large container carrier under arbitrary random wave excitation.

Stochastic Averaging of Roll-Pitch and Roll-Heave Motion in Random Seas

Multi-degree-of-freedom ship motion and ship stability in random seas are of major interest for the development of new advanced intact stability criteria. The purpose of this research is to improve the safety of new ship designs, but the results are relevant also for other engineering systems involving multiple scales. We focus on roll-pitch and roll-heave motion in random seas. The random wave excitation is modeled by a non-white stationary process. This process is derived from a spectral description of the random seaway using traveling effective wave.

Experimental study on subsea pipeline bedding condition assessment using wavelet packet transform

Free span along subsea pipelines is a general problem both in pipeline design phase and during operation of pipelines. Traditionally, subsea pipeline system is inspected over the entire pipeline length with inspection tools, such as side-scan sonar and video/camera carried by remote controlled vehicle. The obvious disadvantage is that this kind of inspection can only be carried out at scheduled intervals. In a previous study [1], a damage indicator, WPECR¯, was proposed based on wavelet packet transform analysis to identify free-spanning damage along subsea pipelines. The reliability and efficiency of the proposed damage indicator is proven with numerical simulated vibration data of pipeline under ambient sea wave forces. To further verify the proposed method, in this paper, two scaled pipe models with different boundary conditions were designed and tested subject to random wave forces in a large wave tank. Waterproof accelerometers were used to collect pipeline responses which were subsequently analyzed for pipeline condition monitoring. The results show that using output-only dynamic responses of pipeline subject to random ambient wave excitations can accurately identify free span locations and length. With a vibration based method, real time monitoring of scouring damage to subsea pipelines is achievable so that any free span longer than allowable span length can be detected at an early stage.

Liquid sloshing dynamics in a barge carrying container subjected to random wave excitation

The sloshing phenomenon of liquid in a partially filled tank mounted rigidly on a barge exposed to random beam waves has been investigated through a well controlled experimental program. Four relative liquid depths, (liquid depth, hs/ length of tank, l) of 0.163, 0.325, 0.488 and 0.585 were considered for the tests. The sloshing oscillation was measured along the length of the tank at predefined locations. The effect of variation of the peak wave excitation frequency on the sloshing oscillation in the frequency domain is studied. The dominant energy is found to be concentrated around lowest nth sloshing mode frequency and, secondary peaks are observed at higher order sloshing frequencies. Odd modes contributions are dominating even modes irrespective of the excitation peak frequency. The sacrifice of second mode is observed while the excitation peak frequency is closer to its primary resonance. DOI: http://dx.doi.org/10.3329/jname.v9i1.7600 Journal of Naval Architecture and Marine Engineering 9(2012) 43-65

Motion responses of barge carrying liquid tank

The interaction between the liquid sloshing in a rectangular tank equipped inside the barge and the barge responses has been investigated through a comprehensive experimental program. The barge was subjected to both regular and random wave excitations under beam sea condition. Three relative fill levels ( h s / l) with liquid fill depth ( h s ) to length of tank ( l) ratio of 0.163, 0.325 and 0.488 were considered. In addition, the barge responses of equivalent dry weight condition corresponding to each fill level were measured to understand the influence of sloshing. While the excitation wave frequency equals to first mode natural sloshing frequency, a noticeable decrease in the sway response has been observed. However, the effect of sloshing oscillation on the heave response is insignificant. A split up of roll resonance was observed for the aspect ratio of 0.163 due to the coupling effect of roll motion and sloshing.

Combined Steady State and Transient Analysis of a Patrol Vessel as Affected by Varying Amounts of Damping and Periodic and Random Wave Excitation

In this paper various techniques of dynamical system analysis are used to analyze the effect of damping on large amplitude nonlinear ship-rolling motion of a patrol vessel. In particular steady state magnification curves, Poincare maps are for harmonic forcing and project phase planes are for random forcing. It has been found that varying amounts of damping substantially affect the vessel’s critical behavior. This is important since most stability regulations ignore damping and solely concentrate on the vessel’s righting ram curve. Moreover roll damping is difficult to predict accurately and small changes in damping may have a significant effect.

A twin unidirectional impulse turbine topology for OWC based wave energy plants

Experimental results from near shore bottom standing OWC based wave energy plants in Japan and India have now been available for about a decade. Historically the weakest link in the conversion efficiency of OWC based wave energy plants built so far has been the bidirectional turbine. This is possibly because a single turbine has been required to deliver power when the plant is exposed to random incident wave excitation varying by a factor of 10. A new topology that uses twin unidirectional turbines (which features a high efficiency spanning a broad range) is proposed. Using the Indian Wave Energy plant as a case study, it is shown that the power output from such a module considerably exceeds existing optimal configurations including those based on a fixed guide vane impulse turbine, linked guide vane impulse turbine or a Well's turbine. A wave to wire efficiency of the order of 50% over the incident range is shown to be feasible in a credible manner by showing the output at all stages of the conversion process. A frequency domain technique is used to compute the OWC efficiency and a time domain approach used for the power module with the turbine pressure being the pivotal variable.