Nonlinear Random Motion Analysis of Coupled Heave-Pitch Motions of a Spar Platform Considering lst-Order and 2nd-Order Wave Loads

In this study, the coupled heave-pitch motion equations of a spar platform were established by considering lst-order and 2nd-order random wave loads and the effects of time-varying displacement volume and transient wave elevation. We generated random wave loads based on frequency-domain wave load transfer functions and the Joint North Sea Wave Project （JONSWAP） wave spectrum, designed program codes to solve the motion equations, and then simulated the coupled heave-pitch motion responses of the platform in the time domain. We then calculated and compared the motion responses in different sea conditions and separately investigated the effects of 2nd-order random wave loads and transient wave elevation. The results show that the coupled heave-pitch motion responses of the platform are primarily dominated by wave height and the characteristic wave period, the latter of which has a greater impact. 2nd-order mean wave loads mainly affect the average heave value. The platform＇s pitch increases after the 2nd-order low frequency wave loads are taken into account. The platform＇s heave is underestimated if the transient wave elevation term in the motion equations is neglected.

Stochastic Analysis of Nonlinear Coupled Heave-Pitch Motion for the Truss Spar Platform

Considering the static stability and the change of the displacement volume, including the influences of higher order nonlinear terms and the instantaneous wave surface, the nonlinear coupled heave-pitch motion was established in stochastic waves. The responses of heave-pitch coupling motion for the Truss Spar platform were investigated. It was found that, when the characteristic frequency of a stochastic wave is close to the natural heave frequency, the large amplitude pitch motion is induced under the parametric-forced excitation, which is called the Mathieu instability. It was observed that the heave mode energy is transferred to pitch mode when the heave motion amplitude exceeds a certain extent. In addition, the probability of internal resonant heave-pitch motion is greatly reduced while the characteristic wave frequency is away from the natural heave frequency.

Time-Domain Simulation for Coupled Motions of Three Barges Moored Side-by-Side in Floatover Operation

Simulating the coupled motions of multiple bodies in the time domain is a complex problem because of the strong hydrodynamic interactions and coupled effect of various mechanical connectors. In this study, we investigate the hydrodynamic responses of three barges moored side-by-side in a floatover operation in the frequency and time domains. In the frequency domain, the damping lid method is adopted to improve the overestimated hydrodynamic coefficients calculated from conventional potential flow theory. A time-domain computing program based on potential flow theory and impulse theory is compiled for analyses that consider multibody hydrodynamic interactions and mechanical effects from lines and fenders. Correspondingly, an experiment is carried out for comparison with the numerical results. All statistics, time series, and power density spectra from decay and irregular wave tests are in a fairly good agreement.

A Coupled Analysis of Nonlinear Sloshing and Ship Motion

Nonlinear interactions among incident wave, tank-sloshing and floating body coupling motion are investigated. The fully nonlinear sloshing and body-surface nonlinear free surface hydrodynamics is simulated using a Non-Uniform Rational B-Spline （NURBS） higher-order panel method in time domain based on the potential theory. A robust and stable improved iterative procedure （Yan and Ma, 2007） for floating bodies is used for calculating the time derivative of velocity potential and floating body motion. An energy dissipation condition based on linear theory adopted by Huang （2011） is developed to consider flow viscosity effects of sloshing flow in nonlinear model. A two-dimensional tank model test was performed to identify its validity. The present nonlinear coupling sway motion results are subsequently compared with the corresponding Rognebakke and Faltinsen （2003）＇s experimental results, showing fair agreement. Thus, the numerical approach presented in this paper is expected to be very efficient and realistic in evaluating the coupling effects of nonlinear sloshing and body motion.

The Nonlinear Bifurcation and Chaos of Coupled Heave and Pitch Motions of a Truss Spar Platform

This paper presents the results from a numerical study on the nonlinear dynamic behaviors including bifurcation and chaos of a truss spar platform. In view of the mutual influences between the heave and the pitch modes, the coupled heave and pitch motion equations of the spar platform hull were established in the regular waves. In order to analyze the nonlinear motions of the platform, three-dimensional maximum Lyapunov exponent graphs and the bifurcation graphs were constructed, the Poincare maps and the power spectrums of the platform response were calculated. It was found that the platform motions are sensitive to wave fre- quency. With changing wave frequency, the platform undergoes complicated nonlinear motions, including 1/2 sub-harmonic motion, quasi-periodic motion and chaotic motion. When the wave frequency approaches the natural frequency of the heave mode of the platform, the platform moves with quasi-periodic motion and chaotic motional temately. For a certain range of wave frequencies, the platform moves with totally chaotic motion. The range of wave frequencies which leads to chaotic motion of the platform increases with increasing wave height. The three-dimensional maximum Lyapunov exponent graphs and the bifurcation graphs reveal the nonlinear motions of the spar platform under different wave conditions.

Modelling erosion of a single rock block using a coupled CFD-DEM approach

Rock block removal is the prevalent physical mechanism for rock erosion and could affect the stability of dam foundations and spillways.Despite this,understanding of block removal is still inadequate because of the complex interactions among block characteristics,hydraulic forces,and erosive processes acting on the block.Herein,based on a previously conducted physical experiment of erosion of a single rock block,the removal processes of two different protruding blocks are represented by a coupled computational fluid dynamics-discrete element model(CFD-DEM)approach under varied flow conditions.Additionally,the blocks could be rotated with respect to the flow direction to consider the effect of the discontinuity orientation on the block removal process.Simulation results visualize the entire block removal process.The simulations reproduce the effects of the discontinuity orientation on the critical flow velocity inducing block incipient motion and the trajectory of the block motion observed in the physical experiments.The numerical results present a similar tendency of the critical velocities at different discontinuity orientations but have slightly lower values.The trajectory of the block in the simulations fits well with the experimental measurements.The relationship between the dimensionless critical shear stress and discontinuity orientation observed from the simulations shows that the effect of block protrusion becomes more dominant on the block incipient motion with the increase of relative protrusion height.To our knowledge,this present study is the first attempt to use the coupled finite volume method(FVM)-DEM approach for modelling the interaction behavior between the block and the flowing water so that the block removal process can be reproduced and analyzed.

Effect of Rigid Pitch Motion on Flexible Vibration Characteristics of a Wind Turbine Blade

Adynamic pitch strategy is usually adopted to improve the aerodynamic performance of the blade of awind turbine.The dynamic pitch motion will affect the linear vibration characteristics of the blade.However,these influences have not been studied in previous research.In this paper,the influences of the rigid pitch motion on the linear vibration characteristics of a wind turbine blade are studied.The blade is described as a rotating cantilever beam with an inherent coupled rigid-flexible vibration,where the rigid pitch motion introduces a parametrically excited vibration to the beam.Partial differential equations governing the nonlinear coupled pitch-bend vibration are proposed using the generalized Hamiltonian principle.Natural vibration characteristics of the inherent coupled rigid-flexible system are analyzed based on the combination of the assumed modes method and the multi-scales method.Effects of static pitch angle,rotating speed,and characteristics of harmonic pitch motion on flexible natural frequencies andmode shapes are discussed.It shows that the pitch amplitude has a dramatic influence on the natural frequencies of the blade,while the effects of pitch frequency and pith phase on natural frequencies are little.

Sloshing-Coupled Ship Motion Algorithm for Estimation of Slosh-Induced Pressures

The effect of coupling between sloshing and ship motions in the evaluation of slosh-induced interior pressures is studied. The coupling between sloshing loads and ship motions is modelled through a hybrid algorithm which combines a potential flow solution based on transient Green function for the external ship hydrodynamics with a viscous flow solution based on a multiphase interface capturing volume of fluid(VOF) technique for the interior sloshing motion. The coupled algorithm accounts for full nonlinear slosh forces while the external forces on the hull are determined through a blended scheme of linear radiationdiffraction with nonlinear Froude-Krylov and restoring forces. Consideration of this level of nonlinearities in ship motions is found to have non-negligible effects on the slosh-coupled responses and slosh-induced loads. A scheme is devised to evaluate the statistical measure of the pressures through long-duration simulation studies in extreme irregular waves. It is found that coupling significantly influences the tank interior pressures, and the differences in the pressures between coupled and uncoupled cases can be as much as 100% or more. To determine the RAO over the frequency range needed for the simulation studies in irregular waves, two alternative schemes are proposed, both of which require far less computational time compared to the conventional method of finding RAO at each frequency, and the merits of these are discussed.

Experimental Study on Coupled Motions of Mother Ship Launching and Recovering of Human-Occupied Vehicle in Regular Waves

The launching and recovery process of a human-occupied vehicle(HOV)faces more complex wave effects than other types of submersible operations.However,due to the nonlinearity between the HOV and its mother ship,difficulties occur in theoretically simulating their coupled motion and hydrodynamics.The coupled motion responses and the load under different regular wave conditions are investigated experimentally in this study.The optimized design of the experimental scheme simulated the launching and recovery process of the mother ship and HOV in regular waves.The attitude sensor performed synchronous real-time measurement of the coupled motion between the mother ship and HOV as well as obtained the load data on the coupled motion under different cable lengths.The results show that models in heading waves mainly lead to the vertical motion of the hoisting point.In beam waves,the transverse and vertical motions of the hoisting point occur in a certain frequency of waves.Under the heading and beam wave conditions,the longer the hoisting cable is,the greater the movement amplitude of the submersible is.Moreover,compared with the condition of the beam waves,the hoisting submersible has less influence on the mother ship under the condition of the heading waves.The findings provide theoretical support for the design optimization of the launching and recovery operation.

Dynamic Positioning Control of Surge−Pitch Coupled Motion for Small-Waterplane-Area Marine Structures

For general dynamic positioning systems,controllers are mainly based on the feedback of motions only in the horizontal plane.However,for marine structures with a small water plane area and low metacentric height,undesirable surge and pitch oscillations may be induced by the thruster actions.In this paper,three control laws are investigated to suppress the induced pitch motion by adding pitch rate,pitch angle or pitch acceleration into the feedback control loop.Extensive numerical simulations are conducted with a semi-submersible platform for each control law.The influences of additional terms on surge−pitch coupled motions are analyzed in both frequency and time domain.The mechanical constraints of the thrust allocation and the frequency characters of external forces are simultaneously considered.It is concluded that adding pitch angle or pitch acceleration into the feedback loop changes the natural frequency in pitch,and its performance is highly dependent on the frequency distribution of external forces,while adding pitch rate into the feedback loop is always effective in mitigating surge−pitch coupled motions.

Numerical analysis of coupled sloshing and motion of a cylindrical FPSO in regular waves

In this paper,the coupled sloshing and motion characteristics of a cylindrical floating production storage offloading(CFPSO)are numerically investigated by means of computational fluid dynamics(CFD)tool.The simulations are performed using an in-house CFD solver naoe-FOAM-SJTU which is developed based on OpenFOAM.The active wave generating-absorbing boundary condition(GABC)is utilized for wave generation and absorption.The stabilized k-omega SST turbulence model are used to avoid excessive eddy viscosity near the free surface.CFPSO with and without partially filled liquid tanks in regular waves with different wave periods are simulated and vertical planar motions such as surge,heave and pitch response amplitude operators(RAOs)are compared.Forces due to liquid sloshing and wave loads are extracted and analyzed.The free surface motions inside liquid tanks in one wave period presented to explain the motion characteristics.

Dynamic Coupled Analysis of the Floating Platform Using the Asynchronous Coupling Algorithm

This paper discusses the numerical modeling of the dynamic coupled analysis of the floating platform and mooring/risers using the asynchronous coupling algorithm with the purpose to improve the computational efficiency when multiple lines are connected to the platform. The numerical model of the platform motion simulation in wave is presented. Additionally, how the asynchronous coupling algorithm is implemented during the dynamic coupling analysis is introduced. Through a comparison of the numerical results of our developed model with commercial software for a SPAR platform, the developed numerical model is checked and validated.

Locomotion control of a rigid-soft coupled snake robot in multiple environments

The versatile motion capability of snake robots offers themselves robust adaptability in varieties of challenging environments where traditional robots may be incapacitated.This study reports a novel flexible snake robot featuring a rigid-flexible coupling structure and multiple motion gaits.To better understand the robot's behavior,a bending model for the soft actuator is established.Furthermore,a dynamic model is developed to map the relationship between the input air pressure and joint torque,which is the model base for controlling the robot effectively.Based on the wave motion generated by the joint coupling direction function in different planes,multiple motion gait planning methods of the snake-like robot are proposed.In order to evaluate the adaptability and maneuverability of the developed snake robot,extensive experiments were conducted in complex environments.The results demonstrate the robot's effectiveness in navigating through intricate settings,underscoring its potential for applications in various fields.

Segmental Kinematic Coupling of the Human Spinal Column during Locomotion

As one of the most important daily motor activities, human locomotion has been investigated intensively in recent decades. The locomotor functions and mechanics of human lower limbs have become relatively well understood. However, so far our understanding of the motions and functional contributions of the human spine during locomotion is still very poor and simultaneous in-vivo limb and spinal column motion data are scarce. The objective of this study is to investigate the delicate in-vivo kinematic coupling between different functional regions of the human spinal column during locomotion as a stepping stone to explore the locomotor function of the human spine complex. A novel infrared reflective marker cluster system was constrncted using stereophotogrammetry techniques to record the 3D in-vivo geometric shape of the spinal column and the segmental position and orientation of each functional spinal region simultaneously. Gait measurements of normal walking were conducted. The preliminary results show that the spinal column shape changes periodically in the frontal plane during locomotion. The segmental motions of different spinal functional regions appear to be strongly coupled, indicating some synergistic strategy may be employed by the human spinal column to facilitate locomotion. In contrast to traditional medical imaging-based methods, the proposed technique can be used to investigate the dynamic characteristics of the spinal column, hence providing more insight into the functional biomechanics of the human spine.

Charge Couple Device-Based Systemfor3-di mensional Real Ti me Positioning on the Assessment of Segmental Range of Motion of Lumbar Spine

Objective： To observe the tested results of the segmental range of motion （ROM） of lumbar spine by charge couple device （CCD）-based system for 3-dimensional real-time positioning （CCD system）, and to analyze its clinical significance. Methods： Seven patients with lumbar joint dysfunction and 8 healthy subjects were tested twice by the CCD-based system with an interval of 10 min. Results： The ROM of the patients was obviously lesser than that of the healthy subjects. The measuring data of segmental ROM of lumbar spine by CCD system is correlated significantly to the same data checked later on the same subjects in every direction of the movements. The differences between two checks are usually less than 1 degree. Conclusion： Specially designed CCD based system for 3-dimensional real-time positioning could objectively reflect the segmental ROM of lumbar spine. The system would be of great clinical significance in the assessment of the biomechanical dysfunction of lumbar spine and the effect of the treatment applied.

Boundary Control of Coupled Nonlinear Three Dimensional Marine Risers

This paper presents a design of boundary controllers implemented at the top end for global stabilization of a marine riser in a three dimensional space under environmental loadings. Based on the energy approach, nonlinear partial differential equations of motion, including bending-bending and longitudinal-bending couplings for the risers are derived. The couplings cause mutual effects between the three independent directions in the riser＇s motions, and make it difficult to minimize its vibrations. The Lyapunov direct method is employed to design the boundary controller. It is shown that the proposed boundary controllers can effectively reduce the riser＇s vibration. Stability analysis of the closed-loop system is performed using the Lyapunov direct method. Numerical simulations illustrate the results.

Monolithic Coupling of the Pressure and Rigid Body Motion Equations in Computational Marine Hydrodynamics

In Fluid Structure Interaction(FSI) problems encountered in marine hydrodynamics, the pressure field and the velocity of the rigid body are tightly coupled. This coupling is traditionally resolved in a partitioned manner by solving the rigid body motion equations once per nonlinear correction loop, updating the position of the body and solving the fluid flow equations in the new configuration. The partitioned approach requires a large number of nonlinear iteration loops per time–step. In order to enhance the coupling, a monolithic approach is proposed in Finite Volume(FV) framework,where the pressure equation and the rigid body motion equations are solved in a single linear system. The coupling is resolved by solving the rigid body motion equations once per linear solver iteration of the pressure equation, where updated pressure field is used to calculate new forces acting on the body, and by introducing the updated rigid body boundary velocity in to the pressure equation. In this paper the monolithic coupling is validated on a simple 2D heave decay case. Additionally, the method is compared to the traditional partitioned approach(i.e. "strongly coupled" approach) in terms of computational efficiency and accuracy. The comparison is performed on a seakeeping case in regular head waves, and it shows that the monolithic approach achieves similar accuracy with fewer nonlinear correctors per time–step. Hence, significant savings in computational time can be achieved while retaining the same level of accuracy.

Reassessment of electromagnetic core-mantle coupling and its implications to the Earth’s decadal polar motion

The observed Earth’s polar motion on decadal time scales has long been conjectured to be excited by the exchange of equatorial angular momentum between the solid mantle and the fluid outer core,via the mechanism of electromagnetic(EM)core-mantle coupling.However,past estimations of the EM coupling torque from surface geomagnetic observations is too weak to account for the observed decadal polar motion.Our recent estimations from numerical geodynamo simulations have shown the opposite.In this paper,we re-examine in detail the EM coupling mechanism and the properties of the magnetic field in the electrically conducting lower mantle(characterized by a thin D '-layer at the base of the mantle).Our simulations find that the toroidal field in the D'-layer from the induction and convection of the toroidal field in the outer core could be potentially much stronger than that from the advection of the poloidal field in the outer core.The former,however,cannot be inferred from geomagnetic observations at the Earth’s surface,and is missing in previous EM torque estimated from geomagnetic observations.Our deduction suggests further that this field could make the actual EM coupling torque sufficiently strong,at approximately 5×1019 Nm,to excite,and hence explain,the decadal polar motion to magnitude of approximately 10 mas.

MODEL OF CENTRIFUGAL EFFECT AND ATTITUDEMANEUVER STABILITY OF A COUPLEDRIGID-FLEXIBLE SYSTEM

The influences of nonlinear centrifugal force to large overall attitude motion of coupled rigid-flexible system was investigated. First the nonlinear model of the coupled rigid-flexible system was deduced from the idea of “centrifugal potential field', and then the dynamic effects of the nonlinear centrifugal force to system attitude motion were analyzed by approximate calculation; At last, the Lyapunov function based on energy norm was selected, in the condition that only the measured values of attitude and attitude speed are available, and it is proved that the PD feedback control law can ensure the attitude stability during large angle maneuver.

MODELING AND ANALYSIS OF A COUPLED RIGID_FLEXIBLE SYSTEM

Correct predictions of the behavior of flexible bodies undergoing large rigid-body motions and small elastic vibrations is a subject of major concern in the field of flexible multibody system dynamics. Because of failing to account for the effects of dynamic stiffening, conventional methods based on the linear theories can lead to erroneous results in many practical applications. In this paper, the idea of 'centrifugal potential field', which induced by large overall rotation is introduced, and the motion equation of a coupled rigid-flexible system by employing Hamilton's principle is established. Based on this equation, first it is proved that the elastic motion of the system has periodic property, then by using Frobenius' method its exact solution is obtained. The influences of large overall rigid motion on the elastic vibration mode shape and frequency are analysed through the numerical examples.