Nonlinear Hydrodynamic Forces Research Articles

Nonlinear couplings in a gear-shaft-bearing system

The present work contributes to the analysis of the interactions between gears, shafts and hydrodynamic journal bearings in geared drives. In contrast to the majority of the models in the literature, the time-varying properties and nonlinearities of gears and bearings are introduced in the simulations. A finite element model is used for the shafts, and a specific gear element is used to account for nonlinear time-varying mesh stiffness as well as tooth shape deviations. The nonlinear hydrodynamic forces are computed with the Reynolds equation for finite-length journal bearings. An iterative Newmark scheme is used to solve simultaneously the motion equations for the shafts, the contact problem for the gears, and the fluid forces in the bearings. The resulting algorithm is applied to a single stage geared system with two shafts, four bearings, a pinion and a gear. Gear-bearing dynamic interactions are demonstrated through the analysis of dynamic gear loads, dynamic bearing loads and bearing displacements. The efficiency of the proposed numerical procedure, the interest of nonlinear models for hydrodynamic bearings and the influence of several parameters ruling the gear assembly are also discussed.

A New Active Tilting-Pad Bearing: Nonlinear Modeling and Feedback Control

This article introduces a new type of active fluid film bearing and its feedback control. In particular, the active adjustment of the angular velocity of the pads of a tilting-pad bearing in response to changes in the operating conditions of the rotating machine is proposed. This is motivated by the observation that there is more control authority in the pad tilting motion than in its radial translation. To this end, a dynamic model for the bearing system is first developed, inclusive of the nonlinear hydrodynamic force for the infinitely short bearing case. A model-based controller is then constructed, based on measurements of the journal position and velocity and pad tilting angles, to ensure that the journal is asymptotically regulated to the bearing center. Numerical simulations illustrate the performance of the active bearing under the proposed control in comparison with the bearing's standard passive mode of operation.

Stability and nonlinear dynamic behavior of drilling shaft system in copper stave deep hole drilling

The stability and nonlinear dynamic behavior of drilling shaft system in copper stave deep hole drilling were analyzed. The effects of the fluctuation of the cutting force, the mass eccentricity and the hydrodynamic forces of cutting fluid could be taken into consideration in the model of drilling shaft system. Based on the isoparametric finite element method, the variational form of Reynolds equation in hydrodynamic fluid was used to calculate nonlinear hydrodynamic forces and their Jacobian matrices simultaneously. In the stability analysis, a new shooting method for rapidly determining the periodic orbit of the nonlinear drilling shaft system and its period was presented by rebuilding the traditional shooting method and changing the time scale. Through the combination of theories with experiment, the correctness and effectiveness of the above methods are verified by using the Floquet theory. The results show that the mass eccentricity can inhibit the whirling motion of drilling shaft to some extent.

Complex nonlinear behaviors of drilling shaft system in boring and trepanning association deep hole drilling

Nonlinear dynamic behaviors of drilling shaft system in boring and trepanning association deep hole drilling are investigated. The effects of the mass eccentricity and the cutting force fluctuating can be taken into consideration in the model of drilling shaft system. Nonlinear hydrodynamic forces are adopted to increase the numerical accuracy. Based on the isoparametric finite element with eight nodal points method, the nonlinear hydrodynamic forces and their Jacobian matrices of compatible accuracy are calculated simultaneously. To solve the nonlinear dynamic orbit of drilling shaft system, the period T of the periodic orbit of the nonlinear drilling shaft system is drawn into the governing equation of the system explicitly by changing the time scale. Then, the period T is taken as a parameter in the iterations of the shooting method. The periodic orbit and its period of the drilling shaft system are determined rapidly by generalized shooting method. The local stability and dynamic behaviors of periodic motion with the change of the drilling shaft design parameters value are obtained by the Floquet theory. The numerical results indicate the rich and complex dynamic behaviors of the drilling shaft system, such as periodic, quasiperiodic, jumped solution, and coexistence of multisolution and can make a good reference for the dynamic design of drilling shaft system in deep hole drilling.

Development of study on the dynamic characteristics of deep water mooring system

To meet the needs of those exploiting deepwater resources, TLP and SPAR platforms are used in some areas and are considered excellent platforms in deep water. However, many problems remain to be resolved. The design of mooring systems is a key issue for deep water platforms. Environmental loads in deep water effect the physical characteristics of mooring line materials. The configuration and analysis of mooring systems involve nonlinearity due to this fluid-solid coupling, nonlinear hydrodynamic forces, and their effects on stability of motion. In this paper, some pivotal theories and technical questions are presented, including modeling of mooring lines, the theory and method of coupled dynamics analysis on the mooring system, and the development of methodologies for the study of nonlinear dynamics of mooring systems. Further study on mooring systems in deep water are recommended based on current knowledge, particularly dynamic parameters of different materials and cable configuration, interactions between seabed and cable, mechanisms of mooring system response induced by taut/slack mooring cables, discontinuous stiffness due to system materials, mooring construction, and motion instability, etc.

On the nonlinear hydrodynamic forces for a ship advancing in waves

In this paper, using a second-order steady-state approach and a three-dimensional (3D) pulsating source distribution method derives the nonlinear hydrodynamic forces on a ship advancing in waves. The nonlinear hydrodynamic forces considered here consist of the mean lateral drifting force and the added resistance, which can be expressed as products of the ship-motion responses, the radiation potential, diffraction potential and the incident-wave potential. All related velocity potentials applied in the calculations are in 3D form. The Series 60 and Marine ship hulls are used for numerical calculations and the results are compared with existing experimental data and two-dimensional (2D) solutions. The comparisons show that the results obtained in the paper generally agree with experimental data well. It is also found that the nonlinear hydrodynamic forces obtained based on the present 3D source distribution methods are indeed improved in some calculations compared with the 2D method, especially for the mean lateral drifting force.

SEISMIC ANALYSIS OF OFFSHORE TRIANGULAR TENSION LEG PLATFORMS

Oil and gas production from deep-water offshore fields represent a major structural engineering challenge for the industry. The tension leg platform (TLP) is a well-established concept for deep-water oil exploration. It is necessary to design an offshore TLP such that it can respond to moderate environmental loads without damage, and is capable of resisting severe environmental loads without seriously endangering the occupants. Seismic analysis of triangular TLP under moderate regular waves is investigated. The analysis considers nonlinearities due to the change in tether tension and nonlinear hydrodynamic drag forces. The coupled response of TLP under moderate regular sea waves due to change in initial pretension in the tethers caused by seismic forces (vertical direction) is then investigated. Seismic forces are imposed at the bottom of each tether as axial forces. The tether tension becomes unbalanced when the hull is under offset position. The vertical component of seismic force is an important item to take into consideration, because it is directly superposed to pretension of tethers. The change in initial pretension due to the vertical component of the earthquake affects the response of the triangular TLP in degrees-of-freedom experiencing such forces. The tether tension varies nonlinearly when the platform is subjected to seismic forces caused by the El Centro earthquake and artificially generated earthquake using Kanai–Tajimi's power spectrum. The response due to earthquakes varies with the intensity of the input ground motion. The seismic response of the triangular TLP exhibits nonlinear behavior in the presence of waves and it is non-proportionately influenced by the wave period and the wave height.

Nonlinear Hydrodynamic Forces on an Accelerated Body in Waves

Wave-drift added mass results from nonlinear interactions between waves and low-frequency oscillatory motions of a floating body, in the presence of incident waves. In previous works, wave-drift damping which is the component of wave-drift force in phase with the velocity of low-frequency oscillations was investigated mainly based on a quasi-steady analysis. However, investigations related to wave-drift added mass, the component in phase with acceleration, were very few. In this paper, wave-drift added mass is derived directly from a perturbation analysis with two small parameters and two time scales, using a Cartesian coordinate system that follows the low-frequency oscillations, dynamic oscillation model has been used. Especially, the method to solve higher-order potentials, which are necessary for evaluation of wave-drift added mass, is presented. Analytical solutions and calculated results of wave-drift added mass, and far field radiation conditions for each order of potentials are obtained. Also, wave-drift added mass of floating bodies has been systematically measured from a slowly forced oscillation test or a free decay test in waves. Experimental results are compared with calculated results. Then, for a supplement, the secular behavior that some velocity potentials show is discussed. Applying a multiple scale perturbation analysis to one of these problems, a nonsecular solution is obtained.

Importance of several nonlinear factors on broaching prediction

In order to realize a more quantitative prediction of broaching and capsizing in following and quartering seas, existing mathematical modelling techniques should be upgraded. Therefore, it is necessary to systematically examine all factors relevant to capsizing in following and quartering seas. To this end, we first attempted to examine the prediction accuracy of wave-induced forces by comparing calculations with captive model experiments. As a result, we found that a wave-induced surge force has a certain nonlinearitiy with respect to wave steepness. The nonlinearity of sway–roll coupling with respect to sway velocity was also found, and our numerical result with these nonlinearities improves the prediction accuracy of the critical ship speed of capsizing in following and quartering seas. The importance of the wave effect on propeller thrust was also examined. We found that this effect is not negligibly small and could improve capsizing predictions in heavy following and quartering seas. Finally, we attempted to investigate the importance of nonlinear heel-induced hydrodynamic forces on numerical predictions of capsizing due to broaching. Here, we propose a new procedure for captive model experiments to obtain hydrodynamic forces with various heel angles up to 90°, and data on heel-induced hydrodynamic forces with respect to heel angle in calm water are provided. We then compare the numerical simulations with the nonlinear heel-induced hydrodynamic forces and without them. These time series comparisons show that the effect of nonlinear heel-induced hydrodynamic forces in calm water is not negligibly small for the case of ship capsizing due to broaching.

Force control of a load through the splash zone

A parallel force and position controller for control of loads through the wave zone in marine operations is proposed in this article. The controller structure has similarities to parallel force/position controllers used in the robotics literature and has been developed based on linear theory and tested in simulations on a system with the highly nonlinear hydrodynamic forces that a load is exposed during water entry in waves. Wave synchronization and parallel force/position control are compared in the simulations. The parallel force/position controller gives significant reductions in the oscillations in the wire force, while hydrodynamic forces on the object are about the same with the two control strategies.

Subharmonic vibrations of a roll tube supported by plain journal bearings

This study presents a dynamic model for describing a journal bearing with circumferential waviness of the shaft journal. The journal bearing model is based on an analytical solution of the Reynolds equation and defines the non-linear hydrodynamic forces that act on the journal bearing. The waviness of the shaft journal is modelled through the use of a Fourier cosine series to describe the shape of the film thickness of the bearing. The waviness of the shaft journal can lead to subharmonic vibrations in which the natural vibration mode of the rotor is excited when the rotation speed is a fraction of the natural frequency of the rotor-bearing system. This phenomenon is investigated in this study utilizing the proposed bearing model in a rotor system simulation. The rotor system under investigation consists of a roll tube, two journal bearings and the supporting structure. The proposed journal bearing model can be used in a general multi-body or rotor dynamics computer code as the interface element between the rotor and the bearing housing. In this study, the model of the journal bearing is implemented in a commercial multi-body simulation software application. This approach enables a detailed analysis of the rotor-bearing system in such a way that the dynamic interaction of the rotor with the bearing is taken into consideration. The simulation result agrees with the experimental measurements obtained from a real roll tube system.

선체 휘핑 해석 전용 프로그램의 개발

선체 휘핑 해석 전용 프로그램은 모멘텀 슬래밍 이론을 사용하여 비선형 유체충격력을 추정하였으며, Timoshenko 보이론에 기초하여 선체 hull girder를 탄성체로 모델링하고 있다. 개발된 프로그램에 대하여 슬래밍 충격력의 효과, 유탄성해석의 효과 그리고, 선속, 파고, 파장 등 각종 설계 변수가 미치는 효과 등에 관한 검증 계산을 수행하였고, 실제 선박 설계에의 적용 예를 보임으로써 본 프로그램의 설계 활용성을 확인하였다. 본 프로그램은 각종 입출력자료의 GUI 처리 및 선박 운동에 대한 시뮬레이션 제공 등 다양한 기능과 특성을 가지고 있다. A special purposed program for ship hull strength analysis considering whipping phenomena is developed. In this program, the non-linear hydrodynamic impact force is considered using the momentum slamming theory and the hull girder is modeled as elastic body on the base of Timoshenko's beam theory. The numerical verifications are conducted in the view points that the effect of slamming impact force, the effect of hydro-elastic formulation, and the effect of various design parameters such as ship speed, wave amplitude, wave length and others. By the application of a real ship design process, the availability of the program is proved. This program has a GUI function for many I/O data process as well as the function to show the 2-D ship motion in the graphic window, and has other available functions for the whipping analysis.

Non-Gaussian Model for Ringing Phenomena in Offshore Structures

Significant interest has been shown in identifying the nonlinear mechanisms that induce a ringing type response in offshore structural systems. This high frequency transient type response has been observed in offshore systems, particularly in tension leg platforms (TLPs). Given the implications of this behavior on the fatigue life of TLP tendons, it is essential that ringing be considered in the overall response evaluation. This study presents two non-Gaussian probabilistic models of nonlinear viscous hydrodynamic wave forces that induce ringing. The response of a single-degree-of-freedom system exposed to these non-Gaussian wave force models is then evaluated using analytical and numerical studies based on the Ito⁁ differentiation rule and the Monte Carlo simulation procedure, respectively. The results demonstrate that the proposed models induce ringing type response in a simplified structure. This study provides a probabilistic framework for modeling ringing type phenomenon which will serve as a building block for more refined hydrodynamic load models.

Dynamic behaviour of square and triangular offshore tension leg platforms under regular wave loads

Among compliant platforms, the tension leg platform (TLP) is a hybrid structure. With respect to the horizontal degrees of freedom, it is compliant and behaves like to a floating structure, whereas with respect to the vertical degrees of freedom, it is stiff and resembles a fixed structure and is not allowed to float freely. The greatest potential for reducing costs of a TLP in the short term is to go through previously applied design approaches, to simplify the design and reduce the conservatism that so far has been incorporated in the TLP design to accommodate for the unproven nature of this type of platform. Dynamic analysis of a triangular model TLP to regular waves is presented, considering the coupling between surge, sway, heave, roll, pitch and yaw degrees of freedom. The analysis considers various nonlinearities produced due to change in the tether tension and nonlinear hydrodynamic drag force. The wave forces on the elements of the pontoon structure are calculated using Airy's wave theory and Morison's equation, ignoring the diffraction effects. The nonlinear equation of motion is solved in the time domain using Newmark's beta integration scheme. Numerical studies are conducted to compare the coupled response of a triangular TLP with that of a square TLP and the effects of different parameters that influence the response are then investigated.

An Aid to Operating Decisions Based on Nonlinear Response of a Crane Barge in Waves

A nine-degree-of-freedom time domain mathematical model was used to numerically simulate crane ship dynamics. This model considers the elasticity and damping of the hoisting rope assembly and includes arbitrary, biangular swing of the suspended hook load coupled with surge, sway, heave, roll, pitch, and yaw motions of the hull. The linear frequency-dependent hydrodynamic response to the hull’s own motions accounted for memory effects that were approximated in the time domain by a finite state space model. The nonlinear hydrodynamic drag force acting on the oscillating hull was quadratically approximated using an empirical drag coefficient. The incident wave force on the hull consisted of the first-order force oscillating at wave frequency and the second-order slowly varying wave drift force. The nonlinear horizontal mooring system restoring force was approximated by a third-order polynomial. To specify operating limits for a shear-leg crane barge in a heavy lift offshore operation, a stochastic analysis was performed based on the system’s simulated dynamic response in an ensemble of natural seaways.

A note on non-linear hydrodynamic force on a floating body

A note on non-linear hydrodynamic force on a floating body

Learning tracking controllers under unknown disturbances for the installation of rigid and flexible structures

During the installation operations of underwater structures, offshore structures such as semisubmersibles, the steady nonlinear hydrodynamic force due to the ocean current, the tidal current, and the hydrodynamic interaction between structures are regarded as major disturbances. Generally, it is difficult and very expensive to collect information about these disturbances in advance, although such knowledge would be important and necessary for precise installation. Because of this situation, we have proposed a learning tracking controller (LTC) for automated tracking/docking operations of offshore structures. The LTC can be operated without any information about current, and can be applied to the installation problem of flexible structures whose flexibility cannot be ignored. In order to confirm the effectiveness of the LTC, basin tests were carried out using experimental models with an unknown current. Two types of basin experiment were executed using an ultrasound ranging system and thrusters. One was trajectory tracking of a semisubmersible, and the other was the precise mating of flexible underwater structures. In both cases the LTC showed good performance, and in the latter case in particular, two flexible models were successfully made to track the given trajectory and dock to the docking targets with an accuracy of ±5 mm, after the trajectory had been tracked six to seven times for learning purposes. No elastic responses were excited in the flexible model.

Nonlinear coupled response of offshore tension leg platforms to regular wave forces

Among the compliant platforms, the tension leg platform (TLP) is a vertically moored structure with excess buoyancy. The TLP is designed to behave in the same way as any other moored structure in horizontal plane, at the same time inheriting the stiffness of a fixed platform in the vertical plane. Dynamic response analysis of a TLP to deterministic first order wave forces is presented, considering coupling between the degrees-of-freedom surge, sway, heave, roll, pitch and yaw. The analysis considers nonlinearities produced due to changes in cable tension and due to nonlinear hydrodynamic drag forces. The wave forces on the elements of the pontoon structure are calculated using Airy's wave theory and Morison's equation ignoring diffraction effects. The nonlinear equation of motion is solved in the time domain by Newmark's beta integration scheme. The effects of different parameters that influence the response of the TLP are then investigated.

Linear and Nonlinear Hydroelastic Analysis of High-Speed Vessels

A linear formulation of ship hydroelasticity is presented. Appropriate body boundary conditions of flexible modes are obtained and the hydroelastic version of high-speed strip theory is established. A nonlinear time-domain simulation method is also presented. The total response is decomposed into linear and nonlinear parts. The linear part is evaluated using appropriate linear potential-flow theory and the nonlinear part comes from the convolution of the impulse response functions of linear ship-fluid system and the nonlinear hydrodynamic forces. Four high-speed vessels with different ship lengths but with similar body plan and internal structural arrangement are used as examples. The calculations of midship bending moments are carried out at different forward speeds and head sea states. The results show that the hydroelastic effect in linear extreme responses is insignificant and that the hydrodynamic damping plays a leading role in the flexible modes when the dynamic amplification of ship hull becomes important. The results also indicate that strong nonlinearity is the most prominent feature of high-speed vessels even in the moderate sea state and must be taken into account. The nonlinear influences are more remarkable in ships at large Froude numbers than in those at small ones, and more important in sagging moment than in hogging moment.

Structural control of a steel jacket platform

This paper deals with the application of certain active and passive control mechanisms to control the dynamic response of a steel jacket platform due to wave-induced forces. The forces are estimated using the nonlinear Morison equation which provides nonlinear self-excited hydrodynamic forces. The influence of these forces on the response of a structure without and with vibration control mechanisms is demonstrated using a steel jacket platform as a simple example.