Radiation Damping Coefficients Research Articles

A time domain approach for the optimal control of wave energy converter arrays

Wave energy converters typically use various control methods to extract energy from ocean waves. The objective of the control system is to optimize the energy extraction process, taking into account the dynamics of the system and the wave conditions. The task of deriving the optimal control laws of wave energy converter arrays for regular and irregular waves using the Pontryagin minimum principle was previously investigated in the literature. The result is a combination between the singular arc and bang-bang control laws. For irregular waves, some complexity arises due to the radiation state-space representation, which requires ignoring the hydrodynamic coupling terms related to the added mass and radiation-damping coefficients; this reduces the computational complexity of the control force but adversely affects the solution's accuracy. Also, the derived control laws are specific to a particular wave condition. Recently, the optimal control of a flexible buoy wave energy converter was derived using the convolution representation for the radiation force. In this work, the optimal control laws of flexible buoy wave energy converters are modified to simulate wave energy converter arrays; then, the results are compared to those obtained by dropping the hydrodynamic radiation coupling terms. Although using a convolution representation adds computational complexity to the optimal control problem, it generates an equation that is generic to any wave condition, can be used with any wave spectrum, and provides an expression for the switching condition.

Collision simulations between an icebreaker and an iceberg considering ship hydrodynamics

This paper examines the motion and structural damage of an icebreaker caused by a collision between an icebreaker and an iceberg based on coupling of potential flow theory with finite element analysis. The hydrodynamic subroutine for Abaqus plug-in HydroQus that was newly developed by present authors was used for the collision simulations. The Korean icebreaker ARAON and spherical shape iceberg were assumed for the collision simulations. The hydrostatic restoring stiffnesses, added masses, radiation damping coefficients, first-order wave excitations obtained through the hydrodynamic frequency response analysis were used for the constants of the Cummins equation of motion. The accuracy of HydroQus was validated from the free-decay simulations where there was a reasonable agreement in the out-of-plane ship motions. Applying each hydro-force one by one, parametric simulations were conducted to analyze the effects of the hydro-forces. Four collision phases were considered for the case subjected to a regular wave load: collision at maximum heave amplitude, maximum heave velocity, maximum pitch amplitude, and maximum pitch velocity. The hydrostatic force and radiation force were not critical components in determining structural damage, while wave loads can significantly increase collision-induced impact force and structural damage.

Hydrodynamic Coefficients of a Floater Near a Partially Reflecting Seawall in the Presence of an Array of Caisson Blocks

Abstract In the present study, surface gravity wave scattering and radiation by a freely floating rectangular buoy placed near a partially reflecting seawall and in the presence of an array of caisson blocks are analyzed. Various hydrodynamic parameters related to the wave scattering and radiation, such as the added mass and radiation damping coefficients, correspond to sway, heave, and roll motions of the floating buoy, horizontal force, vertical force, and moment acting on the floating structure, and horizontal wave force acting on the partially reflecting seawall, are studied for a variety of wave and structural parameters. The study reveals that the resonating pattern in various hydrodynamic coefficients occurs for moderate values of the wavenumber. Furthermore, when the distance between the floater and the sidewall is an integral time of half wavelength, the resonating behavior in the sway, heave, and roll added masses, and associated damping coefficients appears, and the aforementioned hydrodynamic coefficients change rapidly around this zone. These resonance phenomena can be diminished significantly with appropriate positioning of the floater with respect to the sidewall and in the presence of partially reflecting seawall.

Mechanical and acoustic emission properties of vegetable fiber‐reinforced epoxy composites for percussion instrument drums

AbstractPercussion instrument drums are traditionally made of wood, which is becoming less available as environmental concerns grow. In order to replace wood in percussion instruments, four types of epoxy resin composites reinforced with fabrics of hemp, flax, ramie, and jute, respectively, are fabricated using vacuum assisted resin infusion molding. The tensile and flexural properties of the composites are measured and compared with those of ailanthus wood. The results show that the flexural strengths and moduli of the composites are either better or in between to those of the horizontal and vertical strips of the wood. An acoustic test method is developed to test the acoustic performance of the composites and the wood. The results show that the composites have lower acoustic dynamic moduli, acoustic radiation damping coefficients, sound velocity, and higher acoustic impedance than those of the wood due to their lower flexural moduli and higher specific densities. The shapes of the acoustic spectra for the composites are somewhat similar to those of ailanthus. The acoustic performance comprehensive scores of the four composites are also in between those of the vertical and the horizontal strips of ailanthus. It is possible to match the acoustic performance of ailanthus by increasing the flexural modulus and decreasing the specific density of the composites through reducing the yarn crimp, increasing the fiber volume fraction, and introducing microvoids into the composites.

Towing Motion Characteristics of Composite Bucket Foundation for Offshore Wind Turbines

Composite bucket foundation (CBF) is an environmentally friendly form of offshore wind power foundation. By virtue of the air-floating subdivision structure in the bucket, the foundation has a self-floating characteristic and can allow for long-distance air-floating towing and transport. In this study, first, a numerical analysis was performed on the towing motion characteristics of CBF, and the formula for foundation stability was deduced; then, we probed into the variation rules of the natural period of the foundation with draft and analyzed the response amplitude operator (RAO) responses, added mass coefficients, radiation damping coefficients, and exciting forces (moments) of the composite bucket foundation under different drafts and different wave directions. Finally, we clarified the basic hydrodynamic characteristics of the composite bucket foundation in the frequency domain and adopted multiple random wave models to investigate the effects of different wave heights, periods, and spectral peak factors on the towing motion characteristics of composite bucket foundation.

Wave radiation from multiple cylinders of arbitrary cross sections

A semi-analytic model based on linear potential theory is developed to solve wave radiation from an array of truncated cylinders with arbitrary cross sections, free to oscillate independently in water of finite depth. The radiated velocity potential in the interior region below each cylinder and the exterior region extending to infinity are expressed as a series of eigenfunctions in the local cylindrical coordinate system using the method of separation of variables. They are solved subsequently by imposing the continuity condition of the velocity and pressure at the wetted surfaces of the cylinders and the fluid interfaces between adjacent regions. In the process, the Fourier series expansion method is applied to the terms regarding the arbitrary cross sections. The model is successfully validated through a case study with two caissons – excellent agreement is obtained with a numerical model based on the boundary element method. The semi-analytic model is then applied to explore the influence of layout (angle and spacing between the two caissons) on the added-mass and radiation damping coefficients. It is found that the hydrodynamic interaction between the two caissons can increase or decrease the value of the coefficients depending on the wave frequency.

A time domain three-dimensional sono-elastic method for ships’ vibration and acoustic radiation analysis in water

The classical three-dimensional hydroelasticity of ships is extend to include the effect of fluid compressibility, which yields the three-dimensional sono-elasticity of ships. To enable the predictions of coupled transient or nonlinear vibrations and acoustic radiations of ship structures, a time domain three-dimensional sono-elastic analysis method of acoustic responses of a floating structure is presented. The frequency domain added mass and radiation damping coefficients of the ship are first calculated by a three-dimensional frequency domain analysis method, from which a retardation function is derived and converted into the generalized time domain radiation force through a convolution integral. On this basis the generalized time domain sono-elastic equations of motion of the ship hull in water are established for calculation of the steady-state or transient-state excitation induced coupled vibrations and acoustic radiations of the ship. The generalized hydrodynamic coefficients, structural vibrations and underwater acoustic radiations of an elastic spherical shell excited by a concentrated pulsating force are illustrated and compared with analytical solutions with good agreement. The numerical results of a rectangular floating body are also presented to discuss the numerical error resultant from truncation of the upper integration limit in the Fourier integral of the frequency domain added mass coefficients for the retardation function.

Drag, added mass and radiation damping of oscillating vertical cylindrical bodies in heave and surge in still water

Forced heave and surge motion of axisymmetric vertical cylindrical bodies with flat and rounded bases are simulated using the advanced CFD software STAR-CCM+ where the overset method is used so that the mesh local to the body moves within a stationary outer mesh. Viscous effects generating drag, and also influencing added mass and radiation damping, are determined. The Reynolds-Averaged Navier–Stokes (RANS) equations are adopted with different turbulence closure models and the water surface is captured by the volume of fluid (VOF) method. These results are of basic interest but the main motive is to assess appropriate drag coefficients for use with linear diffraction models of wave energy converters (WEC) and we have particular interest in the multi-body WEC M4. A basic dynamical model is set up so that drag, added mass and radiation damping coefficients may be obtained from the CFD results. Added mass and radiation damping coefficients were also obtained from the potential flow solver WAMIT for comparison. Mesh convergence studies were undertaken and while mesh independence was achieved for total force it was not possible for the very small shear force. In the laboratory a free heave decay test was undertaken without mechanical contact for bodies with rounded bases and the inferred drag and added mass coefficients were very close to those from CFD. Some general observations are possible for motion in heave. For the hemispherical base the drag coefficient Cd is very low for small amplitudes but this increases as amplitude is increased. For the rounded base with a flat central area the Cd is larger and for the wholly flat base it is larger again but with values less than 0.35. For a larger geometric scale (times 32) for the hemisphere and round base cases the Cd are generally somewhat reduced. For surge motion the Cd show less variation and are always greater than heave values by at least a factor of 2 which is indicative of effects due to separation and wake generation.

주파수 영역에서 인공감쇠기법을 활용한 복층 유체의 수치조파수조 방사 문제

There are two wave modes induced by an oscillating body on the free surface of a two-layer fluid: the barotropic and baroclinic modes. To investigate the generated waves composed of two modes, a radiation problem involving a heaving rectangular body was solved in a numerical wave tank. A new artificial damping zone scheme was developed and applied in the frequency-domain analysis. The performance of this damping scheme was compared with given radiation boundary conditions for various conditions. The added mass and radiation damping coefficients for the heaving rectangular body were also calculated for various fluid-density ratios.

Wave-induced hydrodynamic responses of a rigid body connected with elastic plates floating on a two-layer fluid

A novel physical model of Very Large Floating Structures (VLFSs) is investigated for a two-layer fluid. This model is composed of a semi-immersed module rigidly connected with two elastic plates, which may provide basic knowledge for the study of multi-module floating structures in a stratified ocean. Under the hypothesis of small-amplitude wave theory, the case with the coupling effects among the wave motion, the vertical deformation of the elastic plates and the rigid body's oscillation is solved by only considering a scattering problem and a radiation one. New inner product relations with orthogonality are suggested to calculate the undetermined coefficients in eigenfunction expansions, which is eligible to yield a convergent result. By this approach, the matching relations are simplified and the numerical calculation speed is remarkably promoted. Exciting forces (caused by the scattering potential), hydrodynamic coefficients for added mass and radiation damping (caused by the direct act of radiation potential) and additional coefficients for added stiffness and added damping (caused by the indirect act of radiation potential via the elastic plates) are obtained. Numerical representations for amplitudes, exciting forces, hydrodynamic coefficients, and additional coefficients are shown, in which Green's theorem is used for ensuring convergence.

Method of measuring the spectral response of technical oils

It is proposed to measure the transmission and fluorescence spectra of technical oils by the method of side illumination of a quartz tube filled with the oil being investigated. This method makes it possible to broaden the range of values of the radiation-damping coefficients that can be determined by using a laser beam that scans along the tube, causing the oil to fluoresce. The fluorescence and transmission spectra of new (unused), oxidized, and contaminated technical oils are investigated. An example is given of choosing a diagnostic parameter for objectively estimating the quality change of the transformer oil.

Development of a time domain boundary element method for numerical analysis of floating bodies’ responses in waves

This article presents the development of a numerical tool for seakeeping simulations of marine systems using a time domain boundary element method based on Rankine sources. The formulation considers two initial boundary value problems defined for the velocity and acceleration potentials, the last being used to avoid numerical problems in calculating the time derivatives of the velocity potential. A fourth-order Runge–Kutta method is used for the time marching of the problem, which consists in the integration of the free surface conditions and body equations of motion. Numerical test cases are presented for bodies with simplified geometries, such as an hemisphere and a circular section cylinder. Exciting forces, added mass and radiation damping coefficients, and motions response amplitude operators are compared to analytical and numerical data, presenting a very good agreement. Furthermore, the numerical method is applied to a floating production storage and Off-loading unit and the results are verified with experimental data carried out in the hydrodynamic calibrator of the University of Sao Paulo. By means of these investigations, we have verified that the developments performed so far are correct and new extensions, therefore, may be planned for more complex applications.

Added mass and wave radiation damping for flow-induced rotational vibrations of skinplates of hydraulic gates

One observed vibration mode for Tainter gate skinplates involves the bending of the skinplate about a horizontal nodal line. This vibration mode can be approximated as a streamwise rotational vibration about the horizontal nodal line. Such a streamwise rotational vibration of a Tainter gate skinplate must push away water from the portion of the skinplate rotating into the reservoir and draw water toward the gate over that portion of the skinplate receding from the reservoir. The induced pressure is termed the push-and-draw pressure. In the present paper, this push-and-draw pressure is analyzed using the potential theory developed for dissipative wave radiation problems. In the initial analysis, the usual circular-arc skinplate is replaced by a vertical, flat, rigid weir plate so that theoretical calculations can be undertaken. The theoretical push-and-draw pressure is used in the derivation of the non-dimensional equation of motion of the flow-induced rotational vibrations. Non-dimensionalization of the equation of motion permits the identification of the dimensionless equivalent added mass and the wave radiation damping coefficients. Free vibration tests of a vertical, flat, rigid weir plate model, both in air and in water, were performed to measure the equivalent added mass and the wave radiation damping coefficients. Experimental results compared favorably with the theoretical predictions, thus validating the theoretical analysis of the equivalent added mass and wave radiation damping coefficients as a prediction tool for flow-induced vibrations. Subsequently, the equation of motion of an inclined circular-arc skinplate was developed by incorporating a pressure correction coefficient, which permits empirical adaptation of the results from the hydrodynamic pressure analysis of the vertical, flat, rigid weir plate. Results from in-water free vibration tests on a 1/31-scale skinplate model of the Folsom Dam Tainter gate are used to demonstrate the utility of the equivalent added mass coefficient.

Linearization of Quadratic Drag to Estimate CALM Buoy Pitch Motion in Frequency-Domain and Experimental Validation

The dynamics of an oil offloading catenary anchor leg mooring (CALM) buoy coupled with mooring and flow lines are directly related to the fatigue life of a mooring system, necessitating an accurate estimate of the buoy hydrodynamic response. Linear wave theory is used for modeling the surface boundary value problem, and the boundary element method is used to solve the fluid-structure interaction between the buoy hull and the incident waves in the frequency-domain. The radiation problem is solved to estimate the added mass and radiation damping coefficients, and the diffraction problem is solved to determine the linear wave exciting loading. The buoy pitch motion is investigated, and linearizations of the quadratic drag/damping term are performed in the frequency-domain. The pitch motion response is calculated by considering an equivalent linearized drag/damping. Quadratic, cubic, and stochastic linearizations of the nonlinear drag term are employed to derive the equivalent drag/damping. Comparisons between the linear and nonlinear damping effects are presented. Time-domain simulations of the buoy motions are performed in conjunction with Morison’s equation to validate the floating buoy response. The time- and frequency-domain results are finally compared with the experimental model test results for validations. The linearization methods applied result in good estimates for the peak pitch response. However, only the stochastic linearization method shows a good agreement for the period range of the incident wave where typical pitch response estimate has not been correctly estimated.

System identification of coupled heave–pitch motion of ships with forward speed in random ocean waves

Ship dynamics in ocean waves involve frequency-dependent added mass and radiation damping which can be estimated either by potential theory-based calculations or by experiments. With uniform forward speed, the added mass and damping matrices become asymmetric. In this paper, we attempt to use a system identification approach, specifically the reverse multiple input–single output (R-MISO) method, for coupled heave–pitch response (two degrees of freedom) of a ship moving with uniform forward speed in random ocean waves. The system mass matrix has both frequency-independent and frequency-dependent components, whereas its damping matrix has only frequency-dependent components. The frequency-dependent components of the mass and damping matrices are asymmetric. The excitation force and moment due to linear monochromatic waves which act on the system are assumed to be known and can be either calculated or obtained from experiments. For numerical illustration, a ship whose hydrodynamic behaviour has been computed by strip theory has been considered. The motion, as well as the loading, is simulated assuming Pierson–Moskowitz spectrum in conjunction with response amplitude operators from the strip theory code, and these results are analysed by the R-MISO method yielding frequency-dependent asymmetric coupled added mass and radiation damping coefficients satisfactorily.

System Identification of a Coupled Two DOF Moored Floating Body in Random Ocean Waves

Dynamics of a large moored floating body in ocean waves involves frequency dependent added mass and radiation damping as well as the linear and nonlinear mooring line characteristics. Usually, the added mass and radiation damping matrices can be estimated either by potential theory-based calculations or by experiments. The nonlinear mooring line properties are usually quantified by experimental methods. In this paper, we attempt to use a nonlinear system identification approach, specifically the reverse multiple input-single output (R-MISO) method, to coupled surge-pitch response (two-degrees-of-freedom) of a large floating system in random ocean waves with linear and cubic nonlinear mooring line stiffnesses. The system mass matrix has both frequency independent and frequency dependent components whereas its damping matrix has only frequency dependent components. The excitation force and moment due to linear monochromatic waves which act on the system are assumed to be known that can either be calculated or obtained from experiments. For numerical illustration, a floating half-spheroid is adopted. The motion as well as the loading are simulated assuming Pierson-Moskowitz (PM) spectrum and these results have been analyzed by the R-MISO method yielding frequency dependent coupled added mass and radiation damping coefficients, as well as linear and nonlinear stiffness coefficients of mooring lines satisfactorily.

Hydrodynamic interaction between two vertical cylinders in water waves

The hydrodynamic interaction between two vertical cylinders in water waves is investigated based on the linearized potential flow theory. One of the two cylinders is fixed at the bottom while the other is articulated at the bottom and oscillates with small amplitudes in the direction of the incident wave. Both the diffracted wave and the rediation wave are studied in the present paper. A simple analytical expression for the velocity potential on the surface of each cylinder is obtained by means of Graf's addition theorem. The wave-excited forces and moments on the cylinders, the added masses and the radiation damping coefficients of the oscillating cylinder are all expressed explicitly in series form. The coefficients of the series are determined by solving algebraic equations. Several numerical examples are given to illustrate the effects of various parameters, such as the separation distance, the relative size of the cylinders, and the incident angle, on the first-order and steady second-order forces, the added masses and radiation-damping coefficients as well as the response of the oscillating cylinder.

The hydrodynamic coefficients for an oscillating rectangular structure on a free surface with sidewall

Based on a two dimensional linear water wave theory, the boundary element method (BEM) is developed and applied to study the heave and the sway problem of a floating rectangular structure in water to finite depth with one side of the boundary is a vertical sidewall and the other boundary is an open boundary. Numerical results for the added mass and radiation damping coefficients are presented. These coefficients are not only depend on the submergence and the width of the structure, but also depend on the clearance between structure and sidewall. Negative added mass and sharp peaks in the damping and added mass coefficients have been found when the clearance with a value close to integral times of half wave length of wave generated by oscillation structure. The important effect of the clearance on the added mass and radiation damping coefficients are discussed in detail. An analytical solution method is also presented. The BEM solution is compared with the analytical solution, and the comparison shows good agreement.

The inherent background in acoustic wave scattering by a submerged target.

A new concept for the background in resonance scattering of acoustic waves by a submerged target is proposed. The idea is that the remainder after removing only the resonances from the target response must be used as the background. The complex wave numbers whose imaginary parts are the attenuation coefficients of elastic bulk waves in target are introduced to damp out the resonances. From the assumption that the structural damping coefficient, the product of the attenuation coefficients of elastic bulk waves, and the outer radius of target, is a large constant compared with the radiation damping coefficients of resonances, a simple mathematical expression is derived for elastic cylindrical shells and named the inherent background. It is shown that the inherent background leads to the impenetrable (i.e., rigid or soft) background in some limiting cases and also describes exactly the background of elastic shell over entire frequencies regardless of shell thickness.

On the heave radiation of a rectangular structure

In this paper, an analytic solution to the heave radiation problem of a rectangular structure is presented. To solve the problem analytically, the nonhomogeneous boundary value problem is linearly decomposed into homogeneous ones, which can be readily solved. To provide further comparisons to the present analytic solution, a boundary element method is also presented to solve the problem. The present analytic solution is compared with the result by Black et al. [(1971)] Radiation and scattering of water waves by rigid bodies. J. Fluid Mech. 46, 151–164], and the boundary element solution, and the comparisons show very good agreements. Upon examination of the present analytic solution, it is shown that the solution satisfies the nonhomogeneous boundary condition in a sense of series convergence. Using the present analytic solution, the generated waves, the added mass and the radiation damping coefficients, as well as the hydrodynamic effects of the submergence and the width of the structure, are investigated.