Wet Natural Frequencies Research Articles

Boundary Element Modelling Aspects for the Hydro-Elastic Analysis of Flexible Marine Propellers

Boundary element methods (BEM) have been used for propeller hydrodynamic calculations since the 1990s. More recently, these methods are being used in combination with finite element methods (FEM) in order to calculate flexible propeller fluid–structure interaction (FSI) response. The main advantage of using BEM for flexible propeller FSI calculations is the relatively low computational demand in comparison with higher fidelity methods. However, the BEM modelling of flexible propellers is not straightforward and requires several important modelling decisions. The consequences of such modelling choices depend significantly on propeller structural behaviour and flow condition. The two dimensionless quantities that characterise structural behaviour and flow condition are the structural frequency ratio (the ratio between the lowest excitation frequency and the fundamental wet blade natural frequency) and the reduced frequency. For both, general expressions have been derived for (flexible) marine propellers. This work shows that these expressions can be effectively used to estimate the dry and wet fundamental blade frequencies and the structural frequency ratio. This last parameter and the reduced frequency of vibrating blade flows is independent of the geometrical blade scale as shown in this work. Regarding the BEM-FEM coupled analyses, it is shown that a quasi-static FEM modelling does not suffice, particularly due to the fluid-added mass and hydrodynamic damping contributions that are not negligible. It is demonstrated that approximating the hydro-elastic blade response by using closed form expressions for the fluid added mass and hydrodynamic damping terms provides reasonable results, since the structural response of flexible propellers is stiffness dominated, meaning that the importance of modelling errors in fluid added mass and hydrodynamic damping is small. Finally, it is shown that the significance of recalculating the hydrodynamic influence coefficients is relatively small. This fact might be utilized, possibly in combination with the use of the closed form expressions for fluid added mass and hydrodynamic damping contributions, to significantly reduce the computation time of flexible propeller FSI calculations.

Forced wet flexural dynamics of spade rudder in propeller slipstream: semi-analytical approach of pivoted FFFF plate

A semi-analytical approach of forced vibration of a trapezoidal, two-way tapered, hollow model spade rudder, is presented. The pivoted rudder is in the propeller slipstream; which causes harmonic hydrodynamic load. The excitation frequency of propeller-induced vibration (PIV) equals the propeller blade passing frequency. The rudder operates at a Reynolds’s number of the order of 108. It has a span of 1 m, root chord of 0.6 m, and tip chord of 0.5 m. It is considered as a hollow Free–Free–Free–Free Kirchhoff’s plate, with the chord section as an NACA0018 profile. The dry vibration is analyzed by the energy-based Galerkin’s method, using closed-form admissible functions in the two perpendicular directions. 3D constant-strength panel method is used to generate the modal added masses, and hence the wet natural frequencies. The drag and the lift coefficient of the rudder, at various angles of attack, are estimated to find the spatial distribution of the total PIV-causing normal force. The mean flow velocity about the rudder is the velocity of advance behind the propeller. The fluid velocity past the aerofoil section depends on the profile thickness and the angle of attack, thereby influencing the local lift. The hydrodynamic pressure varies parabolically along the span. The effect of the wake developed by the propeller, the stern shape, and the rudder angle on the hydrodynamic loading on rudder is included to express the transverse time-varying loading. The wet, forced vibration of the Kirchhoff’s plate is analyzed in MATLAB. The maximum bending stress and twisting sheer at the rudder stock is calculated for various angles of attack.

Free vibration analysis of a functionally graded rectangular plate in contact with a bounded fluid

This study investigates the vibration analysis of a functionally graded (FG) rectangular plate partially in contact with a bounded fluid. The material properties are assumed to vary continuously through the thickness direction according to a simple power-law distribution in terms of volume fraction of material constituents. Wet dynamic transverse displacements of the plate are approximated by a set of admissible trial functions, which are required to satisfy the clamped and simply supported geometric boundary conditions. The fluid velocity potential satisfying fluid boundary conditions is derived, and wet dynamic modal functions of the plate are expanded in terms of finite Fourier series for compatibility requirement along the contacting surface between the plate and the fluid. Natural frequencies of the plate coupled with sloshing fluid modes are calculated using the Rayleigh–Ritz method based on minimizing the Rayleigh quotient. The proposed analytical method is validated with available data in the literature. In the numerical results, the effects of boundary conditions, aspect ratios, thickness ratios, gradient index, material properties of the FG plate, depth of the fluid and dimensions of the tank on the wet natural frequencies are investigated.

On the natural frequency of tidal current power systems—A discussion of sea testing

To study the wet natural frequency (in water) and dry natural frequency (in air) of a tidal current turbine, we conducted a two-year measurement campaign by deploying a full-scale prototype of the system. In this article, a theoretical model is developed and validated with the frequency measurements. It reveals the measured wet natural frequency of the system could approach half that of the dry one. The measurements also show that inflow turbulence is very important in the excitation of system resonances that can lead to system failure. We also briefly discuss how the wet frequency varies over a long period.

Hydroelastic slamming response in the evolution of a flip-through event during shallow-liquid sloshing

The evolution of a flip-through event [6] upon a vertical, deformable wall during shallow-water sloshing in a 2D tank is analyzed, with specific focus on the role of hydroelasticity. An aluminium plate, whose dimensions are Froude-scaled in order to reproduce the first wet natural frequency associated with the typical structural panel of a Mark III containment system, is used. (Mark III Containment System is a membrane-type tank used in the Liquefied Natural Gas (LNG) carrier to contain the LNG. A typical structural panel is composed by two metallic membranes and two independent thermal insulation layers. The first membrane contains the LNG, the second one ensures redundancy in case of leakage.) Such a system is clamped to a fully rigid vertical wall of the tank at the vertical ends while being kept free on its lateral sides. Hence, in a 2D flow approximation the system can be suitably modelled, as a double-clamped Euler beam, with the Euler beam theory. The hydroelastic effects are assessed by cross-analyzing the experimental data based both on the images recorded by a fast camera, and on the strain measurements along the deformable panel and on the pressure measurements on the rigid wall below the elastic plate. The same experiments are also carried out by substituting the deformable plate with a fully stiff panel. The pressure transducers are mounted at the same positions of the strain gauges used for the deformable plate. The comparison between the results of rigid and elastic case allows to better define the role of hydroelasticity. The analysis has identified three different regimes characterizing the hydroelastic evolution: a quasi-static deformation of the beam (regime I) precedes a strongly hydroelastic behavior (regime II), for which the added mass effects are relevant; finally, the free-vibration phase (regime III) occurs. A hybrid method, combining numerical modelling and experimental data from the tests with fully rigid plate is proposed to examine the hydroelastic effects. Within this approach, the measurements provide the experimental loads acting on the rigid plate, while the numerical solution enables a more detailed analysis, by giving additional information not available from the experimental tests. More in detail, an Euler beam equation is used to model numerically the plate with the added-mass contribution estimated in time. In this way the resulting hybrid method accounts for the variation of the added mass associated with the instantaneous wetted length of the beam, estimated from the experimental images. Moreover, the forcing hydrodynamic load is prescribed by using the experimental pressure distribution measured in the rigid case. The experimental data for the elastic beam are compared with the numerical results of the hybrid model and with those of the standard methods used at the design stage. The comparison against the experimental data shows an overall satisfactory prediction of the hybrid model. The maximum peak pressure predicted by the standard methods agrees with the result of the hybrid model only when the added mass effect is considered. However, the standard methods are not able to properly estimate the temporal evolution of the plate deformation.

Wet damping estimation of the scaled segmented hull model using the random decrement technique

Damping is one of the most uncertain factors in structural dynamic problems, and plays a very important role in resonance phenomena such as springing. This paper presents the wet damping estimation of a segmented hull model using a random decrement technique together with continuous wavelet transform. The 16 sea states were grouped together based on the speed of the ship to determine the possible influence of the ship speed on the damping ratio. The measured time histories of the vertical bending moment for each tested sea state were processed using the random decrement technique to derive the free decay signal, from which the damping ratios were estimated. In addition, the autocorrelation functions of the filtered signal were calculated and a comparison was made with the free decay signal obtained from the random decrement technique. The wet damping ratios for each sea state group, as well as precise wet natural frequencies, were estimated using a continuous wavelet transform. The wet natural frequencies derived from the measured signal did not show any significant discrepancy compared to those obtained using the wet hammering test, whereas a significant discrepancy was observed with the damping ratio. The discrepancy of the damping ratio between in calm and moving water might be due to the viscous effects caused by the dramatically different flow patterns and relative velocity between the vibrating structure and surrounding fluid particles.

랜덤 감쇠기법을 이용한 분할모형의 접수 감쇠계수 추정

This paper presents the wet damping estimation of the segmented hull model using the random decrement technique together with the continuous wavelet transform. The tested 16 sea states are grouped together based on the speed of the ship in order to figure out the possible influence of the ship speed on the damping ratio. The measured time histories of vertical bending moment for each tested sea state were processed with random decrement technique to derive the free decay signal, from which the damping ratios are estimated. Also, the autocorrelation functions of the filtered signal were calculated and comparison was made with the free decay signal obtained from the random decrement technique. Then the wet damping ratios for each sea state group, as well as precise wet natural frequencies, are estimated by using continuous wavelet transform. It turned out that the wet natural frequencies derived from the measured signal did not show any significant discrepancy compared with those obtained by wet hammering test, whereas the damping ratio did. It was considered that the discrepancy of the damping ratio between in calm and moving water may be attributed to the viscous effects caused by dramatically different flow pattern and relative velocity between the vibrating structure and surrounding fluid particles.

Investigation into the nonlinear hydroelastic response of an 8600-TEU containership model advancing in regular waves

With the increase in the size of containerships, the lowest wet natural frequencies can be as low as 0.5 rad/s for vessels whose lengths are beyond 300 m. These lower frequencies make the estimation of wave loads when considering the hull flexibility quite challenging. Because no numerical tools are able to account for all of the nonlinear effects properly and there is not enough experimental data available for validation, the extent that current numerical tools can correctly predict the hydroelastic effects is a concern. This paper adopts a nonlinear hydroelastic code, WINSIR, to calculate the nonlinear hydroelastic response of an 8600-TEU containership model advancing in regular waves. The calculated results are compared with test data in terms of the first, second, third, fourth and fifth harmonics of the vertical bending moment amidships. Finally, the uncertainties in the experiments are discussed, and conclusions are given at the end of this paper.

Dynamic response of Kirchhoff’s plates to transient hydrodynamic impact loads

The dynamic analysis of the elastic response of marine structures to intense, localized, transient, hydrodynamic impact loads is important in the design of marine vessels. A dry and wet dynamic analysis of vibration of a square Kirchhoff’ s plate is presented. The dry and wet natural frequencies and modeshapes of the plate have been evaluated by Galerkin’s method. The transient loads are of two kinds: a uniform stretching load and a hydrodynamic impact load (at different deadrise angles). The stretching load sets the plate into small-amplitude high-frequency vibrations. The computationally efficient normal mode analysis has been used to evaluate the dynamic deflections. Three different boundary conditions of the plate have been used. The response of the plate to these loads has been investigated for a wide range of impact speeds and material properties. The response characteristics have been established, namely (a) maximum deflection, (b) maximum dynamic overshoot relative to the static deflection, (c) quasi-static and dynamic zones of the response over the range of time-scales. The dependency of these characteristics on parameters like the deadrise angles, damping ratios, boundary conditions and forcing configurations have also been studied, to aid the designer. A separate study of the modal participation factor is used to establish modal truncation limits for the analysis, with subsequent increases in the computational efficiency.

Dynamic analyses of floating fish cage collars in waves

An extended 3D hydroelasticity theory is used in the frequency domain to predict the dynamic response of 5 by 2 floating fish farm collars in regular waves. Hinge modes in two directions and flexible torsional modes are all considered in the analyses. The dry and wet resonant natural frequencies for each rigid relative motion and the flexible modes are also calculated. The results show that the modes, flexible bending and torsional modes, as well as their interactions, greatly contribute to the dynamic response of the floating collar system. From a design point of view, the effects of the connector rotational stiffness on the dynamic response of the structures are also analysed.

Hydroelastic vibration analysis of liquid-contained rectangular tanks

This paper presents a theoretical analysis for the free vibration of rectangular tanks partially filled with an ideal liquid. Wet dynamic displacements of the tanks are approximated by combining the orthogonal polynomials satisfying the boundary conditions, since the rectangular tanks are composed of four rectangular plates. The classical boundary conditions of the tanks at the top and bottom ends are considered, such as clamped, simply supported, and clamped-free boundary conditions. As the facing rectangular plates are assumed to be geometrically and structurally identical, the vibration modes of the facing plates of the tanks can be divided into two categories: symmetric and antisymmetric modes with respect to the planes passing through the center of the tanks and perpendicular to the free liquid surface. The liquid displacement potentials satisfying the Laplace equation and liquid boundary conditions are derived, and the wet dynamic modal functions of a quarter of the tanks can be expanded by the finite Fourier transform for compatibility requirements along the contacting surfaces between the tanks and liquid. An eigenvalue problem is derived using the Rayleigh-Ritz method. Consequently, the wet natural frequencies of the rectangular tanks can be extracted. The proposed analytical method is verified by observing an excellent agreement with three-dimensional finite element analysis results. The effects of the liquid level and boundary condition at the top and bottom edges are investigated.

Hydroelastic vibration and buckling of rectangular Mindlin plates on Pasternak foundations under linearly varying in-plane loads

Hydroelastic vibration and buckling analysis of horizontal rectangular plates resting on Pasternak foundation and subjected to linearly varying in-plane loads is investigated for different boundary conditions. Fluid is assumed to be inviscid and incompressible. To analyze the interaction of the Mindlin plate with the elastic foundation and fluid system, three displacement components of the plate are expressed in the Ritz method by adopting a set of static Timoshenko beam functions satisfying geometric boundary conditions. Convergence study is first carried out to confirm the stability of the present method. Then, a few comparisons are made for limited cases between present results and the available data in the literature. Finally, parametric study is conducted to highlight the effects of loading factors, fluid levels, foundation stiffness parameters, aspect ratios, thickness to width ratios and different boundary conditions on the critical buckling loads and wet natural frequencies of rectangular plates.

Hydroelastic vibration of a cantilever cylindrical shell partially submerged in a liquid

The dynamic characteristics of a circular cylindrical shell in contact with a liquid are theoretically studied. The cantilevered cylindrical shell with open ends is partially submerged in a liquid which is unbounded in the radial direction, but bounded by a rigid and flat bottom. Since the liquid is assumed to be incompressible and inviscid, the liquid motion can be described as the velocity potentials written in terms of the appropriate Bessel functions for both the inner and the outer liquid regions. The compatibility of the dynamic displacement along the contacting surfaces between the shell and liquid leads the resulting equations, satisfying both the shell and liquid motions, which are solved by using the collocation method. Finally, the Rayleigh–Ritz method is applied to extract the wet natural frequencies and the mode shapes of the liquid-coupled system. The validity of the theoretical method is established with the aid of a commercial finite element computer code. In order to evaluate the dynamic characteristics of the liquid-coupled system, the effects of the submerged depth and the axial gap between the bottom end of the shell and the bottom surface of the liquid are demonstrated.

Free vibration analysis of a circular plate partially in contact with a liquid

This paper provides a theoretical method for a hydroelastic vibration of a clamped circular plate partially in contact with a liquid. It is assumed that the liquid is inviscid and incompressible while the thickness of the plate is small compared to the plate radius, and the gravity effect on the system is neglected. The wet dynamic displacement of the plate is approximated by combining the orthogonal modal functions of a dry circular plate with a clamped boundary condition. The liquid displacement potential satisfying the boundary conditions is derived and the wet dynamic modal functions of the plate are derived by using a compatibility requirement along the contacting surface between the plate and the liquid. The eigenvalue equation for the system is obtained by using the Rayleigh–Ritz method in order to calculate the wet natural frequencies and the mode shapes for the symmetric and the asymmetric modes. The proposed method is verified by observing a good agreement with three-dimensional finite element analysis results. The effects of the liquid thickness and the liquid depth on the natural frequencies are also discussed.

Hydrodynamic loading of microcantilevers oscillating near rigid walls

The vibrations of microcantilevers in atomic force microscopes (AFMs) or radio frequency (RF) switches are strongly influenced by the viscous hydrodynamics of the surrounding fluid in the vicinity of a rigid wall. While prior efforts to model this hydrodynamic loading have focused on squeeze film damping effects at high Knudsen and squeeze numbers, the regimes of low Knudsen and squeeze numbers are also very important for which squeeze film models need to be discarded in favor of unsteady Stokes hydrodynamics. We extend the work of Green and Sader [Phys Fluids 17, 073102 (2005); J. Appl. Phys. 98, 114913 (2005)] and present compact semianalytical formulas for the unsteady viscous hydrodynamic function of slender microbeams oscillating near rigid walls, in terms of key nondimensional numbers. Using these closed-form expressions, it becomes possible to predict easily the wet natural frequencies and quality factors of multiple modes of microcantilevers near rigid walls in diverse applications ranging from AFM in liquids to RF microswitches under ambient conditions. The semianalytical formulas are extensively validated by comparing their predicted wet natural frequencies and quality factors with those based on three-dimensional, transient flow-structure interaction simulations, as well as previous experiments performed in the field by other researchers.

Hydroelastic analysis of fluid storage tanks by using a boundary integral equation method

This paper presents a boundary integral equation method in conjunction with the method of images, in order to investigate the dynamic behaviour ( wet frequencies and associated mode shapes) of fluid containing structures. In the analysis of the linear fluid–structure system, it is assumed that the fluid is ideal, and fluid forces are associated with inertial effects of the contained fluid. This implies that the fluid pressure on the wetted surface of the structure is in phase with the structural acceleration. The in vacuo dynamic properties of the dry structure are obtained by using standard finite element software. In the wet part of the analysis, the fluid–structure interaction effects are calculated in terms of the generalized added mass coefficients by use of the boundary integral equation method together with the method of images in order to impose the Φ=0 boundary condition on the free surface. In this study, three different test cases are considered: (i) a clamped–free cylindrical tank with a rigid bottom; (ii) a flexible circular plate in a rigid cylindrical tank and (iii) a flexible cylindrical shell with flexible end plates (hermetic can). The fluid storage tanks in this investigation are assumed partially or completely filled with water. To assess the influence of the contained fluid on the dynamic characteristics, the wet natural frequencies and associated mode shapes are calculated. The predictions compare well with available experimental and numerical data.

Linear vibration analysis of cantilever plates partially submerged in fluid

Dynamic characteristics, such as natural frequencies and mode shapes, of cantilever plates, partially in contact with a fluid, are investigated. In the analysis of the linear fluid–structure system, it is assumed that the fluid is ideal, and fluid forces are associated with inertial effects of the surrounding fluid. This implies that the fluid pressure on the wetted surface of the structure is in phase with the structural acceleration. Furthermore, the infinite frequency limit is assumed on the free surface. The in vacuo dynamic properties of the plates are obtained by use of a standard finite-element software. In the wet part of the analysis, it is assumed that the plate structure preserves its in vacuo mode shapes when in contact with the surrounding fluid and that each mode shape gives rise to a corresponding surface pressure distribution of the cantilever plate. The fluid–structure interaction effects are calculated in terms of the generalized added-mass values independent of frequency (i.e., infinite frequency generalized added-masses), by use of a boundary-integral equation method together with the method of images in order to impose the Φ = 0 boundary condition on the free surface. To assess the influence of the surrounding fluid on the dynamic characteristics, the wet natural frequencies and associated mode shapes were calculated, and they compared very well with the available experimental data and numerical predictions.

FREE VIBRATION OF A PARTIALLY LIQUID-FILLED AND SUBMERGED, HORIZONTAL CYLINDRICAL SHELL

The dynamic characteristics (i.e., natural frequencies and mode shapes) of a partially filled and/or submerged, horizontal cylindrical shell are examined. In this investigation, it is assumed that the fluid is ideal, and fluid forces are associated with inertial effects only: namely, the fluid pressure on the wetted surface of the structure is in phase with the structural acceleration. The in vacuo dynamic characteristics of the cylindrical shell are obtained using standard finite element software. In the “wet” part of the analysis, it is assumed that the shell structure preserves its in vacuo mode shapes when in contact with the contained and/or surrounding fluid and that each mode shape gives rise to a corresponding surface pressure distribution of the shell. The fluid–structure interaction effects are calculated in terms of generalized added masses, using a boundary integral equation method together with the method of images in order to impose an appropriate boundary condition on the free surface. To assess the influence of the contained and/or surrounding fluid on the dynamic behaviour of the shell structure, the wet natural frequencies and associated mode shapes were calculated and compared with available experimental measurements.

Response Characteristics of Serially Connected Semisubmersibles

One design for a mobile offshore base is to link serially as many as five large semisubmersibles to form a platform long enough to support large aircraft. This paper investigates the linear, wave-induced response characteristics of serially-connected semisubmersibles. A major motivation of this study is to understand more completely the forces required to link semisubmersible modules. The impact of connector strategy and damping on the response, especially the connector forces, is investigated, and the response "modes" which contribute to the connector forces are evaluated in detail. It is shown that the response characteristics can be impacted significantly by the connection strategy, and that connector damping can be a significant source of energy loss when compared to radiation damping. The wet natural frequencies and normal modes are also determined and used to explain the response characteristics of different connection strategies. Although the analyses are based on a specific semisubmersible design, the results provide insight on how other systems of connected semisubmersibles would likely behave.