Wet Natural Frequencies Research Articles

Experimental analysis of bow flare slamming and whipping responses in a ship at different sailing speeds

This study conducted an experimental investigation using a segmented model to determine bow flare slamming loads and whipping responses of a ship in regular waves. The model simulated the distribution of the vertical and horizontal bending stiffnesses of the hull. A continuous wavelet transform was employed to capture temporal variations in the harmonic components. The influences of sailing speeds on the temporal and spatial variations in the slamming pressures in the bow flare were analyzed. Subsequently, the time-frequency characteristics and asymmetry of the global responses under different test conditions are discussed. The random errors in the experimental system were investigated through repeated testing. The findings revealed that a higher sailing speed could significantly influence the magnitude and distribution of slamming pressures on the bow flare, while playing an important role in inducing asymmetric slamming loads under oblique waves. The slamming-induced whipping responses under regular waves caused considerable periodic changes in the strengths of the harmonic components around the wet natural frequency of the model. At high sailing speeds, the whipping responses significantly increased the asymmetry of the vertical bending moment (VBM) at midship.

Feasibility investigation of tuned mass damper for vibration control of curved floating bridge in winds and waves

This paper investigates the feasibility and operability of a tuned mass damper (TMD) to control the lateral vibrations of a curved floating bridge. A curved floating bridge was designed with a bridge girder, 38 columns, and 19 pontoons, which was the same concept proposed for crossing Bjørnafjorden in Norway, except that the cable-stayed bridge was excluded in the present model. TMD was then attached to the mid-length of the bridge girder. Time-domain model was built to consider the girder-column-pontoon interaction based on the Cummins equation for pontoons and the lumped-mass line model for the bridge girder and columns, while TMD was modeled by the mass-spring-damper system. First, the genetic algorithm was employed to optimize the spring and damping coefficients of TMD at a fixed TMD mass under white noise wave excitations, and the fitness function was a summation of the standard deviation of the sway motion of 19 pontoons. Then, its feasibility was evaluated by comparing lateral vibrations and vertical bending moments with and without TMD under various combinations of wind/wave/current loads. Results demonstrate that TMD contributes to a reduction in the lateral vibration when resonance at the second wet natural frequency occurs, which can improve passenger comfort. Wind-induced lateral motions are significantly reduced at the second wet natural frequency while TMD has limited effects under other environmental loads. Moreover, bending moments are governed by higher modes mostly induced by storm waves, which cannot be controlled by TMD effectively.

Mechanics mode characteristics analysis and numerical analytical optimization study of a three-stage centrifugal pump

To increase the accuracy of an analytic method based on the concentrated-mass approach for solving the mechanical operating modal characteristics of complete impellers and impellers with balancing holes in a centrifugal-pump rotor system, we establish a mathematical model and a three-dimensional simulation model for a double-supported three-stage centrifugal-pump rotor system. By comparing the modal results obtained from the two models, we propose a mathematical model that integrates the optimization of mass-distribution methods and bending stiffness correction coefficients, and we derive an optimized mathematical model for rotors with balancing holes. The effectiveness of the optimized mathematical model is validated via three-dimensional simulation, and the influence of balancing holes on the wet modes is analyzed. The results show that the wet natural frequencies are lower than the dry ones and that the wet natural frequencies obtained from analytical calculations are consistently higher than those obtained from three-dimensional simulation results. The optimized mathematical model significantly improves the accuracy of the wet modal characteristics for both complete impellers and impellers with balancing holes, with the errors in the low-order natural frequencies reduced by more than 50%. Under the same hole spacing, the natural frequencies of the three-stage centrifugal pump rotor system decrease with increasing hole diameter slightly. Under the same hole diameter, the natural frequencies of the rotor system increase slightly with the hole spacing. The effect of stiffness changes caused by balancing holes on the natural frequencies is larger than the effect of mass changes.

Experimental study of sloshing characteristics in a rectangular tank with elastic baffles

A series of laboratory experiments are conducted to investigate liquid sloshing in a rectangular tank with a vertical baffle of varying elasticity under horizontal excitations. Image processing techniques are introduced to track baffle displacement, as well as to identify the free surface and the bubble. Dye visualization is employed to reveal the vortex behavior around the baffle. The air entrainment mechanism, including jet impinging, air cavity formation and deformation, and turbulent mixing, is analyzed. Moreover, the study investigates the effect of baffles with different stiffness on suppressing liquid sloshing across a wide range of excitation frequencies. The results show that the peak frequencies of the maximum free surface elevation-frequency curves correspond to the first natural frequency of fluid, the sub-resonance frequency, and the wet natural frequency of the elastic baffle. It is also found that the elastic baffle can effectively reduce liquid sloshing within a certain frequency range and balance the slamming impact both on the wall and the baffle, providing promising prospects for its practical applications.

Analytical method for modal analysis of submerged electric motor stator

Modal analysis of the stator is important for reducing motor noise and diagnosing motor faults. An accurate analytical model is beneficial for the establishment of the relationship between the motor stator parameters and the natural frequencies, which will further aid in optimizing machine designs. Up to now, the existing modal analysis models are mainly aimed at onshore motors. When the motor is submerged in dense fluid, such as those used in the rim-driven thrusters, the vibration behaviors are quite different due to the high characteristic impedance of the fluid. In a bid to analyze the wet natural frequencies of motor stator, this paper proposes analytical models with one, three and five degrees of freedom for different application scenarios based on Donnell, Love and Soedel’s theories. The coupling between the structure and fluid is realized by the acoustic pressure load derived from the acoustic wave function. The resulting non-linear dispersion equations are then solved by Newton’s iterative method. The proposed models are applied to analyze the natural frequencies of motor stators of various sizes and with a wide range of thickness-to-radius and length-to-radius ratios. The results show that the analytical models are insensitive to the stator radius but sensitive to the thickness-to-radius and length-to-radius ratios. For thick and short motor stators, it is necessary to consider the shear deformation and rotatory inertia in the analytical model to compute the natural frequency accurately, especially when the fluid loading is taken into account.

Discrete-module-beam-based hydro-elasticity simulations for moored submerged floating tunnel under regular and random wave excitations

This study investigates the hydro-elastic behaviors of a moored submerged floating tunnel (SFT) using a fully-coupled time-domain hydro-elastic model based on the discrete-module-beam (DMB) method. In frequency domain, multibody hydrodynamic coefficients and wave excitation forces are obtained from 3D potential theory. In the time-domain DMB-based hydro-elastic model, the multibody Cummins equation and coupled stiffness matrix according to the theory of Euler-Bernoulli beam and Saint-Venant torsion are used. Mooring lines are modeled with rod theory and high-order finite-element formulations. The tunnel is then coupled with mooring lines by linear springs at respective connection locations. The present DMB model is validated through comparisons with two independent commercial programs, i.e., quasi-static computation based on the 3D solid model by ABAQUS and dynamic simulation using the semi-empirical Morison model by OrcaFlex. Then, the effects of various design parameters, such as submergence depths, buoyancy-weight ratios (BWRs), and wave conditions, on dynamic responses and mooring tensions are evaluated. The effects of nonlinear snap loadings and first- and second-order wave forces on SFT dynamics in regular and random waves are also investigated. Wet natural frequencies and mode shapes, hydro-elastic lateral and vertical displacements, mooring tensions, and bending moments for various cases are presented and analyzed.

The Scale Effect Study on the Transient Fluid–Structure Coupling Performance of Composite Propellers

In order to predict the fluid–structure coupling performance of a full-scale composite propeller under a wake flow field behind a full ship, a bidirectional transient fluid–structure coupling algorithm was established. The performance includes the transient fluid–structure coupling deformation, structural natural frequency, and unsteady hydrodynamic performance. The results showed that the circumferential non-uniform wake flow field can cause periodic pulsation of the propeller’s hydrodynamic force. The average values of thrust and torque coefficient increase, while the pulsation ratio decreases with the increase in scale. The maximum deformation ratio of fluid–structure coupling is linearly related to the scale ratio. Due to the influence of fluid-added stiffness, the maximum deformation ratio needs to be modified by 3% based on the cantilever plate deflection formula. The first five natural frequencies of dry mode and wet mode decrease with the increasing scale, and the wet natural frequencies of each order decrease by 60~68% compared with dry mode. The fluid–structure coupling hydrodynamic performance still show a periodic pulsation with the phase angle, and its average value increases linearly with the scale ratio, while the pulsation ratio decreases with the power relationship of the scale ratio.

Structure Dynamic Response of Amphibious Aircraft Induced by Water-Taxiing

The significant dynamic response under the combined impact of aerodynamic and hydrodynamic forces could be likely to appear because of the structural flexibility, when taxiing on the water surface for amphibious aircraft. Meanwhile, the modal characteristics of the structure are also affected by the additional motion of water. These require that the influence of the structural elasticity and the coupling effect between water and structure should be considered in dynamic response analysis of water-taxiing. According to the peculiarities of the amphibious aircraft, structural dynamics model is based on the distribution of stiffness and mass, Virtual Mass Theory is utilized to solve the wet modes on the water surface, rational function approximations of unsteady aerodynamic force in time-domain are constructed by the Minimum-State Approximation Formula, and loose coupling method is employed to simulate the hydrodynamic elastic response under the encounter of amphibian with single wave and repeated waves, respectively. Analysis of dynamic characteristics during the water-taxiing of the amphibious aircraft has been achieved in this work. The results show that wet natural frequencies of the aircraft have different degrees of decline compared with the dry frequencies because of the influence of added water on the hull, and the response amplitude of dynamic loads obtained by using the wet modes have some certain extent decrease compared with the dry modes. The dynamic amplitude of different locations changes in different degree relatives to the center of gravity position, which reflects the influence of structural elasticity. Due to the excitation of single wave and repeated waves, the structural vibration amplitude will increase rapidly, but the amplitude shows a certain divergence trend under the action of repeated waves with a given oscillation frequency, which is more severe for structural strength design.

Bio-inspired aquatic propulsion using piezoelectric effect

Underwater propulsion of fishes have inspired many biomimetic structures. Generally, the bio-inspired structures mimics the flapping behaviour of various control surfaces/fins in fishes. The present study mimics the flapping behaviour using a piezoelectric structure. The system is analyzed as a fluid structure interaction problem. The dynamic behaviour of a cantilever beam surrounded by a bounded fluid domain open at top is analyzed. The structure is modeled as a Euler-Bernoulli beam and the fluid is modeled using potential flow theory. The influence of domain size on the wet natural frequencies of the system was analyzed. The dimensions of the fluid domain wherein the variation in wet natural frequencies becomes insensitive were determined. The influence of added mass on the wet natural frequency was parametrized based on Non-dimensional Added Mass Increment (NAVMI) factor. The NAVMI factors were observed to be relatively higher for lower wet modes of the structure. Therefore, the peizo-beam was analyzed by exciting the lower wet modes. The thrust generated at different excitation frequencies were determined using tip velocity of the cantilever beam following Lighthill’s analogy. The results from the study indicated that higher propulsive thrust was produced for lower modes of excitation of the structure.

Optimization of tuned mass damper for seismic control of submerged floating tunnel

This study presents the optimization process of a tuned mass damper (TMD) to mitigate the lateral motions and mooring tensions of a submerged floating tunnel (SFT) under seismic excitations. A time-domain hydro-elastic-simulation model is established to solve the coupled dynamics among the tunnel, TMD, and mooring lines. The wet hydro-elastic natural frequencies of the SFT with mooring are estimated. The spring and damping coefficients of TMD are optimized by using metaheuristic optimization algorithms, i.e., harmony search (HS), genetic algorithm (GA), and particle swarm optimization (PSO). HS is coupled with the time-domain dynamic-simulation model to perform the iterative process of updating the coefficients by HS, running the simulation with updated coefficients, returning the results back to HS. The optimized coefficients obtained by HS are also cross-checked by using GA and PSO with the pre-established objective function, which shows consistent results. Subsequently, the effectiveness of TMD with the optimized parameters is tested for a variety of seismic conditions that can cover the most seismic magnitudes. The time histories, spectra, and statistics of SFT dynamic responses and mooring tensions are systematically analyzed and discussed. As intended, the optimized TMD effectively attenuates the resonant hydro-elastic transient motions of the SFT at its lowest lateral natural frequency. The mooring tensions are also significantly reduced by adopting the optimized TMD, especially in large earthquakes.

Research on Noise Reduction of 3.6 MW Evaporative Cooling Wind Motor Induced by Electromagnetic and Two-Phase Flow Resonance Based on Stator Optimization

In this paper, the vibration frequency of the stator of 3.6 MW fully immersed evaporative cooling permanent magnet semi-direct drive generators (ECPMSDDGs) was analyzed based on the fluid-structure coupling theory and solved by finite element analysis (FEA) simulation. The resonance noise reduction research under the typical working condition induced by two-phase flow and electromagnetic force was studied based on the method of structural optimization. In this paper, a structural optimal design method for the stator of the 3.6 MW ECPMSDDGs was presented. First of all, the frequency and characteristics of electromagnetic force of 3.6 MW ECPMSDDGs under the rated power were analyzed. Secondly, the frequency and characteristics of two-phase flow boiling vibration were analyzed based on the bubble oscillation theory of the two-phase flow and the experiment. Thirdly, the wet modal natural frequency of the stator core cooling structure was analyzed based on the fluid-structure coupling theory and FEA. Finally, the natural wet mode vibration frequency of the stator cooling structure of the 3.6 MW ECPMSDDGs was improved based on the structure optimization. This optimization method could reduce the resonance noise of evaporative cooling motor induced by electromagnetic and two-phase flow. The optimization results showed that the natural wet mode frequency of the stator could be improved by optimizing the radial flow groove and supporting beam under the condition that the effective length of the stator core remained unchanged during the optimization. The noise simulation result showed that the resonance noise of 3.6 MW ECPMSDDGs induced by electromagnetic and two-phase flow could be reduced after the structural optimization.

A numerical method to compute global resonant vibrations of ships at forward speed in oblique waves

Resonant wave-induced vibrations of a ship hull girder, known as “springing”, is an important issue when addressing fatigue aging of the steel structure. To compute resonant wave-induced vibrations, fluid-structure interactions and associated structural and hydrodynamic properties need to be addressed. This paper introduces a time-domain numerical method that predicts higher-order springing taking into account forward speed. Structural dynamics were computed based on a beam element approach that considers vertical and horizontal bending as well as nonuniform torsion. Furthermore, mass and stiffness matrices accounted for strong coupling effects between hull girder bending and torsion. The hydrodynamic solver coupled the fully nonlinear stationary free surface flow with the oscillatory flow and considered geometrical nonlinearities caused by the changing wetted surface due to the incident waves, nonlinear rigid body motions and linear elastic vibrations. Numerical predictions were validated against model tests of a post-Panamax containerships at forward speed. Dry and wet natural frequencies and midship vertical bending, horizontal bending and torsional moments induced by higher order springing vibrations compared favourably to experimental measurements.

The Effect of Key Design Parameters on the Global Performance of Submerged Floating Tunnel under Target Wave and Earthquake Excitations

This study presents a practical design strategy for a large-size Submerged Floating Tunnel (SFT) under different target environments through global-performance simulations. A coupled time-domain simulation model for SFT is established to check hydro-elastic behaviors under the design random wave and earthquake excitations. The tunnel and mooring lines are modeled with a finite-element line model based on a series of lumped masses connected by axial, bending, and torsional springs, and thus the dynamic/structural deformability of the entire SFT is fully considered. The dummy-connection-mass method and constraint boundary conditions are employed to connect the tunnel and mooring lines in a convenient manner. Wave- and earthquake-induced hydrodynamic forces are evaluated by the Morison equation at instantaneous node positions. Several wave and earthquake conditions are selected to evaluate its global performance and sensitivity at different system parameters. Different Buoyancy-Weight Ratios (BWRs), submergence depths, and tunnel lengths (and mooring intervals) are chosen to establish a design strategy for reducing the maximum mooring tension. Both static and dynamic tensions are critical to find an acceptable design depending on the given target environmental condition. BWR plays a crucial role in preventing snap loading, and the corresponding static tension is a primary factor if the environmental condition is mild. The tunnel length can significantly be extended by reducing BWR when environmental force is not that substantial. Dynamic tension becomes important in harsh environmental conditions, for which high BWR and short mooring interval are required. It is underscored that the wet natural frequencies with mooring are located away from the spectral peaks of design waves or earthquakes.

Time-domain coupled analysis of curved floating bridge under wind and wave excitations

A floating bridge is an innovative solution for deep-water and long-distance crossing. This paper presents a curved floating bridge's dynamic behaviors under the wind, wave, and current loads. Since the present curved bridge need not have mooring lines, its deep-water application can be more straightforward than conventional straight floating bridges with mooring lines. We solve the coupled interaction among the bridge girders, pontoons, and columns in the time-domain and to consider various load combinations to evaluate each force's contribution to overall dynamic responses. Discrete pontoons are uniformly spaced, and the pontoon's hydrodynamic coefficients and excitation forces are computed in the frequency domain by using the potential-theory-based 3D diffraction/radiation program. In the successive time-domain simulation, the Cummins equation is used for solving the pontoon's dynamics, and the bridge girders and columns are modeled by the beam theory and finite element formulation. Then, all the components are fully coupled to solve the fully-coupled equation of motion. Subsequently, the wet natural frequencies for various bending modes are identified. Then, the time histories and spectra of the girder's dynamic responses are presented and systematically analyzed. The second-order difference-frequency wave force and slowly-varying wind force may significantly affect the girder's lateral responses through resonance if the bridge's lateral bending stiffness is not sufficient. On the other hand, the first-order wave-frequency forces play a crucial role in the vertical responses.

Time-domain coupled dynamic simulation for SFT-mooring-train interaction in waves and earthquakes

In this study, Submerged Floating Tunnel (SFT)-mooring-train coupled dynamics is solved in the time domain to investigate their dynamic and hydro-elastic interactions under wave and earthquake excitations. The SFT is modeled by the rod-FE (finite element) theory, and it is connected to mooring lines through dummy-connection-mass and linear and rotational springs. A 3D rigid-multi-body dynamic model is developed for train dynamics that consists of seven rigid bodies. The tunnel-train interaction is taken into consideration based on the wheel-rail correspondence assumption and the simplified Kalker linear creep theory. The developed computer simulation program is validated through comparisons with commercial programs and published results when possible. In the case of earthquake-induced dynamics of the coupled system, the effects of soil conditions, tunnel length, mooring interval, seismic-wave propagation, and seaquake are investigated. The magnitudes of the SFT downward motions induced by the moving train are small compared with the motions induced by earthquakes. The earthquake causes transient SFT responses especially at their lowest wet natural frequencies while high-frequency motions are induced by seaquake effect. Structural damping and seismic propagation play an important role in dynamic responses. The interaction of the tunnel and moving train is also evaluated for various train speeds in terms of the derailment and offload factors and riding-comfort criterion. For the given SFT and train designs, the offload factor and riding-comfort criterion can slightly exceed their limits at certain earthquake conditions with the speed as high as 70 m/s, which can be adjusted by reducing train speed.

Fluid Added-Mass Effect on Flexural Vibration of Hemispherical Shell Structure

In this hydroelasticity study, the fluid added-mass effect on a hemispherical shell structure under flexural vibration is investigated. The vibration response of the hemisphere is solved by using a commercial finite element software (ABAQUS) coupled with an in-house boundary element code that models the fluid as potential flow. The fluid-structure interaction is solved as a fully-coupled system by modal superposition to reduce the number of degrees of freedom. The need for an iterative scheme to pass displacement/force information between the two solvers is avoided by direct coupling between the fluid and structure equations. The numerical results on the downward shift in natural frequencies due to added-mass effect compare well with vibration measurements conducted on a stainless-steel bowl with interior and exterior fluid. For water and soap-water solution used in the experiments, the fluid viscosity (varying over a wide range) did not have any significant effect on the wet natural frequencies. This is due to the small viscous boundary layer (milimetre scale) compared to the nominal size of the bowl in centimetres. For such cases, the fluid-added mass only depends on the density of the fluid and the use of potential flow in the numerical model is applicable.

Fast computation for vibration study of partially submerged structures using low resolution hydrodynamic model

A fast computation scheme is presented for simulating fluid–structure interaction of partially submerged structures. Fluid added mass is simulated using the boundary element method, which only requires a mesh at the fluid–structure interface while satisfying the far field boundary conditions in a semi-infinite domain. The boundary element solver is coupled to a finite element solver for structural vibration analysis. To speed up the boundary element simulation, a low resolution and geometrically simplified surface mesh was used. A projection scheme is introduced to map fluid elements to structural elements, allowing proper transfer of displacement and fluid loading between the two solvers. A series of low-resolution meshes for a realistic container ship model were used to study the efficiency of the proposed scheme, comparing the errors incurred in calculating the wet natural frequencies against the speed-up in computation. With a 20 times decrease in the number of boundary elements, the relative error in wet natural frequency calculation was found to be less than 10%, while computation speed-up was found to be almost 500 folds.

The simulation of deformation and vibration characteristics of a flexible hydrofoil based on static and transient FSI

Flexible hydrofoils can be regarded as the simplified lifting bodies of composite propellers, and these hydrofoils are suitable for use in the study of the fluid-structure interaction (FSI) computational method. The hydrodynamic simulation of a rigid cantilevered rectangular hydrofoil is performed first, then the boundary rotational motion and remeshing approach are used for the numerical calculation of the hydrofoil's pitching hydrodynamic performance, and finally the tunnel boundary effect is perfectly simulated via the steps of structured meshing, rotational parameter transforming, and remeshing. The mesh number, boundary layer, orthogonality angle, and turbulent intensity parameters are obtained through comparisons of various models; thus, the Laminar Separation Bubble (LSB) and turbulent transition are captured. Then the static FSI simulation of the flexible hydrofoil is conducted, showing that the lift of the flexible hydrofoil is low and the drag is high for the two-way FSI compared to the one-way FSI. The center of pressure, the maximum deformation, and the stress move toward the leading edge, showing the necessity of the two-way FSI calculation. The transient FSI simulation of the flexible hydrofoil is then studied using the Large Eddy Simulation (LES) turbulence model to calculate the pressure load, which can simulate the turbulence fluctuation as the exciting source of the structural frequency-domain load. The calculated first peak frequencies are 96 Hz and 78 Hz at 4° and 8°, respectively. The modal vibration shapes of the POM hydrofoil are calculated showing the first mode of bending, the second mode of twisting and the higher mode of bend-twist coupling. The calculated wet natural frequencies of the POM hydrofoil and the steel hydrofoil show good agreements with experimental values. As the modal rank increases, the POM hydrofoil's wet modal frequency decrease varies from 53.8% to 29.2%, whereas the steel hydrofoil's wet modal frequency decrease varies from 36.9% to 14.2%.

Modelling a spade rudder as a hollow two-way tapered Kirchhoff’s plate : free dry and wet vibration study with numerical verification

A semi-analytical approach to free dry and wet vibration of a trapezoidal, 2-way tapered, pivoted hollow spade rudder is presented. The rudder is modeled as a hollow Kirchhoff’s plate, with a NACA0018 profile chord section. The rudder pivot is modeled as a combination of a translational spring and a rotational spring. The span-wise and chord-wise non-uniform beam vibration is first analyzed by the Rayleigh-Ritz method, to establish the non-uniform beam modeshapes, which act as admissible functions to the Galerkin’s method for plate vibration. Eigenvalue analysis generates the plate natural frequencies and the plate modeshapes. Alternately, uniform beam modeshapes themselves are used as admissible functions into the Galerkin’s method. Frequencies are analyzed for various pivot positions, taper ratios, and NACA sections. For the wet vibration, constant strength source distribution technique is used to generate the added mass of a 2D aerofoil. Also, 3D panel method is used to generate the modal added masses, and hence the wet natural frequencies. The added mass coefficient is generated for various aerofoil fineness ratios, pivot fixities, taper ratios, aspect ratios.

Free flexural vibration of a partially wet tapered Timoshenko beam with intermittent mass and stiffness variations, eccentric tip mass and non-classical foundation

A compliant tower is modeled as a partially dry, partially tapered, damped Timoshenko beam with the superstructure modeled as an eccentric tip mass, and a non-classical damped boundary at the base. The foundation is modeled as a combination of a linear spring and a torsional spring, along with parallel linear and torsional dampers (Kelvin-Voigt model). The superstructure adds to the kinetic energy of the system without affecting the potential energy, thereby reducing the natural frequencies. The weight of the superstructure acts as an axial compressive load on the beam, reducing its natural frequencies further. The empty space factor due to the truss-type structure of the tower is included. The effect of shear deformation and rotary inertia are included in the vibration analysis; with the non-uniform beam mode-shapes being a weighted sum of the uniform beam mode-shapes satisfying the end condition. The weights are evaluated by the Rayleigh-Ritz (RR) method, and verified using finite element method (FEM). The weight of the superstructure acts as an axial compressive load on the beam. Kelvin-Voigt model of structural damping is included. A part of the structure being underwater, the virtual added inertia is included to calculate the wet natural frequencies. A parametric study is done for various magnitudes of tip mass and various levels of submergence. The computational efficiency and accuracy of the Rayleigh-Ritz method, as compared to the FEA, has been demonstrated. The advantage of using closed-form trial functions is clearly seen in the efficacy of calculating the various energy components in the RR method.