VIV trajectories of an elastically mounted sphere

In this paper, vortex-induced vibrations (VIV) of an elastically mounted sphere with two linear degrees of freedom have been experimentally studied. The dynamic response analysis shows that, for the range of reduced velocity from 2 to 14 here investigated, the sphere exhibits similar amplitudes and frequencies compared to the tethered sphere, and Modes I and II are detected. A new application of the phase average approach allowed extrapolating the periodic trajectory pattern from the experimental trajectories. In particular, more than one dominant harmonic component is observed in the streamwise oscillation, so the sphere trajectories are significantly different from the traditional figure-of-eight shape. The main difference is that these trajectories are not self-intersecting. Based on the experimental observations, an analytical model of the trajectory is built and discussed.     

Plane state of stress and transverse oscillations of a simply supported rectangular plate

This paper deals with a general discussion of the subject matter. Recent literature is analyzed including some incorrect treatments of particular cases. Finally, a variational solution is obtained for the case where a linear distribution of stresses is applied to the plate boundaries. The lower natural frequencies are then tabulated as a function of the governing geometric and mechanical parameters.     

Forced vibrations of a beam elastically restrained against rotation and carrying a spring–mass system

An exact solution for the title problem is obtained in closed form fashion in the case of a Bernoulli–Euler beam. It is assumed that the exciting force is applied to the mass which is elastically mounted on the beam. The mathematical model constitutes a first order approximation to a motor or engine elastically mounted on a structural element. The operation of the machine generates a transverse, sinusoidally varying force. The problem is of basic interest in mechanical, naval and ocean engineering systems from the point of view of the determination of dynamic displacements and stresses; sound radiation calculations, etc. The present problem arose in connection with the mounting of an engine on a structural beam in a small naval vessel and when excessive vibrational level was noted. This study was undertaken in order to understand the physical problem and to correct the mechanical situation     

Forced transverse vibrations of a simply supported rectangular orthotropic plate in the case where the force acts over a plate subdomain

The present study deals with the exact solution of the title problem in the case where a uniformly distributed p₀ cos ωt-type force acts over a rectangular portion of the plate. The problem is of interest in naval and ocean engineering systems where a motor or an engine mounted on a plate or slab induces a dynamic excitation over a finite area. Mechanical designers commonly consider the problem from the point of view of a concentrated dynamic or static force. On the other hand, modelling the structural system as an orthotropic element is of considerable interest in ocean and naval design in view of the ever-increasing use of composite materials and also of the anisotropic characteristics often induced by metallurgical processes.     

Experimental investigation of wave slamming on an open structure supported elastically

The superstructures of marine structures supported by the elastic legs and located in the splash zone will subject to violent wave slamming and vibrate consequently during storms. A series of model tests are carried out to investigate the wave impacting on the open structures supported elastically. Three kinds of models with different natural frequencies are designed. The characteristics of the wave pressures on the three models are compared. The durations of the uplift forces and the corresponding accelerations of the structure during wave impact are analyzed simultaneously. The distributions of the peak impact pressures on the subfaces of the plates with different supporting stiffness are given. The relationship between the uplift force on the three models and the relative clearance are obtained. The spectral properties of the slamming loads on the three different structures are compared. The experimental results indicate that the behaviors of the impact pressures, the uplift forces and accelerations of the plates with small natural frequencies are obviously different from those of the plates with larger natural frequencies within the range of the experimental parameters.     

弹性双柱在规则波作用下振动响应

通过速度势的特征展开方法,建立垂直圆柱对波浪绕射的解析解,得到作用在柱体上的波浪力计算表达式,通过谐波增量平衡法(IHB法),计算研究弹性双柱相对位置对双柱振动响应的影响。设计了弹性双柱体模型试验,数值结果与模型试验结果较吻合,为海洋工程结构振动设计提供一种解决方法。     

Vibration of two elastically mounted cylinders of different diameters in oscillatory flow

Vibration of two elastically mounted cylinders in an oscillatory flow at a Keulegan-Carpenter number of 10 is simulated numerically. The two cylinders are rigidly connected with each other and are allowed to vibrate in the cross-flow direction only. The aim of this paper is to identify the effects of the orientation of the cylinders and the gap between the cylinders on the vibration. The two-dimensional Reynolds-Averaged Navier-Stokes equations are solved to predict the flow and the cylinder vibration is predicted using the equation of motion. When the two cylinders are in a tandem arrangement, a combined single pair flow regime and attached pair flow regime are observed as reduced velocity exceeds 10 and this combined regime and the single pair regime occurs intermittently. Periodic vibration is found when the two cylinders are in a staggered arrangement with a 45° flow attack angle. When the two cylinders are in a side-by-side arrangement, a new single vortex regime is observed. This single vortex remains attached to the cylinder surface and rotates around the cylinder. The intermittent switch between this single vortex regime and the single pair regime are observed.     

Analytical and experimental investigation on transverse vibrations of solid, circular and annular plates carrying a concentrated mass at an arbitrary position with marine applications

This investigation arose from the practical necessity of placing a centrifugal pump rigidly attached to a thin, circular cover plate of a water tank in a medium size ocean vessel. Due to lack of space, it was necessary to locate the system off—center of the circular configuration. It was considered necessary to calculate the fundamental frequency of the coupled system.The first part of the present study is concerned with the determination of the fundamental frequency of vibration of a circular plate carrying a concentrated mass at an arbitrary position, using a variational approach.Numerical results are obtained for the stated problem for several combinations of the intervening geometric and mechanical parameters.An experimental investigation is also performed in the case of clamped plates.Based on the results for solid circular plates, the fundamental frequency of annular plates with a free inner edge and a concentrated mass is also obtained.Circular plates are fundamental structural elements in ocean engineering applications: from off-shore platforms to underwater acoustic transducers. In a great variety of circumstances, they must carry operational systems in an eccentric fashion. Since the dynamic performance is always of interest, one must know at least some of the basic dynamic parameters.     

Forced vibrations of a beam elastically restrained against rotation at the ends in the case of a concentrated load

The title problem is solved using Bernouilli's classical theory and by means of the separation of variables technique.Numerical data is presented for the simply supported beam, simply supported - clamped and clamped - clamped situations for the case of a concentrated load which varies sinusoidally with time.Certainly no claim of originality is made by the authors since the problem is of a classical nature but it hoped that the rather complete tabulation of displacements, bending moments and shear forces amplitudes included in the present study will be of practical use to ocean and naval engineers.     

横向分布载荷作用下简支舟桥甲板的弯曲特性

采用轻质高强铝合金材料设计舟桥的甲板时,其强度计算是初始设计者最为关心的问题之一.本次研究的舟桥甲板采用三层夹心板的结构形式,其横向分布载荷下的挠度和应力的求解问题与一般的单层板不同,虽然可以用有限元法对此进行结构计算,但在初始设计阶段使用该法的计算量大,不方便应用.本文旨在寻求在横向分布载荷作用下,简支条件时,夹心甲板的挠度和应力的解析解,并与有限元计算相比较,为舟桥甲板初始设计提供良好的基础理论.     

Vibrations of a rectangular plate with a free edge in the case of discontinuously varying thickness

Transverse vibrations of a rectangular plate with three edges elastically restrained against rotation while the fourth is free are studied using (a) an analytical approach based on the optimized Rayleigh-Ritz method and (b) the finite element method. The study was motivated by the necessity of finding out how the lower frequencies of vibration were affected by introducing a step discontinuity in the plate thickness. This variation in the plate thickness was practiced in order to obtain a reduction in the total plate weight.     

Vibrations of a rectangular orthotropic plate with a free edge: a comparison of analytical and numerical results

Transverse vibrations of a rectangular orthotropic plate with a free edge while the others are supported according to four different combinations of boundary conditions are considered in the present paper. The presence of a concentrated mass is also taken into account. In view of the difficulty of satisfying exactly the natural boundary conditions at the free edge, the fundamental frequency of vibration is determined by means of the optimized Rayleigh-Ritz method and a pseudo-Fourier expansion which contains, in its argument, the optimization parameters γ_i. The results are in excellent agreement with those obtained by means of a finite element algorithmic procedure.     

Free and damped vibrations of an orthotropic rectangular plate with one edge free while the rest are elastically restrained against rotation

This paper is concerned with the vibrations of an orthotropic rectangular plate with one edge free while the rest are elastically restrained against rotation. The plate is assumed to be thin and its vibration amplitudes small. The plate is in contact with an ideal fluid of infinite extent and the plate-fluid interaction is considered to be quasi-coupled. The calculations are carried out by the Ritz method and the fundamental frequencies for the free and damped vibrations of the rectangular plate are presented. The fluid effect upon the fundamental frequency is briefly discussed.     

The discrete methods for free vibration analyses of an immersed beam carrying an eccentric tip mass with rotary inertia

In this paper, a beam without contact with water is called the “dry” beam and the one in contact with water is called the “wet” beam. For a partially (or completely) immersed uniform beam carrying an eccentric tip mass possessing rotary inertia, the conventional analytical (closed-form) solution is achieved by considering the inertial forces and moments of the tip mass and rotary inertia as the boundary conditions at the tip end of the beam. However, it has been found that the approximate solution for the last problem may be achieved by two techniques: Method 1 and Method 2. In Method 1, the basic concept is the same as the conventional analytical method; but in Method 2, the tip end of the beam is considered as a free end, while the inertial forces and moments induced by the tip mass and rotary inertia are considered as the external loads applied at the tip end of the beam. The main differences between the formulation of Method 1 and that of Method 2 are: In Method 1, the “normal” shapes of the “dry” beam are functions of the frequency-dependent boundary conditions but the external loads at the tip end are equal to zero; On the contrary, in Method 2, the “normal” mode shapes of the “dry” beam are determined based on the zero boundary conditions at the tip end of the beam but the external loads at the tip end due to the inertial effects of the tip mass and rotary inertia must be taken into consideration for the free vibration analysis of the “wet” beam. Numerical results reveal that the approximate solution obtained from Method 2 are very close to that from Method 1 if the tip mass moment of inertia is negligible. Besides, the two approximate solutions are also very close to the associated analytical (closed-form) solution or the finite element solution. In general, it is hoped that there exist several methods for tackling the same problem so that one may have more choices to incorporate with the specified cases. It is believed that the two approximate methods presented in this paper will be significant from this point of view.     

Forced vibration analysis of an offshore tower carrying an eccentric tip mass with rotary inertia due to support excitation

Dynamic responses of structures due to earthquake excitation are the important problems in engineering, thus, the information concerned is plenty. However, most of the literature is relating to the discrete methods, particularly to the finite element method (FEM), and the one relating to the method combining both the “continuous” and “discrete” models is rare. The objective of this paper is to provide some information in this respect. First, the analytical solution for the natural frequencies and normal mode shapes of a “continuous” tower, without contacting water (or “dry” tower), carrying an eccentric tip mass possessing rotary inertia is determined. Next, the partial differential equation of motion for the forced vibration of the tower, contacting water (or “wet” tower), subjected to support excitation is transformed into a matrix equation by using the last natural frequencies and normal modes shape of the freely vibrating dry tower. Finally, the numerical integration method is used to solve the matrix equation to yield the seismic response of the wet tower. In theory, the mode superposition method is correct only if the total number of modes considered approaches infinity, however, numerical results of this paper reveal that superposition of only the lowest six modes will yield excellent results to be very close to the corresponding ones obtained from the conventional FEM. For this reason, the CPU time required by the presented approach is less than 5% of that required by the conventional FEM.     

The virtual mass of a rectangular flat plate of finite aspect ratio

The skin friction of a two-dimensional planing flat plate is made up of two opposing components; a drag force from the flow aft of the stagnation line and an opposing thrust force from the jet flow. This paper is concerned only with the drag term and the wake velocity defect which it causes in the water behind the transom.It is concluded that the skin friction is less than would be expected from a flat plate at ambient static pressure (D_fo say) and is approximately equal to D_fo (1 − C_R), where C_R is the normal force coefficient based on wetted area. The wake velocity decrement due to this drag is found to be significant, particularly for surface piercing propellers.     

Dynamics of an elastically moored floating body by the three-dimensional infinite element method

The three-dimensional problem of the dynamics of a moored floating object under the action of regular waves is solved numerically as a boundary value problem by use of the finite-infinite element method. The cross-sectional shape of the floating body and the mooring arrangements may all be arbitrary. The mathematical formulations of the problem and procedures of the numerical method are presented in this paper. A corresponding computer program WALOAD has been developed, which is capable of computing wave forces on fixed and floating structures. Numerical computations using this program could give very accurate results, even though rather coarse meshes were used. The program is easy to use and is readily applicable in many practical situations.     

板格形导管架桩基码头稳定性简化计算方法

板格形导管架桩基码头是适用于深水域软土地基的新型港口与海岸工程结构。该结构由预制的板格形导管架及桩基构成,承载特性和破坏模式复杂。假定码头结构发生倾覆失稳时,转动点位于前桩轴线上,且距导管架底面距离为L。除结构的自身重量外,综合考虑板格形导管架与周围土体间的摩擦力、桩侧摩阻力、桩侧水平土抗力等,对结构进行极限状态受力分析,建立基于假想支撑点的稳定性分析模型,并通过强度折减法进行计算,求取安全系数。以天津滨海沿岸水文地质条件为背景,利用ABAQUS有限元分析软件,建立板格形导管架桩基码头三维弹塑性有限元模型进行稳定性分析,并与简化方法进行比较。结果表明,通过简化计算方法得到的安全系数与有限元结果较为吻合。     

Wave motion over a submerged breakwater with an upper horizontal porous plate and a lower horizontal solid plate

This study examines the hydrodynamic performance of a modified two-layer horizontal-plate breakwater. The breakwater consists of an upper submerged horizontal porous plate and a lower submerged horizontal solid plate. By means of the matched eigenfunction expansion method, a linear analytical solution is developed for the interaction of water waves with the structure. Then the reflection coefficient, the transmission coefficient, the energy-loss coefficient and the wave forces acting on the plates are calculated. The numerical results obtained for limiting cases are exactly the same as previous predictions for a single submerged horizontal solid plate and a single submerged horizontal porous plate. Numerical results show that with a suitable geometrical porosity of the upper plate, the uplift wave forces on both plates can be controlled at a low level. Numerical results also show that the transmission coefficient will be always small if the dimensionless plate length (plate length versus incident wavelength) exceeds a certain moderate value. This is rather significant for practical engineering, as the incident wavelength varies over a wide range in practice. Moreover, it is found that the hydrodynamic performance of the present structure may be further enhanced if the lower plate is also perforated.