Coupled vibration of a partially fluid-filled cylindrical container with a cylindrical internal body

In the present paper a method is proposed to investigate the effects of a rigid internal body on the coupled vibration of a partially fluid-filled cylindrical container. The internal body is a thin-walled and open-ended cylindrical shell. The internal body is concentrically and partially submerged inside a container. The radial and axial distances between the internal body and the container are filled with fluid. Along the contact surface between the container and the fluid, the compatibility requirement for the fluid–structure interactions is applied and the Rayleigh–Ritz method is used to calculate the natural frequencies and modes of a partially fluid-filled cylindrical container. The fluid domain is continuous, simply connected, and non-convex. The fluid is assumed to be incompressible and inviscid. The velocity potential for fluid motion is formulated in terms of eigenfunction expansions for two distinct fluid regions. The resulting equations are solved by using the Galerkin method. The results from the proposed method are in good agreement with experimental and numerical solutions available in the literature for the partially water-filled cylindrical container without internal body. A finite element analysis is also used to check the validity of the present method for the partially water-filled cylindrical container with internal body. The effects of the fluid level, internal body radius, and internal body length on the natural frequencies of the coupled system are also investigated.     

Free vibration of an elastic bottom plate of a partially fluid-filled cylindrical container with an internal body

An analytical method is developed to consider the free vibration of an elastic bottom plate of a partially fluid-filled cylindrical rigid container with an internal body. The internal body is a rigid cylindrical block that is concentrically and partially submerged inside the container. The developed method captured the analytical features of the velocity potential in a non-convex, continuous, and simply connected fluid domain including the interaction between the fluid and the structure. The interaction between the fluid and the bottom plate is included. The Galerkin method is used for matching the velocity potentials appropriate to two distinct fluid regions across the common horizontal boundary (artificial horizontal boundary). Then, the Rayleigh–Ritz method is also used to calculate the natural frequencies and modes of the bottom plate of the container. The results obtained for the problem without internal body are in close agreement with both experimental and numerical results available in the articles. A finite element analysis is also used to check the validity of the present method in the presence of the internal body. Furthermore, the influences of various variables such as fluid level, internal body radius, internal body length, and the number of nodal diameters and circles on the dynamic behaviour of the coupled system are investigated.     

Coupled vibrations of a partially fluid-filled cylindrical container with an internal body including the effect of free surface waves

Internal bodies (baffles) are used as damping devices to suppress the fluid sloshing motion in fluid-structure interaction systems. An analytical method is developed in the present article to investigate the effects of a rigid internal body on bulging and sloshing frequencies and modes of a cylindrical container partially filled with a fluid. The internal body is a thin-walled and open-ended cylindrical shell that is coaxially and partially submerged inside the container. The interaction between the fluid and the structure is taken into account to calculate the sloshing and bulging frequencies and modes of the coupled system using the Rayleigh quotient, Ritz expansion and Galerkin method. It is shown that the present formulation is an appropriate and new approach to tackle the problem with good accuracy. The effects of fluid level, number of nodal diameters, internal body radius and submergence ratio on the dynamic characteristics of the coupled system are also investigated.     

Free vibration of a sagged cable with attached discrete elements

An algorithm is presented to analyze the free vibration in a system composed of a cable with discrete elements, e.g., a concentrated mass, a translational spring, and a harmonic oscillator. The vibrations in the cable are modeled and analyzed with the Lagrange multiplier formalism. Some fragments of the investigated structure are modeled with continuously distributed parameters, while the other fragments of the structure are modeled with discrete elements. In this case, the linear model of a cable with a small sag serves as a continuous model, while the elements, e.g., a translational spring, a concentrated mass, and a harmonic oscillator, serve as the discrete elements. The method is based on the analytical solutions in relation to the constituent elements, which, when once derived, can be used to formulate the equations describing various complex systems compatible with an actual structure. The numerical analysis shows that, the method proposed in this paper can be successfully used to select the optimal parameters of a system composed of a cable with discrete elements, e.g., to detune the frequency resonance of some structures.     

带裂纹矩形板自由振动解析解

选取带有补充项的双重正弦傅里叶级数作为振型函数通解,来解析研究带裂纹矩形板的自由振动特性。先将带裂纹矩形板分割成若干小矩形板,利用各小矩形板的边界条件,并结合振型函数中待定常数的物理意义,简化得到各小矩形板的振型函数,再结合各板的控制方程、未使用的边界条件、公共边协调条件及本文提出公共自由角点的协调条件,建立求解频率的代数方程组,然后将其转化为广义特征值问题来求解带裂纹矩形板的无量纲频率;最后选取具体参数进行计算并与文献结果对比,吻合良好,证明了本文采用的研究方法以及所提出公共角点协调条件的正确性和合理性。由于该振型函数能满足矩形板的任意边界约束,且其中的待定常数具有明确的物理意义,所以可使矩形板问题的求解统一化、简单化和规律化。     

Nonlinear sloshing of liquid in rigid cylindrical container with a rigid annular baffle: free vibration

In this paper, the free nonlinear sloshing of liquid in a rigid partially liquid-filled cylindrical container with a rigid annular baffle is investigated. The liquid domain is firstly divided into four simple sub-domains so that the liquid velocity potential in each sub-domain has continuous boundary conditions of class \(\hbox {C}^{1}\) . Then, based on the Bateman–Luke variational principle, the equivalent variational formulation of the free boundary problem for the liquid in a baffled container is derived. By introducing the generalized time-dependent coordinates, the surface wave height and the velocity potential are expanded in the form of generalized Fourier series. Substituting the series expression of surface wave height and velocity potential into the variational formulation enables us to obtain the infinite dimensional modal system. Finally, based on the Moiseev asymptotic relations, the infinite dimensional modal system is reduced to a finite dimensional modal system. Good agreements have been achieved by comparing the present results with those obtained from the Gavrilyuk’s solutions. The effects of baffle parameters and initial conditions on the nonlinear sloshing of liquid are discussed in detail.     

四边固支矩形薄板自由振动的哈密顿解析解

在哈密顿体系中利用辛几何方法求解了四边固支矩形薄板自由振动问题的解析解。首先,从基本方程出发,将问题表示成Hamilton正则方程,然后利用辛几何方法导出本征值问题,从而得到本征函数解,使之满足边界条件;再由方程组有非零解的条件,最终推导出四边固支矩形薄板的自振频率方程,得到频率的解析解。计算了不同长宽比情况下四边固支矩形薄板的频率,结果与已有文献完全一致。该解法有望推广至更多尚未得到解析解的矩形板的振动问题。     

Effect of partial elastic foundation on free vibration of fluid-filled functionally graded cylindrical shells

The free vibration characteristics of fluid-filled functionally graded cylindrical shells buried partially in elastic foundations are investigated by an analytical method.The elastic foundation of partial axial and angular dimensions is represented by the Pasternak model. The motion of the shells is represented by the first-order shear deformation theory to account for rotary inertia and transverse shear strains. The functionally graded cylindrical shells are composed of stainless steel and silicon nitride. Material properties vary continuously through the thickness according to a power law distribution in terms of the volume fraction of the constituents. The governing equation is obtained using the Rayleigh–Ritz method and a variation approach. The fluid is described by the classical potential flow theory. Numerical examples are presented and compared with existing available results to validate the present method.     

A semi-analytical method for free vibration of straight orthotropic beams with rectangular cross-sections

On the basis of the two-dimensional elasticity equations with orthotropy, a semi-analytical method is proposed to analyze free vibration of straight beams with rectangular cross-sections. To this end, the state space method is combined with the differential quadrature method so that state equations with respect to state variables at discrete points are derived. The frequency equation for free vibration of straight orthotropic beams is then formulated. Numerical results are presented and compared with that available in the literature. The present method can be used to analyze either shallow or deep orthotropic beams with arbitrary end conditions.     

充液矩形贮箱弹性箱底板应力与变形分析

基于相容拉格朗日-欧拉法,通过对流场与弹性固体间流固耦合作用的分析,得到了矩形贮箱弹性底板流固耦合系统的自由振动微分方程。将伯努利方程与外加激励条件、速度势函数耦合到自由振动方程中,采用迦辽金积分法,给出了矩形贮箱在流体作用下的应力与变形的解析解。讨论了弹性底板的抗弯刚度、结构尺寸、底板材料参数及流体深度等因素对底板应力与变形的影响。研究结果表明:在液体晃动非线性激励作用下,贮箱底板的应力和变形随着液体深度、板长的增大而增大,随着板厚的减小而增大,且成非线性变化关系;底板的变形及应力与底板的材料常数相关,其中板厚的变化对其变形和应力影响要比板长及液体深度的影响显著得多。本文结果可为工程实际中矩形贮箱的设计提供参考。     

Multi-symplectic methods for membrane free vibration equation

In this paper,the multi-symplectic formulations of the membrane free vi- bration equation with periodic boundary conditions in Hamilton space are considered. The complex method is introduced and a semi-implicit twenty-seven-points scheme with certain discrete conservation laws—a multi-symplectic conservation law(CLS),a local energy conservation law(ECL)as well as a local momentum conservation law(MCL)—is constructed to discrete the PDEs that are derived from the membrane free vibra- tion equation.The results of the numerical experiments show that the multi-symplectic scheme has excellent long-time numerical behavior.     

Coupled free,torsional and axial vibration of pre-twisted bars

Summary The present paper deals with the calculation of natural frequencies in a pre-twisted solid with constant cross-section.The axial and torsional displacements of cross-sections are taken into account, as well as the relative inertia forces and couples.The differential equations of the vibration are integrated and their general solution is given.In the paper, by such solution, among the possible cases only the three simplest ones are considered (free-free, fixed-fixed, fixed-free ends) and a numerical example is carried out which enables to evaluate the several orders of magnitude. Other more complex cases will be treated in other papers.

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Sommario Nel presente lavoro viene affrontato il problema della determinazione delle frequenze proprie di vibrazione di un solido svergolato a sezione uniforme.Il problema viene risolto tenendo conto degli spostamenti assiali e torsionali di ogni sezione e delle relative forze e coppie di inerzia.Le equazioni differenziali del moto vengono integrate e ne viene fornita la soluzione generale.Mediante tale soluzione vengono esaminati, fra tutti i diversi casi possibili, i tre casi pratici più semplici (estremità del solido entrambe libere, entrambe incastrate, una libera e l'altra incastrata) e viene inoltre svolto un esempio numerico che consente di ottenere i diversi ordini di grandezza.I casi di maggiore complessità verranno esaminati in successivi lavori.

     

An alternative procedure for the non-linear vibration analysis of fluid-filled cylindrical shells

Using Donnell non-linear shallow shell equations in terms of the displacements and the potential flow theory, this work presents a qualitatively accurate low dimensional model to study the non-linear dynamic behavior and stability of a fluid-filled cylindrical shell under lateral pressure and axial loading. First, the reduced order model is derived taking into account the influence of the driven and companion modes. For this, a modal solution is obtained by a perturbation technique which satisfies exactly the in-plane equilibrium equations and all boundary, continuity, and symmetry conditions. Finally, the equation of motion in the transversal direction is discretized by the Galerkin method. The importance of each mode in the proposed modal expansion is studied using the proper orthogonal decomposition. The quality of the proposed model is corroborated by studying the convergence of frequency–amplitude relations, resonance curves, bifurcation diagrams, and time responses. The parametric analysis clarifies the influence of the lateral and axial loads on the non-linear vibrations and stability of the liquid-filled shell. Finally, the global response of the system is investigated in order to quantify the degree of safety of the shell in the presence of external perturbations through the use of bifurcation diagrams and basins of attraction. This allows one to evaluate the safety and dynamic integrity of the cylindrical shell in a dynamic environment.     

Analyses of dynamic characteristics of a fluid-filled thin rectangular porous plate with various boundary conditions

Based on the classical theory of thin plate and Biot theory, a precise model of the transverse vibrations of a thin rectangular porous plate is proposed. The first order differential equations of the porous plate are derived in the frequency domain. By considering the coupling effect between the solid phase and the fluid phase and without any hypothesis for the fluid displacement, the model presented here is rigorous and close to the real materials. Owing to the use of extended homogeneous capacity precision integration method and precise element method, the model can be applied in higher frequency range than pure numerical methods. This model also easily adapts to various boundary conditions. Numerical results are given for two different porous plates under different excitations and boundary conditions.     

Coupled frequencies of a rectangular hydroelastic system with two fluids

This paper deals with the study of behaviour of an idealized 2D hydroelastic system involving two inviscid liquids with an elastic rectangular container. The main objective is to investigate the influence of the physical parameters on eigenfrequencies and eigenmodes of the system. The study extends the previous results obtained for hydroelastic systems with one fluid. The governing equations describing the behaviour of the system are analyzed by using the concept of normal modes and their solutions presented in the form of infinite series. The expansion coefficients for the velocity potentials are calculated by employing a new inner product that allows the orthogonalisation of the normal modes. An eigenfrequency equation is then derived from the existence condition of a nontrivial solution. The numerical calculations are performed by varying only some relevant parameters.     

Accurate variational approach for free vibration of simply supported anisotropic rectangular plates

The Ritz method is one of the most elegant and useful approximate methods for analyzing free vibration of laminated composite plates. It is simple to use and also straightforward to implement. However, the Ritz method has its own difficulty in determining the natural frequencies of simply supported laminated anisotropic plates. This is caused by the fact that the natural boundary conditions in the vibration of anisotropic plates can never be exactly satisfied by a solution in a variables separable form. As a result, the calculated natural frequencies would be expected to converge to solutions a little higher than true ones. To overcome this difficulty, this paper presents a simple variational formulation with Ritz procedure in which all the natural boundary conditions are implemented in an averaging manner. It is revealed that the proposed method can produce lower upper bound solutions compared with the conventional Ritz method where the geometric boundary conditions can only be satisfied by the assumed deflection functions.     

Transversal oscillations of a fluid in a long cylindrical container with membrane or elastic plate on the free surface

We propose approximate solutions of two-dimensional hydroelastic problems that describe free oscillations of an ideal fluid in a horizontal long cylindrical container with arbitrary symmetric cross section. The free surface of the fluid is covered by a plane membrane or an elastic plate. Using specific examples, we analyze the obtained solutions and the results of computation of frequencies and forms of oscillations of the mechanical system under consideration.