A theoretical computational model of a plate in hypersonic flow

A theoretical model of an elastic panel in hypersonic flow is derived to be used for design and analysis. The nonlinear von Kármán plate equations are coupled with 1st order Piston Theory and linearized at the nonlinear steady-state deformation due to static pressure differential and thermal loads. Eigenvalue analysis is applied to determine the system’s stability, natural frequencies and mode shapes. Numerically time marching the equations provides transient response prediction which can be used to estimate limit cycle oscillation amplitude, frequency and time to onset. The model’s predictive capability is assessed by comparison to an experiment conducted at a free stream flow of Mach 6. Good agreement is shown between the theoretical and experimental natural frequencies and mode shapes of the fluid–structure system. Stability analysis is performed using linear and nonlinear methods to plot stability, flutter and buckling zones on a free stream static pressure vs temperature differential plane.     

Interaction between a simplified soft palate and compressible viscous flow

Fluid–structure interaction in a simplified 2D model of the upper airways is simulated to study flow-induced oscillation of the soft palate in the pharynx. The goal of our research has been a better understanding of the mechanisms of the Obstructive Sleep Apnea Syndrome and snoring by taking into account compressible viscous flow. The inspiratory airflow is described by the 2D compressible Navier–Stokes equations, and the soft palate is modeled as a flexible plate by the linearized Euler–Bernoulli thin beam theory. Fluid–structure interaction is handled by the arbitrary Lagrangian–Eulerian formulation. The fluid flow is computed by utilizing 4th order accurate summation by parts difference operators and the 4th order accurate classical Runge–Kutta method which lead to very accurate simulation results. The motion of the cantilevered plate is solved numerically by employing the Newmark time integration method. The numerical schemes for the structure are verified by comparing the computed frequencies of plate oscillation with the associated second mode eigenfrequency in vacuum. Vortex dynamics is assessed for the coupled fluid–structure system when both airways are open and when one airway is closed. The effect of mass ratio, rigidity and damping coefficient of the plate on the oscillatory behavior is investigated. An acoustic analysis is carried out to characterize the acoustic wave propagation induced by the plate oscillation. It is observed that the acoustic wave corresponding to the quarter wave mode along the length of the duct is the dominant frequency. However, the frequency of the plate oscillation is recognizable in the acoustic pressure when reducing the amplitude of the quarter wave mode.     

各种边界条件平行四边形功能梯度板的动力特性解析解

本文基于斜坐标系,建立起平行四边形功能梯度板的基本微分方程及变分方程,用梁函数组合法对平行四边形及菱形功能梯度板进行动力特性分析,提出了适用于每边任取简支、固定、自由边界之一(包括36种边界)平行四边形功能梯度板固有频率与振型的解析解;在简化情况下,给出了各种边界条件平行四边形功能梯度板各阶固有频率解的统一表达式。     

Dynamic analysis of beam-cable coupled systems using Chebyshev spectral element method

The dynamic characteristics of a beam–cable coupled system are investigated using an improved Chebyshev spectral element method in order to observe the effects of adding cables on the beam. The system is modeled as a double Timoshenko beam system interconnected by discrete springs. Utilizing Chebyshev series expansion and meshing the system according to the locations of its connections,numerical results of the natural frequencies and mode shapes are obtained using only a few elements, and the results are validated by comparing them with the results of a finiteelement method. Then the effects of the cable parameters and layout of connections on the natural frequencies and mode shapes of a fixed-pinned beam are studied. The results show that the modes of a beam–cable coupled system can be classified into two types, beam mode and cable mode, according to the dominant deformation. To avoid undesirable vibrations of the cable, its parameters should be controlled in a reasonable range, or the layout of the connections should be optimized.     

Coupled free vibration analysis of a fluid-filled rectangular container with a sagged bottom membrane

In the present paper, two-dimensional coupled free vibrations of a fluid-filled rectangular container with a sagged bottom membrane are investigated. This system consists of two rigid walls and a membrane anchored along two rigid vertical walls. It is filled with incompressible and inviscid fluid. The membrane material is assumed to act like an inextensible material with no bending resistance. First, the nonlinear equilibrium equation is solved and the equilibrium shape of the membrane is obtained using an analytical formulation neglecting the membrane weight. The small vibrations about the equilibrium configuration are then investigated. Along the contact surface between the bottom membrane and the fluid, the compatibility requirement is applied for the fluid–structure interactions and the finite element method is used to calculate the natural frequencies and mode shapes of the fluid–membrane system. The vibration analysis of the coupled system is accomplished by using the displacement finite element for the membrane and the pressure fluid-finite element for the fluid domain. The variations of natural frequencies with the pressure head, the membrane length, the membrane weight and the distance between two rigid walls are examined. Moreover, the mode shapes of system are investigated.     

二元复合材料板的自由振动: 半解析法

二元复合材料板是超材料板结构中常见的单元之一. 针对由材料参数相差两个量级的基体和嵌入体组成的二元复合材料板, 提出结构自由振动的半解析模型, 并对其振动特性进行了研究. 基于区域分解法和二元材料的分布, 将二维平板分解成两个子区域. 通过在振型函数中附加区域试函数, 来描述复合材料板面内刚度突变引起局部位移和转角的非光滑性. 基于二元复合材料板的基本边界条件和两子区连接处的变形协调条件, 构造了新的振型函数. 基于经典薄板理论, 利用带特殊试函数的里兹法, 求得不同几何构型下二元复合材料板的固有频率和振型, 并研究了嵌入体的尺寸和位置对结构振动特性的影响规律. 通过收敛分析并与有限元仿真结果对比, 验证了本文方法的准确性. 研究结果表明: 传统的全局试函数在分析具有振动局部化的模态时会得到不准确的结果, 而附加区域试函数可以显著提高里兹法的收敛速度以及结果的准确性; 嵌入体位置对低阶固有频率的作用不明显, 却能显著改变低阶振型节线的分布和振动局部化发生的区域.      

COUPLED VIBRATORY CHARACTERISTICS OF A RECTANGULAR CONTAINER BOTTOM PLATE

The natural frequencies of an elastic thin plate placed into a rectangular hole and connected to the rigid bottom slab of a rectangular container filled with fluid having a free surface are studied. The fluid is assumed to be incompressible, inviscid and irrotational, and the effect of surface waves is neglected. An analytical-Ritz method is developed to study the vibratory characteristics of the plate in contact with the fluid. First of all, the exact expression of the motion of the fluid is obtained, in which the unknown coefficients are determined by using the method of separation of variables and the method of Fourier series expansion. Then, the Ritz approach is used to obtain the frequency equation of the system. The vibrating beam functions are adopted as the admissible functions for the wet-mode expansion of the plate, and the added virtual mass incremental (AVMI) matrices are obtained for plates with arbitrary boundary conditions. Finally, a convergence study is carried out and some numerical results are given. The accuracy of AVMI factor solutions is discussed by comparing with the more accurate analytical-Ritz solutions presented in this paper. Furthermore, It is seen that the present method is also suitable for the vibration analysis of rectangular plates in contact with infinite fluid by taking the finite, but larger size fluid domain as an approximation in the computation.     

Stability of a flexible insert in one wall of an inviscid channel flow

A hybrid of computational and theoretical methods is extended and used to investigate the instabilities of a flexible surface inserted into one wall of an otherwise rigid channel conveying an inviscid flow. The computational aspects of the modelling combine finite-difference and boundary-element methods for structural and fluid elements respectively. The resulting equations are coupled in state-space form to yield an eigenvalue problem for the fluid–structure system. In tandem, the governing equations are solved to yield an analytical solution applicable to inserts of infinite length as an approximation for modes of deformation that are very much shorter than the overall length of the insert. A comprehensive investigation of different types of inserts – elastic plate, damped flexible plate, tensioned membrane and spring-backed flexible plate – is conducted and the effect of the proximity of the upper channel wall on stability characteristics is quantified. Results show that the presence of the upper-channel wall does not significantly modify the solution morphology that characterises the corresponding open-flow configuration, i.e. in the absence of the rigid upper channel wall. However, decreasing the channel height is shown to have a very significant effect on instability-onset flow speeds and flutter frequencies, both of which are reduced. The channel height above which channel-confinement effects are negligible is shown to be of the order of the wavelength of the critical mode at instability onset. For spring-backed flexible plates the wavelength of the critical mode is much shorter than the insert length and we show very good agreement between the predictions of the analytical and the state-space solutions developed in this paper. The small discrepancies that do exist are shown to be caused by an amplitude modulation of the critical mode on an insert of finite length that is unaccounted for in the travelling-wave assumption of the analytical model. Overall, the key contribution of this paper is the quantification of the stability bounds of a fundamental fluid–structure interaction (FSI) system which has hitherto remained largely unexplored.     

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.     

Dynamic modelling and active vibration control of a submerged rectangular plate equipped with piezoelectric sensors and actuators

The active vibration control of a rectangular plate either partially or fully submerged in a fluid was investigated. Piezoelectric sensors and actuators were bonded to the plate, and the assumed mode method was used to derive a dynamic model for the submerged plate. The properties of the piezoelectric actuators and sensors, as well as their coupling to the structure, were used to derive the corresponding equations of their behaviour. The fluid effect was modelled according to the added virtual mass obtained by solving the Laplace equation. The natural vibration characteristics of the plate both in air and in water were obtained theoretically and were found to be consistent with the experimental results, and the changes in the natural frequencies resulting from submersion in fluid can be accurately predicted. A multi-input, multi-output positive position feedback controller was designed by taking the natural vibration characteristics into account and was then implemented by using a digital controller. The experimental results show that piezoelectric sensors and actuators along with the control algorithm can effectively suppress the vibration of a rectangular plate both in air and submerged in a fluid.     

A fluid–structure interaction model for stability analysis of shells conveying fluid

In this paper, a fluid–structure interaction model for stability analysis of shells conveying fluid is developed. This model is developed for shells of arbitrary geometry and structure and is based on incompressible potential flow. The boundary element method is applied to model the potential flow. The fluid dynamics model is derived by using an inflow/outflow model along with the impermeability condition at the fluid–shell interface. This model is applied to obtain the flow modes and eigenvalues, which are used for the modal representation of the flow field in the shell. Based on the mode shapes and natural frequencies of the shell obtained from an FEM model, the modal analysis technique is used for structural modeling of the shell. Using the linearized Bernoulli equation for unsteady pressure on the fluid–shell interface in combination with the virtual work principle, the generalized structural forces are obtained in terms of the modal coordinates of the fluid flow and the coupled field equations of the fluid–structure are derived. The obtained model is validated by comparison with results in the literature, and very good agreement is demonstrated. Then, some examples are provided to demonstrate the application of the present model to determining the stability conditions of shells with arbitrary geometries.     

Free vibration of rectangular plates with continuously distributed spring-mass

The vibratory characteristics of rectangular plates attached with continuously and uniformly distributed spring-mass in a rectangular region are studied which may represent free vibration of a human–structure system. Firstly, the governing differential equations of a plate with uniformly distributed spring-mass are developed. When the spring-mass fully occupies the plate, the natural frequencies of the coupled system are exactly solved and a relationship between the continuous system and a series of discrete two degrees-of-freedom system is provided. The degree of frequency coupling is defined. Then, the Ritz–Galerkin method is used to derive the approximate solution, when the spring-mass is distributed on a part of the plate, by using the Chebyshev polynomial series to construct the admissible functions. Comparative studies demonstrate the high accuracy and wide applicability of the proposed method. Finally, the frequency and modal characteristics of the plate partially occupied by distributed spring-mass are numerically analysed. It has been observed that both the natural frequencies and the modes appear in pairs. Moreover, a parametric study is performed for rectangular plates with three edges simply supported and one edge free. The effects of occupation size and position of the distributed spring-mass on natural frequencies of the coupled system are studied in detailed. The present investigation provides an improved understanding of human–structure interaction, such as grandstands or floors occupied by a stationary crowd.     

复合柔性结构全局模态函数提取与状态空间模型构建

随着航空航天等领域中实际工程结构的大型化和柔性化,结构的非线性振动和主动振动控制问题越来越凸显.分析和处理此类结构出现的复杂振动问题的关键在于建立系统的非线性动力学模型与状态空间模型.对于由柔性部件、刚体、连接部件构成的复合柔性结构,由于各部件之间的振动耦合效应,单个柔性部件在悬臂、简支和自由等静定边界下的模态与结构的真实模态有较大差异.为此,本文提出复合柔性结构全局模态的解析提取方法,通过全局模态离散得到系统非线性动力学模型,从而构建状态空间模型.该方法采用笛卡尔坐标描述系统的运动,建立系统的运动方程;结合描述柔性部件的偏微分方程、刚体的常微分运动方程、连接界面处力、力矩、位移和转角的匹配条件以及系统的边界条件,利用分离变量法给出统一形式的频率方程,获取系统的固有频率和解析函数表征的全局模态.这里提出的全局模态提取方法不仅便于复合柔性结构固有频率和全局模态的参数化分析,而且为建立复合柔性结构低维非线性动力学模型和状态空间模型提供了有效的途径,对于推进这类结构的非线性动力学分析与主动振动控制研究具有重要意义.      

HYDROELASTIC VIBRATION OF A CIRCULAR CONTAINER BOTTOM PLATE USING THE GALERKIN METHOD

The free vibration of a flexible thin plate placed into a circular hole and elastically connected to the rigid bottom slab of a circular cylindrical container filled with fluid having a free surface is studied. The liquid is assumed to be incompressible, inviscid and irrotational. The effect of the free surface wave is also taken into account in the analysis. First of all, the exact expression of velocity potential of the liquid movement is derived by a combination of the superposition method and the method of separation of variables. With the help of the Fourier–Bessel series expansion, part of the unknown coefficients in the solution is determined by the consistency condition between the liquid movement and the plate vibration, in the form of integrals associated with the dynamic deflection of the plate. Then, the Galerkin method is applied to derive the eigenfrequency equation of the fluid–plate interaction. Finally, the effects of various parameters and the free surface wave on eigenfrequencies of the fluid–plate system are discussed. As a consequence, the accuracy of the nondimensional added virtual mass incremental (NAVMI) factor solution has also been evaluated by comparing with the more accurate Galerkin solution. It is shown that the proposed method is also applicable to the vibration analysis of circular plates in contact with an infinite liquid by only taking a finite but larger size of liquid to replace the infinite liquid in the computation.     

Numerical analysis on flutter of Busemann-type supersonic biplane airfoil

The Busemann-type supersonic biplane can effectively reduce the wave drag through shock interference effect between airfoils. However, considering the elastic property of the wing structure, the vibration of the wings can cause the shock oscillation between the biplane, which may result in relative aeroelastic problems of the wing. In this research, fluid–structure interaction characteristics of the Busemann-type supersonic biplane at its design condition have been studied. A theoretical two-dimensional structure model has been established to consider the main elastic characteristics of the wing structure. Coupled with unsteady Navier–Stokes equations, the fluid–structure dynamic system of the supersonic biplane is studied through the two-way computational fluid dynamics/computational structural dynamics (CFD/CSD) coupling method. The biplane system has been simulated at its design Mach number with different nondimensional velocities. Different initial disturbance has been applied to excite the system and the effects of the position of the mass center on the system’s aeroelastic stability is also discussed. The results reveal that the stability of the airfoil in supersonic biplane system is decreased compared with that of the airfoil isolated in supersonic flow and such stability reduction effect should be given due attention in practical design.     

Flutter of spring-mounted flexible plates in uniform flow

A fluid–structure interaction (FSI) system is studied wherein a cantilevered flexible plate aligned with a uniform flow has its upstream end attached to a spring mounting. This allows the entire system to oscillate in a direction perpendicular to that of the flow as a result of the mounting׳s dynamic interaction with the flow-induced oscillations, or flutter, of the flexible plate. We also study a hinged-free rotational-spring attachment as a comparison for the heaving system. This variation on classical plate flutter is motivated by its potential as an energy-harvesting system in which the reciprocating motion of the support system would be tapped for energy production. We formulate and deploy a hybrid of theoretical and computational modelling for the two systems and comprehensively map out their linear-stability characteristics at low mass ratio. Relative to a fixed cantilever, the introduction of the dynamic support in both systems yields lower flutter-onset flow speeds; this is desirable for energy-harvesting applications. We further study the effect of adding an inlet surface upstream of the mount as a means of changing the destabilising mechanism from single-mode flutter to modal-coalescence flutter which is a more powerful instability more suited to energy harvesting. This strategy is seen to be effective in the heaving system. However, divergence occurs in the rotational system for low spring natural frequencies and this would lead to its failure for energy production. Finally, we determine the power-output characteristics for both systems by introducing dashpot damping at the mount. The introduction of damping increases the critical speeds and its variation permits optimal values to be found that maximise the power output for each system. The addition of an inlet surface is then shown to increase significantly the power output of the heaving system whereas this design strategy is not equally beneficial for the rotational system.     

Exact solution for free vibration analysis of an eccentric elliptical plate

The method of separation of variables in elliptical coordinates in conjunction with the translational addition theorems for Mathieu functions is used to investigate the free flexural vibrations of a fully clamped thin elastic panel of elliptical planform containing an elliptical cutout of arbitrary size, location, and orientation. The first five natural frequencies are calculated for various plate/cutout aspect ratios and selected cutout location/orientation parameters. Also, a number of representative vibration mode shapes are depicted in graphical form. The accuracy of solutions is demonstrated through proper convergence studies, and the validity of results is established with the aid of a commercial finite element package as well as by comparison with those in the existing literature.     

ANALYTICAL SOLUTION OF THE FREE VIBRATION OF VISCOELASTIC SANDWICH ANNULAR PLATE

基于薄板小挠度理论和Kelvin-Voigt 黏弹性本构方程,建立了黏弹性夹层环形薄板振动控制方程.采用分离变量法计算了内边固支、外边自由黏弹性夹层环形薄板的固有频率和振型,并与有限元计算结果进行比较. 分别讨论了夹心层比和内外半径比对固有频率及衰减系数的影响. 研究表明:系统频率随夹心层厚度增大,先增大后减小,而衰减系数一直增大;系统频率和衰减系数随内外半径比增大而增大.     

密闭腔体声-结构耦合系统的动力灵敏度分析

以密闭空腔为对象,开展了声-结构耦合系统的动力分析和灵敏度计算,为系统性态优化设计提供理论和算法基础。分别把结构和声场进行离散化,推导了声-结构耦合系统的有限元方程,求解了耦合系统的频率和声压级响应。在此基础上,以结构尺寸为设计变量,计算了耦合系统的固有频率和声压级响应的灵敏度,解决了声-结构耦合系统动力灵敏度的数值算法问题。