On the fluid–structure interaction of a splitter plate: vibration modes and Reynolds number effects

Previous work (Eloranta et al. in Exp Fluids 39:841–855, 2005) has shown that flow separation from the trailing edge of a splitter plate in a convergent channel involves a fluid–structure interaction (FSI), which modifies the fundamental instability related to vortex shedding. Under certain conditions, the FSI induces cellular vortex shedding from the trailing edge. This paper reports detailed measurements of the plate vibration mode and studies the effect of the Reynolds number on the FSI. Experimental techniques including laser vibrometer and digital imaging are used to measure the response of the plate and particle image velocimetry is used to measure the flow field in the near wake. Combining data from these techniques, the development of the vibration frequency and mode can be addressed together with the imprint of the vibration mode in the flow. The results show that over most of the Reynolds numbers measured, the plate vibrates in a distinct mode characterized by a spanwise standing wave along the plate trailing edge. The vibration frequency and the spacing between the nodes of the standing wave depend on the Reynolds number. As the Reynolds number is increased, the frequency of the dominant vibration mode does not increase linearly. The plot of the vibration frequency as a function of the Reynolds number shows that the vibration tends to lock to a rather constant frequency over of range of Reynolds numbers. After certain Reynolds number if threshold is exceeded, the frequency jumps to a new level, which also involves a new vibration mode.     

Thermoelastic free vibration of clampe circular plate

An analysis was given for the free vibration of clamped circular plate when temperature and stress fields were coupled. A nonlinear differential equation about time was obtained by using Galerkin’s method. The numerical results of vibration amplitude vs time were compared with the uncoupled case. It is found that if the given initial displacement is small, the effect of thermoelastical coupling will make the natural frequency increase; if the given initial displacement is large, it will be the opposite case. Effects of some different vibration factors are also discussed.     

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。     

Natural vibration of a rectangular plate of composite material with periodically bent structures

The natural vibration of a thin rectangular plate made of a composite material with periodically bent structures is studied. The problem is examined by the finite-element method using the variation principle. The effect of the bending parameters on the natural frequency of the plate is analyzed using the numerical results obtained. Institute of Mathematics and Mechanics, Academy of Sciences of Azerbaijan, Baku, Azerbaijan. Translated from Prikladnaya Mekhanika, Vol. 35, No. 10, pp. 68–73, October, 1999.     

波纹板结构动力学建模及其自由振动特性分析

波纹板结构是高速列车中较为常见的型材结构,因其具有轻质、高强度和高稳定性等一系列优点,所以广泛应用于车体结构轻量化设计。首先,基于一阶剪切变形理论和中厚壳(圆弧壳)理论,采用微分求积有限元法(DQFEM)建立一般边界条件下五自由度波纹板结构的通用动力学分析模型;紧接着对五自由度波纹板结构动力学模型进行收敛性验证,确定最佳微分节点数和惩罚因子的取值;然后,采用有限软件(ABAQUS)验证了建立模型的准确性;最后根据已建立模型,研究波纹板结构参数对其振动特性的影响。结果表明,波纹板结构厚度和固支边界条件会增加其固有频率,提高波纹板结构的稳定性,使得共振频率向高频移动;增加波纹板结构长度、参与耦合圆弧壳半径和参与耦合圆弧壳弧度会降低其固有频率,使得共振频率向低频移动。     

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。     

The effect of a splitter plate on the flow around a finite prism

The effect of a wake-mounted splitter plate on the flow around a surface-mounted finite-height square prism was investigated experimentally in a low-speed wind tunnel. Measurements of the mean drag force and vortex shedding frequency were made at Re=7.4×10⁴ for square prisms of aspect ratios AR=9, 7, 5 and 3. Measurements of the mean wake velocity field were made with a seven-hole pressure probe at Re=3.7×10⁴ for square prisms of AR=9 and 5. The relative thickness of the boundary layer on the ground plane was δ/D=1.5–1.6 (where D is the side length of the prism). The splitter plates were mounted vertically from the ground plane on the wake centreline, with a negligible gap between the leading edge of the plate and rear of the prism. The splitter plate heights were always the same as the heights of prisms, while the splitter plate lengths ranged from L/D=1 to 7. Compared to previously published results for an “infinite” square prism, a splitter plate is less effective at drag reduction, but more effective at vortex shedding suppression, when used with a finite-height square prism. Significant reduction in drag was realized only for short prisms (of AR≤5) when long splitter plates (of L/D≥5) were used. In contrast, a splitter plate of length L/D=3 was sufficient to suppress vortex shedding for all aspect ratios tested. Compared to previous results for finite-height circular cylinders, finite-height square prisms typically need longer splitter plates for vortex shedding suppression. The effect of the splitter plate on the mean wake was to narrow the wake width close to the ground plane, stretch and weaken the streamwise vortex structures, and increase the lateral entrainment of ambient fluid towards the wake centreline. The splitter plate has little effect on the mean downwash. Long splitter plates resulted in the formation of additional streamwise vortex structures in the upper part of the wake.     

Magnetoelastic vibration of a plate

Summary The propagation of plane harmonic waves in a thin flat homogeneous isotropic plate of finite width and infinite length permeated by a constant magnetic field is examined. The frequency equations corresponding to the symmetric and antisymmetric modes of vibration of the plate are obtained, and some limiting cases of the frequency equations are then discussed.     

Analytical solutions of steady vibration of free rectangular plate on semi-infinite elastic foundation

The method of double Fourier transform was employed in the analysis of the semi-infinite elastic foundation with vertical load.And an integral representations for the displacements of the semi-infinite elastic foundation was presented.The analytical solution of steady vibration of an elastic rectangle plate with four free edges on the semi-infinite elastic foundation was also given by combining the analytical solution of the elastic rectangle plate with the integral representation for displacements of the semi- infinite elastic foundation.Some computational results and the analysis on the influence of parameters were presented.     

Dissipation effects on MHD free convection flow past a semi-infinite vertical plate

Viscous and Joule dissipation effects are considered on MHD free convection flow past a semi-infinite isothermal vertical plate under a uniform transverse magnetic field. Series solutions in powers of a dissipation number (=gx/c _p) have been employed and the resulting ordinary differential equations have been solved numerically. The velocity and temperature profiles are shown on graphs and the numerical values of ₁(0)/₀(0) (, temperature function) have been tabulated. It is observed that the dissipation effects in the MHD case become more dominant with increasing values of the magnetic field parameter (=M ²/(Gr _x /4)^1/2) and the Prandtl number.     

Experimental investigation of free convection above a heated horizontal wire

Results are presented of an experimental investigation of the convective plume above a fine horizontal wire, heated by a constant current in air and in water. The temperature distribution in the plume was investigated using the IAB-451 shadow instrument in the diffraction interferometry method. The experimental results are in good agreement with laminar convection theories above a linear heat source. In the air, a comparison was made with the experimental results of other authors.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhicheskoi Fiziki, Vol. 11, No. 2, pp. 169–173, March–April, 1970.The author wishes to express his indebtedness to V. D. Zimin and N. V. Eyzhanov for their assistance with numerical solutions of Eqs. (1.3).     

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.     

Free vibration analysis of a plate on foundation with completely free boundary by finite integral transform method

The theoretical solutions of eigenfrequencies and vibration modes of a rectangular thin plate on an elastic foundation with completely free boundary are derived by using a double finite cosine integral transform method. In the analysis procedure, the elastic foundation is regarded as a Winkler elastic foundation model. Because the basic dynamic elasticity equations of the thin plate on elastic foundation are only used, it is not needed to select the deformation function arbitrarily. Therefore, the solution developed in the present paper is more reasonable and more accurate. To prove the correctness of the solutions, numerical results obtained using the present solutions are compared with those in the literatures.     

Fluid–structure interaction of a splitter plate in a convergent channel

The fluid–structure interaction (FSI) of a splitter plate in a convergent channel flow is studied by measuring both the flow field and the plate vibration. Particle Image Velocimetry (PIV) measurements show that the wake generated by the plate is characterized by cellular vortex shedding. Mean and RMS velocities presented in the plane normal to the main flow direction visualize clearly the cellular structure and related secondary flows. To evaluate the energy and spatial organization of the vortex shedding, spectral and correlation estimation methods are adapted to the PIV data. By presenting the spanwise variation of the streamwise spectra along the trailing edge, the nature of the cellular vortex shedding becomes evident. 2D space-correlation function reveals that the shedding in two neighboring cells occurs in a 180-degree phase shift. The vibration of the plate is studied with Digital Imaging (DI) and Laser Vibrometer (LV). The DI is based on images measured by the PIV system. An image-processing algorithm is used to detect the plate tip location and velocity simultaneously with the estimation of the fluid velocity field. The LV is used for the time-resolved measurement of the plate vibration. The results show that the plate vibrates in a very distinct mode characterized by a spanwise standing wave along the plate-trailing edge. This mode, in turn, causes the cellular vortex shedding.     

Experimental and numerical investigation of a vertical vibration isolator for seismic applications

In near-fault seismic zones, the vertical acceleration experienced during a strong event can be greater than horizontal acceleration. Methods to reduce horizontal acceleration are applied in various forms and are in common use. However, methods to reduce vertical acceleration, and practical protection systems for these applications, remain elusive. One strategy to protect structures, which has been demonstrated to be effective in situations where the excitation is horizontal, is to isolate the structure. For vertical excitations, this is difficult due to the need to maintain sufficient stiffness and strength in the direction of gravitational loads. The need to maintain high stiffness for gravity loading while allowing flexibility for isolation during earthquakes has led to research on the use of High-Static-Low-Dynamic Stiffness Systems (HSLDSS) and in particular Quasi-Zero Stiffness Systems (QZSS), which have zero equivalent stiffness in the equilibrium position. Although effective, the QZSS is sensitive to mistuning and prone to large deformations for relatively small increments in static load for building applications. This paper presents the results of an analytical and experimental study in which a HSLDSS isolation system carrying a payload is subject to vertical base excitation using sinusoidal as well as actual, scaled earthquake signals. Static loading tests are also presented. This isolation system consists of rigid rotating arms, horizontal and vertical springs and a vertical damper. By a suitable selection of parameters this could also serve as a QZSS. Results show that both the QZSS and HSLDSS can significantly reduce the magnification of the force as well as the transmission of the acceleration and that the HSLDSS retains stiffness at the equilibrium position. The numerical model includes friction and is solved using direct integration of the equation of motion. Experimental results from a scale model agree well theoretical predictions.

     

Experimental investigation on the effect of vibration on flow of liquid film in a horizontal duct

Effect of vibration on characteristics of a thin liquid film in a horizontal rectangular duct was studied experimentally in two flow regions viz, prebreak-up and break-up for Re _f =400–900. The lower surface of the duct on which liquid film flows was subjected to external vibration in the range of 25–1000 Hz. The instantaneous film thickness which was recorded by conductivity measurement gauges was analysed statistically. Results of structural changes in liquid film due to external vibration and without it are reported in this paper.