Plate/shell structure topology optimization of orthotropic material for buckling problem based on independent continuous topological variables

The purpose of the present work is to study the buckling problem with plate/shell topology optimization of orthotropic material. A model of buckling topology optimization is established based on the independent, continuous, and mapping method, which considers structural mass as objective and buckling critical loads as constraints. Firstly, composite exponential function (CEF) and power function (PF) as filter functions are introduced to recognize the element mass, the element stiffness matrix, and the element geometric stiffness matrix. The filter functions of the orthotropic material stiffness are deduced. Then these filter functions are put into buckling topology optimization of a differential equation to analyze the design sensitivity. Furthermore, the buckling constraints are approximately expressed as explicit functions with respect to the design variables based on the first-order Taylor expansion. The objective function is standardized based on the second-order Taylor expansion. Therefore, the optimization model is translated into a quadratic program. Finally, the dual sequence quadratic programming (DSQP) algorithm and the global convergence method of moving asymptotes algorithm with two different filter functions (CEF and PF) are applied to solve the optimal model. Three numerical results show that DSQP&CEF has the best performance in the view of structural mass and discretion.

VIBRATION AND ACOUSTIC RADIATION FROM SUBMERGED STIFFENED SPHERICAL SHELL WITH DECK-TYPE INTERNAL PLATE

Based on the motion differential equations of vibration and acoustic coupling system for thin elastic spherical shell with an elastic plate attached to its internal surface,in which Dirac-δ functions are employed to introduce the moments and forces applied by the attachment on the surface of shell,by means of expanding field quantities as Legendre series,a semi-analytic solution is derived for the vibration and acoustic radiation from a submerged stiffened spherical shell with a deck-type internal plate,which has a satisfactory computational effectiveness and precision for an arbitrary frequency range.It is easy to analyze the effect of the internal plate on the acoustic radiation field by using the formulas obtained by the method proposed.It is concluded that the internal plate can significantly change the mechanical and acoustic characteristics of shell,and give the coupling system a very rich resonance frequency spectrum.Moreover,the method can be used to study the acoustic radiation mechanism in similar structures as the one studied here.

Vibration Characteristics and Power Transmission of Coupled Rectangular Plates with Elastic Coupling Edge and Boundary Restraints

Coupled-plate structures are widely used in the practical engineering such as aeronautical,civil and naval engineering etc.Limited works can be found on the vibration of the coupled-plate structure due to the increased mathematical complexity compared with the single plate structure.In order to study analytically the vibration characteristics and power transmission of the coupled-plate structure,an analytical model consisting of three coupled plates elastically restrained along boundary edges and elastically coupled with arbitrary angle is considered,in which four groups of springs are distributed consistently along each edge of the model to simulate the transverse shearing forces,bending moments,in-plane longitudinal forces and in-plane shearing forces separately.With elastic coupling condition and general boundary condition of both flexural and in-plane vibrations taken into account by setting the stiffness of corresponding springs,the double Fourier series solution to the dynamic response of the structure was obtained by employing the Rayleigh-Ritz method.In order to validate the model,the natural frequency and velocity response of the model are firstly checked against results published in literatures and the ANSYS data,and good agreement was observed.Then,numerical simulation of the effects of several relevant parameters on the vibration characteristics and power transmission of the coupled structure were performed,including boundary conditions,coupling conditions,coupling angle,and location of the external forces.Vibration and energy transmission behaviors were analyzed numerically.The results show that the power transmission can be significantly influenced by the boundary restraints and the location of excitation.When the excitation is located at the central symmetry point of the model,the energy flow shows a symmetrical distribution.Once the location deviates from the central symmetry point,the power circumfluence occurs and the vortex energy field is formed at high frequency.

THE LAYOUT OPTIMIZATION OF STIFFENERS FOR PLATE-SHELL STRUCTURES

The plate-shell structures with stiffeners are widely used in a broad range of engineering structures. This study presents the layout optimization of stiffeners. The minimum weight of stiffeners is taken as the objective function with the global stiffness constraint. In the layout optimization, the stiffeners should be placed at the locations with high strain energy/or stress. Conversely, elements of stiffeners with a small strain energy/or stress are considered to be used inefficiently and can be removed. Thus, to identify the element efficiency so that most inefficiently used elements of stiffeners can be removed, the element sensitivity of the strain energy of stiffeners is introduced, and a search criterion for locations of stiffeners is presented. The layout optimization approach is given for determining which elements of the stiffeners need to be kept or removed. In each iterative design, a high efficiency reanalysis approach is used to reduce the computational effort. The present approach is implemented for the layout optimization of stiffeners for a bunker loaded by the hydrostatic pressure. The numerical results show that the present approach is effective for dealing with layout optimization of stiffeners for plate-shell structures.

Modeling and optimal vibration control of conical shell with piezoelectric actuators

In this paper numerical simulations of active vibration control for conical shell structure with dis-tributed piezoelectric actuators is presented.The dynamic equations of conical shell structure are derivedusing the finite element model (FEM) based on Mindlin's plate theory.The results of modal calculationswith FEM model are accurate enough for engineering applications in comparison with experiment results.The Electromechanical influence of distributed piezoelectric actuators is treated as a boundary conditionfor estimating the control force.The independent modal space control (IMSC) method is adopted and theoptimal linear quadratic state feedback control is implemented so that the best control performance withthe least control cost can be achieved.Optimal control effects are compared with controlled responses withother non-optimal control parameters.Numerical simulation results are given to demonstrate the effective-ness of the control scheme.

ON THE STABILITY ANALYSIS OF PLATES AND SHELLS USING A QUADRILATERAL,16-DEGREES OF FREEDOM PLAT SHELL ELEMENT DKQ16

The linear buckling problems of plates and shells were analysed using a recently developped quadrilateral,16-degrees of freedom flat shell element (called DKQ16).The geometrical stiffness matrix was established.Comparison of the numerical results for several typical problems shows that the DKQ16 element has a very good precision for the linear buckling problems of plates and shells.

Wind-induced vibration of single-layer reticulated shell structures

Aiming at the dynamic response of reticulated shell structures under wind load,systematic parameter analyses on wind-induced responses of Kiewitt6-6 type single-layer spherical reticulated shell structures and three-way grid single-layer cylindrical reticulated shell structures were performed with the random simulation method in time domain,including geometric parameters,structural parameters and aerodynamic parameters.Moreover,a wind-induced vibration coefficient was obtained,which can be a reference to the wind-resistance design of reticulated shell structures.The results indicate that the geometric parameters are the most important factor influencing wind-induced responses of the reticulated shell structures;the wind-induced vibration coeffi-cient is 3.0-3.2 for the spherical reticulated shell structures and that is 2.8-3.0 for the cylindrical reticula-ted shell structures,which shows that the wind-induced vibration coefficients of these two kinds of space frames are well-proportioned.

Rotating sandwich cylindrical shells with an FGM core and two FGPM layers: free vibration analysis

The free vibration analysis of a rotating cylindrical shell with an analytical method is investigated. The shell is considered as a sandwich structure, where the middle layer is a functionally graded material(FGM) shell, and it is surrounded by two piezoelectric layers. Considering piezoelectric materials to be functionally graded(FG),the material properties vary along the thickness direction as one innovation of this study.Applying the first-order shear deformation theory(FSDT), the equations of motion of this electromechanical system are derived as the partial differential equations(PDEs) using Hamilton's principle. Then, the Galerkin procedure is used to discretize the governing equations, and the present results are compared with the previously published results for both isotropic and FGM shells to verify the analytical method. Finally, the effects of FGM and functionally graded piezoelectric material(FGPM) properties as well as the thickness ratio and the axial and circumferential wave numbers on the natural frequencies are studied. Moreover, the Campbell diagram is plotted and discussed through the governing equations. The present results show that increasing the non-homogeneous index of the FGM decreases the natural frequencies on the contrary of the effect of non-homogeneous index of the FGPM.

Transverse Vibration Analysis of Concentrated Mass-Loaded Thin Circular Plate Based on Strcutural Circumferential Periodicity

The structural circumferential periodicity of inertial excitation produced by concentrated mass was utilized to establish the mathematical model of thin circular plate carrying eccentric concentrated mass and to analyze its transverse vibration. The fundamental frequency coefficient, natural frequency and mode shape function are determined by this method. A clamped thin circular plate was taken as an example to study the mass effect on the vibrating system.Comparison between the present results and published ones exhibits excellent agreement, which shows that the analytical method in this paper can be used to predict the transverse vibration parameters accurately.

Analytical analysis for vibration of longitudinally moving plate submerged in infinite liquid domain

The vibration of a longitudinally moving rectangular plate submersed in an infinite liquid domain is studied analytically with the Rayleigh-Ritz method. The liquid is assumed to be incompressible, inviscid, and irrotational, and the velocity potential is used to describe the fluid velocity in the whole liquid field. The classical thin plate theory is used to derive mechanical energies of the traveling plate. As derivative of transverse displacement with respect to time in the compatibility condition equation exists, an exponential function is introduced to depict the dynamic deformation of the moving plate. It is shown that this exponential function works well with the Rayleigh- Ritz method. A convergence study shows a quick convergence speed for the immersed moving plate. Furthermore, the parametric study is carried out to demonstrate the effect of system parameters including the moving speed, the plate location, the liquid depth, the plate-liquid ratio, and the boundary condition. Results show that the above system parameters have significant influence on the vibration characteristics of the immersed moving plate. To extend the study, the method of added virtual mass incremental （AVMI） factor is used. The results show good agreement with those from the Rayleigh-Ritz method.

The Coupling Effect of Spatial Reticulated Shell Structure with Cables

The spatial reticulated shell structure with cables (RSC) is a kind of coupling working system, which consists of flexible cables, reticulated shell structure (RS) and tower columns. The dynamic analysis of RSC based on the coupling model was carried out. Three kinds of elements such as the spatial bar element, cable element and beam element were introduced to analyze the reticulated shell, cable and tower column respectively. Furthermore, such parameter influences as structural boundary conditions, grid configuration, the span-to-depth ratio and the arrangement of cable system upon structural dynamics were analyzed. The structural vibration modes can be divided into four groups based on some numerical examples. And the frequencies in the same group are very close while the frequencies in different groups are different from each other obviously. It is clear that the sequence of the appearance of the each mode group heavily depends on the comparative stiffness of the tower column system, RS and cables.

Dynamics of Structural-Inhomogeneous Laminate and Shell Mechanical Systems with Point Constraints and Focused Masses. Part 2. Statement of the Problem of Forced Oscillations, Methods of Solution, Computational Algorithm and Numerical Results

A vibrational formulation, a technique, and an algorithm are proposed for assessing the resonance state of a package of rectangular plates and shells having point bonds and concentrated masses with different rheological properties of deformable elements under the influence of harmonic influences. The viscoelastic properties of elements are described using the linear Boltzmann-Volterra theory. An algebraic system of equations with complex coefficients is obtained, which is solved by the Gauss method. Various problems on steady-state forced vibrations for structurally inhomogeneous mechanical systems consisting of a package of plate and shell systems with concentrated masses and shock absorbers installed in it were solved. A number of new mechanical effects have been discovered associated with a decrease in the maximum resonance amplitudes of the mechanical system as a whole. The concept of “global resonance amplitude” is introduced to study the behavior of the resonance amplitudes of a mechanical system. An analysis of the numerical results showed that the interaction of resonant amplitudes is observed only in structurally inhomogeneous systems (in this case, with elastic and viscoelastic elements) and with a noticeable approximation of the natural frequencies.

VIBRATION ANALYSIS OF ELASTIC PLATE SUBMERGED IN INCOMPRESSIBLE VISCOUS FLUID BY COUPLING FINITE ELEMENT METHOD

The interaction problem of an elastic structure and anincompressible viscous fluid is solved through a computationalprocedure proposed in this paper. The fluid region is considered tobe bounded though without free surface boundary. The discrete fluiddomain is described by the stream- line upwind/Petrov-Galerkin finiteelement method. The arbitrary Lagrangian-Eulerian description isemployed to treat the moving interface between the structure and thefluid. The predictor-multicorrec- tor method is adopted to solve thecoupling finite element equations. The results of three numerical ex-amples show the effectiveness of the procedure.

VIBRATION CHARACTERISTICS OF OFFSHORE CYLINDRICAL TANKS

The vibration characteristics of offshore cylindrical tanks are studied in this paper. This is a typical topic of liquid-shell interaction system. In this paper, a general analytical method is presented by which the axial modal functions of liquid and shell are represented as the same complete orthogonal series for uncoupling the mode of liquid and shell. Simultaneously, the mode functions are expanded into a uniform convergence series and a linear polynomial, so that the problem of convergence and differentiation of mode series is solved. Therefore, the rather exact natural frequency and its corresponding mode of offshore cylindrical tank with different liquid depths, with arbitrary boundary conditions and with intermediate constraints can be obtained.

Nanoscale finite element models for vibrations of single-walled carbon nanotubes: atomistic versus continuum

By the atomistic and continuum finite element models, the free vibration behavior of single-walled carbon nanotubes （SWCNTs） is studied. In the atomistic finite element model, the bonds and atoms are modeled by the beam and point mass elements, respectively. The molecular mechanics is linked to structural mechanics to determine the elastic properties of the mentioned beam elements. In the continuum finite element approach, by neglecting the discrete nature of the atomic structure of the nanotubes, they are modeled with shell elements. By both models, the natural frequencies of SWCNTs are computed, and the effects of the geometrical parameters, the atomic structure, and the boundary conditions are investigated. The accuracy of the utilized methods is verified in comparison with molecular dynamic simulations. The molecular structural model leads to more reliable results, especially for lower aspect ratios. The present analysis provides valuable information about application of continuum models in the investigation of the mechanical behaviors of nanotubes.

复合材料双层球壳与环板耦合结构振动特性研究

以一阶剪切变形理论为基础,采用谱几何法对不同边界条件下复合材料双层球壳与环板耦合结构振动特性分析模型进行构建,其中采用人工虚拟弹簧技术模拟耦合结构的边界条件。根据耦合处的连接关系,通过布置耦合弹簧模拟器对子结构之间的耦合关系进行模拟。应用哈密顿原理对复合材料双层球壳与环板耦合结构的特征方程进行推导,对耦合结构的固有特性和稳态响应进行求解。将所得结果与有限元结果进行对比,验证了计算的正确性。分析了几何参数、材料参数对复合材料双层球壳与环板耦合结构振动响应的影响。

大型船舶加筋板结构焊接裂纹在线检测研究

大型船舶加筋板结构焊接若出现裂纹,会降低加筋板结构强度,为船舶航行安全带来不可预估的风险,为此研究大型船舶加筋板结构焊接裂纹在线检测方法。使用有限元软件Ansys,依据船舶加筋板结构参数,对加筋板结构焊接裂纹模态进行模拟,获取大型船舶加筋板结构焊接裂纹模态,依据该模态,使用改进连续小波分解方式从加筋板结构焊接裂纹模态中获取裂纹对应的小波系数,通过获取裂纹模态小波系数突变点,依据该突变点得到加筋板结构焊接裂纹深度、长度、宽度等在线检测结果。实验结果表明,该方法可有效获取船舶加筋板结构焊接裂纹模态,并依据该模态在线检测到船舶加筋板结构焊接裂纹的长度、宽度等,应用效果较佳。

基于FGMs的声子晶体带隙调控及振动特性研究

为抑制舰艇典型板壳结构低频振动,基于局域共振理论,设计了一种悬臂式声子晶体结构,开展了数值计算并揭示带隙发生机理,探究了各尺寸参数对带隙特性的影响。同时基于功能梯度材料(functional gradient materials, FGMs)思想,提出固有频率梯度组合带隙调控方法,可有效拓宽低频带隙范围。最后,设计振动特性试验并分析误差产生的原因。结果表明:悬臂梁边长越长,带隙宽度变窄,悬臂梁宽度及厚度增加,带隙宽度变宽;固有频率梯度组合研究中,当整体固有频率变化率小于8.1%时,带隙范围有效拓宽。当相邻两者固有频率变化率大于12.27%时,可呈现多频带隙特性;增加悬臂梁安装数量可在固有频率梯度组合的基础上进一步拓宽禁带范围,该研究成果可为船舶板壳结构隔振设计提供参考。

考虑液体晃荡的部分充液圆柱壳受迫振动分析

充液圆柱壳结构广泛存在于在工程中,在外部激励作用下,圆柱壳会与流体产生耦合振动,且未充满的液面也会产生晃荡,这种耦合振动响应分析在工程中具有重要意义。提出了一种求解考虑内部液体小幅晃荡的弹性圆柱壳振动响应的半解析法。首先分别在壳体建立结构坐标系、在自由液面建立液体坐标系,各自用来描述结构及内部液体的运动。基于Flügge壳体理论建立了圆柱壳的运动控制方程。将液体视为无黏、不可压缩的理想流体,根据线性水波理论和液体自由表面运动边界条件,得出了内部液体的速度势函数。通过流固耦合界面的连续性条件,结合坐标变换推导得到流固耦合系统的运动控制方程。研究了壳体在径向点力激励下振动响应,并得出了对应的液面晃荡响应。和有限元法对比验证了方法的正确性,然后改变相关参数讨论了考虑内部液体晃荡的弹性圆柱壳振动响应特性。

板壳结构在声振联合试验下的振动响应耦合分析

目的研究具有工程实践意义的板壳组合结构在声振联合作用下的响应预测方法。方法在噪声试验、振动试验和声振联合试验响应曲线的基础上,分析板壳组合结构在噪声和振动同时激励下的响应耦合规律,并根据噪声试验和振动试验的响应极值包络法,来预测在声振联合试验作用下板壳结构的响应分布。结果声振联合试验响应曲线近似于噪声试验和振动试验的最大值包络线,噪声的面激励和随机振动的基础激励在不同的频率范围内对结构响应起着主要影响。试验件的噪声试验和振动试验响应曲线在给定的频率点出现相交,小于交越频率的声振联合试验响应与振动试验高度吻合,高于交越频率的响应则以噪声试验为主。结论在工程实际中,可以直接利用振动试验和噪声试验的响应数据对声振联合试验的响应进行预测。由于交越频率难以事先获得,因此响应叠加法在实践中更易于实现。