Design of shear-deformable antisymmetric angle-ply laminates to maximize the fundamental frequency and frequency separation

An antisymmetrically laminated angle-ply plate is optimized with the objectives of maximizing the fundamental eigenfrequency and the distance between two consecutive natural frequencies. The formulation includes the contribution of the shear deformation, but neglects the in-plane and rotary inertias. The design variables are the fiber orientations and the thicknesses of individual layers. The design problems are cast into a mathematical programming format and solved by using a quasi-Newton function maximization algorithm. A penalty function method is employed to maximize the fundamental frequency, subject to lower bound constraints on higher-order frequencies. Numerical results are presented for laminates constructed of high-modulus fiber-reinforced materials, and the effects of various problem parameters on the efficiency of the designs are investigated. It is shown that the design variables may not be determined optimally if the effect of shear deformation is neglected. Moreover, it was observed that the classical plate theory leads to erroneous results in optimal material selection problems.     

修正倒谱和动态规划的基频估计算法

基音频率是语音信号处理中的一个重要参数。倍频、半频错误以及清浊音判决的可靠性等问题一直是基频估计中的难点问题。在对语音信号的倒谱进行适当修正的基础上,提出了一种高精度的基频估计算法。该算法根据倒谱、短时能量和短时过零率在清音段和浊音段的不同表现,构造了一个清浊音判决函数,大大提高了清浊音判决精度;然后利用动态规划技术进行基频跟踪。在构造代价函数时．充分考虑了基频连续性的影响,从而使该算法既能有效地避免倍频和半频错误,又能体现出基频的自然加倍和减半。通过与现有的几种效果较好的方法进行对比实验,结果表明该算法具有准确率高、基频轨迹平滑的优点,利用该算法得到的基频轨迹基本不需要进行后期平滑处理。     

结构频带激励下的整体动响应优化

结合频响函数的性质,采用目标频带内各模态峰值响应的 -范数作为整体响应水平的衡量指标,将等效目标函数的灵敏度表示为固有频率和振型灵敏度的函数,进而采用基于梯度的迭代方法求解,实现结构在宽频激励下的整体动响应优化,该方法适合于多频带的响应控制。此外,在频带响应优化模型的基础上,提出了通过增加频率约束,使之能应用于密集频率结构,避免迭代过程中的密频和重频现象。最后通过桁架截面优化和形状优化的数值算例,表明了本文提出方法的合理性和有效性。     

The maximization of fundamental frequency of structures under arbitrary initial stress states

A general technique is proposed to maximize the lowest natural frequency of structures subjected to an arbitrary state of initial stresses. The arbitrary states of initial stresses are represented by nondimensional loading parameters that describe an admissible loading space, i.e. every possible initial stress state lies within the admissible loading space. The key to the proposed optimization strategy is shown to be the concavity of the first natural frequency with respect to variations of the loading parameters within the admissible loading space. A rigorous demonstration is presented to show that, provided buckling has not occurred, all the possible initial stress states must not be considered. Instead, assessment of only a small number of initial stress states must be done in order to guarantee that the first natural frequency does not decrease for all the other initial stress states within the admissible loading space. A minimax optimization technique is used to maximize the lowest natural frequency of a simply supported rectangular plate where the thickness distribution is the design variable and normal and shear initial stress states are considered. Copyright © 2005 John Wiley & Sons, Ltd.     

基于频率灵敏度方法的宏微运动平台连接架结构优化

连接架与驱动设备频率相近而发生共振,将影响宏微运动平台的稳定工作和超精密定位的实现.为避免共振,本文以连接架为研究对象,搭建连接架固有频率有限元分析模型,并对有限元模型进行实验验证.然后,基于频率灵敏度方法对有限元模型结构参数进行摄动,通过分析刚度和第1阶固有频率对结构参数的灵敏度,进行设计变量选择,再以连接架质量为约束条件,将连接架第1阶固有频率倒数作为目标函数,对连接架优化模型进行优化并进行实验验证.通过优化,连接架第1阶固有频率得到了有效的提高,避免了连接架与音圈电机发生共振,从而为宏微运动平台稳定工作和超精密定位提供可靠依据.     

2D optimum distribution of carbon nanotubes to maximize fundamental natural frequency of polymer composite micro-beams

A new two-dimensional (2D) optimum distribution of carbon nanotubes (CNTs) in the longitudinal and thickness directions of a polymer composite micro-beam is obtained to achieve its highest fundamental natural frequency given a weight percent (wt.%) of CNTs. To this end, optimum distribution patterns are first studied individually in the longitudinal and the thickness directions by dividing the micro-beam into multiple segments and multiple layers, respectively. It is assumed that each segment/layer is made of low-viscosity, thermosetting polyester epoxy/amine resin LY-5052 and reinforced by multi-walled carbon nanotubes. A user-defined code, written in the Python language, is compiled with ABAQUS to generate a three-dimensional finite element model of the micro-beam and subsequently to evaluate the optimum CNT distributions under various boundary conditions. It is found that fundamental frequencies of the clamped–free, clamped–guided and clamped–clamped micro-beams can be enhanced up to 19.5%, 16.8% and 14.8%, respectively, by choosing the 2D optimum CNT distribution profile along the longitudinal and the thickness directions of the micro-beam. The results also reveal that the 1D through-axis and 2D optimal CNT distribution patterns depend on the type of boundary condition applied. However, the optimum through-thickness distribution pattern is found to be independent of the type of boundary condition, which can be of high value for practical applications.     

基于变频率区间约束的结构材料优化设计

针对频率约束的结构材料优化问题,基于结构拓扑优化思想,提出变频率区间约束的结构材料优化方法。借鉴均匀化及ICM(独立、连续、映射)方法,以微观单元拓扑变量倒数为设计变量,导出宏观单元等效质量矩阵及导数,进而获得频率一阶近似展开式。结合变频率区间约束思想,获得以结构质量为目标函数、频率为约束条件的连续体微结构拓扑优化近似模型;采用对偶方法求解。通过算例验证该方法的有效性及可行性,表明考虑质量矩阵变化影响所得优化结果更合理。     

Topology optimization for tunnel support in layered geological structures

Tunnel construction commonly proceeds in an environment of layered geological formation. Design for tunnel support relies on the tunnel location and the mismatch of different layers. The present paper proposes a topology optimization method for the design of tunnel support. The design domain is discretized by finite elements. An element is composed of the original rock (hard or soft) and the reinforcing material (rock reinforced by grouting or bolting). The design issue involves the distribution of reinforcements. We model the reinforced host ground by a power‐weighted mixture law. The ratios of two phases in various elements are optimized to minimize the deformation of the tunnel. The method enables a computer‐aided design for the supports of underground tunnels embedded within layered geological structures. The reinforced areas for the tunnel are displayed under the passive geological stress and the active external loading. Four kinds of rock formations are examined. They are homogeneous rock, hard–soft–hard (HSH) sandwich structure, two‐layer structure with the soft rock at the top and the hard rock at the bottom (SH), and the one with the hard rock at the top and the soft rock at the bottom (HS). The simulation reveals the high efficiency of tunnel support by optimizing its topologies. Copyright © 2000 John Wiley & Sons, Ltd.     

Robust topology optimization under multiple independent uncertainties of loading positions

The topology optimization problem of a continuum structure is further investigated under the independent position uncertainties of multiple external loads, which are now described with an interval vector of uncertain-but-bounded variables. In this study, the structural compliance is formulated with the quadratic Taylor series expansion of multiple loading positions. As a result, the objective gradient information to the topological variables can be evaluated efficiently upon an explicit quadratic expression as the loads deviate from their ideal application points. Based on the minimum (largest absolute) value of design sensitivities, which corresponds to the most sensitive compliance to the load position variations, a two-level optimization algorithm within the non-probabilistic approach is developed upon a gradient-based optimization method. The proposed framework is then performed to achieve the robust optimal configurations of four benchmark examples, and the final designs are compared comprehensively with the traditional topology optimizations under the loading point fixation. It will be observed that the present methodology can provide a remarkably different structural layout with the auxiliary components in the design domain to counteract the load position uncertainties. The numerical results also show that the present robust topology optimization can effectively prevent the structural performance from a noticeable deterioration than the deterministic optimization in the presence of load position disturbances.     

The Artificial Bee Colony algorithm in layer optimization for the maximum fundamental frequency of symmetrical laminated composite plates

In this study the layer optimization was carried out for maximizing the lowest (first) fundamental frequency of symmetrical laminated composite plates subjected to any combination of the three classical boundary conditions, and the applicability of the Artificial Bee Colony (ABC) algorithm to the layer optimization was investigated. The finite element method was used for calculating the first natural frequencies of the laminated composite plates with various stacking sequences. The ABC algorithm maximizes the first natural frequency of the laminated composite plate defined as an objective function. The optimal stacking sequences were determined for two layer numbers, twenty boundary conditions and two plate length/width ratios. The outer layers of the composite plate had a stiffness increasing effect, and as the number of clamped plate edges was increased both he stiffness and natural frequency of the plate increased. The optimal stacking sequences were in good agreement with those determined by the Ritz-based layerwise optimization method (Narita 2003: J. Sound Vibration 263 (5), 1005–1016) as well as by the genetic algorithm method combined with the finite element method.     

基于Rayleigh法的独塔非对称悬索桥基频简化算法

为了便于计算独塔非对称悬索桥振动基频,采用Rayleigh法分别推导了一阶正、反对称竖弯及扭转振动基频估算公式,考虑了不对称跨径布置对振动基频的影响,并提出了非对称独塔悬索桥合理的跨径比例。将表征跨径关系的参数k取1即可得到独塔双跨对称悬索桥的基频估算公式,最后通过有限元法验证估算公式的有效性和可靠性。研究结果表明:独塔非对称悬索桥一阶正、反对称竖弯、扭转频率的有限元解和文中解的误差都在10%以内,表明推导的估算公式解与有限元解误差能满足设计阶段的要求,最后讨论了跨径相关系数变化对竖弯和扭转基频的影响并给出了合理边中跨比的建议,该公式可以方便指导独塔悬索桥方案设计和动力计算。     

Fundamental frequency of an orthotropic rectangular plate with an internal centre point support

A method of calculating the fundamental frequency of an orthotropic rectangular plate with a centrally located point support and free edges is presented. The variational equation of motion is derived by applying Hamilton’s principle. The analytical approach determining the fundamental frequency of the plate is developed using the generalised Galerkin method and verified by comparison with the results of the finite element modal analysis. The comparisons of the computational results indicate that the fundamental frequency of the centre-supported plates can be calculated with sufficient accuracy using the analytical technique developed in this work. The approach presented in the paper can be efficiently employed when designing composite rectangular plates for a specified value of the fundamental frequency.     

Sensitivity analysis of truss structures (virtual distortion method approach)

A new approach to structural sensitivity analysis based on the so-called virtual distortion method is presented. The proposed methodology enables the calculation of derivatives for elastic as well as elasto-plastic structures on the basis of knowledge of current strains, permanent plastic deformations and influence matrix, describing interactions between a chosen member and the entire structure. The analytical basis as well as numerical verification of the concept is demonstrated. Advantages of the proposed approach, in the sense of numerical cost, are summarized in conclusions. © 1998 John Wiley & Sons, Ltd.     

Sensitivity analysis of frame structures (virtual distortion method approach)

A novel approach to sensitivity analysis of frame structures based on the virtual distortion method (VDM) has been presented. The sensitivity analysis has been performed for elasto‐plastic material behaviour with respect to selected structural parameters e.g. cross‐sectional area, hardening/softening coefficient and yield stress. Advantages of applying the virtual distortion method to sensitivity analysis have been emphasised i.e. making use of the so‐called influence matrix, constant for the whole analysis and solving only local sets of equations corresponding to plastic locations. Theoretical background as well as the whole variety of examples has been presented. Copyright © 2001 John Wiley & Sons, Ltd.     

基于频率响应灵敏度的车身结构的优化设计

针对车身优化的问题,采用频率响应灵敏度分析来降低车身结构的响应位移。在车身有限元模型的基础上计算频率响应,并求得驾驶员座椅支架处的响应位移曲线,从中找出某频率处的振动峰值;为了降低振动峰值,通过频率响应灵敏度分析选择合适的设计变量进行优化计算。优化后车身关键频率处的响应位移明显减小,提高乘坐的舒适性。     

Optimization of extended-release hydrophilic matrix tablets by support vector regression

Background: In this work, support vector regression (SVR) was applied to the optimization of extended release from swellable hydrophilic pentoxifylline matrix-tablets and compared to multiple linear regression (MLR). Methods: Binary mixtures comprising ethylcellulose and sodium alginate were used as the matrix-former. The matrix-former : drug weight ratio and the percentage of sodium alginate in the matrix-former were the formulation factors (independent variables) and the percentages of drug release at four different time intervals were the responses (dependent variables). Release was determined according to United States Pharmacopeia 31 for 11 pentoxifylline matrix-tablet formulations of different independent variable levels and the corresponding results were used as tutorial data for the construction of an optimized SVR model. Six additional checkpoint matrix-tablet formulations, within the experimental domain, were used to validate the external predictability of SVR and MLR models. Results: It was found that the constructed SVR model fitted better to the release data than the MLR model (higher coefficients of determination, R^?2, lower prediction error sum of squares, narrower range of residuals, and lower mean relative error), outlining its advantages in handling complex nonlinear problems. Superimposed contour plots derived by using the SVR model and describing the effects of polymer and sodium alginate content on pentoxifylline release showed that formulation of optimal release profiles, according to United States Pharmacopeia limitations, could be located at drug : matrix ratio of 1 and sodium alginate content 25% w/w in the matrix-former. Conclusion: The results indicate the high potential for SVR in formulation development and Quality by Design.     

Explicit solution for the natural frequency of structures with partial viscoelastic treatment

The free vibration characteristics of structures with viscoelastic treatment are represented by the complex-valued natural frequencies. The assumed single mode representation associated with the low-frequency stiffness of the viscoelastic treatment is modified by a correction term representing the influence from residual vibration modes. The correction term is eliminated in terms of the corresponding natural frequency associated with the high-frequency stiffness of the viscoelastic treatment, whereby an expression is obtained for the complex-valued natural frequency, which only requires the solution of two real-valued eigenvalue problems.     

Discrete material optimization of general composite shell structures

A novel method for doing material optimization of general composite laminate shell structures is presented and its capabilities are illustrated with three examples. The method is labelled Discrete Material Optimization (DMO) but uses gradient information combined with mathematical programming to solve a discrete optimization problem. The method can be used to solve the orientation problem of orthotropic materials and the material selection problem as well as problems involving both. The method relies on ideas from multiphase topology optimization to achieve a parametrization which is very general and reduces the risk of obtaining a local optimum solution for the tested configurations. The applicability of the DMO method is demonstrated for fibre angle optimization of a cantilever beam and combined fibre angle and material selection optimization of a four‐point beam bending problem and a doubly curved laminated shell. Copyright © 2005 John Wiley & Sons, Ltd.     

液压支架柱窝有限元分析及结构优化

利用ANSYS Workbench软件对液压支架柱窝进行参数化建模,并对其刚度和强度进行校核及多目标结构优化设计。通过计算输入参数的敏感度及响应曲面来确定优化方案。优化结果表明:柱窝在满足刚度和强度要求的同时其质量减少了6%,达到了轻量化的设计目的,获得了理想的结构参数。     

汽车发动机支承参数的灵敏度分析与模糊优化

本文简介发动机支承参数的灵敏度分析与模糊多目标优化方法.实践结果表明,通过“移频、解耦、降低支承处响应力”等多种途径对支承参数进行模糊多目标优化,可以获得颇为满意的综合效果.