Dynamic optimization of framed structures

An analytical procedure for the determination of the least weight structure which satisfies a specific frequency requirement plus upper and lower bounds on the design variables is presented. The design algorithm is an iterative solution of the Kuhn-Tucker optimality criterion. The procedure is to modify an existing design to first obtain the correct structural frequency and then, while the frequency is held constant, to minimize the weight. This is accomplished using gradient equations derived in matrix notation for direct application to the finite element method of analysis.

The most important features of the algorithm are: (a) a small number of design iterations are needed to reach optimal or near-optimal design, (b) structural elements with a wide variety of size stiffness may be used.

The procedure has been completely automated in a computer program. Results of two numerical examples show that the method is convergent and that optimized configurations can be determined in as few as 10 redesign cycles.     

Dynamic characterization of thin-walled laminated channel beams with warping

The dynamic characteristics of a round cornered C-channel beam made of laminated composite material are studied. The sharp corners are rounded for manufacturing considerations. Thin-walled beam theory is used to formulate the coupled vibration of a rounded C-channel beam with fixed-free end conditions. It is shown that cross-sectional properties used for isotropic case are applicable for laminated composites and the material properties needed for the formulation can be obtained using the law of average. A comparative study is conducted to show the advantage of using composites. The effect of radius of corner and warping on the natural frequency is investigated. The functional of the equations of motion is formed using the variational method. The Ritz method has been used to formulate an eigenvalue problem and its frequency equation. The method proposed is systematic. The computerized procedure can be used as a fast design tool in the design of composite channel beam structures.     

具有频率不变波束图的非均匀线列宽带基阵设计

提出一种宽带基阵设计方法，可以在期望的带宽内保持基阵的波束图不随频率变化。该方法利用现有的窄带基阵设计方法得到参考频率下的基阵加权矢量，然后用解析或自适应的方法计算宽带基阵所需的其他频点处的加权矢量。本文还推导了非均匀线列阵阵元位置的表达式。最后，给出了一个设计实例，它可以在十个倍频程内保持波束图基本相同。本文方法对阵元的指向性也没有任何限制，因此具有重要的工程应用价值。     

Axial magnetic field effect on wave propagation in bi-layer FG graphene platelet-reinforced nanobeams

This paper aims to investigate the size-dependent wave propagation in functionally graded (FG) graphene platelet (GPL)-reinforced composite bi-layer nanobeams embedded in Pasternak elastic foundation and exposed to in-plane compressive mechanical load and in-plane magnetic field. The small-scale effects are taken into account by employing the nonlocal strain gradient theory that contains two different length scale parameters. The present two nanobeams are made of multi-composite layers. Each layer is composed of a polymer matrix reinforced by uniformly distributed and randomly oriented GPLs. The GPLs weight fraction is graded from layer to other according to a new piece-wise rule and then four distribution types will be established. Our technique depends on applying the four-variable shear and normal deformations theory to model the wave propagation problem. The equations of motion are obtained using Hamilton principle. These equations are then analytically solved to obtain the wave frequencies and phase velocities of the waves. The calculated results are compared with those published in the literature. The impacts of the length scale parameters, foundation stiffness, in-plane magnetic field, weight fraction of graphene, graphene platelets distribution type and beam geometry on the propagating waves in the FG GPLs nanobeams are discussed in details. It is found that the strength of the composite beams may be enhanced with increasing in the GPLs weight fraction and magnetic field leading to an increment in the phase velocity and wave frequency of the present system.

     

From MEMS devices to smart integrated systems

The smart integrated systems of tomorrow would demand a combination of micromechanical components and traditional electronics. On-chip solutions will be the ultimate goal. One way of making such systems is to implement the mechanical parts in an ordinary CMOS process. This procedure has been used to design an oscillator consisting of a resonating cantilever beam and a CMOS Pierce feedback amplifier. The resonating frequency is changed if the beam is bent by external forces. The paper describes central features of this procedure and highlights the design considerations for the CMOS-MEMS oscillator. The circuit is used as an example of a “VLSI designer” way of making future integrated micromechanical and microelectronic systems on-chip. The possibility for expansion to larger systems is reviewed.     

High-fidelity modeling of MEMS resonators. Part II. Coupled beam-substrate dynamics and validation

A computational multiphysics model of the coupled beam-substrate-electrostatic actuation dynamics of MEMS resonators has been developed for the model-based prediction of Q-factor and design sensitivity studies of the clamped vibrating beam. The substrate and resonator beam are modeled independently and then integrated by enforcing their interface compatibility condition and the force equilibrium to arrive at the multiphysics model. The present model has been validated with several reported single-beam clamped resonators. The validated model indicates that: the anchor loss is primarily engendered through coupling between the resonant modes and the waves propagating through the substrate inner layers; the resonant frequency of the beam decreases up to 5% due to substrate flexibilities interacting with beam at the anchors; and, for a given design the beam mass and its relative compliance with respect to the substrate are key parameters that influence the Q-factor degradation. In addition, the coupled model has also been used to predict the Q-factor of a paired-beam mechanical filter device with high fidelity when compared with the experimentally observed Q-factor.     

Static pull-in instability and free vibration of functionally graded graphene nanoplatelet reinforced micro-sandwich beams under thermo-electrical actuation

This paper investigates the static pull-in instability and free vibration of a multilayer functionally graded graphene nanoplatelet (GPL) reinforced composite (FG-GPLRC) micro-beam sandwiched between two copper layers subjected to a combined action of an electric voltage and a uniform temperature change based on Euler–Bernoulli beam theory. The GPL nanofillers are uniformly dispersed within each individual layer while its weight fraction changes from layer to layer in the multilayer FG-GPLRC micro-beam. The modified Halpin–Tsai model is used to predict the effective Young’s modulus while the rule of mixture is used to determine the effective Poisson’s ratio, mass density and thermal expansion coefficient. The static pull-in voltage and natural frequency of clamped–clamped micro-beams are obtained by employing Galerkin and iterative method. The effects of GPL distribution pattern, weight fraction, geometry and size as well as the geometry of the beam, the temperature change and the total number of layers on the static and dynamic characteristics of the micro-beams are discussed in detail.

     

Investigation of the internal stress effects on static and dynamic characteristics of an electrostatically actuated beam for MEMS and NEMS application

Internal stress is often encountered in fixed–fixed beam based devices with micron or sub-micron length scales during device fabrication or operation. In this paper, we have investigated the effects of internal stress on static and dynamic characteristics of an electrostatically actuated cylindrical beam. The beam has been modelled using Euler–Bernoulli theory including the nonlinearities due to beam stretching and electrostatic forcing. The analysis has been carried out by solving the governing differential equations using a Galerkin based multi-modal reduced order modelling technique. A standard collocation based numerical scheme has also been used to confirm the results of the reduced order method. Our study shows that internal stress significantly influences the static and dynamic characteristics of the beam. We also find that, when compressive internal stress is high, it is important to include higher modes in the reduced order model. A design technique to achieve high resonant frequency stability under temperature variation, for electrostatically actuated beam oscillators, has also been proposed as a result of this investigation.     

Identification and elimination of parasitic shear in a laminated composite beam finite element

A displacement-based finite element for the analysis of laminated composite beams is formulated using strain gradient notation. The definition of the beam’s longitudinal displacement possesses only the independent term (axial displacement) and a term which is linear in the thickness coordinate z. Thus, the finite element is first-order shear deformable. As strain gradient notation is physically interpretable, the contents of the coefficients of the polynomial expansions are identified a priori. Thus, the modeling capabilities as well as modeling deficiencies of the element are identified during the formulation procedure. A single parasitic shear term (spurious) is found to be present in the transverse shear strain expression of the element, which is responsible for locking. This parasitic shear term is also found to be the cause of a qualitative error existing in the representation of transverse shear strain along the length of a typical beam model. As the spurious term has been clearly identified, it can easily be removed to correct the element. The effectiveness of the procedure is shown through numerical analyses performed using the element containing the spurious term and then corrected for it. The beam model is validated by comparing numerical solutions with analytical solutions provided by the minimization of the total potential energy for a given laminated composite beam.     

Crashworthiness design for functionally graded foam-filled bumper beam

Automotive bumper beam is an important component to protect passenger and vehicle from injury and damage induced by severe collapse. Recent studies showed that foam-filled structures have significant advantages in light weight and high energy absorption. In this paper, a novel bumper beam filled with functionally graded foam (FGF) is considered here to explore its crashworthiness. To validate the FGF bumper beam model, the experiments at both component and full vehicle levels are conducted. Parametric study shows that gradient exponential parameter m that controls the variation of foam density has significant effect on bumper beam’s crashworthiness; and the crashworthiness of FGF-filled bumper beam is found much better than that of uniform foam (UF) filled and hollow bumper beam. The multiobjective optimization of FGF-filled bumper beam is also performed by considering specific energy absorption (SEA) and peak impact force as the design objectives, and the wall thickness t, foam densities ρ_f1 and ρ_f2 (foam densities at the end and at mid cross section, respectively) and gradient exponential parameter m as design variables. The Kriging surrogate modeling technique and multiobjective particle swarm optimization (MOPSO) algorithm were implemented to optimize the FGF-filled bumper beam. The optimized FGF-filled bumper beam is of great advantages and it can avoid the harmful local bending behavior and absorb more energy than UF filled and hollow bumper beam. Finally, the optimized FGF-filled bumper beam is installed to a passenger car model, and the results demonstrate that the FGF-filled bumper beam ensures the crashworthiness performance of the passenger car while reduces weight about 14.4% compared with baseline bumper beam.     

A Ritz procedure for optimisation of cylindrical shells, formed by a nearly symmetric and balanced angle-ply composite laminate, with fixed minimum frequency

The paper deals with the problem of optimisation of a cylindrical shell profile under a frequency constraint. The minimum value of the thickness has been established a priori. The structure considered is typical of aerospace craft vessels.The same value of the lowest vibration frequency of the reference cylindrical shell with uniform thickness, has been imposed. That is the minimization procedure of the structure weight must not affect its lowest vibration frequency.Instead of the currently applied finite element method (FEM), Ritz series expansions have been utilized in the analytical developments both for the dynamic variables and for the thickness axial distribution over the shell surface. Lagrange multipliers, together with governing equations and objective function, have been utilized to form the Lagrangian functional, as in the classical Euler–Lagrange method. Imposing the stationary conditions with respect to the Lagrangian degrees of freedom gives a non-linear algebraic equations system, whose solution can be found with an appropriate algorithm.A series of repeated optimisation operations have been performed to arrive at the minimized weight profile, but with the pre-established minimum value of the shell thickness.A simplified nearly symmetric and balanced multilayer composite angle-ply laminate of the shell structure is supposed, as in the case of the uniform thickness reference shell, previously considered for the dynamic analysis. Significant results of some computation application cases can be helpful to evaluate the efficiency of the proposed optimisation procedure applied to cylindrical structures.     

主动约束层阻尼梁有限元建模与动态特性研究

基于弹性、粘弹性和压电材料的本构关系,利用Hamilton原理,推导了主动约束层阻尼梁的有限元动力学模型．结合压电材料的机电耦合特性,采用自感电压的位移反馈,研究了主动约束层阻尼梁的闭环控制特性．求解了主动约束层阻尼简支梁的动态特性如固有频率、模态损耗因子及频率响应特性等．对被动控制、主动控制和主被动混合控制的控制效果进行了分析比较．研究了粘弹性层与约束层厚度等参数对减振控制效果的影响．     

Experimental validation of the utility of structural optimization

This paper addresses the topic of validating structural optimization methods by use of experimental results. The paper describes the need for validating the methods as a way of effecting a greater and an accelerated acceptance of formal optimization methods by practicing engineering designers. The range of validation strategies is defined which includes comparison of optimization results with more traditional design approaches, establishing the accuracy of analyses used, and finally experimental validation of the optimization results. The remainder of the paper describes examples of the use of experimental results to validate optimization techniques. The examples include experimental validation of the following: optimum design of a trussed beam; combined control-structure design of a cable-supported beam simulating an actively controlled space structure; minimum weight design of a beam with frequency constraints; minimization of the vibration response of helicopter rotor blade; minimum weight design of a turbine blade disk; aeroelastic optimization of an aircraft vertical fin; airfoil shape optimization for drag minimization; optimization of the shape of a hole in a plate for stress minimization; optimization to minimize beam dynamic response; and structural optimization of a low vibration helicopter rotor.     

An innovative piezoelectric energy harvester using clamped–clamped beam with proof mass for WSN applications

In this paper a miniature piezoelectric energy harvester (PEH) with clamped–clamped beam and mass loading at the center is introduced which has more consistency against off-axis accelerations and more efficiency in comparison to other cantilever PEH’s. The beams consist of different layers of Si, piezoelectric, and insulators based on MEMS technology that vibrates by applying an external force to the fixed frame. Due to beam vibration, variable stress is applied to the AlN piezoelectric and a potential difference is created at the output terminals. AlN is deposited on clamped–clamped beams in such a way that produce more stress points which cause more power to be generated in comparison to other cantilever beam PEH’s with about same dimensions. A partial differential equations (PDE) describing the flexural wave propagating in the multi-morph clamped–clamped beam are solved as theoretical calculations for inherent frequency estimation and is confirmed by simulation results. The obtained inherent frequency is 42 Hz which with 1 g (g = 9.81 m/s²) acceleration produces 4 V and 80 µW maximum electrical peak power that can be used in the node of low-power consumption wireless sensor node for wireless sensor network (WSN) applications.

     

Chaotic dynamics and forced harmonic vibration analysis of magneto-electro-viscoelastic multiscale composite nanobeam

In this article, the damping forced harmonic vibration characteristics of magneto-electro-viscoelastic (MEV) nanobeam embedded in viscoelastic foundation is evaluated based on nonlocal strain gradient elasticity theory. The viscoelastic foundation consists of Winkler–Pasternak layer. The governing equations of nonlocal strain gradient viscoelastic nanobeam in the framework of refined shear deformable beam theory are obtained using Hamilton’s principle and solved implementing an analytical solution. In addition, a parametric study is presented to examine the effect of the nonlocal strain gradient parameter, magneto-electro-mechanical loadings, and aspect ratio on the vibration characteristics of nanobeam. From the numerical evaluation, it is revealed that the effect of electric and magnetic loading on the natural frequency has a predominant influence.

     

Design optimization with static and dynamic displacement constraints

A design optimization procedure using a sequential linear programming technique is proposed in this paper to design minimum weight structures subjected to frequency response and static displacement constraints. The merit of the proposed approach is that the reanalyses of the static and dynamic responses, as well as the computations of the static and dynamic sensitivity data, are performed in a reduced approximate model. A significant saving of computer time for large scale structures is expected. Two numerical examples show good results of this method.     

磁浮列车底盘梁的有限元优化减重设计

为在已有型号基础上获得更轻型有效的底盘梁结构而保持相对稳定的动力特性,首先利用ANSYS软件建立磁浮列车车体结构的有限元模型,利用Block Lanczos Method进行模态求解并得到结构一阶弹性固有频率.然后以结构一阶弹性固有频率值为约束函数,以车厢底盘梁8个截面上17个主要厚度参数为设计变量,以底盘梁总体积为目标函数,建立磁浮列车底盘梁减重优化的数学模型并进行计算.结果表明,磁浮列车一阶弹性固有模态为前后扭转的振型,在满足模态频率值不小于8.OHz的设计要求下,底盘梁减重20%的设计目标可以达到.说明以一阶弹性固有模态频率值为约束函数的优化方案可行且效果良好.     

A simple method to determine the critical buckling loads for axially inhomogeneous beams with elastic restraint

In this paper, a new and simple approach is presented to exactly calculate the critical buckling loads of beams with arbitrarily axial inhomogeneity. For various end boundary conditions, we transform the governing equation with varying coefficients to linear algebraic equations; then a characteristic equation in critical buckling loads will be obtained. Several examples of estimating buckling loads under typical end supports are discussed. By comparing our numerical results with the exact and existing results for homogeneous and nonhomogeneous beams, it can be found that our method has fast convergence and the obtained numerical results have high accuracy. Moreover, the buckling behavior of a functionally graded beam composed of aluminum and zirconia as two constituent phases is investigated for axially varying material properties. The effects of gradient parameters on the critical buckling loads are elucidated. Finally, we give an example to illustrate the enhancement of the load-carrying capacity of tapered beams for admissible shape profiles with constant volume or weight. The proposed method is of benefit to optimum design of beams against buckling in engineering applications.     

Minimum weight design of stiffened cylinders subjected to pure torsion

A minimum weight design procedure along with actual designs of two typical fuselage type of stiffened circular cylindrical shell geometries subjected to pure torsion is presented. By formulating the weight of the composite shell as the objective function, an optimization technique is adopted to minimize it against general instability. In the design, all other possible failure modes, i.e. panel instability, skin wrinkling, local instability of stringers, yielding of skin and stiffener materials as well as failure mode interactions have been avoided. Typical opened type, like rectangular, tee, I, etc. and closed type, hat stiffener with all possible combinations are studied. For each shell geometry the best suited combination of stiffener geometries-are shown. In the first trial without minimum gauge (WMG), a design is obtained utilizing rectangular stringers and rectangular rings. Then the procedure is extended for other stiffener geometries to obtain a minimum gauge (MG) design. The effect of relaxing the MG on the weight of the stiffened shell is shown graphically and in tabular form which will be of added advantage to the designer while deciding about the MG. Since the smeared technique is employed in the analysis of stiffened shells, an upper bound on the stiffener spacing is initially employed. Then this bound is relaxed and its effect on the minimum weight design is studied for those types of stiffeners which proved economically feasible with bounded spacings. The procedure adopted in this work can be used for any other shell and stiffener geometry.