Visualization of elastic vibrations in solid structures

Although the computation of vibrations in solid structures is routine, visualization of time-dependent vibrations within structures is a relatively undeveloped field. In this paper, visualization methods are applied to vibrations in underwater structures excited by continuous incident sound. Visualization provides a more explicit understanding and reveals features of the underlying elastic-wave behavior not previously known. In particular it is shown that the elastic-wave behavior is dominated by waves whose displacement moves along paths around a center of oscillation. Small-scale rotational waves in this type or turbules, are shown to occur even inside thin shells where their function is to change the direction of flexural displacement as the incident-wave excitation moves over the exterior of the structure. Computed results are presented for canonical shapes such as spheres, and spherical shells. It is expected that the features of elastic-wave behavior will be similar for more general body shapes and for other types of excitation. Visualization results are also presented for sound-power flow around and through a solid sphere. The principal application of the visualization of elastic vibrations is believed to be in noise and vibration control.     

Discussion of finite element formulations of nonlinear beam vibrations

The Lagrange-type, Galerkin, and Ritz-type finite element formulations for large amplitude vibrations of immovably supported slender beams are reexamined. Inconsistency in the definition of frequency or criterion of defining nonlinearity is discussed, and validity of the frequency solution is examined. Improved finite element results by including both longitudinal displacement and inertia in the formulation are presented and compared with available Rayleigh-Ritz continuum solutions.     

Optimal control of thermoelastic beam vibrations by piezoelectric actuation

This paper presents the vibrations suppression of a thermoelastic beam subject to sudden heat input by a single piezoelectric actuator. An optimization problem is formulated as the minimization of a quadratic functional in terms of displacement and velocity at a given time and with the least control effort. The solution method is based on a combination of modal expansion and variational approaches. The modal expansion approach is used to convert the optimal control of distributed parameter system into the optimal control of lumped parameter system. By utilizing the variational approach, an explicit optimal control law is derived and the determination of the corresponding displacement and velocity is reduced to solving a set of ordinary differential equations. Numerical results are presented to demonstrate the effectiveness and the applicability of the proposed method.     

Optimal shakedown design of beam structures

The optimal design of plane beam structures made of elastic perfectly plastic material is studied according to the shakedown criterion. The design problem is formulated by means of a statical approach on the grounds of the shakedown lower bound theorem, and by means of a kinematical approach on the grounds of the shakedown upper bound theorem. In both cases two different types of design problems are formulated: one searches for the minimum volume design whose shakedown limit load is assigned; the other searches for the design of the assigned volume whose shakedown limit load is maximum. The optimality conditions of the four problems above are found by the use of a variational approach; such conditions prove the equivalence of the two types of design problems, provide useful information on the structural behaviour in optimality conditions, and constitute a fifth possible way to determine the optimal design. Whatever approach is used, the strong non-linearity of the corresponding problem does not allow the finding of the analytical solution. Consequently, in the application stage suitable numerical procedures must be employed. Two numerical examples are given.     

Asymptotic non-linear normal modes for large-amplitude vibrations of continuous structures

Non-linear normal modes (NNMs) are used in order to derive accurate reduced-order models for large amplitude vibrations of structural systems displaying geometrical non-linearities. This is achieved through real normal form theory, recovering the definition of a NNM as an invariant manifold in phase space, and allowing definition of new co-ordinates non-linearly related to the initial, modal ones. Two examples are studied: a linear beam resting on a non-linear elastic foundation, and a non-linear clamped–clamped beam. Throughout these examples, the main features of the NNM formulation will be illustrated: prediction of the correct trend of non-linearity for the amplitude-frequency relationship, as well as amplitude-dependent mode shapes. Comparisons between different models—using linear and non-linear modes, different number of degrees of freedom, increasing accuracy in the asymptotic developments—are also provided, in order to quantify the gain in using NNMs instead of linear modes.     

Optimal segmentation of beam and disk structures

Large structures are usually composed of elements by properly designed connections. The optimal design solution in such cases should provide optimal size and number of elements together with optimal connection stiffness. The problem is formulated by assuming the element cost to be a nonlinear function of its size and the cost of connection to depend on its stiffness or transmitted forces. The number of elements and the connection stiffness now constitute the design parameters to be determined. A two-level procedure is proposed for determination of the optimal segmentation for beam and plate structures. Several illustrative examples are presented.     

单帧图像热致光学误差祛除算法研究

在研发浮法玻璃锡槽在线测厚仪的过程中,为了克服高温气体湍流条件下激光传输光线起伏效应对激光测量系统性能的影响.利用基于帧差的像点运动轨迹探测手段,确定的流场内图像模糊、抖动的体现程度与CCD的积分时间成反比;试验采用各种合适窗口的邻域滑动平均法,对线阵CCD数据数列进行修正可消除热致光学误差对视频信号的影响.     

Optimal design of laminated composite beam structures

This work deals with design sensitivity analysis and optimal design of composite structures modelled as thin-walled beams. The structures are treated as a torsion-bending resistant beams. The analysis problem is discretized by a finite element technique. A two-node Hermitean beam element is used. The beam sections are made from an assembly of elements that correspond to flat layered laminated composite panels. Optimal design is performed with respect to the lamina orientations and thickness of the laminates. The structural weight is considered as the objective function. Constraints are imposed on stresses, displacements, critical load and natural frequencies. Two failure criteria are used to limit the structural strength: Tsai-Hill and maximum stress. The Tsai-Hill criterion is also adopted to predict the first-ply-failure loads. The design sensitivity analysis is analytically formulated and implemented. An adjoint variable method is used to derive the response sensitivities with respect to the design. A mathematical programming approach is used for the optimization process. Numerical examples are performed on three-dimensional structures.     

Numerical approximation of flow induced vibrations of channel walls

In this paper the numerical method for solution of an aeroelastic model describing the interactions of air flow with vocal folds is described. The flow is modelled by the incompressible Navier–Stokes equations spatially discretized with the aid of the stabilized finite element method. The motion of the computational domain is treated with the aid of Arbitrary Lagrangian–Eulerian method. The structure motion is described by an equivalent system with two degrees of freedom governed by a system of ordinary differential equations and discretized in time with the aid of an implicit multistep method and strongly coupled with the flow model. The influence of inlet/outlet boundary conditions is studied. The numerical analysis is performed and compared to the related results from literature.     

Thermally induced buckling of laminated composites by a layerwise theory

Thermal buckling analysis of laminated composites is presented by using a layer-wise theory. Governing buckling equations are derived from the variational principle and a finite element method is developed to formulate the problem. Numerical results are obtained and compared with those of other theories addressing the effects of the thickness-to-span ratio, lamination angle, the ratio of thermal expansion coefficients and degree of orthotropy on buckling temperature for antisymmetric angle-ply laminates. It is found that a layer-wise approach may be necessary for more accurate thermal buckling analysis of laminated composites.     

Midpoint difference method for analysing beam structures

The midpoint difference method is organized for analyzing beam structures by approximating the solution of the differential equation of the elastic deflection curve. The method requires the generation and solution of a system of simultaneous, algebraic equations which yield the deflection, slope, bending moment and shear at specified points along the span of the beam. The method is rather simple and well suited for programming on the digital computer.

All of the basic equations necessary for developing an analysis of a beam structure are established and the application of the method is given in the paper. Some general comments are made about the development of a computer program and about the solution of the system of simultaneous equations peculiar to this method of analysis.     

Harmonic analysis of the vibrations of a cantilevered beam with a closing crack

In this article, the vibrational response of a cracked cantilevered beam to harmonic forcing is analysed. The study has been performed using a finite element model of the beam, in which a so-called closing crack model, fully open or fully closed, is used to represent the damaged element. Undamaged parts of the beam are modelled by Euler-type finite elements with two nodes and 2 d.f. (transverse displacement and rotation) at each node. Recently the harmonic balance method has been employed by other researchers to solve the resulting non-linear equations of motion. Instead, in this study, the analysis has been extended to employ the first and higher order harmonics of the response to a harmonic forcing in order to characterize the non-linear behaviour of the cracked beam. Correlating the higher order harmonics of the response with the forcing term the so-called higher order frequency response function (FRFs), defined from the Volterra series representation of the dynamics of non-linear systems, can be determined by using the finite element model to simulate the time domain response of the cracked beam. Ultimately the aim will be to employ such a series of FRFs, an estimate of which in practice could be measured in a stepped sine test on the beam to indicate both the location and depth of the crack, thus forming the basis of an experimental structural damage identification procedure.     

Geometric nonlinear analysis of flexible spatial beam structures

An updated Lagrangian formulation of the spatial beam element is presented for a purely geometric nonlinear analysis in which the geometric stiffness matrix is expressed either by a one-dimensional integration of the stress resultants or by a closed form of element-end forces. A computer code, NACS, is developed based on this formulation which has a number of facilities to meet the special requirements for the analysis of suspension and cable-stayed bridges. Several example problems are reported.     

Plastic instability of beam structures using co-rotational elements

In a previous paper [Comput. Methods Appl. Mech. Engrg. 191 (2002) 1755], the authors have presented a 3D co-rotational elastic beam element including warping effects. This formulation is now further developed in order to incorporate elasto-plastic deformations. The element possesses seven degrees of freedom at each node and can be used to model beams with arbitrary cross-sections. Thus, within the present approach, the centroid and shear center of the cross-section are not necessarily coincident. The main purpose of this element is to model elasto-plastic instability problems. In this context, two methods of branch-switching are tested and discussed. In the first one, the bifurcation point is isolated by successive bisections and the branch-switching is operated by using the eigenvector associated to the negative eigenvalue. In the second one, introduced by Petryk, an energy approach is used to select automatically the stable post-bifurcation path. Six examples, including large displacement and stability problems, are used in order to assess the performances of the element.     

Optimization of composite beam structures using a genetic algorithm

This work examines the possibility of using a to nature adopted, stochastic search method, called genetic algorithm (GA), for optimizing the damping ability of composites.The principles and properties of GAs are briefly described, and an approach to evaluating and optimizing the factor of dampness of composites, using a specific GA combined with an existing FEM program, is given.Some results of optimizing the stacking sequences and the thicknesses of the layers, consisting of different materials, are presented as examples of composite beams in order to demonstrate the efficiency of the method.     

Initially stressed vibrations of a cantilever beam subjected to an intermediate concentrated axial load

A finite element formulation including the effect of shear and rotatory inertia is used to obtain the fundamental frequency of a cantilever beam subjected to an intermediate concentrated axial load. Results in terms of buckling and fundamental frequency parameters as a function of critical load at various intermediate positions are presented in tables for various slenderness ratios of beams.     

Finite element analysis of lateral buckling for beam structures

The application of finite element analysis to lateral buckling problems, locating the critical points and tracing the postbifurcation path, is treated on the basis of a geometrically nonlinear formulation for a beam with small elastic strain but with possibly large rotations. The existing finite element formulations for thin beams are examined in the aspect of application to bifurcation problems, such as lateral buckling, and the choice of an appropriate rotation parameter for representing incremental or variational rotations in finite element formulations is discussed in relation to locating bifurcation points. This is illustrated through several numerical examples and followed by appropriate discussion.     

基于LabVIEW的简支梁振动信号测试分析系统

为了改进振动工程测试实验,改善教学方法,提高实验效果,基于LabVIEW开发了一种针对简支梁的振动信号测试分析系统。下位机采用具有丰富片上资源、较高数据处理能力的C8051F350单片机进行振动数据的采集,并通过RS—232串口与上位机通信,实现信号数据的传输;上位机软件开发基于功能强大的图形化编程虚拟仪器LabVIEW开发平台,完成振动数据采集的实时显示、并对其进行时域、频域分析和数据存储等功能。实际应用证明该系统稳定、可靠,满足实验要求,将LabVIEW应用于实验教学中,大大提高了学生的学习兴趣,达到了提高实验效果和教学质量的目的。