An efficient coupled layerwise theory for dynamic analysis of piezoelectric composite beams

An efficient new coupled one-dimensional model is developed for the dynamics of piezoelectric composite beams. The model combines third order zigzag approximation for the displacement with layerwise approximation of the electric field as piecewise linear for sublayers. By enforcing the conditions of zero transverse shear stress at the top and bottom and its continuity at layer interfaces, the displacement field is expressed in terms of three primary displacement variables and potentials. The governing coupled equations of stress and charge equilibrium and boundary conditions are derived from Hamilton's principle. Analytical solutions are obtained, for free vibrations and forced response under harmonic load, for simply supported hybrid beams and the results are compared with the exact three-dimensional solution and uncoupled first order shear deformation theory solution. The present results show significant improvement over the first order solution and agree very well with the exact solution for both thin and thick hybrid beams. The results demonstrate the capability of the developed theory to adequately model open and closed circuit electric boundary conditions to accurately predict their influence on the response.     

A spectral finite element for analysis of wave propagation in uniform composite tubes

A spectral finite element model (SFEM) for analysis of coupled broadband wave propagation in composite tubular structure is presented. Wave motions in terms of three translational and three rotational degrees of freedom at tube cross-section are considered based on first order shear flexible cylindrical bending, torsion and secondary warping. Solutions are obtained in wavenumber space by solving the coupled wave equation in 3-D. An efficient and fully automated computational strategy is developed to obtain the wavenumbers of coupled wave modes, spectral element shape function, strain-displacement matrix and the exact dynamic stiffness matrix. The formulation emphasizes on a compact matrix methodology to handle large-scale computational model of built-up network of such cylindrical waveguides. Thickness and frequency limits for application of the element is discussed. Performance of the element is compared with analytical solution based on membrane shell kinematics. A map of the distribution of vibrational modes in wavelength and time scales is presented. Effect of fiber angle on natural frequencies, phase and group dispersions are also discussed. Numerical simulations show the ease with which dynamic responses can be obtained efficiently. Parametric studies on a clamped-free graphite-epoxy composite tube under short-impulse load are carried out to obtain the effect of various composite configurations and tube geometries on the response.     

谱有限元数值分析各向异性复合板中导波色散特性

提出谱有限元方法研究层状各向异性复合板中导波的色散特性和波结构。基于三维弹性动力学方程,用有限元方法离散波导截面,波传播方向的位移用简谐波表示,得到了导波色散的特征方程。分析了单层和双层复合板中导波沿不同方向传播的色散特性和波结构,讨论了双层复合板中层厚比对相速度的影响。数值研究结果表明:导波的对称模态沿纤维方向传播时在较宽的频率范围内保持弱色散状态。双层复合板中导波基本模态的相速度在低频时受层厚比的影响较明显,随着频率的增加趋向于相速度较低的材料。数值模拟结果为导波用于复合材料定量无损检测和性能评价提供理论依据。     

A finite element model for sandwich viscoelastic beams: Experimental and numerical assessment

Among the passive control systems for attenuation of vibrations in structures, those that use viscoelastic materials as a damping core in laminated-plate-like components are focused herein. In the present work an assessment of a time-domain formulation for numerical modelling of viscoelastic materials is made. This formulation, which is called Golla–Hughes method (GHM), is based on a second-order time-domain realization of Laplace-domain motion equations. The GHM parameters used in the characterization of a viscoelastic material are experimentally determined and a sandwich GHM-based finite element model is presented and validated through numerical comparisons with classic formulation results. Finally, a time-domain simulation of an experimentally tested sandwich beam is carried out.     

Nonlinear vibration of viscoelastic sandwich beams by the harmonic balance and finite element methods

This paper deals with geometrically nonlinear vibrations of sandwich beams with viscoelastic materials. For this purpose, a new finite element formulation has been developed, in which a zig-zag model is used to describe the displacement field. The viscoelastic behaviour is handled by using hereditary integrals and their relationships with complex moduli. An efficient solution procedure based on the harmonic balance method is also developed. To demonstrate its abilities, various problems of nonlinear vibrations of sandwich beams are considered. First, the results derived from the proposed approach are compared with those of nonlinear dynamic analyses using direct time integration and to experimental data. Then, the influence of the vibration amplitude on the damping properties of sandwich beams is investigated. The effect of an initial axial strain is also examined.     

A finite element for the vibration analysis of Timoshenko beams

A Timoshenko beam finite element is presented which has three nodes and two degrees of freedom per node, namely the values of the lateral deflection and the cross-sectional rotation. The element properties are based on a coupled displacement field; the lateral deflection is interpolated as a quintic polynomial function and the cross-sectional rotation is linked to the deflection by specifying satisfaction of the governing differential equation of moment equilibrium in the absence of the rotary inertia term. Numerical results confirm that this procedure does not preclude convergence to true Timoshenko theory solutions since rotary inertia is included in lumped form at element ends. The new Timoshenko beam element has good convergence characteristics and where comparison can be made in numerical studies it is shown to be generally more efficient than previous elements.     

A nodal discontinuous Galerkin finite element method for nonlinear elastic wave propagation

A nodal discontinuous Galerkin finite element method (DG-FEM) to solve the linear and nonlinear elastic wave equation in heterogeneous media with arbitrary high order accuracy in space on unstructured triangular or quadrilateral meshes is presented. This DG-FEM method combines the geometrical flexibility of the finite element method, and the high parallelization potentiality and strongly nonlinear wave phenomena simulation capability of the finite volume method, required for nonlinear elastodynamics simulations. In order to facilitate the implementation based on a numerical scheme developed for electromagnetic applications, the equations of nonlinear elastodynamics have been written in a conservative form. The adopted formalism allows the introduction of different kinds of elastic nonlinearities, such as the classical quadratic and cubic nonlinearities, or the quadratic hysteretic nonlinearities. Absorbing layers perfectly matched to the calculation domain of the nearly perfectly matched layers type have been introduced to simulate, when needed, semi-infinite or infinite media. The developed DG-FEM scheme has been verified by means of a comparison with analytical solutions and numerical results already published in the literature for simple geometrical configurations: Lamb's problem and plane wave nonlinear propagation.     

Regression curves for vibration transmission across junctions of heavyweight walls and floors based on finite element methods and wave theory

Sound insulation prediction models in European and International Standards use the vibration reduction index to calculate flanking transmission across junctions of walls and floors. These standards contain empirical relationships between the ratio of mass per unit areas for the walls/floors that form the junction and a frequency-independent vibration reduction index. However, calculations using wave theory show that there is a stronger relationship between the ratio of characteristic moment impedances and the transmission loss from which the vibration reduction index can subsequently be calculated. In addition, the assumption of frequency-independent vibration reduction indices has been shown to be incorrect due to in-plane wave generation at the junction. Therefore numerical experiments with FEM, SFEM and wave theory have been used to develop new regression curves between these variables for the low-, mid- and high-frequency ranges. The junctions considered were L-, T- and X-junctions formed from heavyweight walls and floors. These new relationships have been implemented in the prediction models and they tend to improve the agreement between the measured and predicted airborne and impact sound insulation.     

Free vibration analysis of planar curved beams by wave propagation

In this paper, a systematic approach for the free vibration analysis of a planar circular curved beam system is presented. The system considered includes multiple point discontinuities such as elastic supports, attached masses, and curvature changes. Neglecting transverse shear and rotary inertia, harmonic wave solutions are found for both extensional and inextensional curved beam models. Dispersion equations are obtained and cut-off frequencies are determined. Wave reflection and transmission matrices are formulated, accounting for general support conditions. These matrices are combined, with the aid of field transfer matrices, to provide a concise and efficient method for the free vibration problem of multi-span planar circular curved beams with general boundary conditions and supports. The solutions are exact since the effects of attenuating wave components are included in the formulation. Several examples are presented and compared with other methods.     

A basic hybrid finite element formulation for mid-frequency analysis of beams connected at an arbitrary angle

When beams are connected at an arbitrary angle and subjected to an external excitation, both longitudinal and bending waves are generated in the system. Since longitudinal wavelengths are considerably longer than bending wavelengths in the mid-frequency region, the number of bending wavelengths in the beams is considerably larger than the number of longitudinal wavelengths. In this paper, plannar beams connected at arbitrary angles are considered. The energy finite element analysis (EFEA) is employed for modelling the bending behavior of the beams and the conventional finite element analysis (FEA) is utilized for modelling the longitudinal vibration in the beams. Thus, a basic hybrid FEA formulation is presented for mid-frequency analysis of systems that contain two types of energy. The bending vibration is associated with the long members in the system and the longitudinal vibration is associated with the short members. The long members are considered to have high modal overlap and to contain several wavelengths within their dimension, and uncertainty effects are present. The short members contain a small number of wavelengths, and exhibit a low modal overlap. Due to the low modal overlap the resonant frequencies are spaced far apart in the frequency domain, therefore the short members exhibit resonant or non-resonant behavior depending on the frequency of the excitation.In this work, the bending and the longitudinal vibration within the same beam member are treated as a long and as a short member, respectively. A hybrid joint formulation is developed between long and short members. Power reflection and transmission coefficients are derived for each joint. The distribution of the energy throughout the system demonstrates a strong dependency on the power transfer coefficients. Several systems are analyzed by the hybrid FEA and by analytical solutions, and good correlation between them is observed.     

A semi-analytical finite element formulation for modeling stress wave propagation in axisymmetric damped waveguides

A semi-analytical finite element (SAFE) method is presented for analyzing the wave propagation in viscoelastic axisymmetric waveguides. The approach extends a recent study presented by the authors, in which the general SAFE method was extended to account for material damping. The formulation presented in this paper uses the cylindrical coordinates to reduce the finite element discretization over the waveguide cross-section to a mono-dimensional mesh. The algorithm is validated by comparing the dispersion results with viscoelastic cases for which a Superposition of Partial Bulk Waves solution is known. The formulation accurately predicts dispersion properties and does not show any missing root. Applications to viscoelastic axisymmetric waveguides with varying mechanical and geometrical properties are presented.     

谱有限元模拟正交各向异性黏弹性材料中导波传播特性

研究了导波在正交各向异性黏弹性复合板中传播的色散特性、波结构及功率流密度。基于二维平面运动方程,采用谱有限元方法得到了导波色散的特征方程。分析了正交各向异性黏弹性板中各向异性和黏性对能量速度和波结构的影响,以及基底对导波功率流密度的影响。数值研究结果表明:导波沿纤维方向传播的能量速度大,材料的黏性对速度影响较小,但会减小波结构的幅度;在高频时,基底的存在使两个基本模态的功率流密度分别集中到波导的上下表面,形成弱色散、高衰减及无色散、零衰减的表面波。数值模拟结果为导波用于复合材料定量无损检测和性能评价提供理论依据。      

Finite element simulation of wave propagation in an axisymmetric bar

An accurate solution for high-frequency pulse propagation in an axisymmetric elastic bar is obtained using a new finite element technique that yields accurate non-oscillatory solutions for wave propagation problems in solids. The solution of the problem is very important for the understanding of dynamics experiments in the split Hopkinson pressure bar (SHPB). In contrast to known approaches, no additional assumptions are necessary for the accurate solution of the considered problem. The new solution helps to elucidate the complicated distribution of parameters during high-frequency pulse propagation down the bar as well as to estimate the applicability of the traditional dispersion correction used in the literature for the analysis of wave propagation in a finite bar. Due to the dimensionless formulation of the problem, the numerical results obtained depend on Poisson's ratio, the length of the bar and the pulse frequency, and are independent of Young's modulus, the density and the radius of the bar.     

Three-dimensional finite element modeling of guided ultrasound wave propagation in intact and healing long bones

The use of guided waves has recently drawn significant interest in the ultrasonic characterization of bone aiming at supplementing the information provided by traditional velocity measurements. This work presents a three-dimensional finite element study of guided wave propagation in intact and healing bones. A model of the fracture callus was constructed and the healing course was simulated as a three-stage process. The dispersion of guided modes generated by a broadband 1-MHz excitation was represented in the time-frequency domain. Wave propagation in the intact bone model was first investigated and comparisons were then made with a simplified geometry using analytical dispersion curves of the tube modes. Then, the effect of callus consolidation on the propagation characteristics was examined. It was shown that the dispersion of guided waves was significantly influenced by the irregularity and anisotropy of the bone. Also, guided waves were sensitive to material and geometrical changes that take place during healing. Conversely, when the first-arriving signal at the receiver corresponded to a nondispersive lateral wave, its propagation velocity was almost unaffected by the elastic symmetry and geometry of the bone and also could not characterize the callus tissue throughout its thickness. In conclusion, guided waves can enhance the capabilities of ultrasonic evaluation.     

Spectral finite elements for vibrating rods and beams with random field properties

The classical stochastic Helmholtz equation grasps, through the random field of the refraction index, the spatial variability in the mass density but not the variability in elastic moduli or geometric parameters. In contradistinction to this restriction, the present analysis accounts for the spatial randomness of mass density as well as those of elastic properties and cross-sectional geometric properties of rods undergoing longitudinal vibrations and of Timoshenko beams in flexural vibrations. All the material variabilities are described here by random Fourier series with a typical (average) characteristic size of inhomogeneity d, which is either smaller, comparable to, or larger than the wavelength. The third length scale entering the problem, but kept constant, is the rod or beam length. We investigate the relative effects of random noises in all the material parameters on the spectral stiffness matrices associated with rods and beams for a very wide range of frequencies.     

Analytical and finite element prediction of Lamb wave scattering at delaminations in quasi-isotropic composite laminates

This paper presents a theoretical and finite element (FE) investigation of the scattering characteristics of the fundamental anti-symmetric (A₀) Lamb wave at delaminations in a quasi-isotropic (QI) composite laminate. Analytical models based on the Mindlin plate theory and Born approximation are presented to predict the A₀ Lamb wave scattering at a delamination, which is modelled as an inhomogeneity, in an equivalent isotropic model of the QI composite laminate. The results are compared with FE predictions, in which the delamination is modelled as a volume split. The equivalent isotropic model and QI composite laminate are used to investigate the feasibility of the common theoretical approach of modelling the delamination as the inhomogeneity. A good correlation is observed between the theoretical solutions and FE results in the forward scattering amplitudes, but there exists a larger discrepancy in the backward scattering amplitudes. The FE results also show that the fibre direction of the outer laminae has a pronounced influence on the forward and backward scattering amplitudes, which is not predicted by the analytical models.     

平行光管主反射镜组件的非线性有限元分析

为了提高对光学仪器中光机结构分析的精度,引入了非线性分析方法,用非线性有限元法研究了某平行光管主镜组件存在的非线性问题。介绍了产生结构非线性的主要来源,并在接触理论的基础上对主反射镜组件进行了合理的有限元建模。采用接触非线性分析方法对螺钉预紧时反射镜的响应及在自重和温度载荷作用下反射镜的变形进行了分析。将线性和非线性分析结果与实际结果进行了对比,结果表明,采用接触非线性分析方法对反射镜组件进行工程分析减小了分析误差。因此,在工程分析中,对某些接触部位采用非线性分析方法可以提高分析精度,得到的结果更加符合实际。