Geometrically nonlinear free vibrations of the symmetric rectangular honeycomb sandwich panels with simply supported boundaries

Geometrically nonlinear free vibrations of symmetric rectangular honeycomb sandwich panels with simply supported boundaries at the four edges are investigated using the homotopy analysis method (HAM). The honeycomb core of hexagonal cells is modeled as a thick layer of orthotropic material whose parameters of physical and mechanical properties are calculated by the corrected Gibson’s formula. The basic formulation of nonlinear free vibrations has been developed based on the third-order shear deformation plate theory and the nonlinear strain–displacement relation. The equilibrium equations have been obtained using Hamilton’s principle. Effects of axial half-waves, height and height ratio on the nonlinear free vibration response have been investigated for honeycomb sandwich panels.     

Geometrically nonlinear forced vibrations of the symmetric honeycomb sandwich panels affected by the water

Geometrically nonlinear forced vibrations of the symmetric rectangular honeycomb sandwich panels with the four edges simply supported and one surface affected by the water are investigated in this paper using the homotopy analysis method (HAM). The honeycomb core of hexagonal cells is modeled as a thick layer of orthotropic material whose physical and mechanical properties are determined using the Gibson correlations. The effect of water acting on honeycomb panels can be described as added mass, additional damping and additional stiffness coefficients which are obtained by the semi-analytical fluid pressures. The basic formulation of nonlinear forced vibrations has been developed base on the third-order shear deformation plate theory and Green Lagrange nonlinear strain–displacement relation. The equilibrium equations have been obtained using the Hamilton’s principle. Effects of water velocity, height and height ratio on the nonlinear forced vibration response have been studied for the honeycomb sandwich panels.     

四边固支对称蜂窝夹层板主共振非线性动力学计算

应用同伦分析方法(HAM)研究了四边固支对称蜂窝夹层板主共振情况下的非线性动力学特性。将铝基蜂窝芯层等效为一正交异性层, 等效弹性参数由修正后的Gibson方程得出。基于经典叠层板理论(CPT)和几何大变形理论建立了四边固支蜂窝夹层板受横向激振力作用下的受迫振动微分方程, 通过振型正交化将蜂窝夹层板受迫振动微分方程简化成双模态下的动力学控制方程, 得到了主共振情况下的平均方程, 研究了不同结构参数对动力学特性的影响。计算结果表明, 蜂窝夹层板的幅频特性曲线类似单自由度Duffing方程响应曲线, 随着结构参数的增大, 硬特性明显加大并且振幅的峰值明显减小, 所得结论可为蜂窝夹层板的设计和实际应用提供理论依据。     

Impact behavior and energy absorption of paper honeycomb sandwich panels

Dynamic cushioning tests were conducted by free drop and shock absorption principle. The effect of paper honeycomb structure factors on the impact behavior was analyzed. Results of many experiments show that the dynamic impact curve of paper honeycomb sandwich panel is concave and upward; the thickness and length of honeycomb cell-wall have a great effect on its cushioning properties; increasing the relative density of paper honeycomb can improve the energy absorption ability of the sandwich panels; the thickness of paper honeycomb core has an up and down fluctuant effect on the cushioning properties; with the increase of the thickness of paper honeycomb core, the effect dies down; flexible corrugated paperboard as liners can improve the compression resistance and cushioning properties of paper honeycombs. The research results can be used to optimize the structure design of paper honeycomb sandwich panel and material selection for packaging design.     

Generation of transient vibrations on aluminum honeycomb sandwich panels subjected to hypervelocity impacts

This paper presents the results of a set of experiments aimed at discovering the main features of impact-induced vibrations on all-aluminum honeycomb sandwich panels, representative of the GOCE satellite's top floor, which is exposed to the orbital debris environment. The activity focused on the characterization of the vibrations induced in the vicinity of internal payloads by hypervelocity impacts occurring on the vehicle's external shell. More than 30 tests were realized by launching 0.8–2.3 mm aluminum projectiles in the velocity range 4–5.5 km/s on targets with tri-axial accelerometer assemblies mounted on both the front and rear face of the panel, at a nominal distance of 150 mm from the impact point. It was found that a hypervelocity impact produces in both the front and rear side of the sandwich panel a vibration environment which can be described through the shock response spectrum (SRS) of three different types of waves that can be distinguished on the basis of the acceleration direction: out-of-plane, in-plane longitudinal and in-plane shear. The influence of projectile mass and velocity on SRS appeared to vary with frequency, with the most significant difference in the range between ∼10³ and ∼10⁴ Hz. The results of whole experimental set were used to derive an interpolation law through standard techniques of nonlinear fit. The empirical equation obtained makes it possible to predict the near-field vibration environment produced by hypervelocity impacts with debris having given size and velocity, reproducing all the test data with an average uncertainty of ±6 dB.     

四边简支条件下正交各向异性蜂窝夹层板的固有特性分析

以四边简支正交各向异性矩形蜂窝夹层板为研究对象,应用Reissner-Mindlin夹层板剪切理论,在考虑横向剪切变形的基础上,给出了一种将夹层板弯曲控制方程组化为仅含一个位移函数的单一方程的方法,从而获得了四边简支条件下矩形蜂窝夹层板弯曲振动固有频率的精确解,理论结果与数值结果和实验结果取得很好的一致,验证了本文方法的合理性;在此基础上研究了面板、芯层的各项结构和材料设计参数对夹层板其固有频率的影响,并对各设计参数对夹层板固有频率的调控机理进行了分析。研究结果对蜂窝夹层板的结构设计和工程应用具有指导意义。     

Analysis of flexural-flexural-torsional nonlinear vibrations of twisted rotating beams with cross-sectional deplanation

We present results of the investigations on flexural-flexural-torsional nonlinear vibrations of twisted rotating beams described by the system of three nonlinear integro-differential equations expressed in partial derivatives. Cross-sectional deplanation of beams is adjusted to the equations and it is assumed that centre of gravity and shear centre are located at different points. Vibration systems are expressed as a row by free forms of the linear problem. Free vibrations are studied with the help of the Show–Pierre nonlinear normal mode. Translated from Problemy Prochnosti, No. 2, pp. 112–124, March–April, 2009.     

四边简支双曲率蜂窝夹层薄壳自由振动分析

基于Reddy三阶剪切理论,研究了四边简支双曲率蜂窝夹层薄壳的自由振动,以及结构参数对蜂窝夹层薄壳固有频率的影响.将由六边形胞元组成的蜂窝芯层等效为一正交异性层,其等效弹性参数由修正后的Gibson公式得到,应用Reddy三阶剪切理论和Hamilton变分原理推导出四边简支条件下双曲率蜂窝夹层薄壳的频率方程.具体算例表明,采用Reddy三阶剪切理论计算的固有频率精度较高;双曲率蜂窝夹层薄壳的曲率、厚度比及胞元角度对蜂窝夹层壳固有频率有不同程度的影响,其中蜂窝夹层薄壳的固有频率随曲率的增大而增大,随厚度比的增大呈波动变化,随胞元角度的增大而减小.     

Geometrically nonlinear static and free vibration analysis of functionally graded piezoelectric plates

In this paper, nonlinear static and free vibration analysis of functionally graded piezoelectric plates has been carried out using finite element method under different sets of mechanical and electrical loadings. The plate with functionally graded piezoelectric material (FGPM) is assumed to be graded through the thickness by a simple power law distribution in terms of the volume fractions of the constituents. Only the geometrical nonlinearity has been taken into account and electric potential is assumed to be quadratic across the FGPM plate thickness. The governing equations are obtained using potential energy and Hamilton’s principle that includes elastic and piezoelectric effects. The finite element model is derived based on constitutive equation of piezoelectric material accounting for coupling between elasticity and electric effect using higher order plate elements. The present finite element is modeled with displacement components and electric potential as nodal degrees of freedom. Results are presented for two constituent FGPM plate under different mechanical boundary conditions. Numerical results for PZT-4/PZT-5H plate are given in dimensionless graphical forms. Effects of material composition and boundary conditions on nonlinear response are also studied. The numerical results obtained by the present model are in good agreement with the available solutions reported in the literature.     

Finite element analysis of sandwich panels with stepwise graded aluminum honeycomb cores under blast loading

This paper presents details and brief results of an experimental investigation on the response of metallic sandwich panels with stepwise graded aluminum honeycomb cores under blast loading. Based on the experiments, corresponding finite element simulations have been undertaken using the LS-DYNA software. It is observed that the core compression stage was coupled with the fluid–structure interaction stage, and the compression of the core layer decreased from the central to the peripheral zone. The blast resistance capability of sandwich panels was moderately sensitive to the core relative density and graded distribution. For the graded panels with relative density descending core arrangement, the core plastic energy dissipation and the transmitted force attenuation were larger than that of the ungraded ones under the same loading condition. The graded sandwich panels, especially for relative density descending core arrangement, would display a better blast resistance than the ungraded ones at a specific loading region.     

Insight into the shear behaviour of composite sandwich panels with foam core

While sandwich construction offers well-known advantages for high stiffness with light weight, the problem of designing the sandwich structure to withstand shear loading remains an important problem. This problem is more difficult with lower stiffness foam cores under high shear loading because the core is typically the weakest component of the structure and is the first one to fail in shear under the assuming of perfect contact between the skin and the foam core. In the present study, the shear response of the composite sandwich panels with Polyvinylchloride (PVC) foam core was investigated. The PVC H100 foam core is sandwiched between Glass Fiber Reinforced Polymer (GFRP) skins using epoxy resin to build a high performance sandwich panel to be investigated. Experiments have been carried out to characterise the mechanical response of the constituent materials under tension, compression and shear loading. Static shear tests for the sandwich panel reveal that the main failure mode is the delamination between the skin and the core rather than shearing the core itself due to the considerable value of the shear strength of the PVC foam. The Finite Element Analysis (FEA) of the sandwich structure shows that shear response and failure mode can be predicted, but that accurate predictions require a consideration of the non-linear response of the foam core. The results have a direct application in predicting the ability of the sandwich structure to withstand the shear loading.     

Geometrically nonlinear analysis of laminate composite plates and shells using the eight-node hexahedral element with one-point integration

An eight-node hexahedral isoparametric element with one-point quadrature for the geometrically nonlinear static and dynamic analysis of plates and shells of laminate composite materials is formulated and implemented in this work. The element is free of volumetric and shear locking and spurious modes are not detected. Shear locking is avoided using a corotational system for strain and stress components, whereas volumetric locking is controlled using one-point quadrature for the dilatational (or spherical) part of the gradient matrix. The efficiency and potential of the three-dimensional element in the analysis of plates and shells of laminate composite materials undergoing large displacements and rotations are shown solving several numerical examples and comparing with results obtained by other authors using different plate and shells elements.     

四边简支新型类方形蜂窝夹层结构振动特性研究

新型类方形蜂窝是六边形蜂窝的一种过渡形式,对其等效弹性参数和振动特性的研究具有重要意义。采用改进的Gibson公式对比分析了双壁厚与等壁厚类方形蜂窝夹芯的面内等效弹性参数的差异,并应用经典层合板理论分析了不同等效弹性参数下2种壁厚类型的四边简支类方形蜂窝夹层结构的振动特性,基于有限元仿真技术分析了不同壁厚类方形蜂窝夹层结构的振动特性,并与理论分析结果进行对比。结果表明等效弹性参数的数值模拟结果与理论值基本吻合。在蜂窝基本结构参数相同的条件下,双壁厚类方形蜂窝夹芯的面内等效剪切模量、面外刚度和等效密度均比等壁厚类方形蜂窝夹芯大;在低阶振动模态下,双壁厚类方形蜂窝夹层结构的固有频率比等壁厚类方形蜂窝夹层结构的低,在高阶振动模态下,双壁厚类方形蜂窝夹层结构的固有频率比等壁厚类方形蜂窝夹层结构的高;影响夹层结构固有频率的3个主要因素所占权重由大到小依次为蜂窝夹芯yoz面等效剪切模量、蜂窝夹芯等效密度,蜂窝夹芯壁厚。研究结果表明采用经典层结构理论计算得到类方形蜂窝夹层结构的固有频率与数值仿真结果的一致性较好,这进一步证明了采用改进Gibson公式得到的类方形蜂窝夹芯等效弹性参数的正确性,同时证明了将该振动理论运用到一般蜂窝夹层结构研究的可行性,为扩展研究其他类型蜂窝夹层结构振动特性奠定了基础。     

埋微带天线蜂窝夹层结构的力电性能分析

设计并制备了一种共形承载一体化的埋微带天线蜂窝夹层结构。利用有限元方法分析了该结构的力学性能, 与三点弯曲实验比较, 获得较好的一致性。研究了蜂窝层厚度等参数对电性能和力学性能的影响。结果表明, 蜂窝层厚度为8~14 mm时, 共形承载天线不仅具有高增益和低损耗, 还具有较好的力学性能。      

On the free vibration of sandwich panels with a transversely flexible and temperature-dependent core material – Part I: Mathematical formulation

The free vibration analysis of sandwich panels with a core that is flexible and compliant in the vertical direction and with temperature-dependent mechanical properties is presented in two parts. The first part presents the mathematical formulation while the second deals numerically with the effects of the degrading properties of the core on the free vibration response. The analysis is based on the high-order sandwich panel theory approach (HSAPT), and the equations of motions along with the appropriate boundary conditions are derived using the Hamilton’s principle. The study investigates the role of increasing temperature, through the degradation of the mechanical properties of the core, on the free vibration response of structural sandwich panels. The mathematical formulation uses two types of computational models. At first, following the HSAPT approach, the unknowns include the displacements of the face sheets as well as the shear stress in the core. Secondly, it is assumed that the through-thickness distributions of the vertical and horizontal core displacements can be represented as polynomials, following the results of the HSAPT static case, and the effect of the variable mechanical properties are implemented directly.     

On the free vibration of sandwich panels with a transversely flexible and temperature dependent core material – Part II: Numerical study

This paper, the second of two, presents a numerical study of a simply-supported sandwich panel that is based on the mathematical formulation that appears in part I. The solution of the unknowns in the case of a simply-supported panel is based on a trigonometric series solution and it converts the set of PDE’s into an algebraic set of equations that are described by stiffness and mass matrices. In addition, it studies numerically, for a specific sandwich panel construction, the effects of the degradation of the mechanical properties of the core as a result of the thermal field on the free vibration response of the two computational models. The results of the mixed formulation model, denoted by MF, and the displacement formulation, denoted by DF, reveal a significant reduction of the eigenfrequencies as well as a shifting of the eigen-modes from higher modes to lower ones with increasing temperature.     

Free vibrations of anisotropic doubly-curved shells and panels of revolution with a free-form meridian resting on Winkler–Pasternak elastic foundations

The Generalized Differential Quadrature (GDQ) method is applied to study the dynamic behavior of anisotropic doubly-curved shells and panels of revolution with a free-form meridian resting on Winkler–Pasternak elastic foundations. The First-order Shear Deformation Theory (FSDT) is used to analyze the above mentioned moderately thick structural elements. In order to include the effect of the initial curvature from the beginning of the theory formulation a generalization of the kinematical model is adopted for the Reissner–Mindlin and Toorani–Lakis theory. By so doing a generalization of the theory of anisotropic doubly-curved shells and panels of revolution is proposed. Simple Rational Bézier curves are used to define the meridian curve of the revolution structures. The Differential Quadrature (DQ) rule is introduced to determine the geometric parameters of the structures with a free-form meridian. Results are obtained taking the meridional and circumferential co-ordinates into account, without using the Fourier modal expansion methodology. Comparisons between the general formulation and the Classical Reissner–Mindlin and Classical Toorani–Lakis theory are presented. New results are presented in order to investigate the effects of the Winkler modulus, the Pasternak modulus and the inertia of the elastic foundation on the free vibrations of anisotropic shells of revolution with a free-form meridian.