Shear deformable versus classical theories for nonlinear vibrations of rectangular isotropic and laminated composite plates

In the present study, (i) the classical Von Kárman theory, (ii) the first-order shear deformation theory and (iii) the higher-order (third-order) shear deformation theory are compared for studying the nonlinear forced vibrations of isotropic and laminate composite rectangular plates. In particular, the harmonic response in the frequency neighborhood of the fundamental mode of rectangular plates is investigated and the response curves computed by using the three different theories are compared. The boundary conditions of the plates are simply supported with immovable edges. Geometric imperfections are taken into account. Calculations for isotropic and laminated composite plates are presented and results are discussed. For isotropic plates, the frequency-response curves for large-amplitude vibrations obtained by using the three theories are almost coincident. For laminated composite plates, differences arise for relatively thick plates (ratio between the thickness and the edge equal to 0.1), while for thin plates (ratio between the thickness and the edge equal to 0.01), no difference is obtained. For all cases, the first-order shear deformation (with shear correction factor ) and the higher-order shear deformation theories give practically coincident results and differences are observed with respect to the classical Von Kárman theory.     

Analysis of the natural vibrations of circular piezoceramic plates with partial electrodes

The finite-element method is used to analyze the thickness-symmetric vibrations of piezoelectric plates with partial electrodes. The spectra of the natural vibrations at resonance and antiresonance, the dynamic electromechanical coupling coefficient, and the vibration modes of these plates are studied for a wide range of geometric dimensions of both the plates and the partial electrodes. The optimal dimensions of the plates and electrodes, which correspond to the maximal values of the coupling coefficient, are determined. The increase in the coupling coefficient due to the utilization of the partial electrodes is considered for piezoelectric plates made of ceramics of various compositions. It is shown that all piezoceramic compositions can be divided into two groups. For the first group, the utilization of the partial electrodes can increase the coupling coefficient of the thickness vibrations by 7–23%, depending on the vibration mode. For the second group of piezoceramics, the coupling coefficient cannot be increased in this way; in other words, complete electrodes are optimal for the thickness vibrations of plates made of piezoceramics that belongs to the second group.     

An exact analytical approach to the free vibration analysis of rectangular plates with mixed boundary conditions

A comprehensive analytical technique is developed for the free vibration analysis of rectangular plates with discontinuities along the boundaries. For illustrative purposes a solution is obtained for plates with edges partially clamped and partially simply supported and plates with edges partially and partially simply supported. A vast array of first mode eigenvalues is provided for these families of plates. Solutions to the equations are obtained by exploiting a mathematical technique described by the author during an earlier publication. It is shown that eigenvalue matrices are easily generated for a wide range of plates with discontinuities in boundary conditions.     

Dynamic response of composite plates with cut-outs,part I: Simply-supported plates

The effect of square cut-outs on the natural frequencies of square, simply-supported composite plates is investigated. The forced and free dynamic response of plates with cut-outs is formulated. Laminations are assumed to be symmetric about the mid-plane and the plates are considered analytically as homogeneous anisotropic plates. In the method of solution it is assumed that the effect of the cut-out is equivalent to an external loading on the plate. For free vibration, the method leads to an infinite system of frequency equations. Depending upon the accuracy required, a suitable size of the system of frequency equations is selected. Results are given for square, simply-supported composite plates with centrally located square cut-outs for different modulus ratios. A comparison of results obtained from this method for isotropic plates with cut-outs with available literature is made and excellent agreement is obtained.     

Transition scenario to turbulence in thin vibrating plates

A thin plate, excited by a harmonic external forcing of increasing amplitude, shows transitions from a periodic response to a chaotic state of wave turbulence. By analogy with the transition to turbulence observed in fluid mechanics as the Reynolds number is increased, a generic transition scenario for thin vibrating plates, first experimentally observed, is here numerically studied. The von Kármán equations for thin plates, which include geometric non-linear effects, are used to model large amplitude vibrations, and an energy-conserving finite difference scheme is employed for discretisation. The transition scenario involves two bifurcations separating three distinct regimes. The first regime is the periodic, weakly non-linear response. The second is a quasiperiodic state where energy is exchanged between internally resonant modes. It is observed only when specific internal resonance relationships are fulfilled between the eigenfrequencies of the structure and the forcing frequency; otherwise a direct transition to the last turbulent state is observed. This third, or turbulent, regime is characterized by a broadband Fourier spectrum and a cascade of energy from large to small wavelengths. For perfect plates including cubic non-linearity, only third-order internal resonances are likely to exist. For imperfect plates displaying quadratic nonlinearity, the energy exchanges and the quasiperiodic states are favored and thus are more easily obtained. Finally, the turbulent regime is characterized in the light of available theoretical results from wave turbulence theory.     

医用梅花形超声探头声场的分析与计算

本文根据基尔霍夫积分．推导出了用于胎儿监护的梅花形超声多普勒探头声场的具体表达式．对中心大圆晶片发射、周围六个小圆片接收，以及七个相同晶片四发三收两种典型布局的换能器，其轴上声压分布，离辐射面不同距离断面上的声压分布及不同定向平面上的远场指向性进行了数值计算．结果表明，采用四发三收式结构较一发六收式结构具有较大的优越性．计算结果与实际测试结果很好地吻合．本项研究对此类探头的研制、使用及质量评价标准的制订，具有理论指导意义．     

医用梅花形超声探头声场的分析与计算

本文根据基尔霍失积分，推导了出民用于监护的梅花形超声多普勒探头声场的具体表达式。对中心大圆晶片发射，周围六个小圆片接收，以及七个相同晶片四发三收两种典型布局的换能器，其轴上声压分布，离辐射面不距离断面上的声压分布及不同定向平面上的这场指向性进行了数值计算。结果表明，采用四发三收式结构较一发六收式结构具有较大的优越性。计算结果与实际测试结果很好地吻合。     

Polarization transformers for optical anisotropy: II. Transformation of properties of optical anisotropy of reciprocal and nonreciprocal elements

The methods of transformation of an arbitrary phase optical anisotropy using a set of quarter-wave plates are considered. For this purpose, we use a formal analogy between the Jones matrices of these anisotropic elements and the matrices of transformation of the polarization basis states. We consider all types of reciprocal and nonreciprocal optical phase anisotropy. We show that the minimum set of anisotropic elements sufficient for such a transformation is a set of four quarter-wave plates. For nonreciprocal systems, this set should be complemented with Faraday rotators, whose number depends on the initial and final type of the nonreciprocity.     

Free vibration of polar-orthotropic sector plates

The free vibration of ring-shaped polar-orthotropic sector plates is analyzed by the Ritz method using a spline function as an admissible function for the deflection of the plates. For this purpose, the transverse deflection of a sector plate is written in a series of the products of the deflection function of a sectorial beam and that of a circular beam satisfying the boundary conditions. The deflection function of the sectorial beam is approximately expressed by a quintic spline function, which satisfies the equation of flexural vibration of the beam at each point dividing the beam into small elements. The frequency equation of the plate is derived by the conditions for a stationary value of the Lagrangian. The present method is applied to ring-shaped polar-orthotropic sector plates with some combination of boundary conditions, and the natural frequencies and the mode shapes are calculated numerically up to higher modes. This method is very effective for the study of vibration problems of variously shaped anisotropic plates including these sector plates.     

NDT capability of digital shearography for different materials

In this paper, the capabilities of shearography for detecting hole and crack type defects in polymeric and metallic materials using thermal loading were investigated. In order to improve the ability of the identifying defect, the fringe multiplication technique was applied to some suspectable shearography fringe patterns. Based on the test results empirical rules for inspection plates of different material with specific thicknesses were established. For aluminium plates with thicknesses of 3 mm, minimum diameter of a surface breaking hole which is detectable is approximately equal to 2.5–3.0 times the depth of the defect, and for polymer plates with a thickness of 10 mm, the minimum detectable diameter is 0.8–1.3 times of the depth. For polymeric materials results from the increasing temperature period are approximately the same as those from the decreasing temperature period. When the orientation of the image shearing is not perpendicular to the longitudinal direction of the crack type defects, the sensitivity for defect detection is relatively higher than with perpendicular image shearing.     

Origin of Starting Earthquakes under Complete Coupling of the Lithosphere Plates and a Base

The boundary problem of rigid coupling of lithospheric plates modeled by Kirchhoff plates with a base represented by a three-dimensional deformable layered medium is considered. The possibility of occurrence of a starting earthquake in such a block structure is investigated. For this purpose, two states of this medium in the static mode are considered. In the first case, the semi-infinite lithospheric plates in the form of half-planes are at a distance so that the distance between the end faces is different from zero. In the second case, the lithospheric plates come together to zero spacing between them. Calculations have shown that in this case more complex movements of the Earth’s surface are possible. Among such movements are the cases described in our previous publications [1, 2].     

Vibrations of a polygonal plate having orthogonal straight edges by an extended rayleigh-ritz method

An extended Rayleigh-Ritz method is presented for solving vibration problems of a polygonal plate having orthogonal straight edges. The polygonal plate is considered as an assemblage of several rectangular plates. For each element rectangular plate, the transverse displacement is approximated by interpolation functions corresponding to unknown displacements and slopes at the discrete points which are chosen along the edges, and series of trial functions which satisfy homogeneous artificial boundary conditions. By minimizing the energy functional corresponding to the assumed displacement function, the dynamic stiffness matrix of the element rectangular plate, which is similar to that obtained in the finite element method, is derived. The dynamic stiffness matrix of the whole system is obtained by summing up those of the element rectangular plates. Numerical results are presented for the natural frequencies and mode shapes of cantilever L-shaped and T-shaped plates.     

Evolution of the hot image effect in high-power laser system with cascaded Kerr medium

Based on the diffraction theory model for hot image formation, the evolution of the hot image effect in a high-power laser system structured by cascaded Kerr medium with gain and loss is numerically investigated. For a sample simulation system with five pieces of Kerr medium plates arrayed in parallel, the dependences of the peak intensity and the location of the hot image on different parameters of the scatterer and the system are analyzed. It is shown that the possible damage spots caused by the beam peak intensity would be repetitious in the laser system with cascaded medium. Arranging the Kerr medium plates and the downstream optical components in a rational way (for instance increasing the interval between two adjacent medium plates, arranging the Kerr medium plates aperiodically or certain optical components properly) may be helpful to relax or avoid the optical damage caused by the hot image effect.     

FREE VIBRATIONS OF SIMPLY SUPPORTED AND MULTILAYERED MAGNETO-ELECTRO-ELASTIC PLATES

Analytical solutions are derived for free vibrations of three-dimensional, linear anisotropic, magneto-electro-elastic, and multilayered rectangular plates under simply supported edge conditions. For any homogeneous layer, we construct the general solution in terms of a simple formalism that resembles the Stroh formalism, from which any physical quantities can be solved for given boundary conditions. In particular, the dispersion equation that characterizes the relationship between the natural frequency and wavenumber can be obtained in a simple form. For multilayered plates, we derive the dispersion relation in terms of the propagator matrices. The present solution includes all previous solutions, such as piezoelectric, piezomagnetic, and purely elastic solutions as special cases, and can serve as benchmarks to various thick plate theories and numerical methods used for the modelling of layered composite structures. Typical natural frequencies and mode shapes are presented for sandwich piezoelectric/piezomagnetic plates. It is shown clearly that some of the modes are purely elastic while others are fully coupled with piezoelectric/piezomagnetic quantities, with the latter depending strongly upon the material property and stacking sequence. These frequency and mode shape features could be of particular interest to the analysis and design of various “smart” sensors/actuators constructed from magneto-electro-elastic composite laminates.     

Double Symplectic Eigenfunction Expansion Method of Free Vibration of Rectangular Thin Plates

The free vibration problem of rectangular thin plates is rewritten as a new upper triangular matrix differential system. For the associated operator matrix, we find that the two diagonal block operators are Hamiltonian. Moreover, the existence and completeness of normed symplectic orthogonal eigenfunction systems of these two block operators are demonstrated. Based on the completeness, the general solution of the free vibration of rectangular thin plates is given by double symplectic eigenfunction expansion method.     

The effect of magnetic fields and boundary conditions on the shear flow of nematics

The flow of a nematic liquid crystal between plane parallel plates, with one plate moving with uniform velocity relative to the other, is discussed. The apparent viscosity, orientation and velocity profiles are computed forp-azoxyanisole as functions of shear rate and magnetic field for symmetric and asymmetric molecular alignment at the plates. For symmetric homeotropic boundary condition, a magnetic field applied along the flow direction exhibits a threshold reminiscent of a Freedericksz transition in the hydrostatic case. In general the apparent viscosity for the asymmetric boundary condition is less than that for the symmetric case.     

Numerical Investigation on Vibration Performance of Flexible Plates Actuated by Pneumatic Artificial Muscle

This paper theoretically introduced the feasibility of changing the vibration characteristics of flexible plates by using bio-inspired, extremely light, and powerful Pneumatic Artificial Muscle (PAM) actuators. Many structural plates or shells are typically flexible and show high vibration sensitivity. For this reason, this paper provides a way to achieve active vibration control for suppressing the oscillations of these structures to meet strict stability, safety, and comfort requirements. The dynamic behaviors of the designed plates are modeled by using the finite element (FE) method. As is known, the output force vs. contraction curve of PAM is nonlinear generally. In this present finite element model, the maximum forces provided by PAM in different air pressure are adopted as controlling forces for applying for the plate. The non-linearity between the output force and displacement of PAM is avoided in this study. The dynamic behaviors of plates with several independent groups of controlling forces are observed and studied. The results show that the natural frequencies of the plate can be varying and the max amplitude decreases significantly if the controlling forces are applied. The present work also demonstrates the potential of the PAM actuators as valid means for damping out the vibration of flexible systems.     

不同材料参数薄板振动中的热力耦合效应

利用有限Hankel变换法,导出了周界等温-弹性支撑圆薄板在激光束辐照下的轴对称耦合热弹性弯曲振动近似解;针对具有不同弹性模量和热膨胀系数的薄板进行了热力耦合和非耦合弯曲振动的解析和有限元计算与分析。结果表明：热力耦合效应使薄板振动的振幅和周期都有所减小,其程度与材料的性能参数（如弹性模量和热膨胀系数等）密切相关,材料弹性模量和热膨胀系数越大,板振动中的热力耦合效应就越明显。     

The vibration and post-buckling of geometrically imperfect,simply supported,rectangular plates under uni-axial loading,part II: Experimental investigation

The paper describes the experimental part of a theoretical and experimental study of the post-buckling and free vibrational behaviour of thin, rectangular, simply supported plates having initial geometrical imperfection and subject to uni-axially applied, in-plane, compressive loads. The experimental apparatus and procedure used are described. The fundamental natural frequency and central deflection of several plates of different thickness and degree of initial imperfection, subject to loads varying from zero to several times the critical buckling value, are compared with values predicted by using the Rayleigh-Ritz solution described in the companion paper. For one plate, comparisons of theoretically predicted and experimentally measured strains are given. Close agreement is shown to exist between the theoretical and experimental results. An approximate linear relationship between a load-frequency parameter and the central deflection, discussed in the theoretical study, is also shown to exist for the experimental plates.     

Anomalous dispersion of electromagnetic eigenwaves in a sandwich structure with a gap

A system of two different isotropic dielectric plates aligned parallel to each other and separated by a gap is considered. For a sufficiently large gap, electromagnetic eigenwaves in the plates are virtually independent and dispersion curves for the studied system are characterized by an infinite set of intersections. A decrease in the gap thickness leads to coupling between the wave fields and the disappearance of the intersection points due to divergence of the dispersion curves. Each of the dispersion curves of the spectrum transformed under the action of coupling is formed by a set of adjacent portions of initially independent branches corresponding to different plates. A gradual change in the frequency along one of the new dispersion curves results in a periodic displacement of the localization zone of the wave field from one plate to the other.