Decay rates at low and high frequencies for a plate equation with feedback concentrated in interior curves

In this paper we study the stabilization of plate vibrations by means of piezoelectric actuators. In this situation the geometric control condition of Bardos, Lebeau and Rauch [6] is not satisfied. We prove that we have exponential stability for the low frequencies but not for the high frequencies. We give an explicit decay rate for regular initial data at high frequencies while clarifying the behavior of the constant which intervenes in this estimation there function of the frequency of cut n. The method used is based on some trace regularity which reduces stability to some observability inequalities for the corresponding undamped problem. Moreover, we show numerically at low frequencies, that the optimal location of the actuator is the center of the domain Ω. Research supported by the RIP program of Oberwolfach Institut and by the Tunisian Ministry for Scientific Research and Technology (MRST) under Grant 02/UR/15-01. Research supported by the RIP program of Oberwolfach Institut. (Received: September 17, 2003; revised: February 26, 2004)     

Decay rates at low and high frequencies for a plate equation with feedback concentrated in interior curves

In this paper we study the stabilization of plate vibrations by means of piezoelectric actuators. In this situation the geometric control condition of Bardos, Lebeau and Rauch [6] is not satisfied. We prove that we have exponential stability for the low frequencies but not for the high frequencies. We give an explicit decay rate for regular initial data at high frequencies while clarifying the behavior of the constant which intervenes in this estimation there function of the frequency of cut n. The method used is based on some trace regularity which reduces stability to some observability inequalities for the corresponding undamped problem. Moreover, we show numerically at low frequencies, that the optimal location of the actuator is the center of the domain Ω.     

The dynamics of the relationship between spot and futures markets under high and low variance regimes

This investigation is one of the first studies to examine the dynamics of the relationship between spot and futures markets using the Markov‐switching vector error correction model. Three mature stock markets including the U.S. S&P500, the U.K. FTSE100 and the German DAX 30, and two emerging markets including the Brazil Bovespa and the Hungary BSI, are used to test the model, and the differences between the two sets of markets are examined. The empirical findings of this study are consistent with the following notions. First, after filtering out the high variance regime, the futures price is shown to lead the spot price in the price discovery process, as demonstrated by prior studies; conversely, the spot market is more informationally efficient than the futures market under the high variance condition. Second, the price adjustment process triggered by arbitrage trading between spot and futures markets during a high variance state is greater in scale than that based on a low variance state, and the degree of the co‐movement between spot and futures markets is significantly reduced during the high variance state. Third, a crisis condition involved in the high variance state is defined for the two emerging markets, whereas an unusual condition is presented for the three mature markets. Last, the lagged spot–futures price deviations perform as an information variable for the variance‐turning process. However, the portion of the variance‐switching process accounted for by this signal variable is statistically marginal for the three mature markets selected for this study. Copyright © 2008 John Wiley & Sons, Ltd.     

Stability and local bifurcation analysis of functionally graded material plate under transversal and in-plane excitations

In this paper, stability and local bifurcation behaviors for a simply supported functionally graded material (FGM) rectangular plate subjected to the transversal and in-plane excitations in the uniform thermal environment are investigated using both analytical and numerical methods. Three kinds of degenerated equilibrium points of the bifurcation response equations are considered, which are characterized by a double zero eigenvalues, a simple zero and a pair of pure imaginary eigenvalues as well as two pairs of pure imaginary eigenvalues in nonresonant case, respectively. With the aid of Maple and normal form theory, the explicit expressions of transition curves are obtained, which may lead to static bifurcation, Hopf bifurcation and 2-D torus bifurcation. Finally, the numerical solutions obtained by using fourth-order Runge–Kutta method agree with the analytic predictions.     

Application of the technical theory to sandwich shells with a solid polymeric core

The possibilities of using the technical theory for analyzing cylindrical sandwich shells with a core of low-modulus polymeric material are considered. It is shown to be necessary to make assumptions concerning the distribution of the deformations over the elements of the three-layer section and to take account of the shear strains in the core, the flexural rigidity in the longitudinal direction, and the Poisson ratio in determining the forces and moments. The theoretical conclusions have been experimentally confirmed by static tests on a model.All-Union Structural Engineering Correspondence Institute, Moscow. Translated from Mekhanika Polimerov, No. 2, pp. 298–304, March–April, 1973.     

Buckling delamination of a sandwich plate-strip with piezoelectric face and elastic core layers

In this study, the buckling delamination problem of a sandwich plate-strip with a piezoelectric face and elastic core layers is studied. It is assumed that the plate-strip is simply supported and grounded along its two parallel ends and is subjected to uniformly-distributed compressive forces on these ends. Moreover, we suppose that the plate-strip has two interface inner cracks between the face and the core layers and it is also supposed that before the plate-strip is loaded (i.e. in the natural state), the surfaces of these cracks have insignificant initial imperfections. Due to compressive forces acting along the cracks we investigate the evolution of the initial imperfections of the cracks’ surfaces. Hence, the values of the critical buckling delamination force of the considered plate-strip are determined from the criteria, according to which, the considered initial imperfections of the cracks’ surfaces grow indefinitely by the compressive forces. Mathematical modeling of the considered problem is formulated within the scope of the exact nonlinear equations of electro-elasticity in the framework of the piecewise homogeneous body model, the solution of which is found numerically by employing the finite elements method. Numerical results showing the influences of the geometrical and material parameters as well as the coupling of the electrical and mechanical fields on the values of the critical force are presented and analyzed.     

Solvability of physically and geometrically nonlinear problem of the theory of sandwich plates with transversally-soft core

The paper presents a generalized statement of geometrically and physically nonlinear problem of the equilibrium of sandwich plate with transversally-soft core. Generalized statement is formulated as a problem of finding a saddle point of a functional. We investigate the properties of this functional. These properties allow to prove a theorem of solvability of variational problem under consideration.     

Stress-strain state and stability of composite sandwich shells with a scaling zone between the core and facings

A finite element model is presented for analyzing the strength and stability of sandwich shells of arbitrary configuration with an adhesion failure zone between the core and one of the facings. The model is based on the assumptions that both facings are laminated Timoshenko-type composite shells, only transverse shear stresses in the core and normal stresses in the thickness direction have nonzero values, a free slip in the tangential plane in the adhesion failure zone and unilateral contact along the normal are possible, and the prebuckling state in the stability problem is linear. Biquadratic nine-node approximations for all functions and numerical integration were used. The displacements and rotation angles of the normals toward the facings as well as stresses in the core are taken as global degrees of freedom. The algebraic problem is solved using a special step-by-step procedure of determining the contact area in the scaling zone and employing unilateral constraints for some of the unknowns. Numerical examples are also given.Translated from Mekhanika Kompozitnykh Materialov, Vol. 29, No. 5, pp. 640–652, September–October, 1993.     

Free-edge effects in a cylindrical sandwich panel with a flexible core and laminated composite face sheets

In the present study, the static response of cylindrical sandwich panels with a flexible core is investigated. The face sheets are considered as composite laminates with a cross-ply lay-up and the core as a flexible elastic medium. The flexibility of the low-strength core leads to high stress concentrations in terms of peeling stresses between the face sheets and the core at edges of the sandwich panel. To take into account the compressibility of the core and to determine the free-edge stresses of sandwich structures accurately, the Reddy layerwise theory (LWT) is used in this paper. The paper outlines the mathematical formulation, along with a numerical study, of a cylindrical sandwich panel with two simply supported and two free edges under a transverse load. The formulation includes the derivation of field equations along with boundary conditions. A Levy-type solution procedure is performed to determine the distributions of stresses and strains. In the numerical study, first a comparison is made with results from the commercial finite-element software ANSYS to verify the LWT results. Finally, a parametric study is conducted, and the effect caused by varying different parameters, such as the radii of curvature and the core to face sheet thickness ratio, on the results are investigated. The results obtained demonstrate a good agreement between LWT and FEM solutions and show increasing interlaminar stresses in the free edge of the sandwich panel     

On the propagation of Lamb waves in a sandwich plate made of compressible materials with finite initial strains

The propagation of flexural Lamb waves in a prestrained sandwich plate made from compressible highly elastic materials is investigated within the scope of a piecewise homogeneous body model by utilizing TLTEWISB. The mechanical relations of layer materials are described by a harmonic-type potential, and numerical results are obtained for the first and second vibration modes. According to the results, the influence of problem parameters and of the initial stretching strain along the layers on the wave propagation speed is examined. The asymptotic values of the speed are considered in the cases of short and long wavelengths, and the influence of the initial strains on these asymptotic (limit) values are also analyzed. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 44, No. 2, pp. 231–244, March–April, 2008.     

Multi-instability of gap solitons and dynamics of nonlinear excitations in the array of optical fibers

The nonlinear localized excitations and gap multi-instability phenomena in the array of optical fibers are investigated analytically by means of the method of multiple scales combined with the averaging method. The perturbative analysis in nonlinear discrete systems leads to a system of four coupled nonlinear oscillated equations without second members. The method of multiple scales is used to show the different possibilities of instability. The averaging method is considered to show how the angular frequency received the perturbation with the evolution of time.     

Qualitative results on the dynamics of a Berger plate with nonlinear boundary damping

The dynamics of a (nonlinear) Berger plate in the absence of rotational inertia are considered with inhomogeneous boundary conditions. In our analysis, we consider boundary damping in two scenarios: (i) free plate boundary conditions, or (ii) hinged-type boundary conditions. In either situation, the nonlinearity gives rise to complicating boundary terms. In the case of free boundary conditions we show that well-posedness of finite-energy solutions can be obtained via highly nonlinear boundary dissipation. Additionally, we show the existence of a compact global attractor for the dynamics in the presence of hinged-type boundary dissipation (assuming a geometric condition on the entire boundary (Lagnese, 1989)). To obtain the existence of the attractor we explicitly construct the absorbing set for the dynamics by employing energy methods that: (i) exploit the structure of the Berger nonlinearity, and (ii) utilize sharp trace results for the Euler–Bernoulli plate in Lasiecka and Triggiani (1993).We provide a parallel commentary (from a mathematical point of view) to the discussion of modeling with Berger versus von Karman nonlinearities: to wit, we describe the derivation of each nonlinear dynamics and a discussion of the validity of the Berger approximation. We believe this discussion to be of broad value across engineering and applied mathematics communities.     

Multimodal admittance method in waveguides and singularity behavior at high frequencies

This work presents a multimodal method for the propagation in a waveguide with varying height and its relation to trapped modes or quasi-trapped modes. The coupled mode equations are obtained by projecting the Helmholtz equation on the local transverse modes. To solve this problem we integrate the Riccati equation governing the admittance matrix (Dirichlet-to-Neumann operator). For many propagating modes, i.e. at high frequencies, the numerical integration of the Riccati equation shows that the rule is that this matrix has quasi-singularities associated to quasi-trapped modes.     

Isolation and the high/low hierarchy

 Say that a d.c.e. degree d is isolated by a c.e. degree b, if b<d and any c.e. degree c below d is also below b, namely, b is the largest c.e. degree below d. Wu proved that there is a high d.c.e. degree d isolated by a low₂ c.e. degree b. In this paper, we improve this result by making b low, not merely low₂. RID="ID=" <E5>Mathematics Subject Classification (2000):</E5> 03D25, 03D30, 03D35 RID="ID=" <E5>Key words or phrases:</E5> Computably enumerable set &ndash; d.c.e. degree &ndash; Isolation &ndash; High/low hierarchy RID="ID=" Ishmukhametov's research is supported by RFBR grant 01-01-00733, and Wu's research is supported by the Marsden Fund of New Zealand. Wu would like to thank his supervisor, Prof. Rod Downey, for his many helpful suggestions and comments. Received: 31 January 2001 / Published online: 25 February 2002 RID=" ID=" <E5>Mathematics Subject Classification (2000):</E5> 03D25, 03D30, 03D35 RID=" ID=" <E5>Key words or phrases:</E5> Computably enumerable set &ndash; d.c.e. degree &ndash; Isolation &ndash; High/low hierarchy RID=" ID=" Ishmukhametov's research is supported by RFBR grant 01-01-00733, and Wu's research is supported by the Marsden Fund of New Zealand. Wu would like to thank his supervisor, Prof. Rod Downey, for his many helpful suggestions and comments.