Hybrid control of vibrations of a smart von Kármán plate

The suppression of large vibrations of a smart thin elastic rectangular von Kármán’s plate is considered. The plate is subjected to external disturbances and generalized control forces produced by electromechanical feedback. The considered nonlinear initial-boundary value problem is spatially discretized by means of the time spectral method. The implicit Newmark-β iterative method is employed for the time integration of the obtained system of nonlinear equations of motion. Nonlinear controllers are designed, based on a fuzzy inference system. Two numerical algorithms involving a general control of displacement/velocity and a direct control of the Fourier coefficients are proposed. The techniques have been implemented within MATLAB environment with the use of the fuzzy logic toolbox. Numerical examples are presented.     

BEM formulation for von Kármán plates

This work deals with nonlinear geometric plates in the context of von Kármán's theory. The formulation is written such that only the boundary in-plane displacement and deflection integral equations for boundary collocations are required. At internal points, only out-of-plane rotation, curvature and in-plane internal force representations are used. Thus, only integral representations of these values are derived. The nonlinear system of equations is derived by approximating all densities in the domain integrals as single values, which therefore reduces the computational effort needed to evaluate the domain value influences. Hyper-singular equations are avoided by approximating the domain values using only internal nodes. The solution is obtained using a Newton scheme for which a consistent tangent operator was derived.     

Application of the boundary-domain element method to the pre/post-buckling problem of von Kármán plates

Based on the BEM formulations for the finite deflection problem of von-Kármán-type plates, this paper presents an incremental boundary-domain element method for the pre/post-buckling problem of thin elastic plates. As the governing equations involve the coupled in-plane and out-of-plane deformations as the nonlinear terms, the boundary integral equations are formulated in terms of the increment by using the fundamental solutions for the linear parts of the differential operators. Some of the innovations are made in order to improve the accuracy and accelerate the convergence of the solution procedure. The load-incrementation method and also the arc-length-incrementation method are employed for each incremental step. Numerical analysis is carried out and the results are compared with the available analytical solutions to demonstrate the effectiveness of the proposed method.     

A consistent deduction of von Kármán-type plate theories from three-dimensional nonlinear continuum mechanics

Summary Classical and refined plate theories derived from linear continuum mechanics lead to correct results only if the transverse deflection of the plate is small compared to its thickness. In the case of large deflections, geometrical nonlinearities have to be incorporated. For the classical Kirchhoff plate theory, a suitable extension for moderate rotations has been presented by von Kármán in 1911.Starting from the three-dimensional equations of nonlinear continuum mechanics, a family of von Kármán-type plate theories is deduced. For the derivation, the kinematical variables are replaced by a series representation and the principle of virtual displacements is used. It can be shown that most plate theories can be obtained from this type of theory and that the kinematical assumptions must fulfill certain conditions to obtain a solvable system of equations.     

Absolute instability of the von Kármán, Bödewadt and Ekman flows between a rotating disc and a stationary lid

The stability of a family of boundary-layer flows, which includes the von Kármán, B?dewadt and Ekman flows for a rotating incompressible fluid between a rotating disc and a stationary lid, is investigated. Numerical computations with the use of a spectral method are carried out to analyse absolute and convective instability. It is shown that the stability of the system is enhanced with a decrease in distance between the disc and the lid.     

A perturbation solution of von-Kármán circular plates with different moduli in tension and compression under concentrated force

By modifying classical von-Kármán equations, we established bimodular von-Kármán equations of thin plates with different moduli in tension and compression. Adopting central deflection as a perturbation parameter, we used a perturbation method to solve the equations under various boundary conditions, including rigidly clamped, loosely clamped, simply hinged, and simply supported. The relation of load versus central deflection and stress formulas were derived via the perturbation solution obtained. The numerical simulation also shows that the perturbation solution based on central deflection is overall valid. The results indicate that when the compressive modulus of materials is greater than the tensile one, the bearing capacity of the plate will be further strengthened, which should be considered in the analysis and design of plate-like structures with obvious bimodular effect. Moreover, by comparing with the case under uniformly distributed load, the plate-membrane transition under centrally concentrated force presents discontinuity to some extent.     

Determining wake flow parameters for two counterphase synchronized von Kármán vortex streets from configuration stability analysis

The problem of determining the parameters of near wake flow past a pair of cylindrical bluff bodies from one-point spectra of the velocity pulsation is solved in the limit of an inviscid incompressible flow. For this purpose, the stability of wake configuration with respect to infinitesimal perturbations of the equilibrium localization of the vortices is analyzed within the framework of a flow model of two counterphase synchronized von Kármán vortex streets. A necessary condition for the flow stability is determined.     

A general method for analyzing moderately large deflections of a non-uniform beam: an infinite Bernoulli–Euler–von Kármán beam on a nonlinear elastic foundation

The present paper concerns a semi-analytical procedure for moderately large deflections of an infinite non-uniform static beam resting on a nonlinear elastic foundation. To construct the procedure, we first derive a nonlinear differential equation of a Bernoulli–Euler–von Kármán “non-uniform” beam on a “nonlinear” elastic foundation, where geometrical nonlinearities due to moderately large deflection and beam non-uniformity are effectively taken into account. The nonlinear differential equation is transformed into an equivalent system of nonlinear integral equations by a canonical representation. Based on the equivalent system, we propose a method for the moderately large deflection analysis as a general approach to an infinite non-uniform beam having a variable flexural rigidity and a variable axial rigidity. The method proposed here is a functional iterative procedure, not only fairly simple but straightforward to apply. Here, a parameter, called a base point of the method, is also newly introduced, which controls its convergence rate. An illustrative example is presented to investigate the validity of the method, which shows that just a few iterations are only demanded for a reasonable solution.     

Photothermal Radiometry Characterization of Limestone Rocks from the Península of Yucatán

Limestone is a sedimentary rock composed of calcium carbonate with minor amounts of silica, iron oxide, clay, dolomite, and organic material. These types of stones have been used extensively as building materials. Due to this, determination of their thermal properties is of the utmost importance. These properties depend on the microstructure and composition of each type of rock. In this study, the effect of the thermal treatment of three different limestone rocks from the Peninsula of Yucatán were studied, in the range from 100?°C up to 600?°C, using photothermal radiometry. These studies were complemented by the characterization of the crystalline phases using X-ray diffraction and effective porosity measurements performed by the saturation technique. It is shown that the thermal diffusivity, thermal conductivity, and specific heat of the limestone decrease as the temperature increases. This behavior can be related to increases in microcracks and effective porosity due to thermal treatments.     

A discussion on “On the solution of a class of fuzzy system of linear equations”

The paper is a discussion on the results obtained by Salkuyeh in the article (Sādhānā 40(2):369–377, 2015). Salkuyeh considers the solution of the fuzzy system of linear equations (FSLE) and gives numerical examples where the results of FSLE are calculated. It was shown that the results by Salkuyeh are not full solutions and do not satisfy equivalent forms of the analysed FSLE. The paper presents a multidimensional approach to finding the solution of the FSLE. On the basis of the notation of the horizontal fuzzy number, a full solution that satisfies the FSLE and its equivalent forms is obtained.     

A Study of the Isothermal Section in the W-Fe-Co Ternary System at 1380K

The purpose of the present paper isto determine the composition range ofthe θ intermediate phase and establishthe phase equilibria relation in theW-Fe-Co ternary system.The experimentalresult was compared with the calculationof phase diagrams in Guillermet's work.1.Experimental ProcedureThe diffusion couple specimen usedin this investigation was prepared fromthe tungsten bar (99.96 wt-%),electro-     

A Deviation Function Used for the Secondary Realization of the ITS-90 in the Subrange from 83.8033 K to 273.16 K

This article proposes a correlation relation between the resistance ratios of the triple points of argon and mercury. By this relation, the resistance ratio of the triple point of argon can be extrapolated from that of mercury, and a deviation function which is defined in the range from 83.8058 K to 273.16 K can be determined from only the calibration values at the triple points of water and mercury. It is a close approximation to the ITS-90 deviation function in the subrange. Using it, the calibration at the triple point of argon can be saved. Twenty-five standard platinum resistance thermometers are used to check the function. The errors are less than 5 mK. It is sufficient for secondary measurements.     

Conservative interpolation for the boundary integral solution of the Navier–Stokes equations

The Dual Reciprocity Method (DRM) is a technique to transform the domain integrals that appear in the boundary element method into equivalent boundary integrals. In this approach the non-linear terms are usually approximated by mathematical interpolation applied to the convective terms of the form of the Navier–Stokes equations. In this paper we introduce a conservative interpolation scheme that satisfies the continuity equation and performs better than pure mathematical interpolation. The new scheme together with a subdomain variation of the dual reciprocity method allows better approximation of the non-linear terms in the Navier–Stokes equations for moderate Reynolds number. Received: 21 January 2000     

A combined BEM–FEM model for the velocity–vorticity formulation of the Navier–Stokes equations in three dimensions

This paper describes a combined boundary element and finite element model for the solution of velocity–vorticity formulation of the Navier–Stokes equations in three dimensions. In the velocity–vorticity formulation of the Navier–Stokes equations, the Poisson type velocity equations are solved using the boundary element method (BEM) and the vorticity transport equations are solved using the finite element method (FEM) and both are combined to form an iterative scheme. The vorticity boundary conditions for the solution of vorticity transport equations are exactly obtained directly from the BEM solution of the velocity Poisson equations. Here the results of medium Reynolds number of up to 1000, in a typical cubic cavity flow are presented and compared with other numerical models. The combined BEM–FEM model are generally in fairly close agreement with the results of other numerical models, even for a coarse mesh.     

A three-step Oseen correction method for the steady Navier–Stokes equations

We present and analyze a two-grid scheme based on mixed finite element approximations for the steady incompressible Navier–Stokes equations. This numerical scheme aims at the simulations of high Reynolds number flows and consists of three steps: in the first step, we solve a finite element variational multiscale-stabilized nonlinear Navier–Stokes system on a coarse mesh, and then, in the second and third steps, we solve Oseen-linearized and -stabilized problems which have the same stiffness matrices with only different right-hand sides on a fine mesh. We provide error bounds for the approximate solutions, derive algorithmic parameter scalings from the analysis, and present some numerical results to verify the theoretical predictions and demonstrate the effectiveness of the proposed method.     

A variationally separable splitting for the generalized-α method for parabolic equations

We present a variationally separable splitting technique for the generalized-α method for solving parabolic partial differential equations. We develop a technique for a tensor-product mesh which results in a solver with a linear cost with respect to the total number of degrees of freedom in the system for multidimensional problems. We consider finite elements and isogeometric analysis for the spatial discretization. The overall method maintains user-controlled high-frequency dissipation while minimizing unwanted low-frequency dissipation. The method has second-order accuracy in time and optimal rates (h^p+1 in L² norm of u and h^p in L² norm of ∇u) in space. We present the spectral analysis on the amplification matrix to establish that the method is unconditionally stable. Various numerical examples illustrate the performance of the overall methodology and show the optimal approximation accuracy.     

Padé approximants for the numerical solution of singular integral equations

Summary A method for accelerating the convergence of the numerical solution of a singular integral equation, based on Padé Approximants, is given in this paper. At first the general form of the Padé Table and of the “epsilon” algorithm are presented. Taking into consideration the classical quadrature method, based on the Gauss-Jacobi quadrature rule, an approximate formula is derived for the unknown density function of the Cauchy-type singular integral equation or of the equivalent Fredholm integral equation. In this formula applying the “epsilon” algorithm to the solution for the stress intensity factors, the convergence is achieved after a few operations. The number of numerical operations required for the determination of stress intensity factors is considerable reduced, when compared to the number of operations required for a classical type of solution. Illustrative examples are given, indicating the efficiency of the method.     

Thermodynamic properties of (TeO2) n (WO3)1 − n glasses in the range 0–650 K

The heat capacity of (TeO₂) _n (WO₃)_{1 ? n} tellurite glasses with n = 0.75, 0.78, 0.85, and 0.90 has been determined using precision adiabatic calorimetry (6–350 K) and dynamic scanning calorimetry (320–650 K), and the thermodynamic characteristics of their glassy state and devitrification have been evaluated. The experimental heat capacity data have been used to calculate the standard thermodynamic functions of the glassy and supercooled liquid states at temperatures from T → 0 to 650 K: heat capacity C _p ⁰ (T), enthalpy H ⁰(T) — H ⁰(0), entropy S ⁰(T), and Gibbs function G ⁰(T) — H ⁰(0). The character of structural heterodynamicity of the tellurite glasses has been assessed by processing the low-temperature heat capacity data using the multifractal formulation of the Debye theory of heat capacity of solids. The composition dependences of the devitrification temperature and 298.15-K thermodynamic functions have been obtained, and the 298.15-K C _p ⁰ of tellurium dioxide has been estimated. The thermal and thermodynamic properties of the (TeO₂) _n (WO₃)_{1 ? n} tellurite glasses have been compared with those of (TeO₂) _n (ZnO)_{1 ? n} glasses.     

A correlation of ΔK th-value with the exponent,m, in the equation of fatigue crack growth for various steels

Literature (mainly in Japan) relating to fatigue-crack-growth-data at R=0 in an air environment for a wide range of steels is reviewed with particular attention to the threshold stress intensity, K _th. The collected data are analyzed in terms of the exponent, m(the slope of the linear portion of the log(da/dN)-logD relationship) by taking account of microstructure, material strength, fracture toughness and specimen thickness. The mean rate of fatigue crack growth for ductile steels in the range from the intermediate growth rate to threshold level and the relevant threshold values at R=0, K _th0, can be represented asda/dN = 1.700 × 10^-4(K103.6)^m - 10^-6andK _th0 = 103.6(5.88 × 10^-3)^1/m,where (da/dN) and K are measured by the units of mm/cycle and kgf/mm^3/2, respectively. Contrary to this, in the case of extremely brittle steels with K _IC-value below 200 kgf/mm^3/2 (the fracture occurs by the intergranular separation), the relationships are given byda/dN = 2.893 × 10^-5(K/49.94)^m - 2.5 × 10^-7andK _th0 = 49.94(8.64 × 10^-3)^1/m.

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Résumé La littérature, principalement japonaise, relative aux données sur la propagation des fissures de fatigue pour R=0 dans un environnement d'air et pour une large catégorie d'aciers fait l'objet d'une revue avec une attention particulière pour l'intensité de contrainte de seuil, K _th. Les données collectées sont analysées en terme de l'éxposant m (qui représente la pente de la portion linéaire de la relation log(da/dN)-log K, en tenant compte de la microstructure, de la résistance du matériau, de la ténacité à la rupture et de l'épaisseur de l'éprouvette. La vitesse moyenne de propagation d'une fissure de fatigue dans le cas des aciers ductiles dans la fourchette entre la vitesse intermédiaire de fissuration et le niveau critique, et les valeurs correspondantes de seuil à R=0, K _th0 peuvent être représentées par la relation:da/dN = 1.700 × 10^-4(K103.6)^m - 10^-6etK _th0 = 103.6(5.88 × 10^-3)^1/m,où (da/dN) et K sont mesurés en unités de mm/cycle et en kgf/mm^3/2 respectivement. En contraste, dans le cas d'aciers extrèmement fragiles avec des valeurs K _IC en dessous de 200 kgf/mm^3/2 (la rupture se produit par une séparation intergranulaire), ces relations sont données par:da/dN = 2.893 × 10^-5(K/49.94)^m - 2.5 × 10^-7etK _th0 = 49.94(8.64 × 10^-3)^1/m.