Nonlinear bending and buckling for strain gradient elastic beams

Nonlinear bending of strain gradient elastic thin beams is studied adopting Bernoulli–Euler principle. Simple nonlinear strain gradient elastic theory with surface energy is employed. In fact linear constitutive relations for strain gradient elastic theory with nonlinear strains are adopted. The governing beam equations with its boundary conditions are derived through a variational method. New terms are considered, already introduced for linear cases, indicating the importance of the cross-section area, in addition to moment of inertia in bending of thin beams. Those terms strongly increase the stiffness of the thin beam. The non-linear theory is applied to buckling problems of thin beams, especially in the study of the postbuckling behaviour.     

Free vibration analysis of spatial Bernoulli–Euler and Rayleigh curved beams using isogeometric approach

This paper deals with the linear free vibration analysis of Bernoulli–Euler and Rayleigh curved beams using isogeometric approach. The geometry of the beam as well as the displacement field are defined using the NURBS basis functions which present the basic concept of the isogeometric analysis. A novel approach based on the fundamental relations of the differential geometry and Cauchy continuum beam model is presented and applied to derive the stiffness and consistent mass matrices of the corresponding spatial curved beam element. In the Bernoulli–Euler beam element only translational and torsional inertia are taken into account, while the Rayleigh beam element takes all inertial terms into consideration. Due to their formulation, isogeometric beam elements can be used for the dynamic analysis of spatial curved beams. Several illustrative examples have been chosen in order to check the convergence and accuracy of the proposed method. The results have been compared with the available data from the literature as well as with the finite element solutions.     

Identification of crack in a rotor system based on wavelet finite element method

The dynamics and diagnosis of cracked rotor have been gaining importance in recent years. In the present study a model-based crack identification method is proposed for estimating crack location and size in shafts. The rotor system has been modeled using finite element method of B-spline wavelet on the interval (FEM BSWI), while the crack is considered through local stiffness change. Based on Rayleigh beam theory, the influences of rotatory inertia on the flexural vibrations of the rotor system are examined to construct BSWI Rayleigh beam element. The slender shaft and stiffness disc are modeled by BSWI Rayleigh–Euler beam element and BSWI Rayleigh–Timoshenko beam element, respectively. Then the crack identification forward and inverse problems are solved by using surface-fitting technique and contour-plotting method. The experimental examples are given to verify the validity of the BSWI beam element for crack identification in a rotor system. From experimental results, the new method can be applied to prognosis and quantitative diagnosis of crack in a rotor system.     

Fluctuation forces in a three-layer medium with rough boundaries. IV. Calculations in the second order of perturbation theory (Casimir range)

The perturbation theory developed in the previous paper of the series is used to calculate the correction to the Casimir force due to the surface roughness.Scientific Council on the Problem Complex Kibernetika, USSR Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 92, No. 1, pp. 113–118, July, 1992.     

Strain gradient beam model for dynamics of microscale pipes conveying fluid

Based on the strain gradient theory, we present a microstructure-dependent Bernoulli–Euler model to analyze the vibration and stability of microscale pipes conveying fluid. The equation of motion and boundary conditions are derived using Hamilton’s principle. The proposed strain gradient beam model contains three material length scale parameters to capture the size effect. This new model may be reduced to the modified couple stress beam model when two of these three material length scale parameters vanish and may be reduced to the classical beam model in the absence of all the material length scale parameters. From the numerical calculations for micropipes with both ends positively supported, it is found that the natural frequency and the critical flow velocity are size-dependent. The results show that the microscale pipe displays remarkable size effect when its outside diameter becomes comparable to the material length scale parameter, while the size effect is almost diminishing as the diameter is far greater than the material length scale parameter. Moreover, the size effect predicted by the current strain gradient beam model is stronger than that predicted by the modified couple stress beam model, since two other material length scale parameters have been accounted for in the former.     

Applied models of deformation of a three-layered base

Using methods of perturbation theory, some applied models of interactions of strongly nonhomogeneous layered bases of the following forms are constructed: three-layered stack that lies on an absolutely rigid base and is composed from smooth (Problem 1) and linked (Problem 2) layers; and two smooth layers that lie on an elastic half-space. Some applied models of deformation for different relations of elastic parameters of layers are obtained.Translated from Dinamicheskie Sistemy, No. 8, pp. 37–40, 1989.     

Fluctuation forces in a three-layer medium with rough boundaries. III. Aspects of perturbation theory in the Casimir range

The problem of calculating the Casimir force that acts in a system of two metallic plates with rough boundaries separated by an insulating layer is considered. An algorithm for constructing a perturbation series in the small function of the roughness profile is developed.Scientific Council on the Problem Complex Kibernetika, USSR Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 91, No. 3, pp. 474–482, June, 1992.     

Fluctuation Theory for Moran''s Genetics Model

A genetics model of Moran—which includes the Bernoulli–Laplace urn as a special case—is examined to give a picture of the model's fluctuation theory and asymptotics as the number of states grows large, with particular attention to the time of passage between extreme states. These passage times are shown to have means which grow exponentially with the number of states, and when normalised converge weakly to an exponential distribution.     

Analytical analysis of free vibration of non-uniform and non-homogenous beams: Asymptotic perturbation approach

This paper applies the asymptotic perturbation approach (APA) to obtain a simple analytical expression for the free vibration analysis of non-uniform and non-homogenous beams with different boundary conditions. A linear governing equation of non-uniform and non-homogeneous beams is obtained based on the Euler–Bernoulli beam theory. The perturbative theory is employed to derive an asymptotic solution of the natural frequency of the beam. Finally, numerical solutions based on the analytical method are illustrated, where the effect of a variable width ratio on the natural frequency is analyzed. To verify the accuracy of the present method, two examples, piezoelectric laminated trapezoidal beam and axially functionally graded tapered beam, are presented. The results are compared with those results obtained from the finite element method (FEM) simulation and the published literature, respectively, and a good agreement is observed for lower-order beam frequencies.     

Radial Index and Euler Obstruction of a 1-Form on a Singular Variety

A notion of the radial index of an isolated singular point of a 1-form on a singular (real or complex) variety is discussed. For the differential of a function it is related to the Euler characteristic of the Milnor fibre of the function. A connection between the radial index and the local Euler obstruction of a 1-form is described. This gives an expression for the local Euler obstruction of the differential of a function in terms of Euler characteristics of some Milnor fibres.Mathematics Subject Classifications (2000). 14B05, 32S60, 58A10*Partially supported by the DFG-programme “Global methods in complex geometry” (Eb 102/4–2), grants RFBR–04–01–00762, NSh–1972.2003.1     

Points of Positive Density for the Solution to a Hyperbolic SPDE

Using a slightly modified version of Aida–Kusuoka–Stroock's characterization of the points of strictly positive density for an arbitrary Wiener functional, we extend the theorem of Ben Arous–Léandre to solutions of hyperbolic SPDE's. Thus we show that the density f of the law of X_z is positive at y if and only if y can be achieved as S_z(h), where S(h) is the controlled equation corresponding to an element h of the Cameron–Martin space, and S(.)_z is a submersion at h. The proof depends on a convergence result for a sequence X^n, of perturbed processes (defined in terms of a non homogeneous linear interpolation of the Brownian sheet) to the solution X of the corresponding perturbed SPDE.     

Method of solving equations of motion of viscoelastic media

A new method is proposed for solving dynamic problems for viscoelastic media based on the introduction of potential functions and transformation of equations of motion. The equations obtained for potential functions are used for constructing the general solution in the case of the effect of moving loads on viscoelastic media with plane-parallel interfaces. The problem of the propagation of Rayleigh surface waves is solved independently of the form of the kernels of the linear operators; a formula is obtained for determining the velocity of the Rayleigh surface wave with an arbitrary form of the viscoelastic operators. A method of experimental determination of the kernels determining the linear viscoelastic operators is proposed.V. V. Kuibyshev Moscow Civil Engineering Institute. Translated from Mekhanika Polimerov, Vol. 9, No. 3, pp. 429–435, May–June, 1973.     

On Convergence of One-Step Schemes for Weak Solutions of Quantum Stochastic Differential Equations

Several one-step schemes for computing weak solutions of Lipschitzian quantum stochastic differential equations (QSDE) driven by certain operator-valued stochastic processes associated with creation, annihilation and gauge operators of quantum field theory are introduced and studied. This is accomplished within the framework of the Hudson–Parthasarathy formulation of quantum stochastic calculus and subject to the matrix elements of solution being sufficiently differentiable. Results concerning convergence of these schemes in the topology of the locally convex space of solution are presented. It is shown that the Euler–Maruyama scheme,with respect to weak convergence criteria for Itô stochastic differential equation is a special case of Euler schemes in this framework. Numerical examples are given.     

Linear operator perturbation method in minimax control of linear dynamic systems

A perturbation method is proposed for solving some minimax control problems. The method is based on perturbation of eigenvalues of linear operators in a Hilbert space and Raleigh formulas. An explicit solution of the control problem under uncertainty is obtained.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 56, pp. 103–106, 1985.     

Reconstruction of a Source from Observational Data

A procedure is described for determining the strength, position, and startup time of a surface force source acting on an elastic half-space from recorded data on the vertical displacements (seismograms) of a linear oscillator (seismograph). The algorithm is based on a previously published analytical solution of the Lamb problem. Special attention is given to the role of the Rayleigh wavefront as a generator of displacements much larger than in the approach of P waves and SV waves to the seismograph, revealing the Rayleigh wave as the primary indicator of a source's space–time position.     

Effect of surface-active agents on the friction properties of polymers

The effect of surface-active agents on the friction properties of plastics has been investigated in relation to the example of an aqueous solution of alkyl phenol polyethylene glycol ether (OP-10) and polymethyl methacrylate. In the presence of a surface-active agent, as a result of adsorption plasticization, the deformation is localized in a thin plasticized layer without affecting the underlying layers of plastic.Moscow Lenin State Pedagogical Institute, Problem Laboratory of Polymer Physics. Translated from Mekhanika Polimerov, No. 5, pp. 927–929, September–October, 1970.     

Application of the simulated annealing algorithm to the combinatorial optimisation problem with permutation property: An investigation of generation mechanism

Focusing on the generation mechanism of random permutation solutions, this paper investigates the application of the Simulated Annealing (SA) algorithm to the combinatorial optimisation problems with permutation property. Six types of perturbation scheme for generating random permutation solutions are introduced. They are proved to satisfy the asymptotical convergence requirements. The results of experimental evaluations on Traveling Salesman Problem (TSP), Flow-shop Scheduling Problem (FSP), and Quadratic Assignment Problem (QAP) reveal that the efficiencies of the perturbation schemes are different in searching a solution space. By adopting a proper perturbation scheme, the SA algorithm has shown to produce very efficient solutions to different combinatorial optimisation problems with permutation property.     

On the linearization of systems of conservation laws for fluids at a material contact discontinuity

In order to investigate the linearized stability or instability of compressible flows, as it occurs for instance in Rayleigh–Taylor or Kelvin–Helmholtz instabilities, we consider the linearization at a material discontinuity of a flow modeled by a multidimensional nonlinear hyperbolic system of conservation laws. Restricting ourselves to the plane-symmetric case, the basic solution is thus a one-dimensional contact discontinuity and the normal modes of pertubations are solutions of the resulting linearized hyperbolic system with discontinuous nonconstant coefficients and source terms. While in Eulerian coordinates, the linearized Cauchy problem has no solution in the class of functions, we prove that for a large class of systems of conservation laws written in Lagrangian coordinates and including the Euler and the ideal M.H.D. systems, there exists a unique function solution of the problem that we construct by the method of characteristics.     

Note on the uniqueness of subsonic Euler flows in an axisymmetric nozzle

In this paper, we consider the uniqueness of globally subsonic compressible flows through an infinitely long axisymmetric nozzle. The flow is governed by the steady Euler equations and satisfies no-flow boundary conditions on the nozzle walls. We will show that for given mass flux and Bernoulli’s function in the upstream, the subsonic flow is unique in the class of all axisymmetric solutions, which possess the asymptotic behaviors at the far fields. This result extends the uniqueness of solutions in the previous paper Du and Duan (2011) [1].     

Effect of two photon absorption on nonlinear pulse propagation in gain medium

The effects of two photon absorption (TPA) and gain dispersion on soliton propagation in amplified medium are investigated. For finite gain bandwidth, the effect of gain dispersion becomes significant along with TPA and is treated as perturbation in fundamental soliton propagation. Including these perturbing effects an analytical expression of integrated intensity is formulated applying a completely new methodology by adopting Rayleigh’s dissipation function in the framework of variational approach. With classical analogy, the Euler–Lagrange equation in non-conservative system is used to solve the problem analytically. In order to justify the analytical prediction a numerical verification is made by split-step beam propagation method following Ginzburg–Landau equation.