On Aspects of Asymptotics for Plate Equations

In this paper some initial-boundary value problems for plate equations will be studied. These initial-boundary value problems can be regarded as simple models describing free oscillations of plates on elastic foundations or of plates to which elastic springs are attached on the boundary. It is assumed that the foundations and springs have a different behavior for compression and for extension. An approximation for the solution of the initial-boundary value problem will be constructed by using a two-timescales perturbation method. For specific parameter values it turns out that complicated internal resonances occur.     

On Mode Interactions for a Weakly Nonlinear Beam Equation

In this paper an initial-boundary value problem for a weakly nonlinear beam equation with a Rayleigh perturbation will be studied. It will be shown that the calculations to find internal resonances in this case are much more complicated than and differ substantially from the calculations for the weakly nonlinear wave equation with a Rayleigh perturbation as for instance presented in [3] or [7]. The initial-boundary value problem can be regarded as a simple model describing wind-induced oscillations of flexible structures like suspension bridges or iced overhead transmission lines. Using a two-timescales perturbation method approximations for solutions of this initial-boundary value problem will be constructed.     

On aspects of damping for a vertical beam with a tuned mass damper at the top

In this paper, the wind-induced, horizontal vibrations of a vertical Euler–Bernoulli beam will be considered. At the top of the beam, a tuned mass damper (TMD) has been installed. The horizontal vibrations can be described by an initial-boundary value problem. Perturbation methods will be applied to construct approximations of the solutions of the initial-boundary value problem, and it will be shown that the TMD uniformly damps the oscillation modes of the beam. In the analysis, it will be assumed that damping, wind-force, and gravity effects are small but not negligible.     

Multiple reciprocity method with two series of sequences of high-order fundamental solution for thin plate bending

IntroductionIt’swell_knownthatthecomplicatedfundamentalsolution[1,2 ]forHelmholtzequationΔu(x) +k2 u(x) =0　　(x∈Ω:boundedopenregioninR2 )isu (x,y) =-iH(2 )0 (k x-y ) 4,thusit’snotconvenientfornumericalcomputation .IfapplyingthesimplefundamentalsolutionofLaplaceequationu 0 (x ,y) =-ln｜x-y｜(2π) ,theexpressionforthesolutionofequationintheclosedregion Ωisc(y)u(y) + ∫Γu(x) u 0 (x,y) nx -u 0 (x ,y) u(x) n dsx =-k2∫Ωu(x)u 0 (x,y)dΩx.Astherightsideappearstheregionalintegrationinclu…     

On Boundary Damping for a Weakly Nonlinear Wave Equation

In this paper an initial-boundary value problem for a weakly nonlinear string(or wave) equation with non-classical boundary conditions is considered. Oneend of the string is assumed to be fixed and the other end of the string isattached to a spring-mass-dashpot system, where the damping generated by thedashpot is assumed to be small. This problem can be regarded as a rather simple model describing oscillationsof flexible structures such as suspension bridges or overhead transmission lines in a windfield. A multiple-timescales perturbation method will be usedto construct formal asymptotic approximations of the solution. It will also beshown that all solutions tend to zero for a sufficiently large value of thedamping parameter. For smaller values of the damping parameter it will be shownhow the string-system eventually will oscillate.     

An exponentially fitted difference scheme for the hyperbolic-hyperbolic singularly perturbed initial-boundary value problem

In this paper we discuss,an initial-boundary value problem of hyperbolic type with firstderivative with respect to x.The asymptotic solution is constructed and its uniform validityis proved under weader compatibility conditions.Then we develop an exponentially fitteddifference scheme and establish discrete energy inequality.Finally,we prove that thesolution of difference problem uniformly converges to the solution of the original problem.     

Uniqueness in the nonlinear dynamic bending of elastic plates

A uniqueness theorem for the solution of a nonlinear initial-boundary value problem for a time dependent form of von Karman's equations is proven. These equations are two coupled nonlinear fourth order partial differential equations which describe the bending of an elastic plate. The result is proved by the method of energy integrals.     

On a Rayleigh Wave Equation with Boundary Damping

In this paper an initial-boundary value problem for a weakly nonlinear string (or wave) equation with non-classical boundary conditions is considered. One end of the string is assumed to be fixed and the other end of the string is attached to a dashpot system, where the damping generated by thedashpot is assumed to be small. This problem can be regarded as a simple model describing oscillations of flexible structures such as overhead transmission lines in a windfield. An asymptotic theory for a class ofinitial-boundary value problems for nonlinear wave equations is presented. Itwill be shown that the problems considered are well-posed for all time t. A multiple time-scales perturbation method incombination with the method of characteristics will be used to construct asymptotic approximations of the solution. It will also be shown that all solutions tend to zero for a sufficiently large value of the damping parameter. For smaller values of the damping parameter it will be shown how the string-system eventually will oscillate. Some numerical results are alsopresented in this paper.     

Numerical solution of the singularly perturbed problem for the hyperbolic equation with initial jump

In this paper we consider the initial-boundary value problem for a second order hyperbolic equation with initial jump. The bounds on the derivatives of the exact solution are given. Then a difference scheme is constructed on a non-uniform grid. Finally, uniform convergence of the difference solution is proved in the sense of the discrete energy norm.     

On transversal vibrations of an axially moving string with a time-varying velocity

In this paper an initial-boundary value problem for a linear equation describing an axially moving string will be considered for which the bending stiffness will be neglected. The velocity of the string is assumed to be time-varying and to be of the same order of magnitude as the wave speed. A two time-scales perturbation method and the Laplace transform method will be used to construct formal asymptotic approximations of the solutions. It will be shown that the linear axially moving string model already has complicated dynamical behavior and that the truncation method can not be applied to this problem in order to obtain approximations which are valid on long time-scales.     

Multiscale asymptotic homogenization for multiphysics problems with multiple spatial and temporal scales: a coupled thermo-viscoelastic example problem

A systematic approach for analyzing multiple physical processes interacting at multiple spatial and temporal scales is developed. The proposed computational framework is applied to the coupled thermo-viscoelastic composites with microscopically periodic mechanical and thermal properties. A rapidly varying spatial and temporal scales are introduced to capture the effects of spatial and temporal fluctuations induced by spatial heterogeneities at diverse time scales. The initial-boundary value problem on the macroscale is derived by using the double scale asymptotic analysis in space and time. It is shown that an extra history-dependent long-term memory term introduced by the homogenization process in space and time can be obtained by solving a first order initial value problem. This is in contrast to the long-term memory term obtained by the classical spatial homogenization, which requires solutions of the initial-boundary value problem in the unit cell domain. The validity limits of the proposed spatial–temporal homogenized solution are established. Numerical example shows a good agreement between the proposed model and the reference solution obtained by using a finite element mesh with element size comparable to that of material heterogeneity.     

Spatial decay bounds for a class of quasi-linear parabolic initial-boundary value problems

We investigate the decay of solution to a class of quasi-linear parabolic initial-boundary value problems defined in a finite cylinder with non-zero data given only on one end. We determine a time interval on which the solution will exist and derive decay bounds for the solution and its time and cross-sectional derivatives on that time interval.     

On the propagation of pressure waves in saturated porous media

The problem of the propagation of pressure waves through compressible porous media saturated with a slightly compressible fluid is considered. By using Darcy's law the problem is reduced to a mixed initial-boundary value problem for an equation of the heat conduction type with a nonlinear term. The method of quasi-characteristics is used to solve this equation numerically. Solutions of the wave propagation problem for media with different permeability coefficients are presented. A solution of the inverse problem of determining the permeability coefficient using wave-pulse test data is constructed on the basis of a set of solutions of the direct problem.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 81–84, November–December, 1996.     

Point loading on hyperbolic paraboloidal shells

A combination of simple theory and experiment is used to investigated a complex structural problem. The problem is the point loading of hyperbolic paraboloidal shells. The solution is based on an analogy between the response of these shells to a point load and the localized behavior of a flat plate that rests on an elastic foundation. The subgrade modulus of this analogous foundation is derived from the measured deflection of a point load on shell models. The value of the modulus was found to be about one-fifth that of a spherical shell with the same magnitude of principal curvature.     

A Representation Formula for the Velocity Part of 3D Time-Dependent Oseen Flows

We prove a representation formula for the solution of an initial-boundary value problem associated with the 3D time-dependent Oseen system. This formula involves the solution of an integral equation on the lateral boundary of the space-time cylinder. The key point of this article is that the assumptions on the data are chosen in such a way that the formula in question may be used in a theory of the asymptotic behaviour of solutions to the nonstationary 3D Navier–Stokes system with Oseen term (incompressible viscous flow in an exterior domain with nonzero velocity at infinity).     

On the Existence of a Weak Solution of Viscous Incompressible Flow Past a Cascade of Profiles with an Arbitrarily Large Inflow

The paper deals with the analysis of a stationary viscous incompressible flow through a cascade of profiles representing a blade row of a turbine. The initial-boundary value problem for the Navier–Stokes system is formulated in a domain having the form of one spatial period of the cascade. In comparison to previous results, we solve the problem with an arbitrarily large inflow into the turbine. We formulate the “artificial” boundary condition on the outflow (the modification of the so called do-nothing condition) that enables us to prove the existence of a weak solution.     

Singular perturbation of initial-boundary value problems for a class of reaction diffusion systems

In this paper,a class of singularly perturbed initial-boundary value problems for thereaction diffusion systems is considered.Using the theory of differential inequality,weprove that the initial-boundary value problems have a solution and obtain their asymptoticexpansion.     

The second initial-boundary value problem for a quasilinear parabolic equations with nonlinear boundary conditions

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｆａｓｔｄｉｆｆｕｓｉｏｎｅｑｕａｔｉｏｎｏｆｄｉｖｅｒｇｅｎｃｅｆｏｒｍａｓｕｔ =(ａ(ｕ)ｕｘ) ｘ ｂ(ｕ) ｘ ｃ(ｕ)　　 (ａ( 0 ) = ∞ ) ( 1 )ｈａｓｉｍｐｏｒｔａｎｔｐｈｙｓｉｃａｌｂａｃｋｇｒｏｕｎｄ ,ｓｕｃｈａｓ [1 ] .Ｉｎｒｅｃｅｎｔｙｅａｒｓ ,ｓｏｍｅｒｅｓｕｌｔｓａｂｏｕｔ ( 1 )ｈａｖｅｂｅｅｎｏｂｔａｉｎｅｄ .Ｆｏｒｅｘａｍｐｌｅ ,[2 ] ,[3]ｒｅｓｐｅｃｔｉｖｅｌｙｄｉｓｃｕｓｓｅｄｔｈｅＣａｕｃｈｙｐｒｏｂｌｅｍｓｆｏｒｅｑｕａｔｉｏｎ( 1 )ａｎｄｕｔ=(…     

Winkler地基上四边自由正交各向异性矩形薄板弯曲问题的辛叠加解

研究Winkler地基上正交各向异性矩形薄板弯曲方程所对应的Hamilton正则方程, 计算出其对边滑支条件下相应Hamilton算子的本征值和本征函数系, 证明该本征函数系的辛正交性以及在Cauchy主值意义下的完备性, 进而给出对边滑支边界条件下Hamilton正则方程的通解, 之后利用辛叠加方法求出Winkler地基上四边自由正交各向异性矩形薄板弯曲问题的解析解. 最后通过两个具体算例验证了所得解析解的正确性.     

An analytical solution for the laminar flow over a flat plate with similarity preserving suction

An exact analytical solution is presented for the laminar boundary-layer flow over a semi-infinite flat plate subjected to a type of similarity preserving suction. The solution is developed for the case of a plate immersed in either a uniform compressible stream with viscosity proportional to temperature or a uniform incompressible stream with constant viscosity. The problem is formulated in Crocco's variables. It is described by a second-order, non-linear, ordinary differential (and singular) boundary-value problem for the shear stress as a function of the velocity in the boundary layer. A unique solution is shown to exist and to possess a power series representation for all magnitudes of suction. The series is constructed explicitly and provides a transcendental equation for the shear stress at the plate (the important skin friction) which can be solved to any desired accuracy. Examples of upper and lower bounds for the wall shear are presented for several magnitudes of suction and confirm the reasonable accuracy of results obtained heretofore only by numerical solutions of the problem. In addition to the intrinsic value of the technique developed, it can be the basis of accurate checks for the numerical solution of more complex problems.