An exact axisymmetric solution for a simply supported piezoelectric cylindrical shell

Summary Three-dimensional axisymmetric solution is presented for a simply supported piezoelectric cylindrical shell. The variables are expanded in Fourier series to satisfy the boundary conditions at the ends. The solution of the governing differential equations with variable coefficients is constructed as a product of an exponential function and a power series. The coefficients of terms of all degrees in the governing equations are set to zero, yielding a characteristic equation for the exponent and recursive relations for the coefficients of the power series. Results are presented illustrating the effect of thickness parameter of the shell. An inverse problem of inferring the applied temperature from the measured potential difference has been solved. Accepted for publication 26 July 1996     

强非线性动力系统的频率增量法

提出一类强非线性动力系统的暧时频率增量法,将描述动力系统的二阶常微分方程,化为以相位为自变量、瞬廛频率为未知函数的积分方程;用谐波平衡原理,将求解瞬时频率的积分问题,归结为求解以频率增量的Fourier系数为独立变量的线性代数方程组;给出了若干例子。     

弹性功能梯度材料板条中周期裂纹的反平面问题

讨论了弹性功能梯度材料板条中裂纹的反平面问题. 用Fourier 变换方法得到了一个基本解. 这个基本解表示了实轴上一点作用有点位错时引起的影响. 利 用此基本解可得单裂纹和周期裂纹问题的奇异积分方程. 在周期裂纹求解时, 远处裂纹对于中央裂纹的影响作了有效的近似处理. 最后, 给出了数值结果, 它表示了材料性质对于裂纹端应力强度因子的影响.     

Optimal bounded control of stochastically excited MDOF nonlinear viscoelastic systems

A new procedure for designing optimal bounded control of stochastically excited multi-degree-of-freedom (MDOF) nonlinear viscoelastic systems is proposed based on the stochastic averaging method and the stochastic maximum principle. First, the system is formulated as a quasi-integrable Hamiltonian system with viscoelastic terms and each viscoelastic term is replaced approximately by an elastically restoring force and a visco-damping force based on the randomly periodic behavior of the motion of quasi-integrable Hamiltonian system. Thus, a stochastically excited MDOF nonlinear viscoelastic system is converted to an equivalent quasi-integrable Hamiltonian system without viscoelastic terms. Then, by applying stochastic averaging, the system is further reduced to a partially averaged system of less dimension. The adjoint equation and maximum condition for the optimal control problem of the partially averaged system are derived by using the stochastic maximum principle, and the optimal bounded control force is determined from the maximum condition. Finally, the probability and statistics of the stationary response of optimally controlled system are obtained by solving the Fokker–Plank–Kolmogorov equation (FPK) associated with the fully averaged Itô equation of the controlled system. An example is worked out to illustrate the proposed procedure and its effectiveness.     

多自由度参激系统稳定性的数值分析

提出多自由度周期参激系统稳定性的数值直接法。通过将扰动方程表示成状态方程形式,再根据Flo-quet理论将扰动解表示成指数特征分量与周期分量之积,并将其周期分量与系统周期系数展成Fourier级数,导出一系列代数方程,建立矩阵特征值问题,从而由数值求解特征值可直接确定参激系统的稳定性。该方法可用于一般周期参激阻尼系统,特征值矩阵不含逆子阵。应用于斜拉索在支座周期运动激励下的参激振动不稳定性分析,数值结果表明该方法的有效性。     

Sur une solution periodique d'un systeme differentiel singulier a argument retarde: Construction et calcul approche

This study considers a singular ordinary differential system with time-lag and periodic coefficients. A periodic solution is obtained in a unique manner through a uniformly convergent series.The limit of this solution, when the singularity parameter tends to zero, is a periodic solution of the reduced system.Galerkin's method is employed to construct an approximation procedure for obtaining this periodic solution.     

Diffusion in a nonuniform velocity field

The effect of a nonuniform velocity field on the diffusion process is examined. When the local passive admixture transport equation is averaged over the channel cross section, the differential equation for the average concentration over the cross section is obtained in the form of an infinite asymptotic series whose terms are linear combinations of the derivatives of the average concentration with respect to the coordinate and time, while the coefficients depend on the degree of transverse nonuniformity of the velocity field and the radial Péclet number. Estimates show that in most of the cases encountered in practice to ensure that the calculations have the necessary accuracy the series must include derivatives up to the third order. An approximate solution of the averaged equation is found by the method of asymptotic expansions and the initial moments of the residence time distribution function are determined.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 122–128, September–October, 1989.     

On the Parametric Resonance of Columns Carrying Concentrated Masses

Abstract

The effect of various parameters upon the region of dynamic instability of a uniform simply supported column carrying n concentrated masses and subjected to an axial periodic force at one end is presented. The effect of axial inertia of the mass per unit length of the column is also included. This problem is reduced to a coupled system of two homogeneous partial differential equations of second order with periodic coefficients which can be written in the form of a matric differential equation of the Mathieu-Hill type. A solution methodology is developed and successfully demonstrated through numerical examples.     

Theory of electrostatic focusing of intense beams of charged particles

The mathematical formulation of the problem of determining the electrodes for the formation of intense beams of charged particles reduces to the solution of the Cauchy problem for the Laplace equation. One can proceed either by separating the variables [1] or on the basis of the theory of analytic continuation [2–5], This approach can be used for plane or axisymmetric flows. An algorithm for the construction of the analytic solution, which can also be used in the threedimensional case, is given below. It is assumed that the beam boundary coincides with the coordinate surface x¹=0 of an orthogonal system xⁱ (i=1,2,3). The solution is put in the form of a series in x¹ with coefficients dependent on x² and x³, determined from recurrence relations. The case of emission limited by space charge and temperature generally gives rise to difficulties due to the divergence of the series which makes it impossible to calculate the zero equipotential by the indicated method.As an example, the formation of beams with an elliptic cross section is considered in the following cases: (1) periodic variation of the z- component of the velocity; (2) nonmonotonic variation of the potential in one-dimensional flow between planes z=const; (3) a beam accelerated in accordance with a 3/2 law.In the construction of the expansions the conditions on the boundary are satisfied exactly by the first two terms of the series.     

Linearized equations of motion of a viscous fluid in a porous medium of periodic structure

The investigation of flow in essentially inhomogeneous porous systems through the analysis of model periodic structures [1] is considered. In the acoustic approximation, an integrodifferential equation is obtained that describes the motion of a viscous fluid in a rigid porous medium of periodic structure. The velocity vector and pressure are represented in the form of asymptotic series with respect to a small parameter that characterizes the size of the periodicity cell, and the well-known procedure for averaging linearized hydrodynamic equations with small coefficients of viscosity [2, 3] is also used. A solution is presented to the local problem in the periodicity cell for a structure consisting of a doubly periodic system of infinitely long rods of circular section and a compressible viscous fluid that fills the space between them, and also for a structure formed by a system of orthogonal rectilinear channels, filled with viscous fluid, in a solid.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 123–130, March–April, 1988.     

Onsager's reciprocal relations in the linear problem of flow of rarefied gas past a heat conducting body

Symmetry of integral transport coefficients is established on the basis of a linear stationary Boltzmann equation for the problem of flow past a heat conducting body. An expression is obtained for the entropy production in the gas-body system, and this determines the thermodynamic fluxes and forces. Bakanov and Roldugin [2] have considered the problem of motion of a heat conducting sphere at small Knusden numbers Kn, using the symmetry of the Onsager coefficients to construct an asymptotic solution as Kn → 0. In the present paper, a general method is proposed for establishing the Onsager relations that does not require the actual construction of a solution to the problem and is applicable for bodies of arbitrary shape and all values of the Knudsen number.     

EFFICIENT ANALYTIC SERIES SOLUTIONS FOR TWO–DIMENSIONAL POTENTIAL FLOW PROBLEMS

The solution of Laplace's equation for a wide range of spatial domains and boundary conditions is a valuable asset in the study of potential theory. Recently, classical analytic series techniques based on separation of variables have been modified to solve Laplace's equation with both irregular and free boundaries. Computationally the free boundary problem is reduced to an iterative sequence of curve-fitting exercises. At each iteration the series coefficients for a known boundary problem are evaluated numerically. In this paper a new interpolation approach is presented for the estimation of the series coefficients. It has the advantages of providing a conceptually simpler view of the series technique and of estimating the series coefficients significantly faster than alternative approaches. Owing to the choice of basis functions in the truncated series solution, rigorous estimates of the error in the approximation are immediately available. A free boundary problem from steady hillside seepage with irregular boundaries will be used to illustrate the new technique.     

The growth of the thermal boundary layer at the inlet to a circular tube

Summary The problem considered is that of the heat transfer occurring at the inlet to a circular tube. Using the method of a previous paper we solve the energy equation in the form of a power series, instead of transforming the equation into an eigenvalue problem by separation of variables, as is usual in such cases. We thus obtain information concerning the temperature distribution at the inlet to the tube which is not provided by the latter method. A sufficient number of coefficients of the power series has been computed to allow the present solution to be joined to one of the second type, thus completing the solution of the problem for all values of the longitudinal variable.     

Stochastic averaging of n-dimensional non-linear dynamical systems subject to non-Gaussian wide-band random excitations

The averaged generalized Fokker-Planck-Kolmogorov (GFPK) equation for response of n-dimensional (n-d) non-linear dynamical systems to non-Gaussian wide-band stationary random excitation is derived from the standard form of equation of motion. The explicit expressions for coefficients of the fourth-order approximation of the averaged GFPK equation are given in series form. Conditions for convergences of these series are pointed out. The averaged GFPK equation is then reduced to that for 1-d dynamical systems derived by Stratonovich and compared with the closed form of GFPK equation for n-d dynamical systems subject to Poisson white noise derived by Di Paola and Falsone. Finally, this averaged GFPK equation is further reduced to that for quasi linear system subject to non-Gaussian wide-band stationary random excitation. Stationary probability density for quasi linear system subject to filtered Poisson white noise is obtained. Theoretical results for an example are confirmed by using Monte-Carlo simulation for different parameter values.     

Analytic postbuckling solution of a pre-stressed infinite beam bonded to a linear elastic foundation

The postbuckling deflection of an infinite beam that is bonded to a linear elastic foundation and is subjected to an internal compressive stress is analyzed. The nonlinear equilibrium equation that governs the problem considers extensional deformation of the beam. An analytic solution of the nonlinear equilibrium equation is presented and is found to be in good agreement with numerical simulations of the problem. The numerical simulations confirm that for a linear elastic foundation the postbuckling deflection is periodic. The analytic solution shows that the postbuckling wavelength is unaffected by the level of internal stress, and is equal to the wavelength at the critical state.     

Asymptotically Exact Solution of the Problem of Harmonic Vibrations of an Elastic Parallelepiped

An asymptotically exact solution of the classical problem of elasticity about the steadystate forced vibrations of an elastic rectangular parallelepiped is constructed. The general solution of the vibration equations is constructed in the form of double Fourier series with undetermined coefficients, and an infinite system of linear algebraic equations is obtained for determining these coefficients. An analysis of the infinite system permits determining the asymptotics of the unknowns which are used to convolve the double series in both equations of the infinite systems and the displacement and stress components. The efficiency of this approach is illustrated by numerical examples and comparison with known solutions. The spectrum of the parallelepiped symmetric vibrations is studied for various ratios of its sides.     

Analysis of periodic-quasiperiodic nonlinear systems via Lyapunov-Floquet transformation and normal forms

In this paper a general technique for the analysis of nonlinear dynamical systems with periodic-quasiperiodic coefficients is developed. For such systems the coefficients of the linear terms are periodic with frequency ω while the coefficients of the nonlinear terms contain frequencies that are incommensurate with ω. No restrictions are placed on the size of the periodic terms appearing in the linear part of system equation. Application of Lyapunov-Floquet transformation produces a dynamically equivalent system in which the linear part is time-invariant and the time varying coefficients of the nonlinear terms are quasiperiodic. Then a series of quasiperiodic near-identity transformations are applied to reduce the system equation to a normal form. In the process a quasiperiodic homological equation and the corresponding ‘solvability condition’ are obtained. Various resonance conditions are discussed and examples are included to show practical significance of the method. Results obtained from the quasiperiodic time-dependent normal form theory are compared with the numerical solutions. A close agreement is found.     

Nonstationary probability densities of system response of strongly nonlinear single-degree-of-freedom system subject to modulated white noise excitation

The nonstationary probability densities of system response of a single-degree-of -freedom system with lightly nonlinear damping and strongly nonlinear stiffness subject to modulated white noise excitation are studied.Using the stochastic averaging method based on the generalized harmonic functions,the averaged Fokker-Planck-Kolmogorov equation governing the nonstationary probability density of the amplitude is derived. The solution of the equation is approximated by the series expansion in terms of a set...     

The interaction of gas bubbles in an anisotropic elastic solid

The problem of finding the stress field induced in the neighbourhood of two spherical gas bubbles or voids in an anisotropic matrix is formulated in terms of an integral equation for the “transformation stress” in equivalent homogeneous inclusions. An iterative method of solution is outlined, involving the solution of a class of problems for a single spherical inclusion perturbing a polynomial field of stress. Explicit solutions are obtained for polynomials up to second degree. Estimates of the energy of interaction between gas bubbles in α-U and between voids in Mo are deduced as examples, and the results are discussed in relation to earlier calculations and to observations.     

Buckling of a pre-compressed or pre-stretched membrane in shear flow

Membranes enclosing capsules and biological cells undergo periodic compression and stretching due to an imparted hydrodynamic traction as they rotate in a shear flow. Compression may cause transient or permanent buckling manifested by the onset of wrinkled shapes. To study the effect of pre-compression and pre-stretching on the critical conditions for buckling, the response of an elastic circular plate flush mounted on a plane wall and deforming under the action of a uniform tangential load due to an over-passing simple shear flow is considered. Working under the auspices of the theory of elastic instability of plates governed by the linear von Kármán equation, an eigenvalue problem is formulated resulting in a fourth-order partial differential equation with position-dependent coefficients parametrized by the Poisson ratio. Solutions are computed by applying Fourier series expansions to derive an infinite system of coupled ordinary differential equations, and then implementing orthogonal collocation. The solution space is illustrated, critical values for buckling are identified, the associated eigenfunctions representing possible modes of deformation are displayed, and the effect of the Poisson ratio is discussed.