Generalized procedure of the reduction method for vibrating problems of orthotropic plates

A reduction method is proposed for deriving an approximate formula for estimating the natural frequency of an orthotropic plate by using the results of an isotropic one. To justify the method, approximate formulae for estimating the natural frequency of clamped orthotropic rectangular and elliptical plates are derived from results previously reported for clamped isotropic rectangular and elliptical ones, respectively. The accuracy of the approximate formulae is discussed.     

Representation and realization of operational differential equations with time-varying coefficients

An algebraic treatment of operational differential equations with time-varying coefficients is presented in terms of skew rings of differential polynomials defined over a Noetherian ring. Included in this framework are delay differential equations with time- varying coefficients. The operator equations are characterized by transfer matrices which are utilized to construct realizations given by first-order vector differential equations with operator coefficients. It is shown that the realization of matrix equations can be reduced to the realization of scalar equations. Finally, a simple procedure is derived for realizing scalar equations.     

Non-linear dynamic response of circular plates subjected to transient loads

The Chebyshev polynomials have been applied to the large amplitude motions of circular plates under transient loads, with and without damping. The non- linear differential equations are linearized by using Taylor series expansions for one of the terms. It is shown that there is good agreement between the results obtained by the present technique and the available results. The advantage of this technique is essentially due to the fact that the Chebyshev polynomials are rapidly converging polynomials. It is shown that very accurate results can be obtained with only four terms of the Chebyshev series which may not be possible with conventional methods.     

功能梯度夹层板弯曲问题的微分求积法

本文基于一阶剪切变性板理论,运用能量交分得到问题的控制方程以及自然边界条件,并运用二维问题的微分求积法对其进行了求解.可以看出对于线性弯曲问题,微分求积法的收敛性很好.     

Closed plane curves described by finite and infinite sums of rotating vectors

A large group of closed plane curves may be classified as roulettes, including epicycloids, hypocycloids, and related epitrochoids and hypotrochoids. The equation for the roulette in complex polar form shows that any roulette may be described by the vector sum of two vectors of specified constant magnitudes rotating with constant angular velocities. Methods for plotting, and for electronic display of roulettes are described. An equation is derived for a roulette approximation for an N-sided regular polygon. In particular, an application to two-dimensional potential theory is described and illustrated by consideration of the roulette approximation for a square as an equipotential curve, with derivation of equations for equipotential curves in the field surrounding the square. General equations are derived for given closed plane curves with points whose x and y coordinates may separately be expanded in Fourier series as functions of the polar angle, assuming these expansions are valid. It is shown that, in general, a closed plane curve may be considered as being described by an infinite sum of vectors, each rotating in a circle. Simplifying effects of symmetry about a polar axis and/or about the origin are discussed, and methods for harmonic analysis of a given closed plane curve with aid of an electronic calculator are described.     

Standing waves in thin plates tensioned by in-plane loads subject to high frequency excitation

The problem investigated is that of the response of thin plates subjected to pure tone or wide band frequency excitation. A point load is applied perpendicular to the plates. In the case of isotropic or orthotropic plates excited at high frequencies ring shaped standing wave patterns are observed. No classical explanation exists covering this phenomenon. The present analysis indicates that there are two waves radiating from the excitation point with the same frequency, but at different speeds. One wave, a bending wave, is separated from the other one, a membrane wave.     

Modal analysis of turborotors using planar modes—theory

The generalized dynamic equations of motion have been obtained by the direct stiffness method for multimass flexible rotor bearing systems including the effects of gyroscopic moments, disc skew, and rotor acceleration. A set of undamped critical speed mode shapes calculated from the average horizontal and vertical bearing stiffness is used to transform the equations of motion into a set of coupled modal equations of motion. The modal equations are coupled by the generalized bearing coefficients and the gyroscopic moments. An analysis using only undamped critical speeds or decoupled modal analysis assuming proportional damping may lead to erroneous results. This paper presents a rapid method of calculating rotor resonance speeds with their corresponding amplification factors, stability and unbalance response of turborotors. Examples of the application of this modal approach are presented and results are compared to those of other methods such as matrix transfer analysis.     

A three-dimensional solution for plates and laminates

A general three-dimensional solution is presented for statics and dynamics of plates, homogeneous or laminated, of orthotropic materials. The solution is in series form. Using parts of the general solution a variety of problems, especially of rectangular configurations, can be solved. As Mindlin's approximate analysis for vibration of thick plates is often adequate for specific practical purposes, a general solution for Mindlin's analysis is also given.     

纵向槽孔板的应力集中系数

根据Ｈｅｌｌｉｎｇｅｒ－Ｒｅｉｓｓｎｅｒ原理，建立了一种具有一个无外力圆形边界的特殊杂交应力元，其应力场准确满足弹性力学平衡方程、协调方程及无外力圆弧上的边界条件。根据修正的Ｈｅｌｌｉｎｇｅｒ－Ｒｅｉｓｓｎｅｒ原理，建立了另一种具有一个无外力直边界的特殊杂交应力元，其应力场在元上变分满足平衡方程，但准确满足无外力直边上的边界条件。数值算例表明：特殊假定应力元与一般假定位移元这两类元联合，能十分有效地进行纵向倒圆槽孔板的应力集中分析。     

A procedure to obtain the initial amplitude and phase for the Krylov–Bogoliubov method

Using the Krylov–Bogoliubov method for obtaining analytical solutions to systems with small non-linearities, a procedure is employed to determine the initial amplitude and phase in terms of the initial displacement and velocity. Equations representing the time rate of change of amplitude and phase are used directly. Whether the corresponding linear equations of the non-linear system has purely imaginary, complex conjugate or real roots, the same procedure can be applied.An example is given which demonstrates the initial amplitude and phase change for various higher order approximations.     

Dynamic harmonic balance principle and analysis of rocking block motions

The harmonic balance (HB) principle is a powerful and convenient tool for finding periodic solutions in nonlinear systems. In the present paper, this principle is extended to transient processes in systems with one single-valued odd-symmetric nonlinearity and linear plant not having zeros in the transfer function, and named the dynamic HB. Based on the dynamic HB, first the equations for the amplitude, frequency, and amplitude decay of an oscillatory process in the Lure system are derived. It is then applied to analysis of rocking block decaying motions. An example is provided.     

Non-Linear Functional Relations and Superposition Principles in Viscoelasticity

Non-linear constitutive functional equations for isotropic viscoelastic materials with small and finite deformations are derived on the basis of tensorial expansions. These equations express the first-order as well as simple and joint second-order effects. Further, the corresponding non-linear superposition principles are formulated.     

A new direct method based on the Chebyshev cardinal functions for variable-order fractional optimal control problems

In this paper, a new direct method based on the Chebyshev cardinal functions is proposed to solve a class of variable-order fractional optimal control problems (V-OFOCPs). To this end, a new operational matrix (OM) of variable-order (V-O) fractional derivative in the Caputo sense is derived for these basis functions and is used to obtain an approximate solution for the problem under study. In the proposed method, the state and the control variables are expanded in terms of the Chebyshev cardinal functions with unknown coefficients, at first. Then, the OM of V-O fractional derivative and some properties of the Chebyshev cardinal functions are employed to achieve a nonlinear algebraic equation corresponding to the performance index and a nonlinear system of algebraic equations corresponding to the dynamical system in terms of the unknown coefficients. Finally, the method of constrained extremum is applied, which consists of adjoining the constraint equations derived from the given dynamical system and the initial conditions to the performance index by a set of undetermined Lagrange multipliers. As a result, the necessary conditions of optimality are derived as a system of algebraic equations in the unknown coefficients of the state variable, control variable, and Lagrange multipliers. Furthermore, some numerical examples of different types are demonstrated with their approximate solutions for confirming the high accuracy and applicability of the proposed method.     

Non-linear vibration analysis of laminated composite plates resting on non-linear elastic foundations

In this study, the homotopy analysis method (HAM) is used to obtain an approximate analytical solution for geometrically non-linear vibrations of thin laminated composite plates resting on non-linear elastic foundations. Geometric non-linearity is considered using von Karman’s strain-displacement relations. Then, the effects of the initial deflection, ply properties, aspect ratio of the plate and foundation parameters on the non-linear free vibration is studied. Comparison between the obtained results and those available in the literature demonstrates the potential of HAM for the analysis of such vibration problems, whose governing differential equations include the quadratic and cubic non-linear terms. This study shows that only a first-order approximation of the HAM leads to highly accurate solutions for this type of non-linear problems.     

几类混合单调算子方程解存在的唯一性定理

本文利用锥理论和半序方法在Banach空间研究了一类混合单调算子方程解的存在唯一性,把压缩映射推广到了函数及算子的形式,得到一些新的定理,所获结果推广了已知的结论。     

Heat transfer in a dipolar flow through a porous channel

The theory of constitutive equations for dipolar fluids, obtained by Bleustein and Green, is applied to investigate the Couette and Poiseuille flows between parallel plates maintained at constant but different temperatures in addition to being subjected to uniform injection and suction. Explicit expressions for the velocity and the temperature fields are obtained. It is found that different sets of conditions imposed on the flow parameters lead to different expressions for the velocity distribution which are valid only for restricted ranges of the cross-flow Reynolds number R. A table showing the various conditions imposed on the parameters, the corresponding solutions and the ranges of R for which the solution exist is presented. Velocity and temperature profiles for the dipolar and the Newtonian flows are drawn and compared to bring out the important differences resulting from the variations in R and B, the Brinkman number. For the dipolar Couette flow it is found that the value of B at which a transition from cooling to heating of the suction wall occurs always exceeds its corresponding value for Newtonian flow. Tables comparing the rates of heat transfer at the walls are provided for several values of R and B.     

An isoperimetric problem for continuous systems: The aeolian vibration of a conductor span

The equations of motion for perfectly mechanical systems which are restricted to satisfy given constraints are derived through the Lagrangian formulation for continuous media. Hamilton's equations for this situation are derived, and the equations for the balance of energy and angular momentum are generated from fundamental principles. These equations are all specialized to the example of a conductor which is vibrating under Aeolian excitation. Various models for the Hamiltonian density and the constraint energy density due to the wind energy input are also discussed.     

Formulation of stable difference schemes for systems of initial-value partial differential equations

A general system of initial-value partial differential equations is classified into four categories based on the partial differential operators which define the equations. Specific combinations of the operators are termed “invariants” since they are common to all finite difference approximations to the system of equations. The “invariants” are used to a priori determine if one may formulate a stable difference approximation to a system of partial differential equations. This is in essence a numerical existence theory.     

Initial Value Treatment for Torsion of an Elastic Layer

An initial value treatment is presented for a mixed boundary value problem of axisymmetric torsion of an elastic layer. The problem is expressed in terms of dual integral equations which are first reduced to a Fredholm integral equation of second kind with a continuous kernel. An initial value problem is derived for this Fredholm integral equation using invariant imbedding. This constitutes a new analytical approach to a large class of torsion problems.