THE METHOD OF THE RECIPROCAL THEOREM OF FORCED VIBRATION FOR THE ELASTIC THIN RECTANGULAR PLATES(Ⅰ)—RECTANGULAR PLATES WITH FOUR CLAMPED EDGES AND WITH THREE CLAMPED EDGES

In this paper the method of the reciprocal theorem (MRT) is extended to solve the steady state responses of rectangular plater under harmonic disturbing forces. A series of the closed solutions of rectangular plates with various boundary conditions are given and the tables and figures which have practical value are provided.MRT is a simple, convenient and general method for solving the steady stale responses of rectangular plates under various harmonic disturbing forces.The paper contains three parts: (I) rectangular plates with four damped edges and with three clamped edges; (II) rectangular plates with two adjacent clamped edges; (III) cantilever plates.We arc going to publish them one after another.

GENERALIZED RECIPROCAL THEOREMS AND THEIR APPLICATIONS

Generalized reciprocal theorems of non-coupled and coupled systems, which are valid for two deformed bodies with different constitutive relations are established by generalizing the idea of Betti's reciprocal theorem. When the constitutive relations of the two deformed bodies are all alike and linear elastic the generalized reciprocal theorem of non-coupled systems just becomes Betti's. Meanwhile the generalized reciprocal theorems are applied to simulate calculations in elasticity.

An improved correlation of the pressure drop in stenotic vessels using Lorentz's reciprocal theorem

A mathematical model of the human cardiovascu lar system in conjunction with an accurate lumped model for a stenosis can provide better insights into the pressure wave propagation at pathological conditions. In this study, a the oretical relation between pressure drop and flow rate based on Lorentz＇s reciprocal theorem is derived, which offers an identity to describe the relevance of the geometry and the convective momentum transport to the drag force. A voxel based simulator VFLOW VOF3D, where the vessel geome try is expressed by using volume of fluid （VOF） functions, is employed to find the flow distribution in an idealized steno sis vessel and the identity was validated numerically. It is revealed from the correlation flow in a stenosis vessel can that the pressure drop of NS be decomposed into a linear term caused by Stokes flow with the same boundary condi tions, and two nonlinear terms. Furthermore, the linear term for the pressure drop of Stokes flow can be summarized as a correlation by using a modified equation of lubrication the ory, which gives favorable results compared to the numerical ones. The contribution of the nonlinear terms to the pressure drop was analyzed numerically, and it is found that geomet ric shape and momentum transport are the primary factors for the enhancement of drag force. This work paves a way to simulate the blood flow and pressure propagation under dif ferent stenosis conditions by using 1D mathematical model.

A METHOD USING THE RECIPROCAL THEOREM TO SOLVE THE BENDING OF THIN ELASTIC SEMICIRCULAR PLATES

The bending of the thin elastic semicircular plates, because of its complicated boundary conditions, brings some difficulties for us to obtain its solution. This paper applies the reciprocal theorem to propose a general simple convenient method to obtain the transverse deflectional equations of the plates.

RECIPROCAL THEOREM METHOD FOR THE FORCED VIBRATIONOF ELASTIC THICK RECTANGULAR PLATES

In this paper, reciprocal theorem method (RTM) is generalized to solve the problems for the forced vibration of thick rectangular plates based on the Reissner's theory. The paper derives the dynamic basic solution of thick rectangular; and the exact analytical solution of the steady-state responses of thick rectangular plates with three clamped edges and one free edge under harmonic uniformly distributed disturbing forces is found by RTM. It is regarded as a simple, convenient and general method for calculating the steady-stare responses of forced vibration of thick rectangular plates.

APPLICATION OF THE METHOD OF THE RECIPROCAL THEOREM TO FINDING DISPLACEMENT SOLUTIONS OF CUBES

In this paper the method of reciprocal theorem is extended to find solutions of three-D problems of elasticity.First we give the basic solution of the cube with six surfaces fixed as the basic system and then using the reciprocal theorem between the basic system acted on by unit concentrated loads and the actual system with prescribed surface displacements, we find displacement solution of the actual system.

THE METHOD OF THE RECIPROCAL THEOREM OF FORCED VIBRATION FOR THE ELASTIC THIN RECTANGULAR PLATES(Ⅱ)——RECTANGULAR PLATES WITH TWO ADJACENT CLAMPED EDGES

In this paper, applying the method of reciprocal theorem, we give the distributions of the amplitude of bending moments along clamped edges and the amplitude of deflections along free edges of rectangular plates with two adjacent clamped edges under harmonic distributed and concentrated loads.

THE RECIPROCAL THEOREM AND LINEAR ELASTIC VARIATIONAL PRINCIPLES

In this paper classical linear elastic variational principles are systematically derivedfrom the reciprocal theorem and mixed variational principles of variations of boundaryconditions are given.

THE METHOD OF THE RECIPROCAL THEOREM OF FORCED VIBRATION FOR THE ELASTIC THIN RECTANGULAR PLATES(Ⅲ)--CANTILEVER RECTANGULAR PLATES

In this paper,applying the method of the reciprocal theorem,we give the stationary solutions of the forced vibration of cantilever rectangular plates under uniformly distributed harmonic load and concentrated harmonic load acting at any point of the plates,the figures and tables of number value of bending moment and the deflection amplitudes as well.

RECIPROCAL THEOREM METHOD FOR SOLVING THE PROBLEMS OFBENDING OF THICK RECTANGULAR PLATES

In this paper,reciprocal theorem method(RTM) is generalized to solve theof bending of thick rectangular plates based on Reissner’s theory.First,the paper gives the basic solution of the bending of thick rectangular platesand then the exact analytical solution of the bending of thick rectangular plate withthree clamped edges and one free edge under umiformly distributed load is found byRTM, finally, we analyze numerical results of the sohution.

ON THE METHOD OF RECIPROCAL THEOREM TO FIND SOLUTIONS OF THE PLANE PROBLEMS OF ELASTICITY

In this paper the method of reciprocal theorem is extended to find solutions of plane problems of elasticity of the rectangular plates with various edge conditions.First we give the basic solution of the plane problem of the rectangular plate with four edges built-in as the basic system and then find displacement expressions of the actual system by using the reciprocal theorem between the basic system and actual system with various edge conditions.When only displacement edge conditions exist, obtaining displacement expressions by means of the method of reciprocal theorem is actual. But in other conditions, when static force edge conditions or mixed ones exist, the obtained displacements are admissible. In order to find actual displacement, the minimum potential energy theorem must be applied.Calculations show that the method of reciprocal theorem is a simple, convenient and general one for the solution of plane problems of elasticity of the rectangular plates with various edge conditions. Evidently, it is a new method.

Uniqueness,reciprocity theorem,and plane waves in thermoelastic diffusion with a fractional order derivative

In this work, a theory of thermoelasticity with diffusion is taken into consideration by using the methodology of fractional calculus. The governing equations for particle motion in a homogeneous anisotropic fractional order generalized thermoelastic diffusive medium are presented. Uniqueness and reciprocity theorems are proved. The plane wave propagation in the homogeneous transversely isotropic thermoelastic diffusive medium with fractional order derivative is studied. For the two-dimensional problem, there exist a quasi-longitudinal wave, a quasi-transverse wave, a quasi-mass diffusion wave, and a quasi-thermal wave. From the obtained results, the different characteristics of waves, like phase velocity, attenuation coefficient, specific loss, and penetration depth, are computed numerically and presented graphically. Some special cases are also discussed.

Solution of scattering from rough surface with a 2D target above it by a hybrid method based on the reciprocity theorem and the forward-backward method

This paper proposes a hybrid method based on the forward-backward method （FBM） and the reciprocity theorem （RT） for evaluating the scattering field from dielectric rough surface with a 2D target above it. Here, the equivalent electric/magnetic current densities on the rough surface as well as the scattering field from it are numerically calculated by FBM, and the scattered field from the isolated target is obtained utilizing the method of moments （MOM）. Meanwhile, the rescattered coupling interactions between the target and the surface are evaluated employing the combination of FBM and RT. Our hybrid method is first validated by available MOM results. Then, the functional dependences of bistatic and monostatic scattering from the target above rough surface upon the target altitude, incident and scattering angles are numerically simulated and discussed. This study presents a numerical description for the scattering mechanism associated with rescattered coupling interactions between a target and an underlying randomly rough surface.

A Sound Field Separation and Reconstruction Technique Based on Reciprocity Theorem and Fourier Transform

We show a method to separate the sound field radiated by a signal source from the sound field radiated by noise sources and to reconstruct the sound field radiated by the signal source. The proposed method is based on reciprocity theorem and the Fourier transform. Both the sound field and its gradient on a measurement surface are needed in the method. Evanescent waves are considered in the method, which ensures a high resolution reconstruction in the near field region of the signal source when evanescent waves can be measured. A simulation is given to verify the method and the influence of measurement noise on the method is discussed.

Uniqueness theorem,theorem of reciprocity,and eigenvalue problems in linear theory of porous piezoelectricity

The uniqueness theorem and the theorem of reciprocity in the linearized porous piezoelectricity are established under the assumption of positive definiteness of elastic and electric fields. General theorems in the linear theory of porous piezoelectric materials are proved for the quasi-static electric field approximation. The uniqueness theorem is also proved without using the positive definiteness of the elastic field. An eigenvalue problem associated with free vibrations of a porous piezoelectric body is stud- ied using the abstract formulation. Some properties of operators are also proved. The problem of frequency shift due to small disturbances, based on an abstract formulation, is studied using a variational and operator approach. A perturbation analysis of a special ease is also given.

An analogical study of wave equations,physical quantities,conservation and reciprocity equations between electromagnetic and elastic waves

This paper presents an analogical study between electromagnetic and elastic wave fields,with a one-to-one correspondence principle established regarding the basic wave equations,the physical quantities and the differential operations.Using the electromagnetic-to-elastic substitution,the analogous relations of the conservation laws of energy and momentum are investigated between these two physical fields.Moreover,the energy-based and momentum-based reciprocity theorems for an elastic wave are also derived in the time-harmonic state,which describe the interaction between two elastic wave systems from the perspectives of energy and momentum,respectively.The theoretical results obtained in this analysis can not only improve our understanding of the similarities of these two linear systems,but also find potential applications in relevant fields such as medical imaging,non-destructive evaluation,acoustic microscopy,seismology and exploratory geophysics.

Several Fundamental Theorems and Conservative Integrals in Quasicrystal Elasticity Theory

Aim To extend several fundamental theorems of conventional elasticity theory to quasicrystalelasticity theory. Methods The basic governing equations of quasicrystal elasticity theory and Gauss's theorem were applied in the derivation. Results and Conclusion The principle of virtual work, Betti's reciprocal theorem and the uniqueness theorem of quasicrystal elasticity theory are proud, and some conservative integrals in quasicrystal elasticty theory are obtained.

Common Fixed Point Theorem in Intuitionistic Fuzzy Metric Space Using <i>R</i>-Weakly Commuting Mappings

In this paper, we prove a common fixed point theorem in Intuitionistic fuzzy metric space by using pointwise R-weak commutativity and reciprocal continuity of mappings satisfying contractive conditions.

混合边界护顶层力学模型及应用

对于直接顶板稳固性较差的矿山,通常需留设护顶层以防止直接顶板的垮塌。为了保证矿山的安全开采,确定合理的护顶层厚度至关重要。针对房柱法开采的采空区顶板进行结构力学解析,将护顶层视作四边中部自由其余部分固支的薄板,即混合边界薄板;利用功的互等法推导出混合边界薄板的挠曲面方程,构建混合边界护顶层力学模型,在此基础上得到护顶层安全厚度的计算公式。将混合边界薄板与四边固支薄板的变形情况进行对比时发现,混合边界薄板的挠度整体大于四边固支薄板挠度,混合边界自由段长度与最大挠度值呈正相关。将理论分析和数值模拟结合应用在某铝土矿No.9采场上,确定了护顶层安全厚度为1 m,验证了力学模型的准确性。

往复弯曲统一曲率定理证明及应用

关于在矫直领域中广泛应用的往复弯曲统一曲率定理,目前还没有一个简明而直观的数学推导和证明,本文基于载荷-挠度的弯曲理论模型,避开了压弯过程中应力、曲率、微积分方程等方面的复杂计算,重点关注压弯过程中弯曲挠度的变化,根据矫直定理对往复弯曲统一曲率定理进行了严谨的数学推导和证明,丰富了现有的矫直理论,为矫直工艺的制定提供了理论基础。在研究中开发了一种高度可调的垫块,在实际应用中效果显著,利用该垫块可以很方便、可控且稳定地进行矫直操作,在矫直过程中不用去理会每支弯钢的原始弯曲度,一般进行一次反压接触到垫块后卸载即可完成矫直,矫直后精度高并且平直度稳定,能满足各种平直度性能要求。