应用数学和力学

2007 Vol. 28, No. 2

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Scale Effect and the Geometric Shapes of Grains

GUO Hui, GUO Xing-ming

2007, 28(2): 127-134.

Abstract(2692) PDF(737)

Abstract:
The "rule-of-mixture" approach has become one of the widely spread ways to investigate the mechanical properties of nano-materials and nano-structures,and it is very important for the simulation results to exactly compute phase volume fractions.The nanocrystalline(NC) materials were treated as three-phase composites consisting of grain core phase,grain boundary (GB) phase and triple junction phase,and a two-dimensional three-phase mixture regular polygon model was established to investigate the scale effect of NC materials mechanical properties due to the geometrical polyhedron characteristics of crystal grain.For different multi-sides geometrical shapes of grains,the corresponding multi-sides regular polygon model was adopted to obtain more precise phase volume fractions and exactly predict the mechanical properties of NC materials.

Nonlinear Dynamical Behavior of Shallow Cylindrical Reticulated Shells

WANG Xin-zhi, LIANG Cong-xing, HAN Ming-jun, YEH Kai-yuan, WANG Gang

2007, 28(2): 135-140.

Abstract(2527) PDF(786)

Abstract:
By using the method of quasi-shells,the nonlinear dynamic equations of three-dimensional single-layer shallow cylindrical reticulated shells with equilateral triangle cell were founded.By using the method of the separating variable function,the transverse displacement of the shallow cylindrical reticulated shells were given under the conditions of two edges simple support.The tensile force was solving from the compatible equations,a nonlinear dynamic differential equation containing the second and third order is derived by using the method of Galerkin.The stability near the equilibrium point was discussed by solving the Floquet exponent and the critical condition is obtained by using Melnikov function.Existence of the chaotic motion of the single-layer shallow cylindrical reticulated shell is approved by using the digital simulation method and Poincar mapped.

Generalized Synchronization of Continuous Dynamical System

ZHANG Gang, LIU Zeng-rong, MA Zhong-jun

2007, 28(2): 141-146.

Abstract(2712) PDF(670)

Abstract:
Generalized synchronization between two continuous dynamical systems is discussed.By exploring the Liapunov stability theory and constructing appropriately unidirectional coupling term,a sufficient condition for determining the generalized synchronization between continuous systems was proved.Two examples are used to show the effectiveness of this result.

Renewal of Basic Laws and Principles for Polar Continuum Theories(Ⅺ)——Consistency Problems

DAI Tian-min

2007, 28(2): 147-155.

Abstract(2308) PDF(822)

Abstract:
Some consistency problems existing in continuum field theories are briefly reviewed.Three arts of consistency problems are clarified based on the renewed basic laws for polar continua.The first art discussed the consistency problems between the basic laws for polar continua.The second art discussed the consistency problems between the basic laws for polar continua and for other nonpolar continua.The third art discussed the consistency problems between the basic laws for micropolar continuum theories and the dynamical equations for rigid body.The results presented here can helpus get a deeper understanding of the structure of the basic laws for various continuum theories and the interrelations between them.In the meantime,these results obtained also show clearly that the consistency problems could not be solved in the framework of traditional basic laws for continuum field theories.

Analytical Solutions of Steady Vibration of a Free Rectangular Plate on the Semi-Infinite Elastic Foundation

WANG Chun-ling, HUANG Yi, JIA Ji-hong

2007, 28(2): 156-164.

Abstract(2920) PDF(709)

Abstract:
The method of double Fourier transform was employed in the analysis of the semi-infinite elastic foundation with vertical load.And an integral representations for the displacements of the semi-infinite elastic foundation was presented.The analytical solution of steady vibration of an elastic rectangle plate with four free edges on the semi-infinite elastic foundation was also given by combining the analytical solution of the elastic rectangle plate with the integral representation for displacements of the semi-infinite elastic foundation.Some computational results and the analysis of the parameters influence were presented.

Effect of Thermal Diffusion and Electrostatic Force on the Evolution of Wind-Blown Sand Flow

YUE Gao-wei, ZHENG Xiao-jing

2007, 28(2): 165-175.

Abstract(2300) PDF(468)

Abstract:
A theor etical model was suggested to mathematically describe the effect of thermal diffusion from a sand-bed on evolution of a wind-blown sand flow.An upward wind field was engendered by the thermal diffusion and the coupling interaction among the horizontal and upward wind flow,saltating grains,and a kind of electr ostatic force exerted on the grains were considered in this theoretical model.The numerical results show that the effect of the thermal diffusion on the evolution process of wind-blowngrain flow is quite obvious and very similar to the effect of the electrostatic force on the evolution.Notonly the time for the entire system to reach a stea dy state (called the duration time ),the transportrate of gr ains,the mass-flux profiles and the trajectory of saltating grains are affected by the thermal diffusion and the electr ostatic force exerted on saltating grains,but also the wind profiles and the temper ature profiles at the steady state are affected by the wind-blown sand flow.

Variation Principle of Piezothermoelastic Bodies,Canonical Equation and Homogeneous Equation

LIU Yan-hong, ZHANG Hui-ming

2007, 28(2): 176-182.

Abstract(2927) PDF(645)

Abstract:
Combining the symplectic variations theory,the homogeneous control equation and isoparametric element homogeneous formulations for piezothermoelastic hybrid laminates problems were deduced.Firstly,based on the generalized Hamilton variation principle,the non-homogeneous Hamilton canonical equation for piezothermoelastic bodies was derived.Then the symplectic relationship of variations in the thermal equilibrium formulations and gradient equations was considered.The non-homogeneous canonical equation was transformed to homogeneous control equation for solving independently the coupling problem of piezothermoelastic bodies by the incensement of dimensions of the canonical equation.For the convenience of deriving Hamilton isoparametric element formulations with four nodes,one can consider the temperature gradient equation as constitutive relation and reconstruct new variation principle.The homogeneous equation simplifies greatly the solution programs which are often performed to solve non-homogeneous equation and second order differential equation on the thermal equilibrium and gradient relationship.

Lunar Landing Trajectory Design Based on Invariant Manifold

GONG Sheng-ping, LI Jun-feng, BAOYIN He-xi, GAO Yun-feng

2007, 28(2): 183-190.

Abstract(3409) PDF(674)

Abstract:
The low-energy lunar landing trajectory design using the invariant manifolds of restricted three body problem is studied.Considering angle between the ecliptic plane and lunar orbit plane,the four body problem of Sun-Earth-Moon-spacecraft was divided into two three body problems,the Sun-Earth-spacecraft in the ecliptic plane and the Earth-Moon-spacecraft in the lunar orbit plane.Using the orbit maneuver at the place where the two planes and the invariant manifolds intersect,a general method to design low energy lunar landing trajectory was given.It is found that this method can save the energy by 20% compared with the traditional Hohmann transfer trajectory.The mechanism of the method that can save energy was investigated in the point of view of energy and the expression of the amount of energy saved is given.In addition,some rules of selecting parameters with respect to orbit design were provided.The method of energy analysis can be extended to energy analysis in deep space orbit design.

Transverse Vibration Characteristics of Axially Moving Viscoelastic Plate

ZHOU Yin-feng, WANG Zhong-min

2007, 28(2): 191-199.

Abstract(2368) PDF(931)

Abstract:
The dynamic characteristics and stability of axially moving viscoelastic rectangular thin plate are investigated.Based on the two dimensional viscoelastic differential constitutive relation,the differential equations of motion of the axially moving viscoelastic plate were established.Dimensionless complex frequencies of an axially moving viscoelastic plate with four edges simply supported,two opposite edges simply supported and other two edges clamped were calculated by the differential quadrature method.The effects of the aspect ratio,moving speed and dimensionless delay time of the material on the transverse vibration and stability of the axially moving viscoelastic plate were analyzed.

Analytical Solution of a Fractionally Damped Beam by Using Adomian Decomposition Method

LIANG Zu-feng, TANG Xiao-yan

2007, 28(2): 200-208.

Abstract(2779) PDF(708)

Abstract:
The analytical solution of a viscoelastic continuous beam whose damping characteristics are described in terms of a fractional derivative of arbitrary order was derived by means of the Adomian decomposition method.The solution contains arbitrary initial conditions and zero input.For specific analysis,the initial conditions were assumed homogeneous,and the input force was treated as a special process with a particular beam.Two simple cases,step and impulse function responses,were considered respectively.Subsequently,some figures were plotted to show the displacement of the beam under different sets of parameters including different orders of the fractional derivatives.

Stability of Equivariant Bifurcation Problems With Two Types of State Variables and Their Unfoldings in the Presence of Parameter Symmetry

CUI Deng-lan, LI Yang-cheng

2007, 28(2): 209-215.

Abstract(2505) PDF(843)

Abstract:
Based on the contact equivalent relation of smooth map-germs in singularity theory,the stability of equivariant bifurcation problems with two types of state variables and their unfoldings in the presence of parameters symmetry was discussed.Some basic results are obtained.Transversality condition was used to characterize the stability of equivariant bifurcation problems.

Hausdorff Dimension of the Set Generated by Exceptional Oscillations of a Class of N-Parameter Gaussian Processes

LIN Zheng-yan, CHENG Zong-mao

2007, 28(2): 216-224.

Abstract(2330) PDF(775)

Abstract:
A class of N-parameter Gaussian processes were introduced,which are more general than the N-parameter Wiener process.The definition of the set generated by exceptional oscillations of class of these processes was given.And then the Hausdorff dimension of this set was defined.The Hausdorff dimensions of these processes were studied and an exact representative for them was given,which is similar to that for the two-parameter Wiener process by Zacharie (2001).Moreover,the time set considered is a hyperrectangle which is more general than a hyper-square used by Zacharie (2001).For this more general case,a Fernique-type inequality was established and then using this inequality and the Slepian lemma,a Lvy's continuity modulus theorem was shown.Independence of increments is required for showing the representative of the Hausdorff dimension by Zacharie (2001).This property is absent for the processes introduced here,so a different way is to be found.

A Class of Two-Dimensional Dual Integral Equations and Its Application

FAN Tian-you, SUN Zhu-feng

2007, 28(2): 225-230.

Abstract(2542) PDF(531)

Abstract:
Because exact analytic solution was not available,the double expansion and boundary collocation were used to construct an approximate solution for a class of two-dimensional dual integral equations in mathematical physics.The integral equations by this pr ocedure were reduced to infinite algebraic equations.The a ccuracy of the solution lies in the boundary collocation technique.The application of which for some complicated initial-boundary value problems in solidme chanics indicates the method is powerful.

State-Vector Equation With Damping and Vibration Analysis of Laminates

QING Guang-hui, XU Jian-xin, QIU Jia-jun

2007, 28(2): 231-237.

Abstract(3130) PDF(634)

Abstract:
Based on the modified mixed Hellinger-Reissner(H-R) variational principle for elastic bodies with damping,the state-vector equation was directionally derived from the principle.A new solution for the harmonic vibration of simply supported rectangular laminates with damping was proposed by using the precise integration method and Muller method.The general solutions for the free vibration of underdamping,critical damp and overdamping of composite laminates were given simply in terms of the linear damping vibration theory.The effect of viscous damping force on the vibration of composite laminates was investigated through numerical examples.The state-vector equation theory and its application areas are extended.

Stability for the Basic System of Equations of Atmospheric Motion

SHI Wei-hui, XU Ming, WANG Yue-peng

2007, 28(2): 238-244.

Abstract(2935) PDF(884)

Abstract:
The topological characteristics for the basic system of equations of atmospheric motion were analyzed with the help of method provided by stratification theory;It was proved that in the local rectangular coordinate system the basic system of equations of atmospheric motion was stable equations in infinitely differentiable function class;In the sense of local solution,the necessary and sufficient conditions by which the typical problem for determining solution was well posed were also given.Such problems as something about/speculating future from past0 in atmospheric dynamics and how to amend the conditions for determining solution as well as the choice of underlying surface when involving the practical application were further discussed; It is also pointed out that under the usual conditions,three motion equations and continuity equation in the basic system of equations determine entirely the property of this system of equations.

Inequalities Relating to L_p-Version of the Petty’s Conjectured Projection Inequality

WANG Wei-dong, LENG Gang-song

2007, 28(2): 245-252.

Abstract(1855) PDF(774)

Abstract:
Petty's conjectured projection inequality is a famous open problem in convex bodies theory.It was shown that an inequality relating to L_p-version of the Petty's conjectured projection inequality by using the notions of the L_p-mixed volume and the L_p-dual mixed volume,the relation of the L_p-projection body and the geometric body Г-_pK,the Bourgain-Milman inequality and the L_p-Busemann-Petty inequality.In addition,for each origin-symmetric convex body,applying the Jensen inequality and the monotonicity of the geometric body Г-_pK,the reverses of L_p-version of the Petty's conjectured projection inequality and the L_p-Petty projection inequality were given respectively.