应用数学和力学

2007 Vol. 28, No. 7

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Green Quasifunction Method for Vibration of Simply-Supported Thin Polygonic Plates on Pasternak Foundation

YUAN Hong, LI Shan-qing, LIU Ren-huai

2007, 28(7): 757-762.

Abstract(2886) PDF(748)

Abstract:
A new numerical method-Green quasifunction method is proposed. The idea of Green quasifunction method was clarified in detail by considering vibration problem of simply-supported thin polygonic plates on Pasternak foundation. A Green quasifunction was established by using the fundamental solution and boundary equation of the problem. This function satisfies the homogeneous boundary condition of the problem. The mode shape differential equation of vibration problem of simply-supported thin plates on Pasternak foundation was reduced to two simultaneous Fredholm integral equations of the second kind by Green formula. There are multiple choices for the normalized boundary equation. Based on a chosen normalized boundary equation, a new normalized boundary equation can be established such that the irregularity of the kernel of integral equations was overcome. Finally, natural frequency was obtained by the condition that there exists a nontrivial solution in the numerically discrete algebraic equations derived from the integral equations. Numerical results show high accuracy of the Green quasifunction method.

Analytical Solution for Functionally Graded Anisotropic Cantilever Beam Subjected to Linearly Distributed Load

HUANG De-jin, DING Hao-jiang, CHEN Wei-qiu

2007, 28(7): 763-768.

Abstract(3202) PDF(881)

Abstract:
The bending problem of a functionally graded anisotropic cantilever beam subjected to a linearly distributed load is investigated. The analysis was based on the exact elasticity equations for the plane stress problem. The stress function was introduced and assumed in form of a polynomial of the longitudinal coordinate. The expressions for stress components were then educed from the stress function by simple differentiation. The stress function was determined from the compatibility equation as well as the boundary conditions by a skilful deduction. The analytical solution was compared with FEM calculation, indicating a good agreement.

Forced Vibration and Special Effects of Revolution Shells in Turning-Point Range

ZHANG Zhi-liang, CHENG Chang-jun

2007, 28(7): 769-779.

Abstract(2961) PDF(582)

Abstract:
The forced vibration in the turning point frequency range of a truncated revolution shell subject to a membrane drive or a bending drive at its small end or large end was studied by applying the uniformly valid solutions obtained in a previous paper. The vibration in the turning-point range shows a strong coupling between the membrane and bending solutions: either the membrane drive or the bending drive causes motions of both the membrane type and bending type. Three interesting effects characteristic of the forced vibration in the turning point range emerge from the coupling nature: the non-bending effect, the inner-quiescent effect and the inner-membrane-motion-and-outer bendingmotion effect. These effects may have potential applications in engineering.

Lateral Resonances in Initial Stressed 1-3 Piezocomposites

ZHANG Hong-yan, SHEN Ya-peng, YIN Guan-sheng

2007, 28(7): 780-788.

Abstract(2543) PDF(702)

Abstract:
Theoretical analysis of the lateral resonances in 1-3 piezocomposites with poling initial stress was conducted using the Bloch waves theory. Based on the linear piezoelectricity theory, the theoretical formulations including initial stress for the propagation of acoustic plane waves were founded. Numerical calculations were performed to study the effects of the initial stress on the lateral mode frequencies and the stop band. It is found that the lateral mode frequencies increase with the piezoelectricity of the piezocomposites, but decrease with the poling initial stress. The influence of the initial shear stress on the lateral mode frequencies is tiny, thus it can be neglected.

Kink Wave Determined by a Parabola Solution of a Nonlinear Ordinary Differential Equation

LI Ji-bin, LI Ming, NA Jing

2007, 28(7): 789-797.

Abstract(2578) PDF(748)

Abstract:
By finding a parabola solution connecting two equilibrium points of a planar dynamical system, the existence of the kink wave solution for 6 classes of nonlinear wave equations was shown. Some exact explicit parametric representations of kink wave solutions were given. Explicit parameter conditions to guarantee the existence of kink wave solutions were determined.

Control of Chaotic Oscillations of a Satellite

Alexey Bobtsov, Nikolay Nikolaev, Olga Slita

2007, 28(7): 798-804.

Abstract(3042) PDF(488)

Abstract:
Analytical conditions and practical methods of their realization are proposed to solve a problem of a command signal tracking for a nonlinear disturbed system. Nonlinear disturbed plants consisting of linear dynamic block and nonlinear block in feedback were considered. Nonlinear part of the plant and disturbance are unknown and bounded. A possibility of applications of proposed algorithms to control libration angle of satellite was illustrated.

Weighted ?eby》ev-Ostrowski Type Inequalities Involving Functions Whose First Derivatives Belong to a Spaces of the Functions

Arif Rafiq, Nazir Ahmad Mir, Farooq Ahmad

2007, 28(7): 805-810.

Abstract(2552) PDF(678)

Abstract:
On account of the famous ?eby》ev inequality, a rich theory has appeared in some literature. Some new weighted ?eby》ev type integral inequalities via certain integral inequalities for functions whose first derivatives belong to a space of the functions are established. The proofs are of independent interest and provide new estimates on these types of inequalities.

Physical Factors of a Primary Jet Vectoring Control Using Synthetic Jet Actuators

XIA Zhi-xun, LUO Zhen-bing

2007, 28(7): 811-823.

Abstract(2632) PDF(627)

Abstract:
A primary jet vectoring using synthetic jet actuators with different exit configurations was investigated, and the main physical factors influencing jet vectoring were analyzed and summarized. The physical factors of the pressure difference, the location and area of the lower pressure region, the component of the synthetic jet momentum and the entrainment ratio of the synthetic jet flow to primary jet flow directly control the vectoring force and the vectoring angle. Three characteristic parameters of the synthetic jet contribute to the pressure difference and the area of the lower pressure region. Both the extension step and slope angle of the actuator exit have functions of regulating the location of the lower pressure region, the area of the lower pressure region, and the entrainment ratio of the synthetic jet flow to primary jet flow. The slope angle of the actuator exit has additional functions of regulating the component of the synthetic jet momentum. Based upon analyzing the physical factors of jet vectoring control with synthetic jets, the source variables of the physical factors were established. A preparatory control model of jet vectoring using synthetic jet actuator was presented. And it has the benefit of explaining the efficiency of jet vectoring using synthetic jet actuator with source variables at different values. And it indicates the optimal actuator is taking full advantage of the regulating function.

Numerical Study of Dynamic Phase Transitions in Shock Tube

WANG Ping, TANG Shao-qiang

2007, 28(7): 824-832.

Abstract(2895) PDF(621)

Abstract:
Shock tube problem of a Van der Waals fluid with a relaxation model was investigated. In the limit of relaxation parameter tending towards zero, this model yields a specific Riemann solver. Relaxing and relaxed schemes were derived. For an incident shock in a fixed tube, numerical simulations show convergence toward the Riemann solution in one space dimension. Impact of parameters was studied theoretically and numerically. For certain initial shock profiles, nonclassical reflecting wave was observed. In two space dimensions, the effect of curved wave fronts was studied, and some interesting wave patterns were exposed.

Unconventional Hamilton-Type Variational Principles for Nonlinear Elastodynamics of Orthogonal Cable-Net Structures

LI Wei-hua, LUO En, HUANG Wei-jiang

2007, 28(7): 833-842.

Abstract(3106) PDF(888)

Abstract:
According to the basic idea of classical yin-yang complementarity and modern dual-complementarity, in a simple and unified new way proposed by Luo, the unconventional Hamilton-type variational principles for geometrically nonlinear elastodynamics of orthogonal cable-net structures can be established systematically. The unconventional Hamilton-type variational principle can fully characterize the initia-l boundary-value problem of this dynamics. An important integral relation was given, which can be considered as the generalized principle of virtual work for geometrically nonlinear dynamics of orthogonal cable-net structures in mechanics. Based on this relation, it is possible not only to obtain the principle of virtual work for geometrically nonlinear dynamics of orthogonal cable-net structures, but also to derive systematically the complementary functionals for five-field, four field, three-field and two-field unconventional Hamilton-type variational principles, and the functional for the unconventional Hamilton-type variational principle in phase space and the potential energy functional for one-field unconventional Hamilton-type variational principle for geometrically nonlinear elastodynamics of orthogonal cable-net structures by the generalized Legendre transformation given. Furthermore, with this approach, the intrinsic relationship among various principles can be explained clearly.

Three-Point Explicit Compact Difference Scheme With Arbitrary Order of Accuracy and Its Applicatin in CFD

LIN Jian-guo, XIE Zhi-hua, ZHOU Jun-tao

2007, 28(7): 843-852.

Abstract(2547) PDF(1046)

Abstract:
Based on the successive iterative approach in the taylor series expansion method, a threepoint explicit compact difference scheme with arbitrary order of accuracy is derived, and the numerical characteristic of the scheme was studied by the Fourier analysis. Unlike the conventional compact difference schemes which need to solve the equation to obtain the unknown derivatives in each node, the proposed scheme is explicit and can achieve arbitrary order of accuracy in space. Application examples for the convection-diffusion problem with a sharp front gradient and the typical lid-driven cavity flow were given. It is found that the proposed compact scheme is not only simple to implement and economical to use, but also effective to simulate the convection-dominated problem and obtain highorder accurate solution in coarse grid systems.

Markovian Risk Process

WANG Han-xing, YAN Yun-zhi, ZHAO Fei, FANG Da-fan

2007, 28(7): 853-860.

Abstract(2428) PDF(1178)

Abstract:
A Markovian risk process is considered, which is the generalization of the classical risk model. It is proper that a risk process with large claims is modelled as the Markovian risk model. In such a model, the occurrence of claims was described by a point process with it being the number of jumps for a Markov jump process from time 0 to t. The ruin probability of a company facing such a risk model was mainly studied. An integral equation satisfied by the ruin probability was obtained and the bounds for the convergence rate of the ruin probability are given by using a generalized renewal technique.

A Kind of Bivariate Spline Space Over Rectangular Partition and Pure Bending of Thin Plate

WANG Ren-hong, CHANG Jin-cai

2007, 28(7): 861-868.

Abstract(2614) PDF(631)

Abstract:
The mechanical background of the bivariate spline space of degree 2 and smoothness 1 on rectangular partition was presented constructively. Making use of mechanical analysis method, by acting couples along the interior edges with suitable evaluations, the deflection surface was divided into piecewise form, therefore, the relation between a class of bivariate splines on rectangular partition and the pure bending of thin plate was established. In addition, the interpretation of smoothing cofactor and conformality condition from the mechanical point of view was given. Furthermore, by introducing twisting moments, the mechanical background of any spline belonging to the above space was set up.

Alternating Segment Explicit-Implicit Scheme for Nonlinear Third-Order KdV Equation

QU Fu-li, WANG Wen-qia

2007, 28(7): 869-876.

Abstract(2894) PDF(917)

Abstract:
A group of asymmetric difference schemes to approach the Korteweg-de Vries (KdV) equation was given here. Using the schemes and the full explicit difference scheme and the full implicit difference scheme, the alternating difference scheme for solving the KdV equation was constructed. The scheme is linear unconditionally stable by analysis of linearization procedure, and is used directly on the parallel computer. The numerical experiments show that the method has high accuracy.

On a Class of Quasilinear Schr^Ldinger Equations

SHU Ji, ZHANG Jian

2007, 28(7): 877-882.

Abstract(3188) PDF(636)

Abstract:
A type of quasilinear Schr^Ldinger equations in two dimensions are discussed, which describe attractive Bose-Einstein Condensates in physics. By establishing the property of the equation and applying the energymethod, was proved the blowup of the solutions to the Cauchy problem for the equation under certain conditions. At the same time, by the variational method, the a sufficient condition of global existence was got, which is related to the ground state of a classical elliptic equation.