应用数学和力学

2006 Vol. 27, No. 12

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Response of a Parametrically Excited Duffing-Van der Pol Oscillator With Delayed Feedback

LI Xin-ye, CHEN Yu-shu, WU Zhi-qiang, SONG Tao

2006, 27(12): 1387-1396.

Abstract(2713) PDF(777)

Abstract:
The dynamical behaviour of a parametrically excited Duffing-Van der Poloscillator under linear-plus-nonlinear state feedback control with a time delay is concerned. By means of the method of averaging together with truncation of Taylor expansions, two slow-flow equations on the amplitude and phase of response were derived for the case of principal parametric resonance. It is shown that the stability condition for the trivial solution is only associated with the linear terms in the original systems besides the amplitude and frequency of parametric excitation. And the trivial solution can be stabilized by appreciation choice of gains and time delay in feedback control. Different from the case of the trivial solution, the stability condition for nontrivial solutions is also associated with nonlinear terms besides linear terms in the original systems. It is demonstrated that nontrivial steady state responses may lose their stability by saddle-node (SN) or Hopf bifurcation (HB) as parameters vary. The simulations, obtained by numerically integrating the original system, are in good agreement with the analytical results.

Analysis of Breather State in Thin Bar by Using Collective Coordinate

ZHAO Guang-hui, ZHANG Nian-mei, YANG Gui-tong

2006, 27(12): 1397-1404.

Abstract(2911) PDF(517)

Abstract:
Considering Peierls-Nabarro (P-N) force and visco us effect of material, the dynamic behavior of one-dimensional infinite metallic thin barsubjected to axially periodic load was investigated. Governing equation, which was sine-Gordon type equation, was derived. By means of collective-coordinates, the partialequation could be reduced into ordinary differential dynamical system to describemotion of breather. Nonlinear dynamic analysis shows that the amplitude and frequency of P-N force would influence positions of hyperbolic saddlepoints and change subharmonic bifurcation point, while the path to chaos through oddsubhar monic bifurcations remains. Several examples were taken to indicate the effects of amplitude and perio d of P-N force o n the dy namical re sponse o f the bar. The simulatio n states that the area of chaos is half-infinite. This are a incre ases along with enhancement of the amplitude of P-N force. And the frequency of P-N force has similar influence on the system.

Maximal Elements of a Family of Majorized MappingsInvolving a Better Admissible Mapping in Product FC-Spaces and Applications

DING Xie-ping

2006, 27(12): 1405-1416.

Abstract(2662) PDF(925)

Abstract:
A new family of majorized mappings from a topological space into a finite continuous topological space (in short, FC-space) involving a better admissible set-valued mapping was introduced. Some existence theorems of maximal elements for the family of majorized mappings were proved under noncompact setting of product FC-spaces. Some applications to fixed point and system of minimax inequalities were given in product FC-spaces. These theorems improve, unify and generalize many important results in recent literature.

Scattering of Circular Cavity in Right-Angular Planar Space to Steady SH-Wave

SHI Wen-pu, LIU Dian-kui, SONG Yong-tao, CHU Jing-lian, HU Ai-qin

2006, 27(12): 1417-1423.

Abstract(2724) PDF(720)

Abstract:
Complex function method and multi-polar coordinate transformation technology are used here to study scattering of circular cavity in right-angular planar space to SH-wave with out-of-plane loading on the horizontal straight boundary. At first, Green function of right-angular planar space which has no circular cavity was constructed; then the scattering solution which satisfies the free stress conditions of the two right-angular boundaries with the circular cavity existing in the space was formulated, therefore, the total displacement field can be constructed using overlapping principle. An infinite algebraic equations of unknown coefficients existing in the scattering solution field can be gained using multi-polar coordinate and the free stress condition at the boundary of the circular cavity, it can be solved by using limit items in the infinite series which can give a high computation precision. An example was given to illustrate the variations of the tangential stress at the boundary of the circular cavity to different dimensionless wave numbers and the location of the circular cavity and the loading center and the distributing range of out-of-plane loading. The results of the example show the efficiency and the effectiveness of the method introduced here.

Asymptotic Expansions of Zeros for Krawtchouk Polynomials With Error Bounds

ZHU Xiao-feng, LI Xiu-chun

2006, 27(12): 1424-1430.

Abstract(3136) PDF(764)

Abstract:
Krawtchouk polynomials are frequently applied in modern physics. Based on the results which were educed by Li and Wong, the asymptotic expansions of Krawtchouk polynomials were improved by using Airy function, and the uniform asymptotic expansions were got. Furthermore, the asymptotic expansions of the zeros for Krawt chouk polynomials were again deduced by using the property of the zeros of Airy function, and their corresponding error bounds were discussed. The obtained results give the asymptotic property of Krawtchouk polynomials with their zeros, which are better than the results educed by Li and Wong.

Generalized Finite Spectral Method for 1D Burgers and KdV Equations

ZHAN Jie-min, LI Yok-sheng

2006, 27(12): 1431-1438.

Abstract(3064) PDF(781)

Abstract:
A generalized finite spectral method is proposed. The method is of high-order accuracy. To attain high accuracy in time discretization, the fourth-order Adams-Bashforth-Moulton predictor and corrector scheme was used. To avoid numerical oscillations caused by the dispersion term in the KdV equation, two numerical techniques were introduced to improve the numerical stability. The Legendre, Chebyshev and Hermite polynomials were used as the basis functions. The proposed numerical scheme is validated by applications to the Burgers equation (nonlinear convection-diffusion problem) and KdV equation(single solitary and 2-solitary wave problems), where analytical solutions are available for comparison. Numerical results agree very well with the corresponding analytical solutions in all cases.

Non-Linear Dynamic Model Retrieval of Subtropical High Based on Empirical Orthogonal Function and Genetic Algorithm

ZHANG Ren, HONG Mei, SUN Zhao-bo, NIU Sheng-jie, ZHU Wei-jun, MIN Jin-zhong, WAN Qi-lin

2006, 27(12): 1439-1446.

Abstract(2719) PDF(757)

Abstract:
Aiming at the difficulty of accurately constructing the dynamic model of subtropical high, based on the potential height field time series over 500 hPa layer of T106 numerical forecast products, by using EOF(empirical orthogonal function) temporal-spatial separation technique, the disassembled EOF time coefficients series were regarded as dynamical model variables, and dynamic system retrieval idea as well as genetic algorithm were introduced to make dynamical model parameters optimization search, then, a reasonable non-linear dynamic model of EOF time-coefficients was established. By dynamic model integral and EOF temporal-spatial components assembly, a mid/long-term forecast of subtropical high was carried out. The experimental results show that the forecast results of dynamic model are superior to that of general numerical model forecast results. A new modeling idea and forecast technique is presented for diagnosing and forecasting such complicated weathers as subtropical high.

Kuhn-Tucker Condition and the Wolfe Duality of Preinvex Set-Valued Optimization

SHENG Bao-huai, LIU San-yang

2006, 27(12): 1447-1456.

Abstract(2685) PDF(909)

Abstract:
The optimality Kuhn-Tucker condition and the Wolfe duality for the preinvex set-valued optimization are investigated. Firstly, the concepts of alpha-order G-invex set and the alpha-order S-preinvex set-valued function were introduced, from which the properties of the corresponding contingent cone and the alpha-order contingent derivative were studied; Then, the optimality Kuhn-Tucker condition and the Wolfe duality theorem for the alpha-order S-preinvex set-valued optimization were presented with the help of the alpha-order contingent derivative.

Exact Linearization Based Multiple-Subspace Iterative Resolution to Affine Nonlinear Control System

XU Zi-xiang, ZHOU De-yun, DENG Zi-chen

2006, 27(12): 1457-1463.

Abstract(2831) PDF(999)

Abstract:
To the optimal control problem of affine nonlinear system, based on differential geometry theory, feedback precise linearization was used. Then starting from the simulative relationship between computational structural mechanics and optimal control, multiple-substructure method was induced to solve the optimal control problem which was linearized. And finally the solution to the original nonlinear system was found. Compared with the classical linearizational method of Taylor expansion, this one diminishes the abuse of error expansion with the enlargement of used region.

Compactly Supported Non-Tensor Product Form Two Dimension Wavelet Finite Element

JIN Jian-ming, XUE Peng-xiang, XU Ying-xiang, ZHU Ya-li

2006, 27(12): 1464-1476.

Abstract(2853) PDF(632)

Abstract:
Some theorems of compactly supported non-tensor product form two dimension Daubechies wavelet was analysed carefully. Compactly supported non-tensor product form two dimension wavelet was constructed, then non-tensor product form two dimension wavelet finite element was used to solve the deflection problem of elastic thin plate. The error order was researched. A numerical example was given at last.

Poly-Scale Refinable Function and Their Properties

YANG Shou-zhi

2006, 27(12): 1477-1485.

Abstract(2498) PDF(752)

Abstract:
Poly-scale refinable function with dilation factor a was introduced. The existence of solutions of poly-scale refinable equation was investigated. Specially, necessary and sufficient conditions for the orthonormality of solution function phi of a poly-scale refinable equation with integer dilation factor a were established. Some properties of poly-scale refinable function were discussed. Several examples illustrating how to use the method to construct poly-scale refinable function were given.

Nonlinear Galerkin Mixed Element Methods for the Stationary Incompressible Magnetohydrodynamics

LUO Zhen-dong, MAO Yun-kui, ZHU Jiang

2006, 27(12): 1486-1496.

Abstract(2837) PDF(761)

Abstract:
A nonlinear Galerkin mixed element (NGME) method for the stationary incompressible magnetohydrodynamics equations was presented. And the existence and error estimates of the NGME solution were derived.

Analysis of Composite Laminate Beams Using Coupling Cross-Section Finite Element Method

JIANG Wen-guang, John L. Henshall

2006, 27(12): 1497-1505.

Abstract(2973) PDF(493)

Abstract:
Beams and plates manufactured from laminates of composite materials have distinct advantages in a significant number of applications. However, the anisotropy arising from these materials adds a significant degree of complexity, and thus time, to the stress and deformation analyses of such components, even using numerical approaches such as finite elements. The analysis of composite laminate beams subjected to uniform extension, bending, and/or twisting loads was performed by a novel implementation of the usual finite element method. Due to the symmetric features of the deformations, only a thin slice of the beam to be analysed needs to be modelled. Conventional three-dimensional solid finite elements were used for the structural discretization. The accurate deformation relationships were formulated and implemented through the coupling of nodal translational degrees of freedom in the numerical analysis. A sample solution for a rectangular composite laminate beam is presented to show the validity and accuracy of the proposed method.

Analysis of Symmetric Laminated Rectangular Plates in Plane Stress

YANG Duan-sheng, HUANG Yan, REN Xian-hai

2006, 27(12): 1506-1512.

Abstract(2666) PDF(561)

Abstract:
Symmetric laminated plates usually used are anisotropic plates. Based on fundamental equation for anisotropic rectangular plates in plane stress problem, a general analytical solution was established by method of stress function accurately. Therefore it gives the general formula of stress and displacement in plane. The integral constants in general formula can be determined by boundary conditions. This general solution composes the composite solution made by trigonometric function and hyperbolic function which can satisfy the problem of arbitrary boundary conditions along four edges. The algebraic polynomial solutions which can satisfy the problem of boundary conditions at four corners. Consequently this general solution can be used to solve the plane stress problem with arbitrary boundary conditions. For example, a symmetric laminated square plate acted with uniform normal load and tangential load and non-uniform normal load on four edges has been calculated and analyzed.