应用数学和力学

2009 Vol. 30, No. 9

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Free Axisymmetric Vibration of FGM Circular Plates

WANG Yun, XU Rong-qiao, DING Hao-jiang

2009, 30(9): 1009-1014. doi: 10.3879/j.issn.1000-0887.2009.09.001

Abstract(1816) PDF(996)

Abstract:
Based on threedimensional theory,a direct displacement method was presented to investigate free axisymmetric vibration of transversely isotropic circular plates,whose material is functionally graded and its properties obey the exponential law along the thickness of the plate.For two boundary conditions the solution satisfies all basic equations and corresponding boundary condition at every point and thus is three-dimensionally exact.Numerical examples were finally tabulated and compared with those of previous works.The present method can also be extended to the case of arbitrary distribution of the material properties along the thickness of the plate.

Homotopy Analysis Approach to the Duffing-Harmonic Oscillator

FENG Shao-dong, CHEN Li-qun

2009, 30(9): 1015-1020. doi: 10.3879/j.issn.1000-0887.2009.09.002

Abstract(1610) PDF(1252)

Abstract:
The homotopy analysis is performed for the Duffing-harmonic oscillator.The auxiliary parameter in the deformation equation was numerically determined.The response and the frequency of the Duffing-harmonic oscillator were calculated.The analytical results are validated by the direct numerical simulations.

Basic Function Scheme of Polynomial Type

WU Wang-yi, LIN Guang

2009, 30(9): 1021-1032. doi: 10.3879/j.issn.1000-0887.2009.09.003

Abstract(1871) PDF(888)

Abstract:
A new numerical method-Basic Function Method was proposed.This method could directly discrete differential operator on unstructured grids.By using the expansion of basic function to approach the exact function,the central and upwind schemes of derivative were constructed.By using the second-order polynomial as basic function and applying the technique of flux splitting method and the combination of central and upwind schemes to suppress the non-physical fluctuation near the shock wave,the second-order basic function scheme of polynomial type for solving inviscid compressible flow numerically was constructed.Several numerical results of many typical examples for two dimensional inviscid compressible transonic and supersonic steady flow illustrate that it is a new scheme with high accuracy and high resolution for shock wave.Especially,combined with the adaptive remeshing technique,the satisfactory results can be obtained by these schemes.

On the Generalized Mixed Equilibrium Problem in Banach Spaces

ZHANG Shi-sheng

2009, 30(9): 1033-1041. doi: 10.3879/j.issn.1000-0887.2009.09.004

Abstract(1745) PDF(1058)

Abstract:
The purpose is by using hybrid algorithm to find a common element of the set of solutions for a generalized mixed equilibrium problem,the set of solutions for variational inequality problem and the set of common fixed points for a finite family of quasi-Φ-nonexpansive mappings in a uniformly smooth and strictly convex Banach space.By utilizing the results for the study of optimization problem,it shows that the results improve and extend the corresponding results announced recently by many others such as Ceng,Takahashi,Qin,et al.

MHD Natural Convection in Porous Media-Filled Enclosures

F. G. Shehadeh, H. M. Duwairi

2009, 30(9): 1042-1048. doi: 10.3879/j.issn.1000-0887.2009.09.005

Abstract(2001) PDF(711)

Abstract:
The magneto hydrodynamics natural convection heat transfer problem inside a porous media filled inclined rectangular enclosures is investigated numerically.The boundary conditions selected on the enclosure were two adiabatic and two isothermal walls.The governing equations,continuity,Forchheimer extension of Darcy law and energy,were going to be transformed into dimensionless form using a set of suitable variables then solved using a finite difference scheme.The governing parameters are magnetic influence number,Darcy-Rayleigh number,inclination angle,and the aspect ratio of the enclosure.It is found that the magnetic influence number and the inclination angle parameter have pronounced effects on the fluid flow and heat transfer in porous media filled enclosures.

Two-Dimensional Analytical Solution for Compound Channel Flows With Vegetated Floodplains

HUAI Wen-xin, GAO Min, ZENG Yu-hong, LI Dan

2009, 30(9): 1049-1056. doi: 10.3879/j.issn.1000-0887.2009.09.006

Abstract(1960) PDF(928)

Abstract:
A 2-D analytical solution for compound channel flows with vegetated floodplains is presented.For steady uniform flow,the depth-integrated N-S equation was used for analysis.The effects of the vegetation were considered as the drag force item.The secondary currents were also taken into account in the governing equations and preliminary estimation of the secondary current intensity coef-ficient K was discussed.Predicted results for the straight channels and apex cross-section of meandering channels agree well with the experimental data,which shows that the analytical model presented here can be applied to predict the flow in compound channels with vegetated floodplains.

3: 1 Internal Resonance in Multiple Stepped Beam Systems

A. Tekin, E. Özkaya, S. M. BagdatlL

2009, 30(9): 1057-1068. doi: 10.3879/j.issn.1000-0887.2009.09.007

Abstract(1860) PDF(811)

Abstract:
Vibrations of multiple stepped beams with cubic nonlinearities were considered.3:1 internal resonance case was investigated for the system.A general approximate solution of the problem was found using the method of multiple scales,a perturbation technique.The modulation equations of the amplitudes and the phases were derived for two modes.These equations were utilized to determine steady state solutions and their stabilities.It was assumed that external forcing frequency is near to the lower frequency.For numeric part of the study,3:1 ratio in natural frequencies was investigated.These values were observed to be between first and second natural frequencies in cases of clamped-clamped,clamped-pinned support and between second and third natural frequencies in case of pinned-pinned support.Finally,a numeric algorithm was used to solve 3:1 internal resonance.The first mode is externally excited for clamped-clamped,clamped-pinned support and the second mode is externally excited for pinned-pinned support.Then,amplitudes of first and second modes were investigated when the first mode is externally excited.Amplitudes of second and third modes were investigated when the second mode is externally excited.Force-response,damping-response and frequency-response curves were plotted for internal resonance modes of vibrations.Stability analysis was carried out for these plots.

Convolution-Type Semi-Analytic DQ Approach for Transient Response of Rectangular Plates

PENG Jian-she, YANG Jie, YUAN Yu-quan, LUO Guang-bing

2009, 30(9): 1069-1077. doi: 10.3879/j.issn.1000-0887.2009.09.008

Abstract(2047) PDF(835)

Abstract:
The convolution-type Gurtin variational principle is known as the only variational principle,that is,from mathematical point of view,totally equivalent to the initial value problem system.The equation of motion of rectangular thin plates was first transformed to a new governing equation containing initial conditions by using convolution method.A convolution-type semi-analytical DQ approach,which involves differential quadrature (DQ) approximation in space domain and an analytical series expansion in time domain,was proposed to obtain the transient response solution.This approach of-fers the same advantages as Gurtin variational principle and at the same time,is much simpler in the calculation.Numerical results show that it is very accurate,yet computationally efficient for the dynamic response of plates.

Stress Field Analysis on Anti-Plane Orthotropic Bi-Materials of Interface End

LI Jun-lin, WANG Xiao-li

2009, 30(9): 1078-1084. doi: 10.3879/j.issn.1000-0887.2009.09.009

Abstract(1617) PDF(879)

Abstract:
Orthotropic bi-materials anti-plane interface end of flat lap was studied by constructing new stress function and using composite complex function method of material fr acture.The expression of stress fields,displacements fields and stress intensity factor around flat lap interface end,are derived by solving a class of generalize d bi-harmonic equations.The result shows:this type of problems has one singularity,the stress fields has no singularity when two materials constant ratio #>0,the stress field has power singularity,and singularity index trends to -1/2 as # increases.FEM analysis was done to verify correction of the derived equation.

Two-Mode Squeezed State of Nanomechanical Resonators

FAN Rui-qin, YANG Hong-guang, BAI Zhi-ming

2009, 30(9): 1085-1090. doi: 10.3879/j.issn.1000-0887.2009.09.010

Abstract(1299) PDF(685)

Abstract:
A flexible model for the control and measurement of NAMRs was introduced.The free Hamiltonian of the dc-SQUID(direct current superconducting quantum interference device) and the interaction Hamiltonian between these two NAMRs and the dc-SQUID by introducing the annihilation and creation operators under the rotating wave approximation were obtained.The mode of the dcSQUID as a classical flied can be treated.In the Heisenberg picture,the generation of two-mode squeezed states of two nanomechanical resonators is shown by their collective coordinate and momentum operators.

Variational Principle for a Special Cosserat Rod

LIU Dong-sheng, Charles H-T WANG

2009, 30(9): 1091-1099. doi: 10.3879/j.issn.1000-0887.2009.09.011

Abstract(2484) PDF(1136)

Abstract:
Based on the Cosserat theory,the nonlinear models of a rod in 3-dimensional space was described.Using pseudo-rigid body method and variational principle the equations of motion of Cosserat rod including shear deformation were obtained.

Fourier Analysis on Schwarz Domain Decomposition Methods for the Biharmonic Equation

SHANG Yue-qiang, HE Yin-nian

2009, 30(9): 1100-1106. doi: 10.3879/j.issn.1000-0887.2009.09.012

Abstract(1537) PDF(932)

Abstract:
Schwarz methods are an important type of domain decomposition methods.Using the Fourier transform tool,the error propagation matrices and their spectral radii of the classical Schwarz alternating method and the additive Schwarz method for the biharmonic equation were deduced.It not only concisely proves the convergence of the Schwarz methods from a new point of view,but also provides detailed information about the convergence speeds and their dependence on the overlapping size of subdomains.The obtained results are independent of any unknown constant and discretization method,show that the Schwarz alternating method converges twice as quickly as the additive Schwarz method.

Nonlinear Implicit Iterative Method for Solving Nonlinear Ill-Posed Problems

LIU Jian-jun, HE Guo-qiang, KANG Chuan-gang

2009, 30(9): 1107-1116. doi: 10.3879/j.issn.1000-0887.2009.09.013

Abstract(1552) PDF(918)

Abstract:
The implicit iterative method was extended for linear ill-posed operator equations to solve nonlinear ill-posed problems.It shows that under some conditions the error sequence of solutions of the nonlinear implicit iterative method is monotonically decreasing.And with this monotonicity,the convergence of the new method for both of the exact and perturbed equations was proved.

Generalized Elastic Curves in Lorentz Flat Space L⁴

HUANG Rong-pei, SHANG Dong-hu

2009, 30(9): 1117-1124. doi: 10.3879/j.issn.1000-0887.2009.09.014

Abstract(1449) PDF(947)

Abstract:
The extremals of curvature energy actions on non-null curves in 4-dimensional Lorentz-Minkowski space are studied.The motion equations were worked out and three Killing fields along the generalized elastic curves were found.A cylindric coordinate system by using these Killing fields was constructed and the generalized elastic curves by quadratures were expressed.

Discontinuous Penalty Approach With Deviation Integral for Global Constrained Minimization

CHEN Liu, YAO Yi-rong, ZHENG Quan

2009, 30(9): 1125-1134. doi: 10.3879/j.issn.1000-0887.2009.09.015

Abstract(1669) PDF(845)

Abstract:
The discontinuous exact penalty functions is employed to solve constrained minimization problems with the help of integral approach.A general form of constrained deviation integral was provided and its analytical properties was examined.Optimality conditions of the penalized minimization problem was proved as well.In order to implement the algorithm,cross-entropy method and important sampling were used on the basis of Monte-Carlo technique.Numerical tests show that the new algorithm is effective.