应用数学和力学

2005 Vol. 26, No. 8

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Nonlinear Vibration for the Moderate Thickness Rectangular Cracked Plates Including the Coupled Effect of Elastic Foundation

XIAO Yong-gang, FU Yi-ming, ZHA Xu-dong

2005, 26(8): 883-891.

Abstract(2767) PDF(465)

Abstract:
Based on Reissner plate theory and Hamilton variational principle, the nonlinear equations of motion were derived for the moderate thickness rectangular plates with transverse surface penetrating crack on the two-parameter foundation. Under the condition of free boundary, aset of trial functions satisfying all boundary conditions and crack's continuous conditions were proposed. By employing the Galerkin method and the harmonic balance method, the nonlinear vibration equations were solved and the nonlinear vibration behaviors of the plate were analyzed. In numerical computation, the effects of the different location and depth of crack, the different structural parameters of plates and the different physical parameters of foundation on the nonlinear amplitue frequency response curves of the plate were discussed.

Qualitative Analysis of Dynamical Behavior for an Imperfect Incompressible Neo-Hookean Spherical Shell

YUAN Xue-gang, ZHU Zheng-you, CHENG Chang-jun

2005, 26(8): 892-898.

Abstract(2698) PDF(576)

Abstract:
The radial symmetric motion problem was examined for a spherical shell composed of a class of imperfect incompressible hyper-elastic materials, in which the materials may be viewed as the homogeneous incompressible isotropic neo-Hookean material with radial perturbations. A second-order nonlinear ordinary differential equation that describes the radial motion of the inner surface of the shell was obtained. And the first integral of the equation was then carried out. Via analyzing the dynamical properties of the solution of the differential equation, the effects of the prescribed imperfection parameter of the material and the ratio of the inner and the outer radii of the underformed shell on the motion of the inner surface of the shell were discussed, and the corresponding numerical examples were carried out simultaneously. In particular, for some given parameters, it was proved that, there exists a positive critical value, and the motion of the inner surface with respect to time will present a nonlinear periodic oscillation as the difference between the inner and the outer presses does not exceed the critical value. However, as the difference exceeds the critical value, the motion of the inner surface with respect to time will increase infinitely. That is to say, the shell will be destroyed ultimately.

New Algorithm of Coupling Element-Free Galerkin With Finite Element Method

ZHAO Guang-ming, SONG Shun-cheng

2005, 26(8): 899-904.

Abstract(2693) PDF(776)

Abstract:
Through the construction of a new ramp function, the element-free Galerkin method and finite element coupling method were applied to the whole field, and was made fit for the structure of element nodes within the interface regions, both satisfying the essential boundary conditions and deploying meshless nodes and finite elements in a convenient and flexible way, which can meet the requirements of computation for complicated field. The comparison between the results of the present study and the corresponding analytical solutions shows this method is feasible and effective.

Dynamic Stability of Axially Moving Viscoelastic Beams With Pulsating Speed

YANG Xiao-dong, CHEN Li-qun

2005, 26(8): 905-910.

Abstract(2806) PDF(566)

Abstract:
Parametric vibration of an axially moving, elastic, tensioned beam with pulsating speed was investigated in the vicinity of subharmonic and combination resonance. The method of averaging was used to yield a set of autonomous equations when the parametric excitation frequency is twice or the combination of the natural frequencies. Instability boundaries were presented in the plane of parametric frequency and amplitude. The analytical results were numerically verified. The effects of the viscoelastic damping, steady speed and tension on the instability boundaries were numerically demonstrated. It's found that the viscoelastic damping decreases the instability regions and the steady speed and the tension make the instability region drift along the frequency axis.

Interaction of Electric Charges in a Piezoelectric With Rigid External Cracks

HU Yuan-tai, LI Guo-qing, JIANG Shu-nong, HU Hong-ping, YANG Jia-shi

2005, 26(8): 911-920.

Abstract(2810) PDF(516)

Abstract:
Using Stroh's formalism simple explicit expressions of Green's functions for a 2D piezoelectric body with two semi-infinite fixed conductor cracks subjected to a generalized line force were given. The Coulomb force acting on the free line charge due to the piezoelectricity and the distributed bo undary polarization charges was discussed at first. Interactions between two singularities with free charge (s) located in aregion with external cracks were studied, too. The numerical results show that the Coulomb force for two or more singularities with one free charge at least will have much influence on the electr omechanical fields in piezoelectric media when these singularities move closely with each other and therefore cannot be over looked again. The solutions obtained are valid not only for plane and anti-plane problems but also for coupled problems between in-plane and out-of-plane deformations.

Three High-Order Splitting Schemes for the 3D Transport Equation

WANG Shou-dong, SHEN Yong-ming

2005, 26(8): 921-928.

Abstract(2937) PDF(1062)

Abstract:
Two high-order splitting schemes based on the idea of the operators splitting method are given. The three-dimensional advection-diffusion equation was split into several one-dimensional equations that were solved by these two schemes, only three computational grid points were needed in each direction but the accuracy reaches the spatial fourth-order. The third scheme proposed is based on the classical ADI scheme and the accuracy of the advection term of it can reach the spatial fourth-order. Finally, two typical numerical experiments show that the solutions of these three schemes compare well with that given by the analytical solution when the Peclet number is not bigger than 5.

Application of Penalty Function Method in Isoparametric Hybrid Finite Element Analysis

CHEN Dao-zheng, JIAO Zhao-ping

2005, 26(8): 929-936.

Abstract(2723) PDF(521)

Abstract:
By aid of the penalty function method, the equilibrium restriction conditions were intr oduced to the isopar ametric hy brid finite element analysis, and the concrete application course of the penalty function method in three-dimension isoparametric hybrid finite element was discussed. The separated penalty parameters method and the optimal hybrid element model with penalty balance were also presented. The penalty balance method can effectively refrain the par asitical stress on the premise of no additional degrees of freedom. The numeric experiment shows that the prese ntedelement not only is effective in improving greatly the numeric calculation precision of distorted grids but also has universality.

Numerical Study of Periodical Flows of Piezoelectric Valveless Micropump for Biochips

ZHANG Yong-li, WU Jian-kang

2005, 26(8): 937-944.

Abstract(2660) PDF(839)

Abstract:
Shallow water model was employed to approximate the three-dimensional flows of a thin micropump to a two-dimensional thickness-averaged flows. The finite element method and pressure correction algorithm were used to solve the two-dimensional flows of the pump and calculate the pump flow rate. The numerical results indicate that 1) Phase differences in time of flow velocities and backflows occur across section of diffuser connecting to pump chamber; 2) A pair of symmetric vortexes appears inside the pump chamber by the end of suction flow phase; 3) The directional flow rate of the pump is dominated by nonlinearity of Navier-Stokes equations. Quantitative relations of the pump flow rate versus the ratio of diffuser length to width, the ratio of diffuser thickness to width, fluid viscosity and backpressure were also given. Possibly maximal flow rate can be achieved by optimizing the pump parameters.

Linear Quadratic Nonzero Sum Differential Games With Random Jumps

WU Zhen, YU Zhi-yong

2005, 26(8): 945-950.

Abstract(2643) PDF(625)

Abstract:
The existence and uniqueness results of the solutions for one kind of forward-backward stochastic differential equations with Brownian motion and Poisson process as the noise source were given under the monotone conditions. Then these results were applied to get the explicit form of the open-loop Nash equilibrium point for nonzero sum differential games problem with random jump by the solution of the forward-backward stochastic differential equations.

Nonlinear Dynamics Response of Casing Pipe Under Combined Wave-Current

TANG You-gang, GU Jia-yang, ZUO Jian-li, MIN Jian-qin

2005, 26(8): 951-956.

Abstract(2909) PDF(626)

Abstract:
The vortex-induced nonlinear vibration of casing pipes in the deep water was studied considering the loads of current and combined wave-current. The vortex-induced vibration equation of a casing pipe was set up considering the beam mode and Morison's nonlinear fluid loads as well as the vortex-excited loads. The approach of calculating vortex-excited nonlinear vibration by Galerkin's method was proposed. The natural vibration frequencies and modes were obtained, and the response including primary resonance induced by current and the composite resonance under combined wave-current for the 170 m long casing pipe in the 160 m depth of water were investigated. The results show that the dynamics response of casing pipe obviously increases, and the complicated response behaviors of casing pipe are described under combined wave-current.

Analysis of a Screw Dislocation Inside a Circular Inclusion With Interfacial Cracks in Piezoelectric Composites

ZHENG Jian-long, FANG Qi-hong, LIU You-wen

2005, 26(8): 957-964.

Abstract(2550) PDF(538)

Abstract:
The electroelastic interaction of a screw dislocation inside a circular inclusion with interfacial cracks in piezoelectric composite materials under anti-plane shear and in-plane electric loads at infinity is investigated. The general solution to this problem was obtained by means of Riemann- Schwarz's symmetry principle integrated with analysis of singularities of corresponding complex potentials. As a typical example, closed form expressions of the complex potentials and electroelastic field components in the matrix and inhomogeneity regions were derived explicitly when the interface contains a single crack. The image force acting on the screw dislocation was calculated by using the generalized Peach-Koehler formula. The influence of interfacial crack geometry and piezoelectric material property combinations upon the image force was discussed in detail. The results show that interfacial crack has significant perturbation effect on the image force and the equilibrium position of the screw dislocation. The presence of the interfacial crack can change the direction of the image force when the length of the crack goes up to a critical value. The obtained explicit solutions can be used as Green's functions to study the problem on the interaction between interfacial cracks and arbitrary shape crack inside the inclusion. The present solutions can lead to previously known results as the special case.

Partition of Unity Finite Element Method for Short Wave Propagation in Solids

LI Xi-kui, ZHOU Hao-yang

2005, 26(8): 965-971.

Abstract(3043) PDF(657)

Abstract:
A partition of unity finite element method for numerical simulation of short wave propagation in solids is presented. The finite element spaces were constructed by multiplying the standard isoparametric finite element shape functions, which form a partition of unity, with the local subspaces defined on the corresponding shape functions, which include a priori knowledge about the wave motion equation in trial spaces and approximately reproduce the highly oscillatory properties within a single element. Numerical examples demonstrate the performance of the proposed partition of unity finite element in both computational accuracy and efficiency.

h-Adaptivity Analysis Based on Multiple Scale Reproducing Kernel Particle Method

ZHANG Zhi-qian, ZHOU Jin-xiong, WANG Xue-ming, ZHANG Yan-fen, ZHANG Ling

2005, 26(8): 972-978.

Abstract(2429) PDF(637)

Abstract:
An h-adaptivity analysis scheme based on multiple scale reproducing kernel particle method was proposed, and two nodes refinement strategies were constructed using searching-neighbor-nodes (SNN) and local-Delaunay-triangulation (LDT) techniques, which were suitable and effective for h-adaptivity analysis on 2-D problems with the regular or irregular distribution of the nodes. The results of multiresolution and h-adaptivity analyses on 2-D linear elastostatics and bending plate problems demonstrate that the improper high-gradient indicator will reduce the convergence property of the h -adaptivity analysis, and that the efficiency of the LDT node refinement strategy is better than SNN, and that the presented h-adaptivity analysis scheme is provided with the validity, stability and good convergence property.

Dynamic Modeling for Airship Equipped With Ballonets and Ballast

CAI Zi-li, QU Wei-dong, XI Yu-geng

2005, 26(8): 979-987.

Abstract(2684) PDF(765)

Abstract:
Total dynamics of an airship is modeled. The body of an airship was taken as a submerged rigid body with neutral buoyancy, i. e., buoyancy with value equal to that of gravity, and the coupled dynamics between the body with ballonets and ballast was considered. The total dynamics of the airship was firstly derived by Newton-Euler laws and Kirchhoff's equations. Furthermore, by using Hamiltonian and Lagrangian semi-direct product reduction theories, the dynamics was formulated as a Lie-Poisson system, and also an Euler-Poincar system. These two formulations can be exploited for the control design using energy-based methods for Hamiltonian or Lagrangian system.

New Simple Smooth Merit Function for Box Constrained Variational Inequalities and Damped Newton Type Method

Ulji, CHEN Guo-qing

2005, 26(8): 988-996.

Abstract(2633) PDF(643)

Abstract:
By introducing a smooth merit function for the median function, a new smooth merit function for box constrained variational inequalities (BVIs) was constructed. The function is simple and has some good differential properties. A damped Newton type method was presented based on it. Global and local superlinear/quadratic convergence results were obtained under mild conditions, and the finite termination property was also shown for the linear BVIs. Numerical results suggest that the method is efficient and promising.

Probabilistic Models for the Long Fatigue Crack Growth Rates of LZ50 Axle Steel

ZHAO Yong-xiang, HE Chao-ming, YANG Bing, HUANG Yu-zhong, GAO Qing, WU Ping-bo

2005, 26(8): 997-1002.

Abstract(2486) PDF(707)

Abstract:
Experimental study is performed on the probabilistic models for the long fatigue crack growth rates (da/dN) of LZ50 axle steel. An equation for crack growth rate was derived to consider the trend of stress intensity factor range going down to the threshold and the average stress effect. The probabilistic models were presented on the equation. They consist of the probabilistic da/dN- delta K relations, the confidence-based da/dN-delta K relations, and the probabilistic- and confidence-based da/dN-delta K relations. Efforts were made respectively to characterize the effects of probabilistic assessments due to the scattering regularity of test data, the number of sampling, and both of them. These relations can provide wide selections for practice. Analysis on the test data of LZ50 steel indicates that the present models are available and feasible.

Cusp Catastrophe Model of Instability of Pillar in Asymmetric Mining

LI Jiang-teng, CAO Ping

2005, 26(8): 1003-1008.

Abstract(2851) PDF(581)

Abstract:
A simplified mechanical model of pillar-hang wall was established in asymmetric mining and instability of the system was discussed by means of potential energy principle and cusp catastro phe theory. The necessary-sufficient condition and the jump value of displacement of pillar and the re leased energy expressions were derived, which established foundation for quantifying of the instability of system. The results show that instability of the system is related to load and its stiffness distribu tion. The critical load increases with the increasing relative stiffness, and the system is more stable. On the contrary, the instability of system is likely to occur, and the released energy is larger in insta bility process, and the harm is more tremendous accordingly. Furthermore, an example was calculat ed, and the estimated results correspond to practical experience, which provide basis for mining order and arranging stope.