应用数学和力学

2006 Vol. 27, No. 2

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Numerical Study for the Resonant Triad Interaction and Secondary Instability in a Two-Dimensional Supersonic Boundary Layer

DONG Ya-ni, ZHOU Heng

2006, 27(2): 127-133.

Abstract(2921) PDF(824)

Abstract:
Direct numerical simulation was done for a supersonic boundary layer, to see if the mechanism for the generation of sub-harmonic waves, similar to those for the incompressible flows, existed in the process of laminar-turbulent transition. The results show that mechanisms of both resonant triad and secondary instability do exist. Discussions were made on whether these two mechanisms are really important in laminar-turbulent transition.

New Cyclic J-Integral for Low-Cycle Fatigue Crack Growth

HU Hong-jiu, GUO Xing-ming, LI Pei-ning, XIE Yu-jun, LI Jie

2006, 27(2): 134-143.

Abstract(2761) PDF(676)

Abstract:
The constitutive equation under low-cycle fatigue (LCF) was discussed, and a two-dimensional (2-D) model for simulating fatigue crack extension was put forward in order to propose a new cyclic J-integral. The definition, primary characteristics, physical interpretations and numerical evaluation of the new parameter were investigated in detail. Moreover, the new cyclic J-integral for LCF behaviors was validated by the compact tension (CT) specimens, results show that the calculated values of new parameter can correlate well with LCF crack growth rate, during constant-amplitude loading. In addition, the phenomenon of fatigue retardation was explained through the viewpoint of energy based on the concept of new parameter.

Effects of Constant Excitation on Local Bifurcation

WU Zhi-qiang, CHEN Yu-shu

2006, 27(2): 144-150.

Abstract(2271) PDF(556)

Abstract:
The effects of the constant excitation on the local bifurcation of the periodic solutions in the 1:2 internal resonant systems were analyzed based on the singularity theory. It is shown that the constant excitation make influence only when there exist nonlinear terms, in the oscillator with lower frequency. Besides acting as main bifurcation parameter, the constant excitation, together with coefficients of some nonlinear terms, may change the values of unfolding parameters and the type of the bifurcation. Under the non-degenerate cases, the effect of the third order terms can be neglected.

Renewal of Basic Laws and Principles for Polar Continuum Theories(Ⅹ)—Master Balance Law

DAI Tian-min

2006, 27(2): 151-158.

Abstract(2308) PDF(825)

Abstract:
Through a comparison between the expressions of master balance laws and the conservation laws derived by the use of Noether. s thorem a unified master balance law and six physically possible balance equations for micropolar continuum mechanics are naturally deduced. Among them, by extending the well-known conventional concept of energy-momentum tensor, the rather general conservation laws and balance equations named after energy-momentum, energy-angular momentum and energy-energy are obtained. It is clear that the forms of the physical field quantities in the master balance law for the last three cases could not be assumed directly perceived through the intuition. Finally, some existing results are reduced immediately as special cases.

Nonlinear Stability of Balanced Rotor Due to the Effect of Ball Bearing Internal Clearance

BAI Chang-qing, XU Qing-yu, ZHANG Xiao-long

2006, 27(2): 159-169.

Abstract(2302) PDF(781)

Abstract:
Stability and dynamic characteristics of a ball bearing-rotor system are investigated under the effect of the clearance in the ball bearing. Different clearance values were assumed to calculate the nonlinear stability of periodic solution with the aid of the Floquet theory. Bifurcation and chaos behavior were analyzed with variation of the clearance and rotational speed. It is found that there are three routes to unstable periodic solution. The period-doubling bifurcation and the secondary Hopf bifurcation are two usual routes to instability. The third route is the boundary crisis, a chaotic attractor occurs suddenly as the speed passes through its critical value. At last, the instable ranges for different internal clearance values were described. It is useful to investigate the stability property of ball bearing-rotor system.

Some Problems on the Jump Conditions of Shock Waves in 3-Dimensional Solids

LI Yong-chi, YAO Lei, HU Xiu-zhang, CAO Jie-dong, DONG Jie

2006, 27(2): 170-176.

Abstract(2466) PDF(775)

Abstract:
Based on the general conservation laws in continuum mechanics, the Eulerian and Lagrangian descriptions of the jump conditions of shock waves in 3-dimensional solids were presented respectively. The implication of the jump conditions and their relations between each other, particularly the relation between the mass conservation and the displacement continuity, were discussed. Meanwhile the shock wave response curves of the shock waves in 3-dimensional solids, i. e. the Hugoniot curves were analysed, which provide the foundation to study the coupling effects of shock waves in 3-dimensional solids.

Symplectic Duality System on the Plane Magnetoelectroelastic Solids

YAO Wei-an, LI Xiao-chuan

2006, 27(2): 177-185.

Abstract(2811) PDF(578)

Abstract:
By means of the generalized variable principle of magnetoelectroelastic solids, the plane magnetoelectroelastic solids problem was derived to Hamiltonian system. In symplectic geometry space, which consists of origin variables, displacements, electric potential and magnetic potential, and their duality variables, lengthways stress, electric displacement and magnetic induction, the effective methods of separation of variables and symplectic eigenfunction expansion were applied to solve the problem. Then all the eigen-solutions and eigen-solutions in Jordan form on eigenvalue zero can be given, and their specific physical significations were showed clearly. At last, the special solutions were presented with uniform loader, constant electric displacement and constant magnetic induction on two sides of the rectangle domain.

Frequency Domain Criteria for Robust D-Stability of MIMO Systems Based on LMI Method

LI Hai-bin, WANG Zhi-zhen, WANG Long, LI Zhao-ping, LI Er-xiao

2006, 27(2): 186-192.

Abstract(2834) PDF(725)

Abstract:
The problem of checking robust D-stability of multi-in and multi-out (MIMO) systems is studied. Three system models were introduced, i. e. multilinear polynomial matrix, polytopic polynomial matrix and feedback system model. Furthermore, the convex property of each model with respect to the parametric uncertainties was estabilished respectively. Based on this, sufficient conditions for D-stability were expressed in terms of linear matrix inequalities (LMIs) involving only the convex vertices. Therefore, the robust D-stability was tested by solving an LMI optimal problem.

Progressive Fragment Modeling of the Failure Wave in Ceramics Under Planar Impact Loading

YAO Guo-wen, LIU Zhan-fang, HUANG Pei-yan

2006, 27(2): 193-198.

Abstract(2472) PDF(806)

Abstract:
Polycrystalline ceramics have heterogeneous meso-structures which result in high singularity in stress distribution. Based on this, a progressive fragment model was proposed which describes the failure wave formation and propagation in shocked ceramics. The governing equation of the failure wave is characterized by inelastic bulk strain with material damage and fracture. And the inelastic bulk strain consists of dilatant strain from nucleation and expansion of microcracks and condensed strain from collapse of original pores. Numerical simulation of the free surface velocity was performed in good agreement with planar impact experiments on 92.93% aluminas at China Academy of Engineering Physics. And the longitudinal, lateral and shear stress histories upon the arrival of the failure wave were predicted, which present the diminished shear strength and lost spall strength in the failed layer.

Harmonic Component Extraction From a Chaotic Signal Based on EMD Method

LI Hong-guang, MENG Guang

2006, 27(2): 199-203.

Abstract(2380) PDF(849)

Abstract:
A novel approach of signal extraction of a harmonic component from a chaotic signal gen erated by a Duffing oscillator is proposed. Based on empirical mode decomposition(EMD) and con cept that any signals were composed of a series of the simple intrinsic modes, the harmonic compo nents were extracted from the chaotic signals. Simulation results show the approach is satisfactory.

Capillary Effect on the Vertically Excited Surface Wave in a Circular Cylindrical Vessel

JIAN Yong-jun, E Xue-quan, ZHANG Jie

2006, 27(2): 204-210.

Abstract(2715) PDF(522)

Abstract:
In a vertically oscillating circular cylindrical container, singular perturbation theory of two-time scale expansions was developed in inviscid fluids to investigate the motion of single free surface standing wave including the effect of surface tension. A nonlinear slowly varying amplitude equation, which incorporates cubic nonlinear term, external excitation and the influence of surface tension, was derived from potential flow equation. The results show that when forced frequency is lower, the effect of surface tension on mode selection of surface wave is not important. However, when forced frequency is higher, the surface tension can not be neglected. This proved that the surface tension causes free surface returning to equilibrium location. In addition, due to considering the effect of surface tension, the theoretical result much more approaches to experimental results than that of no surface tension.

Perforation of Plastic Spherical Shells Under Impact by Cylindrical Projectiles

NING Jian-guo, SONG Wei-dong

2006, 27(2): 211-216.

Abstract(2435) PDF(584)

Abstract:
The objective is to study the perforation of a plastic spherical shell impacted by a cylindrical projectile. First, the deformation modes of the shell were given by introducing an isometric transformation. Then, the perforation mechanism of the shell was analyzed and an analytical model was advanced. Based on Hamilton principle, the governing equation was obtained and solved using Runge-Kuta method. Finally, some important theoretical predictions were given to describe the perforation mechanism of the shell. The results will play an important role in understanding the perforation mechanism of spherical shells impacted by a projectile.

Nonclassical Potential Symmetries and Invariant Solutions of Heat Equation

QIN Mao-chang, MEI Feng-xiang, XU Xue-jun

2006, 27(2): 217-222.

Abstract(2373) PDF(679)

Abstract:
Some nonclassical potential symmetry generators and group-invariant of heat equation and wave equation were determined. It is shown that new explicit solutions of conserved equations can be constructed by using the nonclassical potential symmetry generators which are derived from their adjoit system. These explicit solutions cannot be obtained by using the Lie or Lie-B_3/4 cklund symmetry group generators of differential equations.

Numerical Simulation of Three Dimensional Gas-Particle Flow in a Spiral Cyclone

YI Lin, WANG Can-xing

2006, 27(2): 223-229.

Abstract(2456) PDF(788)

Abstract:
The three-dimension gas-particle flow in a spiral cyclone is simulated numerically. The gas flow field was obtained by solving the three-dimension Navier-Stokes equations with Reynolds stress model (RSM). It is shown that there are two regions in the cyclone, the steadily tangential flow in the spiral channel and the combined vortex flow in the centre. Numerical results for particles trajectories show that the initial position of the particle at the inlet plane substantially affects its trajectory in the cyclone. The particle collection efficiency curves at different inlet velocities were obtained and the effects of inlet flow rate on the performance of the spiral cyclone were presented. Numerical results also show that the increase of flow rate leads to the increase of particles collection efficiency, but the pressure drop increases sharply.

Analyses of Pulse Wave Propagation in Arteries

PAN Yi-shan, JIA Xiao-bo, CUI Chang-kui, XIAO Xiao-Chun

2006, 27(2): 230-236.

Abstract(2600) PDF(1188)

Abstract:
The blood vessel was regarded as the elasticity tube. And the tissue was considered to re strict the blood vessel wall. The rule of pulse wave propagation in blood vessel was studied. The vis cosity of blood, the elastic modulus of blood vessel and the radius of tube that influenced the pulse wave propagation were analyzed. The result comparison that considered the viscosity of blood with another result shows the viscosity of blood that influences the pulse wave propagation can not be ne glected. The speed of propagation augments with the accretion of the elastic modulus. When the press value of blood stream heightens and diameter of blood vessel reduces, the press of blood stream also heightens, the speed of pulse wave also augments. These results will contribute to making use of the information of pulse wave to analyse and auxiliarily diagnose some causes of human disease.

Set Value Mapping and Its Application

LI Ting

2006, 27(2): 237-242.

Abstract(2925) PDF(1314)

Abstract:
The dynamics of set value mapping is considered. For the upper semi-continuous set value maps, the existence of attractors under some conditions and the upper semi-continuity of attractors under the perturbation were proved. Its application in numerical simulation of differential equation was also considered. The upper semi-continuity of attractors in set value maps under the perturbation was used to show the reasonable of subdivision algorithm and interval arithmetic in numerical simulation of differential equation.

Invariant Form and Integral Invariants on Kähler Manifolds

ZHANG Rong-ye

2006, 27(2): 243-252.

Abstract(3128) PDF(887)

Abstract:
The important notions and results of the integral invariants of Poincar and Car tan-Poincar and the relationship between integral invariant and invariant form established first by E. Cartan in the classical mechanics are generalized to Hamilton mechanics on Kähler manifold by the theory of modern geometry and advanced calculus, to get wider and deeper related results.