力学学报

Assessment of second-order velocity-slip boundary conditions of the navier-stokes equations

Chong Xie, Jing Fan

For micro-scale gas flows, the Navier-Stokes equations with first-ordervelocity slip boundary conditions give results that agree with experimental data in the slipregime, but differ obviously in the transitional regime. Second-ordervelocity-slip boundary conditions were introduced to improve the performanceof the Navier-Stokes equations in the transitional regime. This paperconsiders two-dimensional gas flows through microchannels for which theNavier-Stokes solutions based on different second-order velocity-slipboundary conditions suggested by Cercignani, Deissler, Beskok andKarniadakis, respectively, are compared with the kinetic results given bythe information preservation (IP) method, the direct simulation Monte Carlo(DSMC) method, and experimental data. It is shown that the Cerciganani modelperforms best among the three second-order models we examined, and its massflow rate agrees with the DSMC and IP results even at the Knudsen number of0.4. However, a careful examination of the slip velocities and velocitydistributions at and around the channel surfaces given by the Cercignanimodel demonstrates that they considerately deviate from those given by theDSMC and IP methods at the Knudsen number of 0.1, that is generally regardedas a critical value to divide the slip and transitional regimes.

2007, 23(1): 1-6. doi: 10.6052/0459-1879-2007-1-2005-577

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A model for the scattering of long waves by slotted breakwaters in the presence of currents

Zhenhua Huang, M.S. Ghidaoui

Slotted breakwaters have been used to provide economicalprotection from waves in harbors where surface waves and currents may co-exist. Inthis paper, the effects of currents on the wave scattering by slotted breakwatersare investigated by using a simple model. The model is based on a long waveapproximation. The effects of wave height, barrier geometry and current strength onthe reflection and transmission coefficients are examined by the model. The modelresults are compared with recent experimental data. It is found that boththe wave-following and wave-opposing currents can increase the reflectioncoefficientand reduce the transmission coefficient. The model can be used to study theinteraction between long waves and slotted breakwaters in coastal waters.

2007, 23(1): 1-9. doi: 10.6052/0459-1879-2007-1-2006-240

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Numerical analysis of acoustic radiation vortical modes in a spatially evolving supersonic plane shear layer

Qing Shen, Qiang Wang, Fenggan Zhuang

Disturbance modes and flow structures are numericallyanalyzed for a two-dimensional spatially evolving supersonic plane freeshear layer at Mc = 1.2. The compressible disturbance Navier-Stokesequations are solved by using a modified MacCormack scheme, which is athird-order method in both temporal and spatial accuracies. Three primaryharmonic-wave disturbances with different frequencies are superimposed uponthe mean transverse velocities at inflow sections, and then their nonlineardevelopments are investigated by using the direct numerical simulation (DNS)method. Moreover, a spatial linear stability theory is also introduced, and itis shown that the induced disturbance waves obtained by the DNS methodare ofacoustic radiation vortical modes. The results from analyses of disturbanceparameters and eigenfunctions reveal that the acoustic radiation vorticalmode is an outer mode consisting of fast and slow modes, radiating in thesupersonic disturbance convective Mach number side in the shear layer, witha series of expansion and compression fans. Single-frequency forcingdisturbance can produce a multi-mode mixed disturbance wave without phasedifference, while the slow mode is dominant in the shear layer for a naturaldisturbance.

2007, 23(1): 7-14. doi: 10.6052/0459-1879-2007-1-2005-442

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On the refracted patterns produced by liquid vortices

Yasser Aboelkassem, Georgios H. Vatistas

2007, 23(1): 11-15. doi: 10.6052/0459-1879-2007-1-2006-122

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A finite element/finite volume mixed solver on hybrid grids

Lixin He, Laiping Zhang, Hanxin Zhang

The Discontinuous Galerkin (DG) finite element methods (FEM)have shown to be of high-accuracy for simulating complex flows with shock waves,especially viscous effects near boundary layers. However,they require more CPU time and memory storage than finite volume methods. Onthe other hand, the finite volume methods face thedifficulty of predicting the heat flux over complex geometries, especiallyon unstructured grids. An optimal choice is to combine the two kinds ofmethods to take all their advantages.So in this paper, a finite element/finite volume mixed solver is presented.Within the mixed solver, the previous DG-FEM solver on non-orthogonal gridsis used near the boundary layers to capture the viscous effects, while thefinite volume solver is adopted in the outer field to save the CPU time and memory storage. The numerical flux on the interface of FE/FV solvers is solvedconservatively to guarantee the transformation of FE/FV solvers smoothly.The mixed solver is validated by two hypersonic cases, e.g. hypersonic flowsover a blunt cone and double-ellipsoids. The computational results, includingflow patterns and heat flux distributions, show good agreements withexperimental data, and the comparison on CPU time and memory storagedemonstrates the higher efficiency over the finite element solver.

2007, 23(1): 15-22. doi: 10.6052/0459-1879-2007-1-2005-604

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Stokes' first problem for the fourth order fluid in a porous half

T. Hayat, F. Shahzad, M. Ayub

In this study, the flow of a fourth order fluidin a porous half space is modeled. By using the modified Darcy's law, the flow overasuddenly moving flat plate is studied numerically. The influence of variousparameters of interest on the velocity profile is revealed.

2007, 23(1): 17-21. doi: 10.6052/0459-1879-2007-1-2006-138

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Experimental studies on SL scaling law and spatial scales of coherent structures in near-wall turbulent boundary layer

Wei Zhao

In this paper, a PIV system is used to measure the scales oflarge scale coherent structures in the turbulent boundary layer of a flatplate. The streamwise and spanwise scales of coherent structures atRe_{\theta }=628.5 and 1032.9 are obtained by wavelet analysis andtraditional statistic methods. Multi-scalespatial vector distributions of {\pmb u}', {\pmb v}' and \d {\pmbv}' / \d {\pmb x} obtained by wavelet analysis in theturbulent boundary layer at Re_{\theta }=628.5 are used to verify the SLscaling law and the self-similarity law. It is found that at asingle scale of the flow field, the three statistical structures are matchedwell with the SL law and the self-similarity law, respectively, but theparameters of the SL law are not unique, but varying with flow scales.From the relation between S_{5}, and l (the distance), inertia ranges are foundnot the same for the SL scaling law and the self-similarity law, butvarying also with the scale.

2007, 23(1): 23-36. doi: 10.6052/0459-1879-2007-1-2005-215

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Flow resistance in compound channels and its prediction methods

Kejun Yang, Shuyou Cao, Xingnian Liu

A series of experiments was carried out in a largesymmetric compound channel composed of a rough main channel and roughfloodplains to investigate the resistance characteristics of inbank andoverbank flows. The effective Manning, Darcy-Weisbach, Chezy coefficientsand the relative Nikuradse roughness height were analyzed. Manydifferent representative methods for predicting the composite roughness weresystematically summarized. Besides the measured data, a vast number oflaboratory data and field data for compound channels were collected and usedto check the validity of these methods for different subsection divisionsincluding the vertical, horizontal, diagonal and bisectional divisions. Thecomputation showed that these methods resulted in big errors in assessingthe composite roughness in compound channels, and the reasons were analyzedin detail. The error magnitude is related to the subsection divisions.

2007, 23(1): 23-31. doi: 10.6052/0459-1879-2007-1-2006-017

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Wave dynamic processes in cellular detonation reflection from wedges

Zongmin Hu, Zonglin Jiang

When the cell width of the incident detonation wave (IDW) is comparable to or larger than the Mach stem height, self-similarity will fail during IDW reflection from a wedge surface. In this paper, the detonation reflection from wedges is investigated for at the wave dynamic processes occurring in the wave front, including transverse shock motion and detonation cell variations behind the Mach stem. A detailed reaction model is implemented to simulate two-dimensional cellular detonations in stoichiometric mixtures of H2/O2 diluted by Argon. The numerical results show that the transverse waves, which cross the triple point trajectory of Mach reflection, travel along the Mach stem and reflect back from the wedge surface, control the size of the cells in the region swept by the Mach stem. It is the energy carried by these transverse waves that sustain the triple-wave-collision of higher frequency within the over-driven Mach stem. In some cases, local wave dynamic processes and wave structures play a dominant role in determining the pattern of cellular record, leading to the fact that the cellular patterns after the Mach stem exhibit some peculiar modes.

2007, 23(1): 33-41. doi: 10.6052/0459-1879-2007-1-2005-431

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Development of reduced chemical reaction kinetic model for hydro-carbon fuel combustion

Weiqi Qian, Shunhua Yang, Baoguo Xiao, Jialing Le

A method based on quasi-steady state approximation (QSSA) isused to construct a reduced chemical kinetic model for the ignitionand combustion of Hydro-carbon fuel, and a software package named SPARCK(Software Package for Reduction of Chemical Kinetics) is developed.Firstly, this method and software is applied to reduce the detailedelementary chemical kinetic model GRI2.11 for the combustion of methane,and a reduced global reaction kinetic model containing 14 species and 10 globalreactions is obtained. Secondly, when this method and software is applied tothe detailed elementary chemical kinetic model of ethylene combustion whichcontaining 51 species and 365 elementary reactions, a reduced globalreaction kinetic model involving 20 species and 16 global reactions canbe obtained. Finally, for the combustion of heptane, a reduced global reactionkinetic model involving 26 species and 22 global reactions is obtained from adetailed mechanism having 160 species and 1540 elementary reactions by thismethod. After using these reduced kinetic models to numerically simulate thetypical fuel ignition process in the shock tube, it can be seen that thesereduced models can represent the ignition mechanism of the detailed kineticmodels quite well and with good accuracies.

2007, 23(1): 37-45. doi: 10.6052/0459-1879-2007-1-2005-275

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Bifurcation of thermocapillary convection in a shallow annular pool of silicon melt

Yourong Li, Lan Peng, Shuangying Wu, Nobuyuki Imaishi

In order to understand the nature of surface patterns on siliconmelts inindustrial Czochralski furnaces, we conducted a series of unsteadythree-dimensional numerical simulations of thermocapillary convections inthin silicon melt pools in an annular container. The pool is heated from theouter cylindrical wall and cooled at the inner wall. Bottom and top surfacesare adiabatic. The results show that the flow is steady and axisymmetric atsmall temperature difference in the radial direction. When the temperaturedifference exceeds a certain threshold value, hydrothermal waves appear andbifurcation occurs. In this case, the flow is unsteady and there are twopossible groups of hydrothermal waves with different number of waves, whichare characterized by spoke patterns traveling in the clockwise andcounter-clockwise directions. Details of the flow and temperaturedisturbances are discussed and number of waves and traveling velocity of thehydrothermal wave are determined.

2007, 23(1): 43-48. doi: 10.6052/0459-1879-2007-1-2005-470

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Reflection and transmission of an internal solitary wave over a step in stratified fluid

Gang Wei, Yunxiang You, Guoping Miao, Xueming Qin

Transmission, reflection and fission of an internal solitarywave incident upon a step in a two-layer fluid system are investigatedanalytically based on the matched-asymptotic expansion andthe Green function. The reduced boundary condition relevant to the effect ofthe step-topography on Boussinesq equations is derived by applying theconformal mapping theory and solving the singular Fredholm integralequation. A problem of the `initial' value for KdV evolution equation isformulated. The explicit expressions for transmitted and reflected waves aregiven by the inverse scattering method. It follows that there exist obviouseffects of step height, density ratio and thickness ratio of upper- tolower-layer on the amplitudes of transmitted and reflected waves and theirnumber of fission. It is also found that when the upper layerthickness is larger than the lower layer one, the amplitude of the reflectedwave monotonously increases with theincrease of step height is before the critical point, thenmonotonously decreases, and it is the other way round for thetransmitted wave. The phase of the reflected wave on the convex step isjust opposite to the incident wave, and its maximum amplitude can approachseveral folds of the incident one. The reflected wave on the concave step can evolveinto a single solitary wave in certain stratified situations, which differfrom the oscillating decay tail in the single layer fluid system.

2007, 23(1): 45-53. doi: 10.6052/0459-1879-2007-1-2005-295

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Lane changing analysis for two-lane traffic flow

Tieqiao Tang, Haijun Huang, S.C. Wong, Rui Jiang

In this paper, the two-lane traffic are studied by using thelane-changing rules in the car-following models. The simulation show thatthe frequent lane changing occurs when the lateral distance in car followingactivities is considered and it gives rise to oscillating waves. Incontrast, if the lateral distance is not considered (or consideredoccasionally), the lane changing appears infrequently and soliton wavesoccurs. This implies that the stabilization mechanism no longer functionswhen the lane changing is permitted. Since the oscillating and solitonwaves correspond to the unstable and metastable flow regimes, respectively,our study verifies that a phase transition may occur as a result of the lanechanging.

2007, 23(1): 49-54. doi: 10.6052/0459-1879-2007-1-2006-282

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Elasto-plastic analysis for 2d structures with truss-like materials

Lin Liu, Jun Yan, Gengdong Cheng

The elasto-plastic analysis of structures composed oftruss-like materials takes much time and considerable resourcesin modeling and numerical calculation if all struts are taken intoconsideration. The main purpose of this paper is to speed up the solution ofthis class of problems. The unit cell is simplified as a truss model accordingto the large ratio of strut's length to the section size. Numericalhomogenization is carried out due to the periodic arrangement of cells inspace. The original problem is thereby transformed to two interrelatedproblems of two different scales: a nonlinear elastic continuum computation inmacro-scale and several elasto-plastic analyses of small-scale truss systemsin micro-scale. Monotone load, non-monotone load, regular macro-structureand irregular macro-structure with imperfect unit cells are, respectively,considered in two numerical examples. In comparison with the results ofthe actual structures, the proposed method is found to enjoy the same precision but to take less time. At last the applicable conditions of the proposed method are discussed.

2007, 23(1): 54-62. doi: 10.6052/0459-1879-2007-1-2005-407

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Electromagnetoelastic behaviors of functionally graded piezoelectric

Hongliang Dai, Yiming Fu, J.H. Yang

Analytical studies on electromagnetoelastic behaviors arepresented for the functionally graded piezoelectric material (FGPM) solidcylinder and sphere placed in a uniform magnetic field and subjected to theexternal pressure and electric loading. When the mechanical, electric andmagnetic properties of the material obey an identical power law in theradial direction, the exact displacements, stresses, electric potentials andperturbations of magnetic field vector in the FGPM solid cylinder and sphereare obtained by using the infinitesimal theory of electromagnetoelasticity.Numerical examples also show the significant influence of materialinhomogeneity. It is interesting to note that selecting a specific value ofinhomogeneity parameter can optimize the electromagnetoelasticresponses, which will be of particular importance in modern engineeringdesigns.

2007, 23(1): 55-63. doi: 10.6052/0459-1879-2007-1-2006-127

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The reproducing kernel particle method for numerical analysis of high-speed impact process

Guangming Zhao, Shuncheng Song, Xianjie Yang

The Reproducing Kernel Particle Method, a new type ofmeshless method, is adopted in this study, for numericalsimulations of high-speed impact processes. The relatedcontrol equations are obtained and the Bordner-Partom constitutive modelis used for large strain and high strain-rate when the material is under ahigh-speed impact. A new method is proposed for the treatment of interfaces,and the Velocity Collocation Method(VCM) is used to meet the speed requirement of the interface and its boundary.Numerical simulations show that the Reproducing KernelParticle Method can correctly simulate the high-speed impact processwith velatively high degree of accuracy.

2007, 23(1): 63-69. doi: 10.6052/0459-1879-2007-1-2005-404

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Finite element simulation on the mechanical properties of MHS materials

Z.Y. Gao, Tongxi Yu, D. Karagiozova

Finite element simulations are carried out to examine the mechanical behavior of the metallic hollow sphere (MHS) material during their large plastic deformation and to estimate the energy absorbing capacity of these materials under uniaxial compression. A simplified model is proposed from experimental observations to describe the connection between the neighboring spheres, which greatly improves the computation efficiency. The effects of the governing physical and geometrical parameters are evaluated; whilst a special attention is paid to the plateau stress, which is directly related to the energy absorbing capacity. Finally, the empirical functions of the relative material density are proposed for the elastic modulus, yield strength and plateau stress for FCC packing arrangement of hollow spheres, showing a good agreement with the experimental results obtained in our previous study.

2007, 23(1): 65-75. doi: 10.6052/0459-1879-2007-1-2006-198

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Constitutive model of frozen soil with damage and numerical simulation of the coupled problem

Jianguo Ning, Zhiwu Zhu

On the basis of the microscopic mechanics of compositematerial, an elastic constitutive model for frozen soil containing damage ispresented. For the frozen sand soil with different ice contents and underdifferent temperature conditions, the result calculated by the constitutivemodel agrees with the actually measured stress-strain curve. Afternumerically simulating the coupled problem of water, temperature and stressfields of channel frozen and frozen soil foundation by means ofa program, a more accurate and practical temperature field isobtained, which is consistent with the prediction and tests inliterature. It is shown that relatedphysical properties of frozen soil and a quantitativerelationship between them can be obtained.

2007, 23(1): 70-76. doi: 10.6052/0459-1879-2007-1-2006-177

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The third dimensionless parameter in the penetration dynamics of rigid projectiles

Xiaowei Chen, Xiaoli Li, Yuze Chen, Haijun Wu, Fenglei Huang

The present paper defines a third dimensionless parameter,i.e., the damping function {\xi}, besides the impact function I andgeometry function N of a projectile introduced early, in the penetrationdynamics of rigid projectiles. It only depends on the interaction of theprojectile and the target materials and is independent of the projectile geometry. Ageneral penetration drag, which contains the terms of damping effect and thedummy mass of the projectile induced by the deceleration effect, is adopted inthe formulation. Dimensionless formula for the depth of penetration (DOP) isobtained with only three parameters I, N and {\xi} for generalconvex shapes of various hard projectiles. Different geometry parameters arealso presented for some common shapes of warheads. Theoretical predictionsfor DOP dependent/independent of the damping function {\xi}, showgood agreement with the published test data for differentprojectiles and impact velocities as well as different targets.

2007, 23(1): 77-84. doi: 10.6052/0459-1879-2007-1-2006-069

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Topology optimization of material microstructures using strain energy-based prediction of effective elastic properties

Weihong Zhang, Gaoming Dai, Fengwen Wang, Shiping Sun, Hicham Bassir

An alternative strain energy method is proposed for the prediction of effective elastic properties of orthotropic materials in this paper. The method is implemented in the topology optimization procedure to design cellular solids. A comparative study is made between the strain energy method and the well-known homogenization method. Numerical results show that both methods agree well in the numerical prediction and sensitivity analysis of effective elastic tensor when homogeneous boundary conditions are properly specified. Two dimensional and 3 dimensional microstructures are optimized for maximum stiffness designs by combining the proposed method with the dual optimization algorithm of convex programming. Satisfactory results are obtained for a variety of design cases.

2007, 23(1): 77-89. doi: 10.6052/0459-1879-2007-1-2006-086

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A microstructure-based damage constitutive model for pearlitic materials

Based on the laminar microstructure formed by ferrite andcementite with very fine interlamellar spacing, and the morphology ofmicrodefects in the ferrite, cementite and interface, a unifieddamage evolution is proposed by making use of the work dissipated on damage.It is then embedded in the constitutive model of each phase and a damageelastoplastic constitutive model is obtained for a single pearlitic colony.The damage constitutive description for pearlitic materials is formulatedusing the Hill's self-consistent scheme by assuming that a pearliticmaterial element is an aggregate of numerous cells of pearlitic colonieswith randomly distributed orientations. It is significant that the obtainedconstitutive description contains explicitly the interlamellar spacing as amicrostructure parameter, which easily accounts for the better comprehensivemechanical properties of the pearlitic materials with smaller interlamellarspacing. The constitutive behavior of pearlitic steel is simulated, and comparedwith the experimental results.

2007, 23(1): 85-92. doi: 10.6052/0459-1879-2007-1-2006-229

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A new simple method of implicit time integration for dynamic problems of engineering structures

Jun Zhou, Youhe Zhou

This paper presents a new simple method of implicit time integration with two control parameters for solving initial-value problems of dynamics such that its accuracy is at least of order two along with the conditional and unconditional stability regions of the parameters. When the control parameters in the method are optimally taken in their regions, the accuracy may be improved to reach of order three. It is found that the new scheme can achieve lower numerical amplitude dissipation and period dispersion than some of the existing methods, e.g. the Newmark method and Zhai's approach, when the same time step size is used. The region of time step dependent on the parameters in the new scheme is explicitly obtained. Finally, some examples of dynamic problems are given to show the accuracy and efficiency of the proposed scheme applied in dynamic systems.

2007, 23(1): 91-99. doi: 10.6052/0459-1879-2007-1-2006-167

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Calculation of stress intensity factors by boundary element method based on erdogan fundamental solutions

Cheng Su, Chun Zheng

The Erdogan fundamental solutions for infinitecracked plates are introduced in this paper. The spline fictitious boundaryelement method is then proposed and formulated for analysis of mode I andmixed mode (mode I and II) problems based on the above fundamentalsolutions. The proposed method is further applied to analyze certaincrack problems, in which the computation accuracy, convergence rate and theversatility of the method are demonstrated in details.

2007, 23(1): 93-99. doi: 10.6052/0459-1879-2007-1-2006-040

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Application of euler midpoint symplectic integration method for the solution of dynamic equilibrium equations

Yufeng Xing, Rong Yang

The dynamic equilibrium equations {\pmb M} \ddot {\pmb x} +{\pmb C}\dot {\pmb x} + {\pmb K \pmb x} = {\pmb R} are solved by the Euler midpoint implicit integration method. The properties ofJacobi matrix of the algorithm are discussed in detail, and it is shown that Jacobi matrix independent of the external load vector{\pmb R} is symplectic if {\pmb C} = 0, and the amplitude of alleigenvalues of symplectic matrix are equal to unity.It is proved that the Newmark method with \delta = 0.5 and \alpha =0.25 is just the Euler midpoint implicit integration method; and for aconservative system, it is a structure-preserving algorithm, which meansthat the energy of the system is preserved through the solution process.Numerical analyses are carried out to illustrate the advantages ofthe symplectic algorithm in the solution ofnon-conservative systems.The accuracy of structure-preserving algorithm is not sensitive to theratio of the frequency of the external force to that of the system,while the accuracy of Newmark algorithm with \delta \ne 0.5 issensitive to that ratio.

2007, 23(1): 100-105. doi: 10.6052/0459-1879-2007-1-2006-026

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Improved suboptimal Bang-Bang control of aseismic buildings with variable friction dampers

Hongnan Li, Jun Li, Gangbing Song

One of the challenges in civil engineering is to find an innovative means of suppressing the structural vibration due to earthquake and wind loadings. This paper presents an approach for effectively suppressing vibrations of a structure with variable friction damper using a new Bang-Bang control input. A continuous function of story velocities is used to represent the improved control to reduce chatter, high frequency switching and avoid instability. With a genetic algorithm, the amplitudes of control and preloading friction forces individually prescribed in the controller and damper are optimized for enhancing the seismic performance of buildings. The control strategy for the friction damper is proposed to for three story building with one variable friction damper installed at the first story for seismic reduction. The numerical results indicate that a better reduction of peak response accelerations of floors can be achieved than those of the unmodified controller, and the adaptability of the control system is also improved greatly by comparison with the reduction ratios of the structural response energy excited by different earthquake intensities.

2007, 23(1): 101-109. doi: 10.6052/0459-1879-2007-1-2005-601

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A variational principle form for initial value problems in analytical mechanics

Lifu Liang, En Luo, Xiaojiu Feng

According to relationsbetween generalized forces and generalized displacements,convolution are performed between the governingequations of initial value problems in the primary space and the corresponding virtual quantities, and results are added algebraically.A variational principle form for initial valueproblems in analytical mechanics are then established in the original space, i.e.the variational principles and generalized variational principlesin convolution form for initialvalue problems in analytical mechanics are established in the originalspace. The stationary conditions of the variational principles andgeneralized variational principles are deduced. In the meantime, the variational integral method is generalized intothe convolutional variational integral method. Using these variationalprinciples and generalized variational principles for initial value problemsin analytical mechanics, we can establish models of finite elementmethod and other approximate calculation method and can also find exactsolutions of initial valueproblems in analytical mechanics and transform differentialequations into algebraic equations.

2007, 23(1): 106-111. doi: 10.6052/0459-1879-2007-1-2006-027

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Nonlinear formulation for flexible multibody system with large deformation

Jinyang Liu, Jiazhen Hong

In this paper, nonlinear modeling for flexible multibody system with large deformation is investigated. Absolute nodal coordinates are employed to describe the displacement, and variational motion equations of a flexible body are derived on the basis of geometric nonlinear theory, in which both the shear strain and the transverse normal strain are taken into account. By separating the inner and the boundary nodal coordinates, the motion equations of a flexible multibody system are assembled. The advantage of such formulation is that the constraint equations and the forward recursive equations become linear because the absolute nodal coordinates are used. A spatial double pendulum connected to the ground with a spherical joint is simulated to investigate the dynamic performance of flexible beams with large deformation. Finally, the resultant constant total energy validates the present formulation.

2007, 23(1): 111-119. doi: 10.6052/0459-1879-2007-1-2006-113

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Modal perturbation method for obtaining complex modal characteristics of non-proportional damping systems

Menglin Lou, Yaoqing Fan

An approximateapproach based on the direct modal perturbation method is suggested foranalyzing the complex modal characteristics of non-proportional dampingstructure systems. The proposed approach uses the expansion of complex Ritz vectors. The procedure is in two steps. First, thesubspace of the solution based on the complex vectors is formed by using theconventional real modes of the structure. Then, a set of solutions ofnon-linear complex algebraic equations is established to displace thesolution of the complex eigen-value equations. The results of two numericalexamples show that the suggested method simplifies the solving procedure andis of high precision and efficiency. It can be used for the solution of thecomplex modal characteristics and dynamic responses of non-proportionaldamping structure systems.

2007, 23(1): 112-118. doi: 10.6052/0459-1879-2007-1-2006-170

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Wave propagation characteristics of cylindrical shells with circumferential surface crack

Xiang Zhu, Tianyun Li, Yao Zhao, Jingxi Liu

In this paper, thewave propagation characteristics of infinite cylindrical shells withcircumferential surface crack are investigated. The vibration of thecylindrical shell is described by Fl\"{u}gge's shell equations and the wavepropagation approach is used to solve the equations. Inconsideration of the crack's three modes: opening, sliding and tearingmodes andtheir combinations, the surface crack is modeled by distributed line springs.The local compliance matrix at the cracked region is then deduced by usinglinear elastic fracture mechanics, and the relationships between theadditional general displacements and the general forces are obtained. To verify the accuracy of the theoretical model and the wavepropagation method proposed, the axisymmetric free vibration of a finiteshell with a circumferential surface crack is investigated first, thenatural frequencies of the finite cracked shell are calculated by usingtheanalytical method and finite element method, respectively. The resultsbetween the two methods show a good agreement. Then an infinite shell with acircumferential surface crack is investigated. For a given incident wave,the amplitudes of the reflected wave and transmitted wave are obtainedaccording to the continuity conditions of displacements and inner forcesat both sides of the crack. The relationships between the transmitted wavecoefficient and the crack's depth as well as the frequency areobtained. The numerical results indicate that the existence of the crackchanges the vibration wave in the shell substantially and the amplitudes ofthe transmitted wave and reflected wave are closely related to the crack'sdepth and the frequency. The results also show that the coefficients ofthe transmitted wave decrease with the increase of the crack's depth. Thisresearch provides a theoretical basis for the further crack detection incylindrical shells based on the vibration method.

2007, 23(1): 119-124. doi: 10.6052/0459-1879-2007-1-2005-560

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Non-probabilistic reliability-based topology optimization of continuum structures

Yangjun Luo, Zhan Kang

It is of great importance to incorporate uncertainties intostructural topology optimization problems. Based on the definition of thenon-probabilistic reliability index, this paper proposes a mathematicalmodel for topology optimization of 3D continuum structures withnon-probabilistic reliability constraints on deformation. In this model,the uncertainties in material properties, geometrical dimensionsand loading conditions are accounted for. The Performance Measure Approach(PMA) is adopted for transformation of the constraints. In this context, theadjoint variable scheme for sensitivity analysis of the target performanceis discussed, which facilitates a mathematical programming solution of theoptimization problem. Numerical investigations illustrate the applicabilityand the validity of the present model as well as the proposednumerical techniques. The computational results show thatnon-probabilistic reliability-based topology optimization may yield morereasonable material layouts than the conventional deterministic approach.

2007, 23(1): 125-131. doi: 10.6052/0459-1879-2007-1-2006-181

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Analysis to grazing bifurcation in linear vibro-impact system with N-dimensions

Sijin Zhang, Libiao Zhou, Qishao Lu

Grazing bifurcation is an important dynamical behavior ofa vibro-impact system and is usually analyzed by choosing the impact plane as thePoincar\'{e} section. However, this plane sometimes does not meet thetransverse intersection condition of Poincar\'{e} section, especially whilegrazing motion or chaos take place. Moreover, the bifurcation of impactnumber instead of period of the motion is considered in former cases. Thebifurcations with time evolution are more attractive for a vibro-impactsystem.In this paper, the Poincar\'{e} map of period-n motions with single-impactis set up for a linear vibro-impact system by using a fixed phase plane asthe Poincar\'{e} section here. Based on analysis of the Poincar\'{e} map,the grazing bifurcation conditions and bifurcation equations are determinedfor the vibro-impact system, and a vibro-impact system with single DOF isused as an example to testify the obtained analytical result. A numericalsimulation is carried out for the bifurcation diagram of the vibro-impactsystem, which agrees with analytical results very well. This method canbe used tocalculate not only the parameters of grazing bifurcation, but also those ofany period-$n$ motions, for a linear vibro-impact system.

2007, 23(1): 132-136. doi: 10.6052/0459-1879-2007-1-2006-169

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Remarks on the ambiguity in multiple scales method

Yanmao Chen, Jike Liu

The method of multiple scales (MMS), developed forsystems with small non-linearities, is one of the most widely usedperturbation methods. Only particular solutions are sought for the higherorder approximate equations by using the ordinary MMS. An observation ismade inthis paper that the MMS works well only for the approximate solutions ofthe first two orders, while gives rise to a paradox in obtaining the thirdorder approximate solution of van der Pol equation. Taking the famous vander Pol equation as an illustrative example, it is proven that neglectingthe first order harmonic of the first order approximate solution may make the derivative sequence of the second order mixed partial derivativenot commutable. This leads to the ambiguity of the MMS and anothermathematical paradox. Unlike the ordinary MMS, the general solutioncontaining the first harmonic is adopted for the first order approximateequation, and then the ambiguity and the paradox are both eliminated. Theapproximate solutions are obtained by the proposed method and compared withthe numerical solutions. It is shown that the present technique is valid.

2007, 23(1): 137-140. doi: 10.6052/0459-1879-2007-1-2006-293

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