#### Table of Content

18 January 2007, Volume 23 Issue 1
Research paper
Assessment of second-order velocity-slip boundary conditions of the navier-stokes equations
Chong Xie Jing Fan
2007, 23(1):  1-6.  DOI: 10.6052/0459-1879-2007-1-2005-577
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For micro-scale gas flows, the Navier-Stokes equations with first-order velocity slip boundary conditions give results that agree with experimental data in the slip regime, but differ obviously in the transitional regime. Second-order velocity-slip boundary conditions were introduced to improve the performance of the Navier-Stokes equations in the transitional regime. This paper considers two-dimensional gas flows through microchannels for which the Navier-Stokes solutions based on different second-order velocity-slip boundary conditions suggested by Cercignani, Deissler, Beskok and Karniadakis, respectively, are compared with the kinetic results given by the information preservation (IP) method, the direct simulation Monte Carlo (DSMC) method, and experimental data. It is shown that the Cerciganani model performs best among the three second-order models we examined, and its mass flow rate agrees with the DSMC and IP results even at the Knudsen number of 0.4. However, a careful examination of the slip velocities and velocity distributions at and around the channel surfaces given by the Cercignani model demonstrates that they considerately deviate from those given by the DSMC and IP methods at the Knudsen number of 0.1, that is generally regarded as a critical value to divide the slip and transitional regimes.
A model for the scattering of long waves by slotted breakwaters in the presence of currents
Zhenhua Huang M.S. Ghidaoui
2007, 23(1):  1-9.  DOI: 10.6052/0459-1879-2007-1-2006-240
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Slotted breakwaters have been used to provide economical protection from waves in harbors where surface waves and currents may co-exist. In this paper, the effects of currents on the wave scattering by slotted breakwaters are investigated by using a simple model. The model is based on a long wave approximation. The effects of wave height, barrier geometry and current strength on the reflection and transmission coefficients are examined by the model. The model results are compared with recent experimental data. It is found that both the wave-following and wave-opposing currents can increase the reflection coefficient and reduce the transmission coefficient. The model can be used to study the interaction between long waves and slotted breakwaters in coastal waters.
Numerical analysis of acoustic radiation vortical modes in a spatially evolving supersonic plane shear layer
Qing Shen Qiang Wang Fenggan Zhuang
2007, 23(1):  7-14.  DOI: 10.6052/0459-1879-2007-1-2005-442
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Disturbance modes and flow structures are numerically analyzed for a two-dimensional spatially evolving supersonic plane free shear layer at Mc = 1.2. The compressible disturbance Navier-Stokes equations are solved by using a modified MacCormack scheme, which is a third-order method in both temporal and spatial accuracies. Three primary harmonic-wave disturbances with different frequencies are superimposed upon the mean transverse velocities at inflow sections, and then their nonlinear developments are investigated by using the direct numerical simulation (DNS) method. Moreover, a spatial linear stability theory is also introduced, and it is shown that the induced disturbance waves obtained by the DNS method are of acoustic radiation vortical modes. The results from analyses of disturbance parameters and eigenfunctions reveal that the acoustic radiation vortical mode is an outer mode consisting of fast and slow modes, radiating in the supersonic disturbance convective Mach number side in the shear layer, with a series of expansion and compression fans. Single-frequency forcing disturbance can produce a multi-mode mixed disturbance wave without phase difference, while the slow mode is dominant in the shear layer for a natural disturbance.
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
2007, 23(1):  15-22.  DOI: 10.6052/0459-1879-2007-1-2005-604
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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. On the other hand, the finite volume methods face the difficulty of predicting the heat flux over complex geometries, especially on unstructured grids. An optimal choice is to combine the two kinds of methods 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 grids is used near the boundary layers to capture the viscous effects, while the finite 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 solved conservatively to guarantee the transformation of FE/FV solvers smoothly. The mixed solver is validated by two hypersonic cases, e.g. hypersonic flows over a blunt cone and double-ellipsoids. The computational results, including flow patterns and heat flux distributions, show good agreements with experimental data, and the comparison on CPU time and memory storage demonstrates the higher efficiency over the finite element solver.
Stokes' first problem for the fourth order fluid in a porous half
2007, 23(1):  17-21.  DOI: 10.6052/0459-1879-2007-1-2006-138
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In this study, the flow of a fourth order fluid in a porous half space is modeled. By using the modified Darcy's law, the flow over a suddenly moving flat plate is studied numerically. The influence of various parameters of interest on the velocity profile is revealed.
Experimental studies on SL scaling law and spatial scales of coherent structures in near-wall turbulent boundary layer
Wei Zhao
2007, 23(1):  23-36.  DOI: 10.6052/0459-1879-2007-1-2005-215
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In this paper, a PIV system is used to measure the scales of large scale coherent structures in the turbulent boundary layer of a flat plate. The streamwise and spanwise scales of coherent structures at Re_{\theta }=628.5 and 1032.9 are obtained by wavelet analysis and traditional statistic methods. Multi-scale spatial vector distributions of {\pmb u}', {\pmb v}' and \d {\pmb v}' / \d {\pmb x} obtained by wavelet analysis in the turbulent boundary layer at Re_{\theta }=628.5 are used to verify the SL scaling law and the self-similarity law. It is found that at a single scale of the flow field, the three statistical structures are matched well with the SL law and the self-similarity law, respectively, but the parameters 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 found not the same for the SL scaling law and the self-similarity law, but varying also with the scale.
Flow resistance in compound channels and its prediction methods
Kejun Yang Shuyou Cao Xingnian Liu
2007, 23(1):  23-31.  DOI: 10.6052/0459-1879-2007-1-2006-017
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A series of experiments was carried out in a large symmetric compound channel composed of a rough main channel and rough floodplains to investigate the resistance characteristics of inbank and overbank flows. The effective Manning, Darcy-Weisbach, Chezy coefficients and the relative Nikuradse roughness height were analyzed. Many different representative methods for predicting the composite roughness were systematically summarized. Besides the measured data, a vast number of laboratory data and field data for compound channels were collected and used to check the validity of these methods for different subsection divisions including the vertical, horizontal, diagonal and bisectional divisions. The computation showed that these methods resulted in big errors in assessing the composite roughness in compound channels, and the reasons were analyzed in detail. The error magnitude is related to the subsection divisions.
Wave dynamic processes in cellular detonation reflection from wedges
Zongmin Hu Zonglin Jiang
2007, 23(1):  33-41.  DOI: 10.6052/0459-1879-2007-1-2005-431
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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.
Development of reduced chemical reaction kinetic model for hydro-carbon fuel combustion
Weiqi Qian Shunhua Yang Baoguo Xiao Jialing Le
2007, 23(1):  37-45.  DOI: 10.6052/0459-1879-2007-1-2005-275
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A method based on quasi-steady state approximation (QSSA) is used to construct a reduced chemical kinetic model for the ignition and 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 detailed elementary chemical kinetic model GRI2.11 for the combustion of methane, and a reduced global reaction kinetic model containing 14 species and 10 global reactions is obtained. Secondly, when this method and software is applied to the detailed elementary chemical kinetic model of ethylene combustion which containing 51 species and 365 elementary reactions, a reduced global reaction kinetic model involving 20 species and 16 global reactions can be obtained. Finally, for the combustion of heptane, a reduced global reaction kinetic model involving 26 species and 22 global reactions is obtained from a detailed mechanism having 160 species and 1540 elementary reactions by this method. After using these reduced kinetic models to numerically simulate the typical fuel ignition process in the shock tube, it can be seen that these reduced models can represent the ignition mechanism of the detailed kinetic models quite well and with good accuracies.
Bifurcation of thermocapillary convection in a shallow annular pool of silicon melt
Yourong Li Lan Peng Shuangying Wu Nobuyuki Imaishi
2007, 23(1):  43-48.  DOI: 10.6052/0459-1879-2007-1-2005-470
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In order to understand the nature of surface patterns on silicon melts in industrial Czochralski furnaces, we conducted a series of unsteady three-dimensional numerical simulations of thermocapillary convections in thin silicon melt pools in an annular container. The pool is heated from the outer cylindrical wall and cooled at the inner wall. Bottom and top surfaces are adiabatic. The results show that the flow is steady and axisymmetric at small temperature difference in the radial direction. When the temperature difference exceeds a certain threshold value, hydrothermal waves appear and bifurcation occurs. In this case, the flow is unsteady and there are two possible groups of hydrothermal waves with different number of waves, which are characterized by spoke patterns traveling in the clockwise and counter-clockwise directions. Details of the flow and temperature disturbances are discussed and number of waves and traveling velocity of the hydrothermal wave are determined.
Reflection and transmission of an internal solitary wave over a step in stratified fluid
Gang Wei Yunxiang You Guoping Miao Xueming Qin
2007, 23(1):  45-53.  DOI: 10.6052/0459-1879-2007-1-2005-295
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Transmission, reflection and fission of an internal solitary wave incident upon a step in a two-layer fluid system are investigated analytically based on the matched-asymptotic expansion and the Green function. The reduced boundary condition relevant to the effect of the step-topography on Boussinesq equations is derived by applying the conformal mapping theory and solving the singular Fredholm integral equation. A problem of the `initial' value for KdV evolution equation is formulated. The explicit expressions for transmitted and reflected waves are given by the inverse scattering method. It follows that there exist obvious effects of step height, density ratio and thickness ratio of upper- to lower-layer on the amplitudes of transmitted and reflected waves and their number of fission. It is also found that when the upper layer thickness is larger than the lower layer one, the amplitude of the reflected wave monotonously increases with the increase of step height is before the critical point, then monotonously decreases, and it is the other way round for the transmitted wave. The phase of the reflected wave on the convex step is just opposite to the incident wave, and its maximum amplitude can approach several folds of the incident one. The reflected wave on the concave step can evolve into a single solitary wave in certain stratified situations, which differ from the oscillating decay tail in the single layer fluid system.
Lane changing analysis for two-lane traffic flow
Tieqiao Tang Haijun Huang S.C. Wong Rui Jiang
2007, 23(1):  49-54.  DOI: 10.6052/0459-1879-2007-1-2006-282
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In this paper, the two-lane traffic are studied by using the lane-changing rules in the car-following models. The simulation show that the frequent lane changing occurs when the lateral distance in car following activities is considered and it gives rise to oscillating waves. In contrast, if the lateral distance is not considered (or considered occasionally), the lane changing appears infrequently and soliton waves occurs. This implies that the stabilization mechanism no longer functions when the lane changing is permitted. Since the oscillating and soliton waves correspond to the unstable and metastable flow regimes, respectively, our study verifies that a phase transition may occur as a result of the lane changing.
Elasto-plastic analysis for 2d structures with truss-like materials
Lin Liu Jun Yan Gengdong Cheng
2007, 23(1):  54-62.  DOI: 10.6052/0459-1879-2007-1-2005-407
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The elasto-plastic analysis of structures composed of truss-like materials takes much time and considerable resources in modeling and numerical calculation if all struts are taken into consideration. The main purpose of this paper is to speed up the solution of this class of problems. The unit cell is simplified as a truss model according to the large ratio of strut's length to the section size. Numerical homogenization is carried out due to the periodic arrangement of cells in space. The original problem is thereby transformed to two interrelated problems of two different scales: a nonlinear elastic continuum computation in macro-scale and several elasto-plastic analyses of small-scale truss systems in micro-scale. Monotone load, non-monotone load, regular macro-structure and irregular macro-structure with imperfect unit cells are, respectively, considered in two numerical examples. In comparison with the results of the 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.
Electromagnetoelastic behaviors of functionally graded piezoelectric
Hongliang Dai Yiming Fu J.H. Yang
2007, 23(1):  55-63.  DOI: 10.6052/0459-1879-2007-1-2006-127
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Analytical studies on electromagnetoelastic behaviors are presented for the functionally graded piezoelectric material (FGPM) solid cylinder and sphere placed in a uniform magnetic field and subjected to the external pressure and electric loading. When the mechanical, electric and magnetic properties of the material obey an identical power law in the radial direction, the exact displacements, stresses, electric potentials and perturbations of magnetic field vector in the FGPM solid cylinder and sphere are obtained by using the infinitesimal theory of electromagnetoelasticity. Numerical examples also show the significant influence of material inhomogeneity. It is interesting to note that selecting a specific value of inhomogeneity parameter can optimize the electromagnetoelastic responses, which will be of particular importance in modern engineering designs.
The reproducing kernel particle method for numerical analysis of high-speed impact process
Guangming Zhao Shuncheng Song Xianjie Yang
2007, 23(1):  63-69.  DOI: 10.6052/0459-1879-2007-1-2005-404
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The Reproducing Kernel Particle Method, a new type of meshless method, is adopted in this study, for numerical simulations of high-speed impact processes. The related control equations are obtained and the Bordner-Partom constitutive model is used for large strain and high strain-rate when the material is under a high-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 Kernel Particle Method can correctly simulate the high-speed impact process with velatively high degree of accuracy.
Finite element simulation on the mechanical properties of MHS materials
Z.Y. Gao Tongxi Yu D. Karagiozova
2007, 23(1):  65-75.  DOI: 10.6052/0459-1879-2007-1-2006-198
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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.
Constitutive model of frozen soil with damage and numerical simulation of the coupled problem
Jianguo Ning Zhiwu Zhu
2007, 23(1):  70-76.  DOI: 10.6052/0459-1879-2007-1-2006-177
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On the basis of the microscopic mechanics of composite material, an elastic constitutive model for frozen soil containing damage is presented. For the frozen sand soil with different ice contents and under different temperature conditions, the result calculated by the constitutive model agrees with the actually measured stress-strain curve. After numerically simulating the coupled problem of water, temperature and stress fields of channel frozen and frozen soil foundation by means of a program, a more accurate and practical temperature field is obtained, which is consistent with the prediction and tests in literature. It is shown that related physical properties of frozen soil and a quantitative relationship between them can be obtained.
The third dimensionless parameter in the penetration dynamics of rigid projectiles
Xiaowei Chen Xiaoli Li Yuze Chen Haijun Wu Fenglei Huang
2007, 23(1):  77-84.  DOI: 10.6052/0459-1879-2007-1-2006-069
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The present paper defines a third dimensionless parameter, i.e., the damping function {\xi}, besides the impact function I and geometry function N of a projectile introduced early, in the penetration dynamics of rigid projectiles. It only depends on the interaction of the projectile and the target materials and is independent of the projectile geometry. A general penetration drag, which contains the terms of damping effect and the dummy mass of the projectile induced by the deceleration effect, is adopted in the formulation. Dimensionless formula for the depth of penetration (DOP) is obtained with only three parameters I, N and {\xi} for general convex shapes of various hard projectiles. Different geometry parameters are also presented for some common shapes of warheads. Theoretical predictions for DOP dependent/independent of the damping function {\xi}, show good agreement with the published test data for different projectiles and impact velocities as well as different targets.
Topology optimization of material microstructures using strain energy-based prediction of effective elastic properties
Weihong Zhang Gaoming Dai Fengwen Wang Shiping Sun Hicham Bassir
2007, 23(1):  77-89.  DOI: 10.6052/0459-1879-2007-1-2006-086
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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.
A microstructure-based damage constitutive model for pearlitic materials
2007, 23(1):  85-92.  DOI: 10.6052/0459-1879-2007-1-2006-229
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Based on the laminar microstructure formed by ferrite and cementite with very fine interlamellar spacing, and the morphology of microdefects in the ferrite, cementite and interface, a unified damage 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 damage elastoplastic constitutive model is obtained for a single pearlitic colony. The damage constitutive description for pearlitic materials is formulated using the Hill's self-consistent scheme by assuming that a pearlitic material element is an aggregate of numerous cells of pearlitic colonies with randomly distributed orientations. It is significant that the obtained constitutive description contains explicitly the interlamellar spacing as a microstructure parameter, which easily accounts for the better comprehensive mechanical properties of the pearlitic materials with smaller interlamellar spacing. The constitutive behavior of pearlitic steel is simulated, and compared with the experimental results.
A new simple method of implicit time integration for dynamic problems of engineering structures
Jun Zhou Youhe Zhou
2007, 23(1):  91-99.  DOI: 10.6052/0459-1879-2007-1-2006-167
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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.
Calculation of stress intensity factors by boundary element method based on erdogan fundamental solutions
Cheng Su Chun Zheng
2007, 23(1):  93-99.  DOI: 10.6052/0459-1879-2007-1-2006-040
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The Erdogan fundamental solutions for infinite cracked plates are introduced in this paper. The spline fictitious boundary element method is then proposed and formulated for analysis of mode I and mixed mode (mode I and II) problems based on the above fundamental solutions. The proposed method is further applied to analyze certain crack problems, in which the computation accuracy, convergence rate and the versatility of the method are demonstrated in details.
Application of euler midpoint symplectic integration method for the solution of dynamic equilibrium equations
Yufeng Xing Rong Yang
2007, 23(1):  100-105.  DOI: 10.6052/0459-1879-2007-1-2006-026
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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 of Jacobi 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 all eigenvalues 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 a conservative system, it is a structure-preserving algorithm, which means that the energy of the system is preserved through the solution process. Numerical analyses are carried out to illustrate the advantages of the symplectic algorithm in the solution of non-conservative systems. The accuracy of structure-preserving algorithm is not sensitive to the ratio of the frequency of the external force to that of the system, while the accuracy of Newmark algorithm with \delta \ne 0.5 is sensitive to that ratio.
Improved suboptimal Bang-Bang control of aseismic buildings with variable friction dampers
Hongnan Li Jun Li Gangbing Song
2007, 23(1):  101-109.  DOI: 10.6052/0459-1879-2007-1-2005-601
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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.
A variational principle form for initial value problems in analytical mechanics
Lifu Liang En Luo Xiaojiu Feng
2007, 23(1):  106-111.  DOI: 10.6052/0459-1879-2007-1-2006-027
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According to relations between generalized forces and generalized displacements, convolution are performed between the governing equations of initial value problems in the primary space and the corresponding virtual quantities, and results are added algebraically. A variational principle form for initial value problems in analytical mechanics are then established in the original space, i.e. the variational principles and generalized variational principles in convolution form for initial value problems in analytical mechanics are established in the original space. The stationary conditions of the variational principles and generalized variational principles are deduced. In the meantime, the variational integral method is generalized into the convolutional variational integral method. Using these variational principles and generalized variational principles for initial value problems in analytical mechanics, we can establish models of finite element method and other approximate calculation method and can also find exact solutions of initial value problems in analytical mechanics and transform differential equations into algebraic equations.
Nonlinear formulation for flexible multibody system with large deformation
Jinyang Liu Jiazhen Hong
2007, 23(1):  111-119.  DOI: 10.6052/0459-1879-2007-1-2006-113
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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.
Brief Report
Modal perturbation method for obtaining complex modal characteristics of non-proportional damping systems
Menglin Lou Yaoqing Fan
2007, 23(1):  112-118.  DOI: 10.6052/0459-1879-2007-1-2006-170
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An approximate approach based on the direct modal perturbation method is suggested for analyzing the complex modal characteristics of non-proportional damping structure systems. The proposed approach uses the expansion of complex Ritz vectors. The procedure is in two steps. First, the subspace of the solution based on the complex vectors is formed by using the conventional real modes of the structure. Then, a set of solutions of non-linear complex algebraic equations is established to displace the solution of the complex eigen-value equations. The results of two numerical examples show that the suggested method simplifies the solving procedure and is of high precision and efficiency. It can be used for the solution of the complex modal characteristics and dynamic responses of non-proportional damping structure systems.
Wave propagation characteristics of cylindrical shells with circumferential surface crack
Xiang Zhu Tianyun Li Yao Zhao Jingxi Liu
2007, 23(1):  119-124.  DOI: 10.6052/0459-1879-2007-1-2005-560
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In this paper, the wave propagation characteristics of infinite cylindrical shells with circumferential surface crack are investigated. The vibration of the cylindrical shell is described by Fl\"{u}gge's shell equations and the wave propagation approach is used to solve the equations. In consideration of the crack's three modes: opening, sliding and tearing modes and their combinations, the surface crack is modeled by distributed line springs. The local compliance matrix at the cracked region is then deduced by using linear elastic fracture mechanics, and the relationships between the additional general displacements and the general forces are obtained. To verify the accuracy of the theoretical model and the wave propagation method proposed, the axisymmetric free vibration of a finite shell with a circumferential surface crack is investigated first, the natural frequencies of the finite cracked shell are calculated by using the analytical method and finite element method, respectively. The results between the two methods show a good agreement. Then an infinite shell with a circumferential surface crack is investigated. For a given incident wave, the amplitudes of the reflected wave and transmitted wave are obtained according to the continuity conditions of displacements and inner forces at both sides of the crack. The relationships between the transmitted wave coefficient and the crack's depth as well as the frequency are obtained. The numerical results indicate that the existence of the crack changes the vibration wave in the shell substantially and the amplitudes of the transmitted wave and reflected wave are closely related to the crack's depth and the frequency. The results also show that the coefficients of the transmitted wave decrease with the increase of the crack's depth. This research provides a theoretical basis for the further crack detection in cylindrical shells based on the vibration method.
Non-probabilistic reliability-based topology optimization of continuum structures
Yangjun Luo Zhan Kang
2007, 23(1):  125-131.  DOI: 10.6052/0459-1879-2007-1-2006-181
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It is of great importance to incorporate uncertainties into structural topology optimization problems. Based on the definition of the non-probabilistic reliability index, this paper proposes a mathematical model for topology optimization of 3D continuum structures with non-probabilistic reliability constraints on deformation. In this model, the uncertainties in material properties, geometrical dimensions and loading conditions are accounted for. The Performance Measure Approach (PMA) is adopted for transformation of the constraints. In this context, the adjoint variable scheme for sensitivity analysis of the target performance is discussed, which facilitates a mathematical programming solution of the optimization problem. Numerical investigations illustrate the applicability and the validity of the present model as well as the proposed numerical techniques. The computational results show that non-probabilistic reliability-based topology optimization may yield more reasonable material layouts than the conventional deterministic approach.
Analysis to grazing bifurcation in linear vibro-impact system with N-dimensions
Sijin Zhang Libiao Zhou Qishao Lu
2007, 23(1):  132-136.  DOI: 10.6052/0459-1879-2007-1-2006-169
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Grazing bifurcation is an important dynamical behavior of a vibro-impact system and is usually analyzed by choosing the impact plane as the Poincar\'{e} section. However, this plane sometimes does not meet the transverse intersection condition of Poincar\'{e} section, especially while grazing motion or chaos take place. Moreover, the bifurcation of impact number instead of period of the motion is considered in former cases. The bifurcations with time evolution are more attractive for a vibro-impact system. In this paper, the Poincar\'{e} map of period-n motions with single-impact is set up for a linear vibro-impact system by using a fixed phase plane as the Poincar\'{e} section here. Based on analysis of the Poincar\'{e} map, the grazing bifurcation conditions and bifurcation equations are determined for the vibro-impact system, and a vibro-impact system with single DOF is used as an example to testify the obtained analytical result. A numerical simulation is carried out for the bifurcation diagram of the vibro-impact system, which agrees with analytical results very well. This method can be used to calculate not only the parameters of grazing bifurcation, but also those of any period-$n$ motions, for a linear vibro-impact system.
Remarks on the ambiguity in multiple scales method
Yanmao Chen Jike Liu
2007, 23(1):  137-140.  DOI: 10.6052/0459-1879-2007-1-2006-293
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The method of multiple scales (MMS), developed for systems with small non-linearities, is one of the most widely used perturbation methods. Only particular solutions are sought for the higher order approximate equations by using the ordinary MMS. An observation is made in this paper that the MMS works well only for the approximate solutions of the first two orders, while gives rise to a paradox in obtaining the third order approximate solution of van der Pol equation. Taking the famous van der Pol equation as an illustrative example, it is proven that neglecting the first order harmonic of the first order approximate solution may make the derivative sequence of the second order mixed partial derivative not commutable. This leads to the ambiguity of the MMS and another mathematical paradox. Unlike the ordinary MMS, the general solution containing the first harmonic is adopted for the first order approximate equation, and then the ambiguity and the paradox are both eliminated. The approximate solutions are obtained by the proposed method and compared with the numerical solutions. It is shown that the present technique is valid.