力学学报

ADVANCES IN RESEARCH ON BIRKHOFFIAN MECHANICS

Mei Fengxiang, Wu Huibiny, Li Yanmin, Chen Xiangwei

The Birkhoffian mechanics is a natural development of Hamiltonian mechanics, and is a new development stage of analytical mechanics.This mechanics can be widely applied to mechanics, physics and engineering.In this paper, the formation and development of Birkhoffian mechanics, especially the achievements in the last twenty years are summarized.First, to start with the relevant paragraphs of Birkhoff's "Dynamical Systems", the origin of the Birkhoffian mechanics is narrated.Next, the formations and developments of the basic principle, the Pfaff-Birkhoff principle, and the basic equations, Birkhoff equations for the mechanics, are described.After that, a brief introduction to some special problems of the mechanics is given, including constrained Birkhoffian systems, integration methods for Birkhoff equations, inverse problems of Birkhoffian dynamics, stability of motion of Birkhoff equations, geometric methods for Birkhoffian systems, global analysis of Birkhoffian systems and so on.Finally, some suggestions are put forward for the future research of Birkhoffian mechanics.

2016, 48(2): 263-268. doi: 10.6052/0459-1879-15-193

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NUMERICAL STUDY ABOUT AERODYNAMIC CHARACTERISTICS AND FLOW FIELD STRUCTURES FOR A SKIN OF AIRFOIL WITH ACTIVE OSCILLATION AT LOWREYNOLDS NUMBER

Liu Qiang, Liu Zhou, Bai Peng, Li Feng

For the poor aerodynamic performance of airfoil at low Reynolds number, the paper targets the active oscillation for flexible skin of airfoil in order to improve its aerodynamic characteristics and flow field structures.Roe method with preconditioning technique was used to solve the unsteady compressible N-S equations and simulate the flow for NACA4415 airfoil at low Reynolds number.The aerodynamic force characteristics and laminar flow separation structures were compared by time-average and unsteady methods when the skin is static or oscillating.Preliminary studies indicate when the flexible skin actively oscillates with appropriate amplitude and frequency, the time-average lift and drag characteristics increase significantly.The separation bubble structures transform from trailing-edge laminar separation bubble to classic long laminar separation bubble.The separation position moves downstream and the separation region reduces.On this basis, the paper studied the alterations of the unsteady flow structures and pressure coefficient distributions more detailed during one period with the two states of the skin.The flow field structures and the pressure distributions keep steady in the front part of separation region at a static skin.The flow approximates steady separation and the unsteady flow phenomena like Karman vortex streets only appear at the trailing edge.Otherwise, for the oscillating skin, the unsteady vortexes generate near the separation position, subsequently move, and then shed along the airfoil surface.The flow presents unsteady separation and displays a wide range of pressure oscillation.Owing to the oscillating skin, the fluid movements are closer to the wall.The large scale laminar flow separation phenomenon is suppressed apparently.

2016, 48(2): 269-277. doi: 10.6052/0459-1879-15-188

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UNCERTAINTY AND GLOBAL SENSITIVITY ANALYSIS OF HYPERSONIC CONTROL SURFACE AEROTHERMOELASTIC

Ye Kun, Ye Zhengyin, Qu Zhan, Wu Xiaojin, Zhang Weiwei

Considering that the uncertainty of hypersonic aerothermodynamics prediction affects the reliability of aerothermoelastic analysis, a parameterized model for temperature distribution is therefore proposed.Based on this model, uncertainty and global sensitivity analysis on aerothermodynamics of hypersonic control surface aerothermoelastic are conducted.In the present analysis method, temperature distribution of the control surface is first obtained by solving NS equation and then parameterized.Using Monte Carlo simulation(MCS) method and spare grid numerical integration(SGNI) method to generate samples for analyzing uncertainly and global sensitivity and then analyzing all the samples, aerothermoelastic analysis is carried out as following:To get temperature distribution by the sample, then to analyze structural modal under the effect of structure thermal stress and material property, interpolate structural mode to the aerodynamic grid, and then to analyze aeroelasticity of the control surface in state space based on CFD local piston theory.Under two fly conditions, the calculation results show that:(1) With M=5 and H=15 km, the variation coefficient of natural frequency and flutter analysis is 5.83%,(2) With M=6 and H=15 km, variation coefficient of natural frequency of the structure and flutter analysis is 8.84%, and the global sensitivity of the two uncertainty parameters is about 50% under the two conditions.And the coupling of two parameters is about 0%, which is very small.Comparing with MCS method, SGNI method can be used to improve the efficiency of uncertainty analysis significantly.

2016, 48(2): 278-289. doi: 10.6052/0459-1879-14-406

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OPTIMIZATION AND ANALYSIS OF THE LEADING EDGE SHAPE FOR HYPERSONIC AIRPLANES BASED ON DOE METHODS

Hu Shouchao, Cui Kai, Li Guangli, Xiao Yao, Situ Ming

By taking a Blend-Wing-Body configuration with dual flanking inlets layout as the baseline, an aerodynamic shape optimization study for the leading edge of the vehicle is carried out.An increment-based method is developed to parameterize the leading edge, and the uniform design method is taken as the optimization driver.Totally 31 configurations with different leading edge shapes are generated, and then the aerodynamic performance of all configurations are evaluated by computational fluid dynamic simulation in the cruising conditions of Mach 6, flight altitude 26 km, and flight angle of attack 4°.Numerical results show that the relative variations of the lift and the drag coefficients in whole design space are 21.3% and 31.8%, respectively.Consequently, the relative variation of the lift-to-drag ratio is about 10.63%.In contrast, the maximal value of the relative position transition for the longitudinal pressure center is only 3.87%.Besides, the results also indicate that there is a proportional relationship between the vertical project area and the lift or drag coefficients. Finally, the effect of the leading edge deformation on the aerodynamic characteristics is analyzed on the basis of the results, and some primary rules are concluded.A concave shape of the leading edge is benefit to the improvement of the lift-to-drag ratio, while a convex one leads to a large lift coefficient.

2016, 48(2): 290-299. doi: 10.6052/0459-1879-15-231

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INVESTIGATION OF DRAG FORCES FOR FLEXIBLE RISERSUNDERGOING VORTEX-INDUCED VIBRATION IN SHEARED FLOW

Song Leijian, Fu Shixiao, Yu Dapeng, Ren Tie, Zhang Mengmeng

The drag forces of a flexible riser undergoing vortex-induced vibration(VIV) in sheared flow were investigated using a scale model test.The mean drag forces were determined based on beam theory using the strains measured by the fiber bragg grating strain sensors of the riser.The distributions of the drag coefficients along the riser and Reynolds numbers(Re), and the VIV amplification of the drag coefficient, were studied, and a new empirical model to estimate the drag coefficient on a flexible riser undergoing VIV was proposed.The results show that VIV can amplify the drag coefficient, resulting in the local drag coefficient of the riser up to 3.2.For Re values from 1.0×10⁴~1.2×10⁵, the mean drag coefficient value was between 1.3 and 2.0, and decreased as Re increased.Furthermore, the proposed empirical prediction model, which accounts for the effects of current, the VIV dominant modal number and the frequency in the cross-flow direction, can predict riser drag coefficients under VIV accurately at high Re values.

2016, 48(2): 300-306. doi: 10.6052/0459-1879-15-309

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INVESTIGATION IN HYDRODYNAMICS OF A CIRCULAR CYLINDER WITH THE NEW SUPPRESSING SHROUD FOR VORTEX-INDUCED VIBRATION

Wu Yingxiang, Lin Liming, Zhong Xingfu

Through model experiment and numerical simulation, the hydrodynamics of the circular-section cylinder with a suppressing shroud for vortex-induced vibration is investigated.The model experiment for the vortex-induced vibration is carried out for the pendulum with the harmonic and conic-like radial disturbances at different incoming flow velocities.As for the simulation for the harmonic and conic disturbances, the hydrodynamic parameters, like drag, lift and vortex-shedding frequency, varied with the wavelength and wave steepness are studied at Reynolds numbers from 10² to 10⁵.The model experiment has shown that the amplitude with the shroud does be reduced at the velocity at which the synchronization of a straight circular cylinder is occurred, but obviously increased at higher velocities.Numerical results have shown that the hydrodynamics of the harmonic disturbance is similar to that of the conic disturbance.And variations of the drag, lift and vortex-shedding frequency are similar for the different Reynolds numbers.With the increasing wave steepness, the drag is generally increased, while the lift is reduced firstly and then increased in most cases, and vortexshedding frequency is generally decreased.

2016, 48(2): 307-317. doi: 10.6052/0459-1879-14-300

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STUDY OF PRESSURE DISTRIBUTION OF A FIXEDWALL UNDER LOW-SPEED JET IMPINGEMENT USING PRESSURE SENSITIVE PAINT

Wu Di, Feng Lihao, Wang Jinjun

Pressure sensitive paint(PSP) is a pressure measurement technology based on oxygen quenching mechanism of luminescence.Compared with conventional techniques, PSP technique has lower cost and higher spatial resolution.In this work, the PSP technique platform is constructed.It consists of the hardware for measurement and the software for data processing.A novel calibration method for PSP is developed and validated, which is based on the relation between intensity and pressure at the stagnation point of the impinging wall.The influences of Reynolds numbers and impingement heights between the orifice and the wall on the pressure distribution are analyzed.When the ratio value of impingement heights h/d is smaller than 6, the pressure distribution remains the same at different heights.When the value of h/d is greater than 6, the peak pressure decreases while the half width increases linearly with the increase of the impingement height.Pressure distribution curves are self-similar at different impingement heights when they are normalized by the peak pressure and half width.Further study shows that the pressure distributions are rarely influenced by the Reynolds number at the same impingement height.

2016, 48(2): 318-326. doi: 10.6052/0459-1879-15-277

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THE PRIMARY STABILITY ANALYSIS OF HARTMANN BOUNDARY LAYER

Dong Shuai, Lin Dianji, Lü Yukun

A thin Hartmann boundary layer will be formed near the wall when the conducting fluid is passing through a vertical magnetic field.The property of Hartmann boundary layers is vital to design and operation of equipments in electromagnetic metallurgy and thermonuclear fusion cooling system.This stability problem is investigated by non-modal stability analysis method.Through solving the governing equations of disturbances and adjoint field variables iteratively, the amplification and spatial distributions of primary perturbations are obtained.The effects of magnetic field on the optimal perturbation amplification G_max, spanwise wavenumber β_opt and time t_opt are analyzed, and the interaction between two opposite Hartmann boundary layers is considered as well.Results indicate that the optimal initial perturbations are in the form of streamwise vortices, which is symmetric or antisymmetric with respect of the normal direction.When the Hartmann number Ha is larger(Ha>10), the initial perturbations of symmetric and antisymmetric vortices are amplified equally, and the two opposite Hartmann boundary layers can be considered as independent from each other.In this case, the dependence of optimal perturbation amplification G_max on the square of local Reynolds number R is obtained, and the corresponding optimal spanwise wavenumber β_opt and time t_opt are proportional to the Hartmann number Ha.When the Hartmann number Ha is smaller(Ha<10), the antisymmetric vortices are more unstable, and the perturbation amplifications G_max is larger than that of symmetric vortices.There is still a kind of interaction between the two opposite Hartmann boundary layers to influence the stability of the flow field.

2016, 48(2): 327-335. doi: 10.6052/0459-1879-15-179

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THE COMPARISON OF WEIGHTED COMPACT SCHEMES AND WENO SCHEME

Zhang Shuhai

Linear compact scheme and weighted essentially non-oscillatory(WENO) scheme are two typical high order numerical schemes.They have their own merits and drawbacks.Linear compact scheme has high order accuracy order, high resolution and low dissipation, which is a nice numerical scheme for multi-scale flow.However, it can not compute the flow with strong shock wave.WENO scheme is a robust high order shock capturing scheme.But the dissipation is high.The resolution for short wave is not ideal.In recent years, a series of weighted compact schemes were developed by the combination of linear compact scheme and WENO scheme.In this paper, we systematically compare the properties of weighted compact schemes and WENO scheme including the construction, the capability to capture strong shock wave, the resolution, the convergence and paralleling efficiency.Our study shows that weighted compact schemes have no obvious superiority in the computation of aerodynamics to WENO scheme.

2016, 48(2): 336-347. doi: 10.6052/0459-1879-15-190

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ARBITRARY LAGRANGIAN EULERIAN SIMULATION OF FREE PISTON COMPRESSION TUBE

Li Haiyan, Li Zhihui, Lü Zhiguo, Luo Wanqing, Chang Yu

One of the approach obtaining high-enthalpy flow environments is the pulse facility driven by free piston, including free piston shock tunnel and free piston expansion tube.When using a free piston compression tube as the driver part of an expansion tube or a shock tunnel, the performance of such a facility will be determined to a great extent by its driving ability.Numerical simulation of the piston motion and flow properties in the free piston compression tube has been performed by means of an arbitrary Lagrangian Eulerian approach(ALE).Moving mesh strategy was used for the piston boundary adaptation.The motion of mesh and fluid was solved by coupling method.A dual time-step approach developed in computational fluid dynamics was introduced in the construction of ALE time-integrator.The normal vector and surface area of the mesh element for dynamic mesh were verified to satisfy the geometric conservation law(GCL). Both the comparison of piston positions with those of Eulerian method and the comparison of piston base pressures, piston velocities and piston positions with those of theoretical model based on simple wave assumption are very well.This work established the base on which numerical simulation of the flow at different parts for free piston shock tunnel and for free piston expansion tube will be carried out at the next step such as compressor, shock tube and nozzle, etc.

2016, 48(2): 348-352. doi: 10.6052/0459-1879-14-398

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MECHANICAL ANALYSIS OF A DROPLET SPREADING ON THE DISCRETE TEXTURED SURFACES

Jiao Yunlong, Liu Xiaojun, Liu Kun

The spreading of a droplet on a solid surface is of obvious importance in a number of practical situations, such as lubrication and coating.In this paper, droplet spreading behaviors on the surfaces with four discrete textures were studied through numerical simulations and wetting experiments.Mechanical analysis on the movement of triple contact line was conducted from interface mechanics in micro-scale.We also introduce a new parameter, texture wetting factor θ^*, to characterize the influence of texture type on wettability.Results show that the discrete textures on the substrates provide excess contact area between liquid and solid, and the spreading fringe partly penetrates into the space among the textures, which leads to the increase of curvature, and the increase of Laplace pressure in the droplet provides excess driving force and accelerates the liquid as it approaches.Droplet on the grooved surface shows spreading anisotropy.In addition, simulation analysis indicates that the pinning effect of contact line has a close relationship with surface roughness and texture type, the greater the surface roughness, the greater the pinning effect, and the pinning effect on the grooved surface also shows directivity.

2016, 48(2): 353-360. doi: 10.6052/0459-1879-15-257

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MESOSCOPIC DAMAGE BEHAVIORS OF PLAINWOVEN CERAMIC COMPOSITE UNDER IN-PLANE SHEAR LOADING

Guo Hongbao, Wang Bo, Jia Purong, Yang Chengpeng

The in-plane shear mechanical behaviors and mesoscopic damage behaviors of plain woven SiC/SiC and C/SiC composites were investigated with Iosipescu specimens.The materials' monotonic and hysteresis stress-strain responses in the various directions were obtained by tests, and the shear damage properties of these two materials were analyzed comparatively.It is found that the shear damage evolutions of materials are affected by residual thermal stress significantly.SEM(scanning electron microscope) photos of fracture surfaces indicate that the bridging fibers withstand significant bending load and deflection.Therefore a fiber bending-bearing mechanism was proposed.And combined with the crack closure effect, the shape of hysteresis loops was elucidated well.By considering the characteristic of mesoscopic shear damage modes, two damage variables were defined to character the shear damage evolution of material.And the shear damage evolution models of these two materials were developed.The results show that the initiation stress for matrix cracks in the 45°direction of 2D-C/SiC composite is much lower than that of 2D-SiC/SiC composite, but the initiation stress values for matrix cracks in the 0°/90°direction are the same for these two materials.

2016, 48(2): 361-368. doi: 10.6052/0459-1879-15-133

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MICROSTRUCTURE EVOLUTION DURING TENSION DEFORMATION OF SEMI-CRYSTALLINE POLYMER

Duan Fangli, Jin Yikuang, Yan Shidang

Molecular dynamics simulation with coarse-grained model of polyvinyl alcohol was used to investigate the structure of semicrystalline polymer through melt-cooling process.The relationship between microstructure and macroscopic mechanical behavior was then investigated to reveal the microscopic mechanism of semicrystalline polymer during uniaxial tension.The stress-strain behavior comprised elastic stage, yield stage, strain softening stage and strain hardening stage.Several important structural evolution forms were investigated:reorientation of molecular chains, slipping of PVA molecules in crystalline region, disturbed crystalline region(crystal to amorphous) and disentanglement of PVA molecules in the amorphous region.The stress-strain behaviors in the crystal region and amorphous region were investigated respectively. The stress varied in two regions during uniaxial tension, which mainly caused by various microstructure evolution in different stages.In the elastic stage, the main microstructure evolution was the reorientation of molecular chains.In the strain softening stage, the slipping behavior of folded chains in the crystalline region and the disentanglement of the PVA molecules in the amorphous region were the main structural evolution forms.The stress in the crystal region in this stage was larger than that in the amorphous region, because keeping slipping behavior of folded chains in crystalline region was harder than to deform in the amorphous region.In the strain hardening stage, the deformation of amorphous region was more difficult than crystal slipping, in other words, the disentanglement of PVA molecules need more energy.The stress in this stage increased which mainly led to the mechanical behavior of strain hardening.In conclusion, the coordinated microstructure evolution contributed to macroscopic mechanical behavior during tension in spite of the variation of the main microstructure evolutions in different stages.

2016, 48(2): 369-377. doi: 10.6052/0459-1879-15-345

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A GENERALIZED MULTISCALE FINITE ELEMENT METHOD FOR DYNAMIC ANALYSIS OF HETEROGENEOUS MATERIAL

Zhuo Xiaoxiang, Liu Hui, Chu Xihua, Xu Yuanjie

Almost all natural materials, as well as industrial and engineering materials, have multiscale features.This paper presents a generalized multiscale finite element method for dynamic analysis of heterogeneous materials.In this method, multiscale base functions, which can reflect the internal heterogeneity of materials within the coarse-scale element, are constructed by using the static condensation method and penalty function method.And these functions, which are different from those in the traditional extended multiscale finite element method(EMsFEM), are obtained by matrix operations analytically rather than solving elliptic problem many times on the sub-grid domain numerically.The main steps of this proposed method are described as follows.Firstly, a single heterogeneous unit cell will be equivalent into a macroscopic element by virtue of the constructed multiscale base functions.Then, the stiffness matrix of heterogeneous structure can be calculated on the macroscopic scale.Thus, the macroscopic nodal displacements of coarse-scale mesh can be obtained.Finally, the microscopic nodal displacements of sub grids within the unit cell can be calculated by using the numerical base functions once again.This generalized multiscale finite element method is a new expansion of the EMsFEM, which can simulate the mechanical behavior of heterogeneous unit cell with more complex geometric configurations. The static problem, generalized eigenvalue problem and transient response problem of the heterogeneous material are then simulated.It can be found that the calculation results of the generalized multiscale finite element method maintain highly consistent with those from the traditional FEM.Compared with the traditional FEM, the present multiscale method has significantly improved the computational efficiency while ensuring the computational accuracy, which has a good application potential.

2016, 48(2): 378-386. doi: 10.6052/0459-1879-15-211

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A DUAL BOUNDARY INTEGRAL EQUATION METHOD BASED ON DIRECT EVALUATION OF HIGHER ORDER SINGULAR INTEGRAL FOR CRACK PROBLEMS

Li Jun, Feng Weizhe, Gao Xiaowei

Compared with finite element method, boundary element method has special advantages in solving the problems of fracture mechanics.The existing methods mainly include the subdomain method and dual boundary integral equation method.This paper presents an improved dual boundary integral equation method to evaluate stress intensity factors for two and three-dimensional crack problems.The method uses a pair of boundary integral equations, in which the traditional displacement boundary integral equation is collocated on the external boundary and the traction boundary integral equation is collocated on one of the crack surfaces.The relative crack opening displacements(CODs) are introduced as unknowns on the crack surface, and the evaluating results of CODs are used to evaluate the stress intensity factors(SIFs) of crack directly.The method uses a direct method to evaluate the hypersingular integral appeared in traction boundary integral equation.For crack tip elements, three kinds of interpolation functions for CODs are provided, and two of these are constructed in the present study.Two-point formula is used to evaluate the SIFs.Some examples are given to verify the correctness of the presented method, compared with the existing exact solution or reference solution, to show that this method can get high precision of the calculation results.

2016, 48(2): 387-398. doi: 10.6052/0459-1879-15-342

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PRECISE SYMPLECTIC TIME FINITE ELEMENT METHOD AND THE STUDY OF PHASE ERROR

Zhu Shuai, Zhou Gang, Liu Xiaomei, Weng Shilie

Hamiltonian system is one kind of important dynamical systems.Many kinds of symplectic methods were proposed for Hamiltonian systems, such as SRK, SPRK, multi-step method, generating function method and so on.The numerical methods for Hamiltonian can not be satisfied the character properties(symplectic and energy-preserving) at the same time.Recently, time finite element method was proposed for Hamiltonian systems, which was symplectic and could keep the conservation of energy.However, the mentioned methods have phase-drift(orbit deviation).For longtime simulation, the accuracy decay a lot.Precise Symplectic Time Finite Element Method is proposed for Hamiltonian systems(HPD-FEM), which could keep the conservation of Hamiltonian function and the structure of symplectic of Hamiltonian systems, as finite element method does.Meanwhile, HPD-FEM can highly decrease the phase error compared to FEM.HPD-FEM has fewer phase-error than the determined schemes aimed at decreasing the phase drift, such as:FSJS, RKN, and SPRK.FSJS, RKN and SPRK cannot keep the Hamiltonian function of Hamiltonian system, while HPD-FEM can keep the energy conservation.For the systems with different frequencies or the stiff systems, HPDFEM can simulate the signals both high and low frequency, with big time step.During the computation, HPD-FEM does not increase the cost of computation.Numerical experiment shows the validity of HPD-FEM.

2016, 48(2): 399-405. doi: 10.6052/0459-1879-15-272

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OPTIMIZATION DESIGN OF CONDUCTIVE PATHWAYS FOR COOLING A HEAT GENERATING BODY WITH HIGH CONDUCTIVE INSERTS

Chen Wenjiong, Liu Shutian, Zhang Yongcun

It is an importance to optimize the position of high conductive material in a heat generating body with low thermal conductivity and high heat generation for reducing the internal temperature.A key problem is how to design the structures with high conductive material in the heat generating body for temperature control and heat collection.In this paper, a new method based on topology optimization is proposed to design the conductive pathways with embedded high conductive materials.Based upon SIMP approach, an artificial material model with thermal conductivity and heat generation rate is suggested and the relative densities of the high conductive material are taken as design variables. With the minimum heat potential capacity as the objective, a topology optimization model for designing high conductive inserts is formulated and the corresponding solving method is developed.Two topology optimization configurations of high conductive surfaces and high conductive inserts are given in the numerical examples to show be much different.The capability of heat dissipation for the high conductive inserts is much better than that of the high conductive surfaces.It indicates that it is necessary to consider the impact of the conductive pathways on the layout of heat source.Several numerical examples are given to demonstrate the effectiveness and validity of the proposed optimization method.

2016, 48(2): 406-412. doi: 10.6052/0459-1879-15-270

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ANALYTICAL SOLUTIONS OF THE GENERALIZED PROBABILITY DENSITY EVOLUTION EQUATION OF THREE CLASSES STOCHASTIC SYSTEMS

Jiang Zhongming, Li Jie

As a gradually improving and developed method, generalized probability density evolution equation(GDEE) provides a new methodology for the analysis and control of stochastic dynamic system.A variety of numerical methods such as finite difference method and meshfree method were introduced to solve the generalized probability density evolution equation.However, the analytical solution of the GDEE corresponding to typical stochastic systems is scarce relatively.In this paper, the explicit solutions of the GDEE corresponding to three classes of nonlinear stochastic systems including Van der Pol oscillator, Riccati system, and Helmholtz oscillator with random parameters are studied by using Lie group method.The results not only can be the benchmark of numerical methods, but also can provide more information for the further research.

2016, 48(2): 413-421. doi: 10.6052/0459-1879-15-221

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PRIMARY RESONANCE OF DUFFING OSCILLATOR WITH FRACTIONAL-ORDER PID CONTROLLER BASED ON VELOCITY FEEDBACK

Niu Jiangchuan, Shen Yongjun, Yang Shaopu, Li Sujuan

Compared with the traditional integer-order PID controller, the fractional-order PID(FOPID) controller may present much better control performance due to its two surplus adjustable parameters.The primary resonance of Duffing oscillator with FOPID controller of velocity feedback is investigated by the averaging method, and the approximately analytical solution is obtained.The effects of the parameters in FOPID controller on the dynamical properties are studied and characterized by some equivalent parameters.The proportional component of FOPID controller is characterized in the form of equivalent linear damping.The integral component of FOPID controller is characterized in the form of the equivalent linear damping and equivalent linear stiffness.The differential component of FOPID controller is characterized in the form of the equivalent linear damping and equivalent mass.Those equivalent parameters could distinctly illustrate the effects of the parameters in FOPID controller on the dynamical response.The amplitude-frequency equation for steady-state solution and associated with the stability condition are also presented.A comparison of the analytical solution with the numerical results is made, and their satisfactory agreement verifies the correctness of the approximately analytical results.Finally, the effects on the amplitude-frequency performance of the coefficients and the orders in FOPID controller are analyzed, and the control performances of fractional-order and integer-order PID controller are compared. The results show that the control performance of FOPID controller is better than the traditional integer-order counterpart for controlling the vibration of the primary resonance of Duffing oscillator, when the coefficients of the two controllers are the same.

2016, 48(2): 422-429. doi: 10.6052/0459-1879-15-332

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ROBUST MARKET-BASED CONTROL METHOD FOR LINEAR STRUCTURE

Song Jianzhu, Li Hongnan

It is inevitable to ignore some factors during modeling, and thus lead to some model errors.So designing controllers based on uncertain non-precise model is very important.With the efforts of researchers, online system H_∞ control has yield great achievement.However, it still has some problems, such as complex solving process, weak conservation, complex structures and high order controllers.The present study uses computational structural mechanics and optimal control theory to try to solve these problems.Induced norm is an eigenvalue problem in structural mechanics, i.e.elastic and stable Euler critical force or eigenvalue of structural vibration.Precise integration with extended Wittrick-Williams(W-W) algorithm is used to calculate induced norm by introducing the concept of interval mixed energy.And then this method is introduced to Market-Based Control theory and the MBC robust control for linear system is proposed.In the robust control, the stability and parameter identification are validated through Lyapunov direct method.Finally, a high controlled structure is simulated to compare the performance of the MBC robust control and the H_∞ robust algorithm.The results indicate that the induced norm cannot reach critical force, and otherwise the control input will be infinite.MBC robust control is better than H_∞ robust control algorithm, and has better strain capacity and shorter online calculation time.MBC robust control can be used in high or long-span structures.

2016, 48(2): 430-436. doi: 10.6052/0459-1879-15-334

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POWERED GRAVITY ASSIST IN THREE DIMENSIONS

Jia Jianhua, Lü Jing, Wang Qi

Applying an impulsive thrust during a close encounter with a celestial body can significantly improve the efficiency of the swing-by maneuver.In the existing literature, no analysis can be carried out when the impulse velocities are bigger than 1% of the orbital velocity of the spacecraft.To solve this problem, powered gravity assist is studied applying an arbitrary impulse with any magnitude and direction.The three-dimensional powered gravity assist maneuver based on the patched-conics approximation can be identified by eight independent parameters, in which five specify the three-dimensional gravity assist and the other three specify the magnitude and the direction of the impulse.Multiple reference frames are established to describe the trajectories before and after the impulse.Using the method of coordinate transformation and hyperbolic orbit dynamics, a set of new analytical equations are derived, including the variation in velocity, angular momentum, energy and inclination of the spacecraft due to the maneuver as a function of the eight parameters.These equations developed here are verified by numerical integrations, using the circular restricted threebody problem.Finally, the influences of the parameters on the orbit of spacecraft are discussed based on the above equations, and some conclusions about the optimal direction to apply the impulse are given.The results show that the optimal direction of the impulse is not parallel to the velocity of the spacecraft, and the orbital inclination is significantly influenced by the direction of the impulse.

2016, 48(2): 437-446. doi: 10.6052/0459-1879-15-218

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GIVING DYNAMIC RESPONSE BOUNDS UNDER UNCERTAIN EXCITATIONS-A NON-RANDOM VIBRATION ANALYSIS METHOD

Jiang Chao, Liu Ningyu, Ni Bingyu, Han Xu

A non-random vibration analysis method is proposed in this paper, which calculates the dynamic response bounds of vibrational systems under time-variant uncertain excitations.It provides a prominsing alternative computational tool for uncertain vibration analysis in case of lack of experimental information and the corresponding reliability design in the future.The non-probabilistic convex model process, rather than traditional stochastic process, is used to describe uncertain dynamic excitations because the former needs only the bound information instead of precise probability distribution at any time point and therefore dependence on large sample size is weakened effectively.Based on the convex model process, non-random vibration analysis algorithms are formulated to obtain dynamic response bounds of SDOF system and MDOF system under time-variant uncertain excitations, respectively.Besides, corresponding Monte Carlo method is proposed to verify accuracy of the response bounds calculated and provide a general analytical tool for non-random vibration analysis.Finally, the feasibility of the non-random vibration analysis method is validated by several numerical examples.The proposed non-random vibration analysis method could provide a promising supplement for random vibration theory, and thereby plays an important role in structural uncertain dynamic analysis and reliability design.for engineering problems.

2016, 48(2): 447-463. doi: 10.6052/0459-1879-15-244

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EFFECT OF WELLBORE GAS-LIQUID TWO-PHASE FLOWON WELL TEST OF FRACTURED HORIZONTAL WELLS IN TIGHT GAS RESERVOIR

Ouyang Weiping, Zhang Mian, Sun Hu

Multistage fracturing horizontal well technology is one of the most commonly used methods for the exploitation of tight gas.Well testing of fractured horizontal wells in the tight gas reservoir is often associated with water rate.Wellbore gas-liquid two-phase flow will increase the flow resistance of wellbore fluid, which increases the effect of wellbore fluid flow on the well test interpretation.In order to understand the effect of wellbore gas-liquid two-phase flow on well test and improve the precision of well test interpretation of fractured horizontal wells producing water in tight gas reservoir, this paper presents a coupling well test model including gas-liquid two-phase flow in wellbore and seepage flow in tight reservoir.Well test type curves, pressure fields and fracture flow rate of fractured horizontal wells considering wellbore gas-liquid two-phase flow are obtained by using numerical solving method.The results show that the gas-liquid two-phase flow will increase the pressure and pressure derivative value in the well test type curves.It also will cause that pressure diffusion near the window point is faster than that away from the window point and flow rate of fractures near the window point is greater than that of fractures away from the window point.Hence, it can be concluded that the results interpreted by well test analysis may have large error without consideration for the wellbore gas-liquid two-phase flow in the fractured horizontal well test model.The assumption that the horizontal wellbore has infinite conductivity will cause the skin factor obtained by curve matching to be too large.The pressure of the test point considered as the pressure of window point will cause the initial formation pressure obtained by matching to be too small.The well test model of fractured horizontal wells considering wellbore two-phase flow proposed in this paper will provide an important technical support to interpret the well test data of tight gas wells producing water correctly.

2016, 48(2): 464-472. doi: 10.6052/0459-1879-15-184

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A NUMERICAL STUDY ON THERMAL INTERNAL BOUNDARY LAYER OVER A COASTAL CITY

Liang Tinghao, Yu Xiping

In the coastal thermal internal boundary layer is very common, and the warmed air becomes unstable to induce a vertical advection of the air mass.Over a coastal city, the atmospheric flow is strongly affected by the complex morphology of the city, which is an alternation of the street canyon and the building.This leads to some special features of the coastal thermal internal boundary layer.A large-eddy simulation is carried out to study the thermal internal boundary layer flow over a coastal city in this study.The buildings in coastal cities are modelled as boxes with given geometry and known thermal properties.The immersed boundary method is used to represent the resistance of the buildings on the air flow.The numerical model thus captures more details of the coastal city as compared with the conventional regional models in which urban canopy is considered as the surface with a uniform property.Three cases are considered in the present study for a comparative study of the influence of urban resistance and land heating on the atmospheric flow.The computational results show that the spatial variation of the turbulence quantities is regular in a scale of the buildings. It is also demonstrated that the thermal buoyancy and the strong shearing effects interact with each other to intensify the turbulence in the boundary layer.Compared with the natural coast thermal boundary layer under the same thermal conditions, thermal internal boundary layer over a coastal city develops much faster.

2016, 48(2): 473-481. doi: 10.6052/0459-1879-15-056

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INVESTIGATION ON FILLING HEIGHT OF EMBANKMENT ON UNSATURATED SOFT GROUND WITH STIFF CRUST

Zhang Changguang, Zhao Junhai, Dai Yan

The stiff crust commonly distributed in the soft ground area has a good capacity for load transmission in embankment engineering, and its beneficial effect should be brought into a full play.Analytical formulae for both the critical edge load and critical load of unsaturated soft ground are deduced with taking into account of the stress dispersion and self-weight berm of stiff crust.It is assumed that the strength of unsaturated soils is governed by the unified shear strength formulation and a hyperbolic model of suction angle.Then, analytical solutions of the critical edge filling height and critical filling height of embankment are presented, whose applicable conditions and implementation steps are also proposed.Finally, the influences of various parameters involved are conducted on the critical filling height of embankment.The obtained analytical formulae have good comparability and are validated.In addition, the necessity and rationality of the proposed applicable conditions are highlighted.The findings of the study illustrate that, the stiff crust has an overall influence, and the changes about thickness and Young's modulus of stiff crust could be simultaneously considered;the effect of lateral pressure coefficient is remarkable, and thus a real lateral pressure coefficient would be determined by in-situ test;moreover, the differences are significant with different strength criteria, and the effect of intermediate principal stress should be fully considered on the applicable conditions of formulation and filling height of embankment;the effect of high suction is different as compared with that of low suction.

2016, 48(2): 482-494. doi: 10.6052/0459-1879-15-321

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THEORETICAL ESTIMATION OF INTERPOLATION BIAS ERROR IN DIGITAL IMAGE CORRELATION

Su Yong, Zhang Qingchuan, Xu Xiaohai, Gao Zeren, Cheng Teng

The popularity of digital image correlation technique have pointed out the urgent need to establish standard assessment criterion of speckle pattern quality, namely, the development of standard procedure to assess the metrological performance of various digital speckle patterns.The magnitude of digital image correlation calculation error due to subpixel interpolation(interpolation bias error) is an important parameter to evaluate the quality of speckle.However, there is no available method to estimate interpolation bias efficiently at present.In this paper, frequency method is employed to obtain the analytical expression of interpolation bias error.Band-limited and sinusoidal approximation forms are attained when sampling theorem is satisfied.The sinusoidal variation of interpretation bias error with respect to sub-pixel shift is explained.Based on sinusoidal approximation form of interpolation bias, this work introduces the concept of interpolation bias kernel.Interpolation bias kernel, which characterizes frequency bias response of specific speckle frequency, is exploited to decide the merits of the interpolation algorithm for correlation matching algorithm.Based on these theoretical results, this paper presents a method to estimate the interpolation bias error by speckle spectrum and interpolation bias kernel.This simple and effective algorithm has obvious speed advantage compared to traditional translation methods, and the simulation is conducted to verify this method.This work explains the inherent nature of interpolation bias and solves the problem of efficient interpolation bias estimation.This work could be used in interpolation optimization and filter size selection and contribute to the establishment of speckle quality assessment criterion as well.

2016, 48(2): 495-510. doi: 10.6052/0459-1879-15-166

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REVIEW OF THE NINTH NATIONAL SYMPOSIUM ON FLUID MECHANICS FOR YOUNG SCHOLARS

Luo Zhenbing, Sun Mingbo, Zhang Panfengy

The paper brief introduced the ninth national symposium on fluid mechanics for young scholars.Reports of the symposium were reviewed, and some constructive suggestions were put forward.

2016, 48(2): 511-517. doi: 10.6052/0459-1879-16-054

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