应用数学和力学

An EEP Adaptive Strategy of the Galerkin FEM for Axially Forced Vibration of Bars

XING Qinyan, YANG Qinghao, LU Chenyu, YANG Xing

2019, 40(9): 945-956. doi: 10.21656/1000-0887.400051

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Abstract:
Based on the successful applications of the element energy projection (EEP) adaptive method for the static problems of bars and the dynamic equations for discrete systems, a strategy was proposed to adaptively solve the axially forced vibration problems of bars in both the time dimension and the space dimension. In this strategy, the continuous space-time Galerkin finite element method (FEM) was used. Based on the idea of semi-discretization, through discretization in space first, the governing partial differential equations of the model problem were transformed into a system of 2nd-order ordinary differential equations with initial boundary conditions, which were called dynamic equations for discrete systems hereinafter. These dynamic equations for discrete systems were then solved with the proposed EEP adaptive FEM in the time domain. After that, the EEP super-convergent formula for dynamic displacements in the space direction was derived, with which errors of the conventional Galerkin FEM solutions were estimated and the corresponding adaptive analysis method was established. Finally, the presented EEP adaptive strategy gave dynamic displacements with high accuracy point-wisely satisfying the pre-specified error tolerance, together with the automatically produced space-time mesh. The basic idea, the key technologies and the implementation strategy were elaborated. Representative numerical examples including seismic wave input demonstrate effectiveness and reliability of the method.

Numerical Simulation of Droplets’Dynamic Wetting Process With the Phase Field Method

LI Jiayu, ZENG Zhong, QIAO Long

2019, 40(9): 957-967. doi: 10.21656/1000-0887.400129

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Abstract:
Dynamic wetting phenomena of droplets are widely observed in nature and industrial production, the numerical research of which needs a solution of singularity and a correct model of the dynamic contact angle. Based on the phase field method (PFM) and the modified dynamic contact angle model, a 2-phase flow numerical method with dynamic wetting effects was developed, and the related program was realized on the OpenFOAM platform. The dynamic process of droplets impacting on a wall was simulated, and the comparison of the computation results of different contact angle models was conducted. The results indicate that the contact angle model influences the dynamic wetting process simulation significantly; the results of the proposed method are in good agreement with those of the experiment, which shows the proposed method is effective in the simulation of dynamic wetting phenomena.

Dynamic Analysis of the Nonlinear Vibration Absorber-Energy Harvester Integration Model With Inerters

DONG Yanchen, ZHANG Yewei, CHEN Liqun

2019, 40(9): 968-979. doi: 10.21656/1000-0887.400049

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Abstract:
In order to solve the vibration and energy supply problems in aerospace engineering, an nonlinear vibration absorber integrated into an energy harvester with inerters was designed. The nonlinear vibration absorber was built based on the nonlinear energy sink (NES) in which the traditional inertial components were replaced with inerters, and the energy harvester based on the giant magnetostrictive material (GMM) was integrated into this device. This NES-I-GMM device achieves vibration absorption and energy collection simultaneously. Under the background of the whole satellite vibration reduction, the modeling, simulation and analysis were carried out, and the effects of vibration reduction and energy collection were investigated through numerical computation. The numerical results show that, the NES-I-GMM device works well in vibration reduction and vibration energy collection under reasonable parameters.

Properties of the Eigen Solution of Taut Strings With Concentrated Damping

ZHENG Gang, LI Zhangyu, GUO Zengwei, ZHANG Xiaodong

2019, 40(9): 980-990. doi: 10.21656/1000-0887.400023

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Abstract:
By means of the Dirac delta function, the free-vibration equation of motion for taut strings with concentrated damping, namely the damping hybrid string system, was established and solved. The analytic solution to the eigen problem was obtained for the system with only one single damping dashpot at the midspan, and the properties of the eigen value and the eigen function were analyzed. The dynamic behaviors of the damping hybrid string system, including the frequency-damping relationship, the decay ratio and the full suppression of the motion at the optimal damping, which distinctly differentiate the hybrid system from a continuous system or a discrete system, were identified: 1) The frequency of the hybrid string system is independent of its damping ratio; 2) The decay ratios keep the same for different orders of eigen functions; 3) The decay ratios approach infinity when the damping ratio equals 2, which indicates any damped vibration of the system will be fully suppressed.

A Transfer Matrix Algorithm for Vertical Free Vibration of Suspension Bridges

ZHONG Hao, XIANG Tianyu

2019, 40(9): 991-999. doi: 10.21656/1000-0887.390221

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Abstract:
A transfer matrix algorithm was proposed for the solution of vertical free vibration of suspension bridges. First, a solution scheme was presented for the governing differential equation of vertical free vibration of suspension bridges, in which the analytic expression of structural free vibration was deduced directly. On this basis, the transfer matrix of the vertical free vibration was obtained. With this method, the problem of vertical free vibration of both the uniform-cross-section and the nonuniform-cross-section suspension bridges was studied. The research results indicate that the transfer matrix algorithm is efficient for the analysis of the vertical free vibration of suspension bridges, and has specific advantages in dealing with the nonuniform-cross-section problem. The work makes a good reference for the dynamic analysis of suspension bridges in the concept design stage.

Nonlinear Near Inertial Waves With Complete Coriolis Effects

DU Chunyao, YANG Liangui, ZHANG Yongli, ZHANG Ruigang

2019, 40(9): 1000-1010. doi: 10.21656/1000-0887.390280

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Abstract:
Based on the original basic equations of atmospheric motion under the action of the complete Coriolis force, through the scale analysis, the multi-scale method and the perturbation expansion method were used to derive the Korteweg-de Vries equation satisfying the amplitude evolution of the atmospheric near inertial wave at the mid-high latitudes. The results of the evolution equation show that, the influence of the horizontal component of the Coriolis parameter on the nonlinear near inertial wave mainly lies in the correction of the dispersion effect and the interaction with the elementary stream. The physical mechanism of atmospheric near inertial wave motion at the mid-high latitudes under the action of the complete Coriolis force was theoretically explained.

Study on Effects of Interfacial Arc Cracks on Cracking Strengths of Concrete

YIN Yueming, LI Zongli, LI Dongqi, Lü Congcong

2019, 40(9): 1011-1024. doi: 10.21656/1000-0887.390330

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Abstract:
Owing to water evaporation, dry shrinkage, bleeding and inconsistent deformation between aggregates and mortar, cracks will inevitably occur in the interfacial transition zone between aggregates and mortar and significantly affect the cracking strength of concrete. From the viewpoint of meso-scopic mechanism, concrete is regarded as a 2-phase composite material of coarse aggregates and cement mortar, and the interfacial transition zone is considered as a contact layer in the analysis. Firstly, in view of the interaction between aggregate particles in concrete, the far-field external load on the representative volume element (RVE) of concrete was simplified as an equivalent load on an infinite matrix containing a single aggregate with the interaction direct derivative (IDD) method. Then the equivalent external load was transformed into principal stresses, and the stress intensity factor (SIF) of the interfacial arc crack was obtained based on the fracture mechanics theory. The compound power law was chosen to judge the cracking of concrete, and the variation law of the cracking strength of concrete was studied. In comparison with the FEM, the analytical SIF solution of the interfacial arc crack has verified validity. The parametric analysis results show that, the arc crack is most likely to open when it is perpendicular to the maximum principal stress or at an angle of 45° to the minimum principal stress. With the increase of the crack length, the tensile and compressive cracking strengths decrease first and then increase, and they both have the most unbeneficial crack lengths. The cracking strength increases with the aggregate volume fraction and decreases with the aggregate particle size. For a small crack length, increasing the elastic modulus of aggregates can improve the cracking strength. The cracking strength will increase in the case of the same-sign stresses around aggregates, and on the contrary, will decrease in the case of different-sign stresses.

General Solutions of Elastodynamics for 2D Isotropic Porous Media

WANG Yuning, ZHAO Baosheng

2019, 40(9): 1025-1034. doi: 10.21656/1000-0887.390235

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Abstract:
With the general equations of solid displacement and fluid velocity, the general solution of elastodynamics for 2D isotropic porous media were studied. Through introduction of 4 undetermined functions, the equations of motion, fluid velocity and continuity were formulated and divided into 2 parts of expansion wave and torsion wave. Thus, the general solutions expressed with 3 quasi harmonic functions were obtained under the Lur’e operator theory. With the solid displacement and fluid velocity independent of the time, the general steadystate solutions for 2D isotropic porous media were given, and their completeness was proved.

Analytical Solution for 1D Consolidation of Unsaturated Soil Under Constant Load and DoubleSide Drainage

HE Zizhou, ZHOU Fengxi

2019, 40(9): 1035-1047. doi: 10.21656/1000-0887.390292

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Abstract:
Based on the elastic theory for porous media and the principle of the effective stress expressed with intergranular suction stresses, the coupled partial differential governing equation for unsaturated soil consolidation was established. For the 1D consolidation problem, the analytical solution for the consolidation of unsaturated soil under constant load and doubleside drainage was obtained through the Laplace integral transform. The effects of soil saturation on the excessive pore water pressure, the effective stress and the soil settlement were analyzed with numerical examples. The results show that, the higher the initial saturation of the soil is, the faster the pore water pressure will dissipate and the faster the effective stress will increase.

Analysis of Water Contents in Uranium Packaging Containers

SHENG Junjie, LI Jia, ZHANG Yuqing, LI Shuyong, LIAO Bin

2019, 40(9): 1048-1058. doi: 10.21656/1000-0887.390310

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Abstract:
The calculation method and changing rules for water contents in uranium packaging containers were studied based on the corrosion theory of uranium, the Henry law and the permeation and leakage theory. The results show that, the water permeation and leakage are proportional to the partial pressure difference across the seal and the permeation is related to the permeability coefficient of the material. The water mass absorbed by organic materials is dependent not only on the saturated water content but also on the relative humidity. The water consumed by corrosion reaction of uranium with water is a function of the time, the uranium mass and the reacting surface area of the specimen. The water contents in uranium packaging containers are mainly influenced by the water permeation, the relative humidity of environment and the water absorption in organic materials, but is less influenced by the water leakage.

2019 Vol. 40, No. 9