应用数学和力学

2018 Vol. 39, No. 5

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Non-Stationary Radom Vibration Analysis of Coupled Pipeline-Soil Systems Under Earthquake

ZHAO Yan, JIA Tian, ZHOU Ruipeng

2018, 39(5): 493-505. doi: 10.21656/1000-0887.380333

Abstract(1756) PDF(523)

Abstract:
According to the non-stationary and coherent characteristics of seismic load, a frequency-domain method for random vibration analysis of underground pipeline-soil structures was proposed. The Fourier-Stieltjes integral was used to describe the non-stationary stochastic process, and the time-dependent characteristics of the amplitude and frequency components of the seismic load were depicted by its kernel function; the exponential decay function was used to describe the spatial distribution of seismic load. Based on the combination of the pseudo-excitation method and the Fourier analysis technique in the frequency domain, the closed-form solution of the response evolution power spectrum of the pipeline-soil structure was derived under the coherent and non-stationary random loading, and the frequency-domain relation between the input and the output of the random vibration was established. In numerical examples, the proposed method was compared with the traditional calculating methods, and the correctness and validity of the method were illustrated. Furthermore, the mechanism of random vibration behavior of the pipeline with changing soil parameters and different end constraints were studied.

A Singularity Analysis on Dynamics of Symmetric Cross-Ply Composite Sandwich Plates Under 1∶2 Resonance

GUO Yuhong, ZHANG Wei, YANG Xiaodong

2018, 39(5): 506-528. doi: 10.21656/1000-0887.380190

Abstract(1413) PDF(575)

Abstract:
Inner resonance is a typical nonlinear dynamic behavior, and the symmetric crossply composite sandwich plates have been widely used in aerospace. The studies about inner resonance of such sandwich plates have both theoretical and engineering significances. Based on the dynamic equations for the sandwich-plates, of which the boundary conditions were simply supported on 4 sides, the transverse and inplane excitations were both considered. The average equations in the polar form were obtained with the multiscale method, and the algebraic equations in the steady state form were derived through the average equations. The singularity theory was utilized to investigate 1∶2 resonant bifurcations of the symmetric crossply sandwich plates. Based on the algebraic equations in the steady state form, the restricted tangent space was obtained for the bifurcation equations with 2 tuning parameters, an inplane excitation and a transverse excitation. Then the algebraic equations were simplified under strong equivalence, and the normal form of the algebraic equations were obtained in nondegenerate cases. The singularity theory were generalized for the general nonlinear dynamic equations with 2 state variables and 4 bifurcation parameters, and the 18 universal unfoldings of bifurcation equations with codimension 4 were obtained in the case of 1∶2 internal resonance. The transition sets in the parameter plane and the bifurcation diagrams were depicted. The relationships between the tuning parameters and the exciting parameters were determined when bifurcation, hysteresis, and double limit points happened. The numerical results indicate that the vibration modes in different bifurcation regions are different.

Dynamic Response Analysis of Viscoelastic Multilayered Foundation in the Cartesian Coordinate System

KOU Lei, XU Jianguo, WANG Bo

2018, 39(5): 529-537. doi: 10.21656/1000-0887.380155

Abstract(1376) PDF(504)

Abstract:
Based on the governing viscoelastic equations in the Cartesian coordinate system, the analytical element stiffness matrices in the integral transform domain of 3D problems of viscoelastic foundation under axisymmetric and non-axisymmetric dynamic loads were derived with the Fourier-Laplace transform, the decoupling transformation, the differential equations theory and the matrix theory. The global stiffness matrix was assembled in view of the boundary conditions and the continuity between adjacent layers, the solution in the integral transform domain was obtained from the algebraic equations involving the global stiffness matrix, and the solution in the physical domain was acquired through the inverse Fourier-Laplace transform. Through a corresponding computer program, comparison of the dynamic responses of viscoelastic multilayered foundation between the proposed algorithm and the existing method proves the validity of the former.

Elasticity Solutions for Functionally Graded Circular and Annular Plates Subjected to Boundary Forces and Moments

ZHANG Ying, MEI Jing, CHEN Ding, YANG Bo

2018, 39(5): 538-547. doi: 10.21656/1000-0887.380275

Abstract(1572) PDF(465)

Abstract:
On the basis of the generalized England-Spencer plate theory, the 3D elastic fields of functionally graded circular and annular plates subjected to boundary forces were studied. The elastic constants of the material can be arbitrarily continuously variable along the thickness direction. Expressions of 4 complex potentials for solving the problem were given with the complex variable function method in which the undetermined constants were solved from the cylindrical boundary conditions of the plate. When the inner radius of the annular plate approaches zero, solutions of the annular plate reach to those of the corresponding circular plate. Numerical examples conducted show the effects of the material gradient, the load type and the thickness to radius ratio on the static responses of functionally graded plates.

Complex Variable Function Solutions for Stress and Displacement Distributions in Surrounding Rock of Deep Buried Highway Tunnels

LIU Shuhong, WANG Zihao, ZHEN Weigang

2018, 39(5): 548-557. doi: 10.21656/1000-0887.380137

Abstract(1491) PDF(523)

Abstract:
In order to optimize the design of highway tunnels and ensure the safety of construction, mechanical behavior of rock shall be made clear during highway tunnel excavation. With the complex variable method, firstly, the exterior domain of the tunnel was transformed into a unit-circle domain through the conformal mapping function; then 2 stress functions were derived with the Cauchy integral formula and the residue theorem; thereby, closed-form plane strain solutions of the surrounding rock were obtained for stresses and displacements. The curved-wall horseshoe-shaped cross section was adopted, and the distributions of stresses and excavation displacements along the tunnel boundary and the coordinate axes were calculated respectively with mathematical software MATLAB. To verify the accuracy of the stress and displacement distributions derived with analytical solutions, a 2D plane strain model was established with finite element software ANSYS. Comparison between the numerical results and the approximate analytical results shows good agreement. The results show that, the maximum hoop stress occurs at the arch foot, the maximum horizontal displacement occurs at the hance, the largest settlement and uplift are at the centers of the vault and the invert, respectively. Normal stresses along coordinate axes change obviously near the tunnel, and the maximum stress doesn’t always occur at the tunnel boundary. The maximum stress approaches the applied load at a distance less than 10 m from the tunnel boundary. The largest displacement occurs at the tunnel boundary, and gradually reaches zero with an increasing distance from the tunnel boundary.

A New Well Test Analysis Method for Multi-Stage Fractured Horizontal Wells With Angle Faults

LIU Qiguo, XU Youjie, LIU Yicheng, QI Shengzhi

2018, 39(5): 558-567. doi: 10.21656/1000-0887.380297

Abstract(1094) PDF(501)

Abstract:
Closed faults exist in some regions of eastern China, and they have great influence on the pressure characteristics of oil wells. Based on the basic principles of seepage mechanics, firstly, the bottom pressure solution of infinite-conductivity vertical fractured wells in the Laplace space was obtained by means of the basic theory of point source function and the Laplace integral transform. Combined with the conductivity function, the pressure solution of finite-conductivity vertical fractured wells was obtained. Secondly, the solution of multi-stage fractured horizontal wells with different angle faults was obtained with the mirror reflection principle and the pressure drop superposition principle. The pressure solution in the real space was given through the Stehfest numerical inversion, and the typical pressure as well as the pressure-derivative log-log curves were drawn. The results show that, the typical characteristic curve is divided into 8 flow stages, the pressure derivative is a horizontal line of a 0.5×360°/θ in the phase of fault reflection, and the smaller the vertical distance from the horizontal well center to the fault is, because the pressure-derivative curve characteristics of the radial flow are covered by the boundary reflection characteristic curve, the earlier the fault reflection time will be. Moreover, the larger the spacing between adjacent fractures is, the more obvious the early mirror flow characteristic curve will be; the larger the fracture number is, the lower the early pressure and pressure-derivative curve will be; and the larger the bilinear flow stage pressure derivative curve is, the lower conductivity will be. The accurate evaluation of reservoirs with closed fault boundaries and the reliable calculation of relevant parameters can be made through the proposed model.

A Method of Fluid-Solid Coupling Dynamics for Tube Bundle Vibration and Collision in a Cylinder Fluid Domain

YUE Qianbei, LIU Jubao, LUO Min, ZHANG Qiang

2018, 39(5): 568-583. doi: 10.21656/1000-0887.380265

Abstract(1598) PDF(804)

Abstract:
In offshore platforms, nuclear power stations and oil fields, the marine risers and heat exchangers are liable to vibrations induced by cross flow and consequent collision between bundle tubes, and even fail in the flow field. This is a typical problem in the fluid-solid coupling dynamics of flow-induced vibration and collision. But the relevant research has rarely been reported. In view of the nonlinear dynamics problem of flow-induced vibration and collision in the tube bundle, the 3D tube bundle and the unsteady fluid in a cylinder were selected as the research objects. The consistent conditions of displacement and velocity and the load balancing were considered at the fluid-solid coupling interface. The contact boundary between bundle tubes and the change of topological structure were also studied at the interface. The mechanical model and algorithm were established for flow-induced vibration and collision between bundle tubes in the cylinder fluid. The examples show that, when the elastic tubes collide with each other, the fluid pressure is equal at the contact point, and the fluid flow faster at the side stream surface of the bundle tubes.

Global Exponential Stability of Complex-Valued Neural Networks With Proportional Delays

ZHANG Lei, SONG Qiankun

2018, 39(5): 584-591. doi: 10.21656/1000-0887.380257

Abstract(1992) PDF(601)

Abstract:
The global exponential stability of complex-valued neural networks with proportional delays was investigated. By means of the vector Lyapunov function theory, the homomorphic mapping theorem, the M-matrix theory and the inequality technique, a delay-independent sufficient condition was obtained to ensure the existence, uniqueness and global exponential stability of the considered neural networks.

Traveling Waves of a Delayed Differential System in a Lattice

CAO Huarong, WU Shiliang

2018, 39(5): 592-610. doi: 10.21656/1000-0887.380165

Abstract(1548) PDF(491)

Abstract:
The qualitative properties of traveling waves of a delayed differential system in a lattice with a quiescent stage were addressed. Under monostable and quasi-monotone assumptions, the existence of the traveling wave solutions were first established. Then, the asymptotic behavior, monotonicity and uniqueness of all wave profiles were proved. The exponential asymptotic stability of all non-critical traveling fronts was finally proved.

Traveling Wave Solutions for Nonlocal Dispersal SIR Models With Spatio-Temporal Delays

ZOU Xia

2018, 39(5): 611-630. doi: 10.21656/1000-0887.380118

Abstract(1312) PDF(466)

Abstract:
In view of the infected individuals with the ability to move freely and spread disease, the traveling wave solutions for nonlocal dispersal SIR models with spatiotemporal delays were investigated. The threshold dynamics was determined by means of the basic reproduction number and the minimal wave speed. Firstly, based on Schauder’s fixed point theorem, the existence of fixed points on the cone was proved through construction of an invariant cone of the initial function on a bounded region. Then, the nonexistence of traveling wave solutions was verified through the twosided Laplace transform. Since the minimum propagation velocity of the traveling wave solution had important practical significance to control the disease transmission, the influences of the nonlocal diffusion term and the delay on the minimum wave velocity of the disease were discussed.