应用数学和力学

2015 Vol. 36, No. 9

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Vibration Characteristics of an Axially Moving Variable Length Beam With a Tip Mass

MA Guo-liang, XU Ming-long, CHEN Li-qun, DING Hu

2015, 36(9): 897-904. doi: 10.3879/j.issn.1000-0887.2015.09.001

Abstract(1617) PDF(1242)

Abstract:
A semi-analytical method and a numerical method were used to investigate the vibration characteristics of an axially moving variable length (velocity) beam with a tip mass. First, the equation of transverse free vibration for the axially moving Euler beam was simplified. The eigenequation was derived with the complex modal analysis method. Moreover, the frequency equation was obtained under the boundary conditions with a tip mass. The numerical method was used to calculate the natural frequencies and modal shapes. Then, the equation of transverse free vibration was also derived with the finite element method (FEM). The complex eigenvalues and eigenvectors were obtained as solutions to the complex matrix equation, and the complex modal displacements were given through combination with the shape functions. Finally, the results from these 2 methods were comparatively analyzed. The numerical example illustrates that different velocities and tip masses influence the beam vibration characteristics significantly. The calculated results from the 2 methods are close to each other and effective.

THz Wave Propagation in Carbon Nanotube Arrays Under the Symplectic System

ZHAO Peng, DENG Zi-chen, ZHANG Yu

2015, 36(9): 905-913. doi: 10.3879/j.issn.1000-0887.2015.09.002

Abstract(1734) PDF(1195)

Abstract:
For a planar waveguide filled with periodic parallel finite-length carbon nanotube array, we got the dielectric properties of the parallel carbon nanotube array based on the equivalent medium model for parallel carbon nanotube arrays while ignoring the spatial dispersion but considering the electromagnetic wave propagation loss, respectively, and led the electromagnetic wave propagation in the waveguide into the Hamilton system with the ideal conductive boundary conditions, then we used the symplectic theory framework to solve the eigenvalue equations for the electromagnetic wave propagation and obtain the dispersion relationships. According to the analysis, it shows that the fundamental mode for the electromagnetic waveguide can't propagate within a narrow spectrum, however, the fundamental mode propagates smoothly with very low loss elsewhere, which makes the carbon nanotube array a waveguide material with better propagation characteristics in a wide spectrum than traditional materials.

Dynamic Deflection Influence Line of Bridge Subjected to VehicleBridge Random Vibration

XU Wen-tao, ZHANG Jian-bo, CHEN Yong-jie

2015, 36(9): 914-923. doi: 10.3879/j.issn.1000-0887.2015.09.003

Abstract(1501) PDF(1296)

Abstract:
Traditional static influence line was developed into dynamic field, and dynamic influence lines of deflections in the mid span of a simply-supported bridge and a three-span elastically supported bridge were studied by considering the effect of interaction between the bridge, the bridge random surface roughness and the vehicle. To obtain the mean value and standard deviation of dynamic influence lines in the mid span of bridge, deterministic harmonic excitations were derived from bridge surface roughness with pseudo excitation method, and the equation of vehicle-bridge system was solved with precise integration method. Based on 3σ method, deterministic values range of dynamic deflections were obtained. Finally, the random characteristics of dynamic influence lines were analyzed with numerical examples, and the effects of vehicle velocity and bridge random surface roughness on dynamic influence line were discussed. Then the difference between the simply-supported bridge and the three-span elastically supported bridge was studied upon dynamic influence line.

Thermal Buckling of Thin Spherical Shells Under Interaction of Uniform External Pressure and Uniform Temperature

LI Chen, TIAN Xue-kun, WANG Hai-ren, MIAO Ya-nan

2015, 36(9): 924-935. doi: 10.3879/j.issn.1000-0887.2015.09.004

Abstract(1433) PDF(741)

Abstract:
The thermal buckling equation for thin spherical shells was deduced on the basis of axisymmetric thin spherical shell buckling equation derived with the tensor method. The thermal buckling equation involving the coupling of uniform external pressure and temperature was expressed in terms of displacement. The thin spherical shell buckling of minimum potential energy functional was also established according to the virtual work principle. 3 thermal buckling problems for simply supported hemispherical shells were analyzed with the Ritz method. The following 3 conclusions are drawn: 1) The critical value of uniform external pressure on condition that the temperature does not exceed the critical buckling level. 2) The buckling critical temperature value on condition that the uniform external pressure is 0. 3) The buckling critical temperature value on condition that the uniform external pressure does not exceed the critical level.

Equivalent Micropolar Thermoelastic Analysis of Thermal Bending for Grid Sandwich Beams

ZHANG Rui, SHANG Xin-chun

2015, 36(9): 936-944. doi: 10.3879/j.issn.1000-0887.2015.09.005

Abstract(1894) PDF(738)

Abstract:
The thermal bending of the grid sandwich beam was considered to be equivalent to the deformation of the micropolar thermoelastic beam under thermal load, and the control equations of thermal deformation for the equivalent micropolar thermoelastic beam were established based on the plane micropolar thermoelastic theory, with the expression of the thermal displacements given. The material parameters of the equivalent micropolar thermoelastic beam were obtained with the cell energy equivalence method. The example cantilever grid sandwich beam’s thermal bending deformations calculated according to the proposed analytical equivalent micropolar beam model and the numerical Ansys FEM were compared. The results from the proposed analytical method are perfectly close to those from the numerical mothod, which validates that the equivalent micropolar thermoelastic beam is a simple and effective model to simulate the thermal deformations of grid sandwich beams.

Analytical Solutions to Problems of 1D Orthorhombic Quasicrystal With SemiInfinite Cracks

GAO Jian, LIU Guan-ting

2015, 36(9): 945-955. doi: 10.3879/j.issn.1000-0887.2015.09.006

Abstract(1333) PDF(615)

Abstract:
The anti-plane elasticity problem of the 1D orthorhombic quasicrystal with a semi-infinite crack penetrating along the quasiperiodic direction was investigated through introduction of a new generalized conformal mapping and with the generalized complex variable method. The analytical solutions of the stress fields and the stress intensity factors under the action of the uniform out-of-plane shear load on the partial crack surface were obtained. In addition, this method was applied to solve the plane elasticity problem of the 1D orthorhombic quasicrystal with a semi-infinite crack penetrating perpendicular to the quasiperiodic direction and the analytical solutions were derived. Under the condition of higher symmetry, the analytical solutions to the corresponding problem of the 1D tetragonal quasicrystal were also obtained.

A Non-Splitting PML for Elastic Waves in Polar Coordinates and Its Finite Element Implementation

ZHOU Feng-xi, CAO Xiao-lin, Mark B. Jaksa

2015, 36(9): 956-969. doi: 10.3879/j.issn.1000-0887.2015.09.007

Abstract(1592) PDF(771)

Abstract:
In the solving of the elastic wave equations with the numerical approximation techniques, the absorbing boundary conditions had been widely used to truncate the infinite-space simulation to a finite-space one. The perfect matched layer (PML) technique as an absorbing boundary condition had exhibited excellent absorbing efficiency in the forward simulation of the elastic wave equation formulated in rectangular coordinates. Based on the stretched coordinate concept, an advanced non-splitting-field perfect matched layer (non-splitting PML) equation for elastic waves was formulated in the polar coordinate system. Through the introduction of integrated complex variables in the radial direction into the auxiliary functions, the PML formulation was extended in polar coordinates in view of the 2nd-order elastic wave equation with displacements as basic unknowns. In addition, aimed at the time-domain cases and with the finite-element method for space discretization, the finite-element time-domain (FETD) scheme in standard displacement-based formulation was presented. The scheme for the special cases in axisymmetric polar coordinates was also given. The effectiveness and validity of the present non-splitting PML formulation are demonstrated with several numerical examples.

Modeling of Hydraulic Fracturing of Concrete Gravity Dam Considering Fluid-Structure Interaction

WANG Ke-feng, ZHANG Qing, XIA Xiao-zhou

2015, 36(9): 970-980. doi: 10.3879/j.issn.1000-0887.2015.09.008

Abstract(1624) PDF(736)

Abstract:
High-pressure hydraulic fracturing (HF) is an important part of the safety-assessment of high concrete dams. Fluid-structure interaction during the progress is the keypoint to accurately predict crack growth path and risk level. In this paper, the extended finite element method (XFEM) was used for numerical simulation of HF of concrete gravity dam. Water pressure distribution model was obtained from experimental results of the hydraulic fracturing test carried out by Brühwiler and Saouma, which embodied the coupling relationship between water pressure and crack width. The method loading water pressure on crack surfaces was also given in XFEM program and then hydraulic fracturing of gravity dam was simulated. Results show that: XFEM is a very convenient and effective tool to simulate hydraulic fracture propagation. In XFEM, crack can propagate without re-meshing and crack width can be easily gained with the enriched freedom degrees. When fluid-structure interaction is in consideration, the extension angle is bigger and the crack growth length is shorter than that in the condition without considering fluid-structure interaction.

Effects of Ion Concentration on Electroosmotic Flow and Micromixing in Microchannels

YANG Da-yong, WANG Yang

2015, 36(9): 981-989. doi: 10.3879/j.issn.1000-0887.2015.09.009

Abstract(1655) PDF(1326)

Abstract:
Electroosmotic flow is widely used to transport and mix fluids in microfluidic chips. A variable model for the ion concentration gradient effects on the electroosmotic flow and micromixing in microchannels was presented. The effects were investigated numerically with the finite element method. The impacts of the zeta potential and the dielectric constant on the flow field and concentration field were also analyzed. The micromixing efficiency in the microchannel was evaluated quantitatively. The results show that the flow field is inhomogeneous, and the distribution of the ion concentration will be asymmetric in the microchannel while the zeta potential and the dielectric constant vary with the ion concentration. When the concentration of the electrolyte solution is approximate to 1 mol/L, the solution essentially couldn't be driven into the microchannel. The micromixing efficiency decreases with the ion concentration difference between the electrolyte solutions, and the larger the difference is, the shorter the distance is needed to reach perfect mixing.

Quasi-Periodic Solution and Its Asymptotic Behavior for Camassa-Holm Equation

WANG Zhen, QIN Yu-peng, ZOU Li, MA Rui-fang, ZHU Gui-xun

2015, 36(9): 990-1002. doi: 10.3879/j.issn.1000-0887.2015.09.010

Abstract(1847) PDF(779)

Abstract:
Many researchers have paid attention to the shallow water wave model Camassa-Holm (CH) equation over the last two decades. The one-periodic solution of CH equation based on the Hirota bilinear method had been presented in our previous work. In this paper, we offer quasi-periodic solution in genus two and its asymptotic behavior. First, we have rearranged the parameters appeared in the bilinear equation system, such as the coordinate transformation, extended bilinear form and Riemman theta function and so on. Then quasi-periodic solution of CH equation is presented, which is expressed by Riemann theta function in genus two. Second, asymptotic behavior of the quasi-periodic solution is discussed. It can be shown that this solution will degenerate into its two-soliton solution.

Asymptotic Travelling Wave Soliton Solutions for Nonlinear Disturbed Generalized NNV Systems

SHI Juan-rong, WU Qin-kuan, MO Jia-qi

2015, 36(9): 1003-1010. doi: 10.3879/j.issn.1000-0887.2015.09.011

Abstract(1432) PDF(623)

Abstract:
A class of nonlinear disturbed generalized NNV (NizhnikNovikovVeselov) system was addressed with a simple and valid technique. Firstly, the soliton solution to the corresponding typical differential system was obtained by means of the undetermined coefficient method. Secondly, a generalized functional equation was built and variationally calculated, and the corresponding Lagrange multiplier was derived according to the variation principle. Thereby, a special variational iteration relation expression was constructed. Then, the asymptotic travelling wave soliton solution for the original nonlinear disturbed generalized NNV system was attained successively. Finally, through an example, the proposed approximate analysis method is proved to be convenient and effective.