应用数学和力学

2014 Vol. 35, No. 11

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An Improved Symplectic Integration for Rigid Body Dynamics in Terms of Unit Quaternions

XU Xiao-ming, ZHONG Wan-xie

2014, 35(11): 1177-1187. doi: 10.3879/j.issn.1000-0887.2014.11.001

Abstract(1366) PDF(1418)

Abstract:
An identity transformation between the time derivative of quaternions and angular velocity was introduced into the kinetic energy term, according to the theory of quaternionbased rigid body dynamics. This proposed approach yielded a nonsingular mass matrix. Combined with the analytical structural mechanics, a new symplectic integration scheme with 4 formulations, was proposed. In practice, the discrete variational principle of the action function was employed to replace the relevant quaternion differential equations for the proposed method. Correspondingly, the unit length constraint was met explicitly by means of the algebraic constraint at the integration grid points. The numerical results show that the new scheme avoids the severe periodical nutation errors for the special cases of steady precession of a gyro top, which is a puzzling phenomenon in recent researches. In addition, the new scheme presents an impressive improvement of accuracy for the general cases as well.

Steady-State Periodic Responses of a Viscoelastic Buckled Beam in Nonlinear Internal Resonance

XIONG Liu-yang, ZHANG Guo-ce, DING Hu, CHEN Li-qun

2014, 35(11): 1188-1196. doi: 10.3879/j.issn.1000-0887.2014.11.002

Abstract(1073) PDF(879)

Abstract:
Nonlinear vibration of a hinged-hinged viscoelastic buckled beam subjected to primary resonance in the presence of internal resonance was investigated. The governing integro-partial differential equation was derived via introduction of coordinate transform for the non-trivial equilibrium configuration, with the viscoelastic constitutive relation taken into account. Based on the Galerkin method, the governing equation was truncated to a set of infinite ordinary differential equations and the condition for internal resonance was obtained. The multiple scales method was applied to derive the modulation-phase equations. Steady-state periodic solutions to the system as well as their stability were determined. The numerical examples were focused on the nonlinear phenomena, such as double-jump and hysteresis. The generation and vanishing of a double-jumping phenomenon on the amplitude-frequency curves were discussed in detail. The Runge-Kutta method was developed to verify the accuracy of results from the multiple scales method.

Corrected Reciprocal Theorem of Works for Bending Thin Plates and Its Application

FU Bao-lian

2014, 35(11): 1197-1209. doi: 10.3879/j.issn.1000-0887.2014.11.003

Abstract(1135) PDF(896)

Abstract:
It was discovered that the 2 main premises in the proposition of Betti’s reciprocal theorem of works for bending thin plates, ‘1 bending thin plate’ and ‘action of 2 sets of forces’, were contradictory to each other because any one of the 2 sets of forces might change ‘1 bending thin plate’ to a different one. The contradiction results in the fact that Betti’s reciprocal theorem of works for bending thin plates is one with error in logic. Based on the analysis of the contradiction, a corrected reciprocal theorem was proposed, in which the correct proposition of the reciprocal theorem for bending thin plates was given. The corrected reciprocal theorem provides a theoretical foundation for the reciprocal method of work, which makes a novel and powerful way to the analysis of bending plates.

Symmetric Solitary Waves and Their Existence Conditions in Cubic Nonlinear Microstructured Solids

Naranmandula, Ereduncang

2014, 35(11): 1210-1217. doi: 10.3879/j.issn.1000-0887.2014.11.004

Abstract(955) PDF(725)

Abstract:
In view of the macroscale cubic nonlinear effect, the microscale cubic nonlinear effect and the microscale dispersion effect of solid materials, a new model for the longitudinal wave propagation in 1D microstructured solids was established based on the modified Mindlin theory. The qualitative analysis method was applied to the dynamical system, the existence of symmetric bell and anti-bell type solitary waves in the cubic nonlinear microstructured solid was proved under appropriate conditions, and the existence conditions of the 2 solitary waves were given. The microscale cubic nonlinear effect on the bell and anti-bell type solitary waves was analyzed with the numerical method. The results indicate that the widths of the 2 solitary waves decreases (or increases) with the rise (or fall) of the microscale nonlinear effect while the amplitudes of the 2 solitary waves remain unchanged.

Lagrange High Order Cell-Centered Conservative Scheme in Cylindrical Geometry

GE Quan-wen

2014, 35(11): 1218-1231. doi: 10.3879/j.issn.1000-0887.2014.11.005

Abstract(924) PDF(672)

Abstract:
A Lagrange high order cell-centered conservative scheme in cylindrical geometry was presented for gas dynamics. The high order volume weighting subcell force in cylindrical geometry and the high order area weighting subcell force in cylindrical geometry were introduced by means of the MUSCL type method to construct 2 Lagrange high order cell-centered conservative schemes in cylindrical geometry. The vertex velocities and the numerical fluxes through the cell interfaces were evaluated in a consistent manner due to an original solver located at the nodes. The volume weighting scheme satisfies the momentum conservation and energy conservation, but does not surely keep the 1D spherical symmetry. The area weighting scheme satisfies the energy conservation and preserves the 1D spherical symmetry. 2 numerical tests were conducted. The results demonstrate that the new scheme is a high order one with satisfactory validity and accuracy.

Reconstructed Formulas Calculating Bursting Pressures of the Special Spherical Pressure Vessels Based on Experimental Data

LIU Ai-qun, YIN Yi-hui, LIU Xing-fu

2014, 35(11): 1232-1238. doi: 10.3879/j.issn.1000-0887.2014.11.006

Abstract(1090) PDF(1059)

Abstract:
With several traditional empirical formulas, the bursting pressures of the special spherical pressure vessels were calculated, comparison between the calculated and tested results was performed, and the characteristics of the related error ranges were analyzed. Based on the experimental data of the bursting pressure, several reconstructed formulas were given and the accuracy of their calculated results was examined in two ways. One way was to compare the results with the experimental data of the same kind of special spherical vessels; the other was to compare the results with the experimental data of the general spherical vessels. It turns out that the results of the reconstructed formulas are far more accurate than those of the traditional empirical formulas, which implies that the reconstructed formulas are practically useful in design of the discussed special spherical pressure vessels, and the related reconstruction method is reasonable and helpful to the research of similar problems.

Numerical Simulation of Shale Hydraulic Fracturing Based on the Extended Finite Element Method

ZENG Qing-dong, YAO Jun

2014, 35(11): 1239-1248. doi: 10.3879/j.issn.1000-0887.2014.11.007

Abstract(1533) PDF(1591)

Abstract:
In view of fluid flow in cracks and rock deformation, the mathematical model for shale hydraulic crack propagation was established. The crack flow field and the rock stress field were solved with the finite element method and the extended finite element method respectively, and the two fields were coupled through the Picard iteration. The presented model gave results consistent with those of the classic model, which verified correctness of the former. Based on the model, the effects of the rock elasticity modulus, Poisson’s ratio and injection rate on the crack geometry, and the dynamic process of a hydraulic crack approaching a natural crack at an arbitrary angle, were simulated. The numerical results show that the elasticity modulus and injection rate have significant influence on the crack geometry, while the Poisson’s ratio has little effect; the more brittle the shale is, the longer and narrower the hydraulic crack will grow; the greater the principal stress difference and the approaching angle are, the easier the hydraulic crack crosses the natural crack; there is a relatively large decrease in the crack width at the intersection between a hydraulic crack and a natural crack; the extended finite element method avoids mesh reconstruction and refinement during computation, and reduces the computing time. The presented model provides an effective theoretical tool for the shale fracturing design.

Research on the Meshless Solving Algorithm for 3D Steady ConvectionDiffusion Problems

ZHANG Xiao-hua, DENG Ji-heng

2014, 35(11): 1249-1258. doi: 10.3879/j.issn.1000-0887.2014.11.008

Abstract(1099) PDF(793)

Abstract:
The meshless method is a numerical algorithm for the simulation of flow fields in complicated shapes and solves fluid mechanics problems without grids. In order to improve the computation efficiency of meshless methods based on the Galekrin weak integration form for solving 3D steady convection-diffusion problems, a meshless shape function was proposed based on convex-polyhedral nodal influence domain in the discrete space. Then with a properly selected factor of nodal influence radius, the node-searching process was avoided and the bandwidth of the stiffness matrix for the system was reduced. With a factor of nodal influence radius at 1.01 during the calculation, the shape function of the meshless method almost possesses interpolation properties and the imposition of essential boundary conditions is simplified as that for the FEM. The numerical results of 2 exemplary steady convection-diffusion problems for 3D cubic regions show that: compared with the traditional meshless methods, the present meshless method based on convex-polyhedral nodal influence domain enables the computing time to be reduced by up to 42% without impairing the calculation accuracy. Finally, in the cases that both the computation efficiency and the accuracy are highly demanded, this meshless method based on convex-polyhedral nodal influence domain is suggested for the solution of 3D steady convection-diffusion problems.

Dynamic Simulation of Liquid Sloshing Characteristics for Tank Trucks in Lateral Movement

ZHAO Shu-en, ZHAO Ling-he

2014, 35(11): 1259-1270. doi: 10.3879/j.issn.1000-0887.2014.11.009

Abstract(1301) PDF(1066)

Abstract:
Aimed at the problem of lateral liquid sloshing in tank trucks partially filled with liquid, based on the multiphase flow model and the VOF method, the liquid sloshing dynamics was simulated for tank trucks in the cases of highspeed turning or emergency avoidance. The effects of the number of baffle plates, baffle configuration, liquid filling ratio and lateral acceleration on the dynamic characteristics of liquid sloshing were studied. The numerical results show that the longitudinal baffle plates reduce the lateral liquid impact force on the tank wall significantly and the largerarea baffle plates achieve the better reduction effect. The lateral sloshing falls quickly with the liquid filling ratio, while the liquid sloshing increases and the truck lateral stability goes down with the lateral acceleration.

Reliability-Based Design Optimization With the RBF Neural Network Model

LI Gang, MENG Zeng

2014, 35(11): 1271-1279. doi: 10.3879/j.issn.1000-0887.2014.11.010

Abstract(1015) PDF(933)

Abstract:
The reliability-based design optimization (RBDO) provides a trade-off between cost and safety given a lot of uncertain factors, such as structure sizes, material properties and external loads. The traditional RBDO approaches were computationally expensive owing to the nesting bilevel optimization. Thereafter, the decoupling approach and single loop approach were proposed to overcome this obstruction. However，the computational demand of these 2 types of approaches was still too huge for complex engineering problems. Here a new adaptive RBDO approach was presented based on the RBF neural network theory. The latin hypercube sampling method was used to construct the surrogate model. The error metrics were used to validate the RBF neural network model and the surrogate model was updated adaptively. Compared with the previous 4 popular RBDO algorithms, the presented method is efficient and robust.

Periodic Orbits of Electric Particles Sporting in Neutral Sheet Magnetic Field Without Dawn-Dusk Electric Field

CHEN Li-juan, LU Shi-ping

2014, 35(11): 1280-1286. doi: 10.3879/j.issn.1000-0887.2014.11.011

Abstract(737) PDF(792)

Abstract:
In order to describe the dynamic characteristics of the electric particles sporting in neutral sheet magnetic field without dawn-dusk electric field, a nonlinear motion model was proposed. Based on the Mawhin’s continuation theorem, the existence of periodic solutions to a class of nonlinear problems was discussed, and wherery, the problem of periodic solution of electric particles sporting in neutral sheet magnetic field without dawn-dusk electric field was investigated. Under the given initial conditions, a result about the existence of periodic orbits of the model was obtained. Furthermore, based on our result, other dynamic behaviours of the model, such as the homoclinic orbits can are to discussed.