应用数学和力学

2014 Vol. 35, No. 12

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A Methodology Based on FEM and Duhamel Integration for Bridges Subjected to Moving Loads

ZHU Dan-yang, ZHANG Ya-hui

2014, 35(12): 1287-1298. doi: 10.3879/j.issn.1000-0887.2014.12.001

Abstract(1230) PDF(1089)

Abstract:
Based on the finite element (FE) method and Duhamel integration, a numericalanalytical combined method for the dynamic response problem of an FE bridge under moving loads was proposed, and the conditions of resonance and cancellation for the bridge subjected to multiple moving loads were derived. The FE modes of the bridge structure were first computed and then converted into an analytical form constructed over all the elements of the bridge deck with the element shape functions. The analytical dynamic responses of the bridge were derived from Duhamel integration, and transformed into a simple integration and a summation of the previous results through elimination of the time variable from the integration, which enables the computation process more efficient. The proposed approach has the versatility of the FE method in dealing with structures of arbitrary configurations and the special efficiency and convenience of the analytical method in dealing with moving loads. In the numerical examples, the present method is verified with the Newmark method and the traditional analytical method based on a simply supported beam bridge and a 3span continuous beam bridge. The results show that the accurate solution of the FE structures subjected to moving loads is obtained with the present method, and no approximation is introduced during the computation process. The conditions of resonance and cancellation are discussed for the problems with multiple moving load, and the influence of the load distances on the responses of the bridge is also revealed.

Nonlinear Dynamic Behavior of the Satellite Rendezvous and Docking Based on the Symplectic Runge-Kutta Method

LI Qing-jun, YE Xue-hua, WANG Bo, WANG Yan

2014, 35(12): 1299-1307. doi: 10.3879/j.issn.1000-0887.2014.12.002

Abstract(1482) PDF(894)

Abstract:
The simulation of the satellite rendezvous and docking is one of most important problems for space platforms and so on. The nonlinear dynamic behavior of the satellite rendezvous and docking was investigated. According to the energy principle, the Lagrange function was given; then, the generalized coordinates, generalized momentum and Legendre transformation were introduced to derive the Hamilton equations; both the symplectic Runge-Kutta method and the 4th-order Runge-Kutta method were comparatively used to solve the Hamilton equations. Through numerical analysis, it is easily found that the natural properties of the nonlinear dynamic system are well preserved with the symplectic RungeKutta method, especially in the long-time chasing cases. The proposed symplectic method is applicable to the related astrodynamic problems.

Deployment Dynamics for Solar Wings With Joint Friction

DUAN Liu-cheng, LI Hai-quan, LIU Xiao-feng, CAI Guo-ping

2014, 35(12): 1308-1319. doi: 10.3879/j.issn.1000-0887.2014.12.003

Abstract(1621) PDF(743)

Abstract:
The deployment and locking process of solar wings of the freefloating spacecrafts was studied. The dynamic model describing the deployment and locking process was established with the method of forward recursive formulation. The contribution of joint friction to the dynamic equations was derived based on the virtual power principle. The closed dynamic equations for the system was constructed through supplementation of the dynamic balance equations for the bodies with the frictional joints’ effects. The validity of the proposed model was verified through comparison of the simulation results with those of the ADAMS software. Research results show that the proposed model effectively describes the deployment and locking process of solar wings, the joint friction has considerable influence on the dynamic behavior of the mutibody system, and the PD control method substantially restrain the drift of the spacecraft attitude caused by the deployment of solar wings.

Dynamic Response of the Submerged Floating Tunnel Under Random Seismic Excitation

DONG Man-sheng, LI Man, LIN Zhi, TANG Fei, JIANG Shu-ping

2014, 35(12): 1320-1329. doi: 10.3879/j.issn.1000-0887.2014.12.004

Abstract(1157) PDF(819)

Abstract:
The submerged floating tunnel stayed by tension legs was simplified as the superposition of an elastic beam and an elastically supported rigid beam, the motion equations of forced vibration for the floating tunnel tube based on the BernoulliEuler beam theory were presented, then the nonlinear term of the dynamic equations was linearized equivalently. The pseudo excitation was input to simulate the random seismic impact and study the response of the submerged floating tunnel tube, with the displacement power spectrum at the middle point given and discussed detailedly. The displacement power spectrum analysis indicates that the displacement response of the tube decreases with the spring stiffness of the supporting tension legs and the damping coefficient of the energy dissipation devices, of which the former is of much higher significance.

Vibration Responses of Rubber Tires With Fractional Damping Under Stochastic Excitation

FAN Yuan-qin, XU Wei, HAN Qun, YANG Yong-ge

2014, 35(12): 1330-1340. doi: 10.3879/j.issn.1000-0887.2014.12.005

Abstract(1144) PDF(985)

Abstract:
The vertical dynamic responses of rubber tires with fractional damping under the KanaiTajimi noise excitation were investigated with the stochastic averaging method. Firstly, the earthquake wave was approximated with the KanaiTajimi noise, and the differential equation for tire vibration was established through combination of the point contact model with the fractional derivative model. Then, the stochastic averaging method was used to solve the stationary probability density analytically. In turn, validity of the proposed method was verified against the MonteCarlo numerical simulation results. The probability density was applied to determine the mean values and variances of vibration displacements of the 2 kinds of tires made of polybutadiene and butyl B252 rubbers, respectively. The results show that the mean value and variance of vibration displacement increase with the rubber’s storage modulus and decrease with its dissipation modulus. That means, the lower the rubber’s storage modulus is or the higher its dissipation modulus is, the better the vibration damping effect of the resulting tire will be. The work provides a theoretical reference for the design and manufacture of rubber tires.

Identification of Temperature-Dependent Thermal Conductivity for 2-D Transient Heat Conduction Problems

ZHOU Huan-lin, XU Xing-sheng, LI Xiu-li, CHEN Hao-long

2014, 35(12): 1341-1351. doi: 10.3879/j.issn.1000-0887.2014.12.006

Abstract(1440) PDF(1244)

Abstract:
The temperature-dependent thermal conductivity was identified for 2-D transient heat conduction problems with the boundary element method (BEM). The nonlinear governing equations were transformed into linear ones through the Kirchhoff transformation. The BEM was used to build the numerical model for the 2-D transient heat conduction problems. The inversion parameters were defined as the optimization variables. The quadratic sum of residual errors between the calculated temperature values and the measured temperature values at the measuring points was regarded as the objective function. The complex variable differentiation method was employed to compute the gradient matrix of the objective function. The gradient-regulation method was developed to optimize the objective function. Effects of the time step length, the number of measuring points and the random noise on the inverse results were discussed. With decrease of the time step length and increase of the number of the measuring points, the converging rate quickens. With decrease of the random noise, the results grow accurate. Numerical tests show the effectiveness and stability of the presented method.

Basic Equations of the Unstressed State Control Method Based on Co-Rotational Formulation

WEI Shao-yang

2014, 35(12): 1352-1362. doi: 10.3879/j.issn.1000-0887.2014.12.007

Abstract(1128) PDF(910)

Abstract:
The basic theory of unstressed state control method was addressed through introduction of the principle of co-rotational formulation. Based on the co-rotational formulation analysis, the differences of the reference frame between the traditional incremental method and the unstressed state control method were discussed, and the basic equations of the unstressed state control method were derived. The resulting formulas show that the co-rotational formulation procedure is appropriate for analysis of the basic equations of the unstressed state control method, and the derivation process has a clear mechanics concept. Moreover, case analysis demonstrates validity and advantage of the co-rotational formulation in application of the unstressed state control method.

Computational Fluid Dynamics Numerical Simulation of an Ultrasonic Velocimeter

GUAN Hui, SUN Xue-jin, XIONG Ying, WEI Ke-jing, YANG Qi-dong

2014, 35(12): 1363-1372. doi: 10.3879/j.issn.1000-0887.2014.12.008

Abstract(1070) PDF(740)

Abstract:
An ultrasonic velocimeter is a device with ultrasonic transmitter-receiver to detect the time between transmission and reception of ultrasonic waves and to calculate the velocity of fluid flow. The current domestic studies mainly focus on how to eliminate the measurement errors, but spare little attention to the influence of the model structure on the measuring wind field. In order to investigate the accuracy of a model velocimeter, the impact of the model structure on the wind velocity in the measuring wind field was addressed, the computational fluid dynamics (CFD) method was used to simulate the flow fields of different incoming flows from low speed to high, and to calculate the velocity profiles in the central area of the velocimeter and the average velocities in different cross sections. Consequently, the interference effect of the model velocimeter structure on the velocity in the central area of measurement was evaluated. The research results show that the velocities measured on the plane at the tops of measuring balls are the most accurate in spite of low or high incoming flow velocities. Therefore, the CFD method is proved to be a powerful tool for the model design of high-accuracy velocimeters.

Research on the AUSMV Scheme for 1D Gas-Liquid Two-Phase Flow Drift Flux Models

XU Chao-yang, MENG Ying-feng, WEI Na, LI Gao, WAN Li-ping

2014, 35(12): 1373-1382. doi: 10.3879/j.issn.1000-0887.2014.12.009

Abstract(1809) PDF(884)

Abstract:
Application of the AUSMV (advection upstream splitting method V) scheme was extended from gas dynamics to transient 1D isothermal gas-liquid two-phase pipe flow problems. The method of numerical flux for the DFM (drift flux model) constructed with the AUSMV scheme and treatment of boundary cells were stated for the simulations. The numerical calculation method of 2nd-order accuracy in time and space was obtained with the classical Runge-Kutta method and the monotonous MUSCL (monotone upstream-centred schemes for conservation laws) technique combined with the Van Leer limiter. The numerical examples including the Zuber-Findlay shock tube problem and the variable mass flow problems with complex slip relations were conducted and comparatively discussed. The results indicate that the proposed AUSMV scheme, with advantages of high efficiency, high precision and low effects of dissipation and dispersion, accurately details the discontinuities of 1D gas-liquid two-phase flow problems under low flow velocity conditions.

Perturbation Method for a Class of High-Order Nonlinear Reaction Diffusion Equations With Double Parameters

WANG Wei-gang, XU Yong-hong, SHI Lan-fang, MO Jia-qi

2014, 35(12): 1383-1391. doi: 10.3879/j.issn.1000-0887.2014.12.010

Abstract(1177) PDF(940)

Abstract:
The model for a class of highorder nonlinear reaction diffusion singularly perturbed problems with double parameters was addressed. With the singular perturbation method, the structure of the solution to the problem was discussed in the cases of double related small parameters. Firstly, the outer solution to the boundary value problem was given. Secondly, the variable of multiple scales was introduced to obtain the boundary layer correction term for the solution. Then the stretched variable was applied to the boundary neighborhood to get the initial layer correction term. Finally, the theorem of differential inequalities was constructed and the uniformly valid asymptotic expansion of the solution to the problem was proved. The proposed method possesses the advantages of convenient use and high accuracy.

Influence of Double Parameters on the Equilibrium Stability of Generalized Hamilton Systems

CHEN Xiang-wei, LI Yan-min, MEI Feng-xiang

2014, 35(12): 1392-1397. doi: 10.3879/j.issn.1000-0887.2014.12.011

Abstract(1081) PDF(941)

Abstract:
Influence of double parameters on the equilibrium stability of generalized Hamilton systems with additional terms was studied. Firstly, a generalized Hamilton system with additional terms was considered as a gradient system under certain conditions. Secondly, the characteristics of the gradient system was used to study the equilibrium stability and its dependance on the two parameters of the system. Thirdly, the stability domain was given in the parameter plane. The results show that the equilibrium of the system is likely to be stable, or asymptotically stable, or even unstable with the change of the two parameters, and the range of parameters corresponding to each equilibrium state is given.