应用数学和力学

2017 Vol. 38, No. 12

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Attitude Control of Targets Captured by Tethered Space Robots Based on the Quaternion Theory

XU Fang-nuan, WANG Bo, DENG Zi-chen, LI Qing-jun, WEI Yi

2017, 38(12): 1309-1318. doi: 10.21656/1000-0887.380168

Abstract(1489) PDF(1550)

Abstract:
For better understanding of tethered space robot systems to be used in the future onorbit service, the attitude control problem of targets captured by tethered space robots was investigated. Firstly, based on the quaternion theory, the governing equations for the combined body of the target and the tethered space robot were constructed. Secondly, a projected RungeKutta method was developed for the issue of constraint systems, and this method is explicit and can preserve the norm of quaternions. Finally, a numerical example was given to validate the proposed model and the numerical method, where 3 kinds of control strategies were compared in terms of control effectiveness and energy consumption.

Optimal Attitude Control of Underactuated Spacecrafts With the Gauss Pseudospectral Method

YI Zhong-gui, GE Xin-sheng

2017, 38(12): 1319-1330. doi: 10.21656/1000-0887.380013

Abstract(1145) PDF(549)

Abstract:
The attitude and orientation of modern spacecrafts generally can be controlled by 3 orthogonal reaction momentum wheels. The attitude control of underactuated spacecrafts with only 2 momentum wheels and failed out of complete 3-axis controlling torques, was investigated. The control problem was converted to a nonholonomic motion planning problem of a drift-free system with zero angular momentum. The attitude motion planning problem of a spacecraft with 2 momentum wheels was converted to a nonlinear programming problem (NLP) with the Gauss pseudospectral method, and then solved with the SQP algorithm. Through numerical simulation the optimal control met the design requirements of zero boundaries, so the wheels can be controlled easily by servomotors; the planned attitude curves were almost identical to the numerically integrated solutions; the number of suitable collocation points was found through balancing among 3 factors: the target function value, the running time and the solution error. The results show that the proposed method is effective for the optimal attitude control of underactuated spacecrafts.

Noether Symmetry of Automotive Electromagnetic Suspension Systems and Its Application

CUI Xin-bin, FU Jing-li

2017, 38(12): 1331-1341. doi: 10.21656/1000-0887.380060

Abstract(1144) PDF(603)

Abstract:
The Noether symmetry of vehicle vibration systems with electromagnetic suspension was studied, and the conserved quantity of the system was given. Furthermore, with the conserved quantity, the symmetry solution of the system was obtained. In the form of energy, the Lagrangian equations under different vibration modes were built. With the chosen displacement coordinates as the generalized coordinates, the Noether symmetries of the system under different vibration modes were studied, the corresponding Noether identities, Killing equations and generalized Noether theorems were given. The conserved quantity of the system was applied so that a new method for solving vehicle vibration system responses was proposed. Then this method was used in the calculation of a specific vehicle vibration system, and the displacement response curves and velocity response curves of the system in the cases of swerving, braking, accelerating and so on can be obtained. The calculation results agree well with the empirical data.

Construction of a Parametric Derivative-Free Iterative Method With Memory for Dynamic System Analysis

WANG Ting, TANG Shuo

2017, 38(12): 1342-1358. doi: 10.21656/1000-0887.370350

Abstract(978) PDF(466)

Abstract:
According to the usual practice that 2-step iterative methods with derivative are transformed into derivative-free schemes, a more general 2-step derivative-free iterative method was proposed. For this method the optimal order of convergence was ensured by the weight value. By means of the self-accelerating parameter and the Newton interpolation polynomial, the 2-parameter and 3-parameter iterative schemes with memory were obtained. Some of the existing 2- and 3-parameter iterative methods with memory were compared with the proposed method. The attraction domains of several schemes were presented, and the performances of several iterative schemes were compared.

Principles of Minimum Potential Action and Stationary Complementary Action With Dual and Triple Mixed Variables for Linear Elastodynamics of Finite Displacement Theory and the Application

FU Bao-lian

2017, 38(12): 1359-1376. doi: 10.21656/1000-0887.380005

Abstract(1188) PDF(507)

Abstract:
2 new concepts, potential action and complementary action, were first introduced into the variational principles for linear elastodynamics. On the basis of the concept of potential action, the principle of minimum action (Hamilton’s principle) was renamed to the principle of minimum potential action. In terms of the concept of complementary action, the principle of stationary complementary action was proposed for the first time. Next, the principles of minimum potential action and stationary complementary action with dual mixed variables of displacement and stress were derived in view of the boundary condition changes by means of the reciprocal theorem. And then, through the application of the relations between the strain energy density and the complementary energy density to the above 2 principles with dual mixed variables, the principles of potential action and complementary action with triple mixed variables of displacement, stress and strain were derived. Finally, the generalized principles of potential action and complementary action were given with the Lagrange multiplier method, in the meantime, the principle of minimum potential action with dual mixed variables of large deflection beams was applied to the calculation of a bending cantilever beam under forced vibration.

A Parallel Discontinuous Galerkin FEM for Solving Compressible Navier-Stokes Equations

MA Xin-rong, DUAN Zhi-jian, XIE Gong-nan, LIU San-yang

2017, 38(12): 1377-1388. doi: 10.21656/1000-0887.380077

Abstract(1400) PDF(629)

Abstract:
Based on unstructured grids, discontinuous Galerkin finite element methods (DGFEM) are very suited to realize high-order approximations of Navier-Stokes equations, but are rather demanding in computing resources. In order to improve the computational efficiency of the DGFEM, an efficient parallel algorithm on distributed-memory multicomputers coupled with the multigrid strategy based on the GMRES+LU-SGS procedure was presented here. The domain decomposition method was employed to handle meshes properly and make each processor maintain load balancing. Moreover, the LU-SGS and the local time stepping techniques were used to accelerate the convergence of the solution of Navier-Stokes equations. Numerical tests were conducted for viscid turbulence flow problems around the RAE2822 airfoil and over the M6 wing. The parallel acceleration is near to a linear convergence and up to the ideal solutions. The results indicate that the proposed parallel algorithm reduces computation time significantly and allocates memory reasonably with advantages of high acceleration and efficiency, and is very suited for coarse-grained scientific computation of MIMD models.

Dynamical Behavior Analysis of a Class of Complex-Valued Neural Networks With Time-Varying Delays

XU Xiao-hui, SONG Qian-kun, ZHANG Ji-ye, SHI Ji-zhong, ZHAO Ling.

2017, 38(12): 1389-1398. doi: 10.21656/1000-0887.380015

Abstract(1140) PDF(544)

Abstract:
The dynamical behavior of a class of complex-valued Cohen-Grossberg neural networks with time-varying delays was studied. It was supposed that the activation functions satisfied the Lipschitz condition and the amplification functions had only the lower bounds. The sufficient conditions ensuring the existence and the uniqueness of the equilibrium point of the system were acquired by means of the M matrix and the homeomorphic mapping. Furthermore, based on the vector Lyapunov function method and the inequality technique the criteria were obtained to judge the mode exponential stability of the equilibrium point of the system. The form of the obtained sufficient conditions is simple, and is easy to be verified in practice. The presented results generalize the existing ones. Finally a numerical example through simulation was given to verify the correctness and feasibility of the obtained results.

Characterizations of E-Borwein Properly Efficient Solutions

TANG Li-ping, YANG Yu-hong

2017, 38(12): 1399-1404. doi: 10.21656/1000-0887.380238

Abstract(1028) PDF(580)

Abstract:
Based on the idea of Borwein properly efficient solutions, a new concept of approximate Borwein properly efficient solutions for vector optimization problems was proposed via free disposal sets. Equivalent relations were established between the approximate Borwein properly efficient solutions and the E-Benson properly efficient solutions.

Generalized Solutions to Nonlinear Nonlocal Singularly Perturbed Parabolic Initial-Boundary Problems With Two Parameters

FENG Yi-hu, MO Jia-qi

2017, 38(12): 1405-1411. doi: 10.21656/1000-0887.380008

Abstract(1099) PDF(457)

Abstract:
A class of generalized parabolic equation singular perturbation problems were considered. Firstly, under suitable conditions, a class of nonlinear nonlocal generalized parabolic equation initial-boundary value problems with two parameters were raised. Secondly, the existence of solutions to corresponding problems was proved. Next, from the Fredholm integral equation, the outer solutions to the initial-boundary value problems were found, and the boundary and initial layer terms were structured by means of the theory of functional analysis, the stretched variables and the multiscale methods, respectively. Then the formal asymptotic expansion of the problem was obtained. Finally, according to the fixed point theorem, the uniform validity of the asymptotic expansion of generalized solutions to the corresponding nonlinear nonlocal initial-boundary value problems was proved.

An Epidemic Model With Dual Delays in View of Media Coverage

LIAO Shu, YANG Wei-ming

2017, 38(12): 1412-1424. doi: 10.21656/1000-0887.380025

Abstract(1277) PDF(715)

Abstract:
A multi-delay epidemic model in view of media coverage was established and analyzed. By means of the corresponding characteristic equation roots, the stability of the system was studied under 5 different time delay conditions, and the existence of the Hopf bifurcation was discussed. Furthermore, for a basic reproduction number greater than 1, the system’s uniform persistence was proved based on the persistence theory. At last, numerical simulations were conducted to verify the analytical predictions and evaluate the effects of media coverage and time delays on the control of emerging infectious diseases.