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Table of Content

    20 July 2016, Volume 52 Issue 4
    Two Kinds of Gradient Representations for Nielsen Equations
    MEI Fengxiang,WU Huibin,LI Yanmin
    2016, 52(4):  588-591.  DOI: 10.13209/j.0479-8023.2016.078
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    The two kinds of generalized gradient systems are proposed and the characteristics of the two systems are studied. The conditions under which the Nielsen equations can be considered as one of the two generalized gradient systems are obtained. The characteristics of the generalized gradient systems can be used to study the stability of solution of the Nielsen equations. Some examples are given to illustrate the application of the results.

    The Affection of Symmetry Reduction to the Numerical Integration for Holonomic System
    LIU Shixing, XING Yan, LIU Chang, GUO Yongxin
    2016, 52(4):  592-596.  DOI: 10.13209/j.0479-8023.2016.077
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    This paper researches the effection of symmetry reduction to the numerical integration for holonomic system. Through numerical experiment, there is not essential effection for numerical integrator when system is reduced to lower dimension. But under the reduced system, it can effectively decrease the difficulty of writing program and the time of computation. So for the complicated dynamical system, it should be firstly reduced by symmetry methods and obtain the dynamical system with less degree of freedom, then the dynamical nature of system can be researched under the reduced system.

    Lagrange Equation Applied to Continuum Mechanics
    FENG Xiaojiu, LIANG Lifu
    2016, 52(4):  597-607.  DOI: 10.13209/j.0479-8023.2016.076
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    How to apply the Lagrange equation to the continuous medium mechanics has been a theoretical issue of academic circles. Using variational derivative concepts and operational rules, the properties of variational derivative in Lagrange equation are studied. The Lagrange equation is applied to linear elastic dynamics and nonlinear elastic dynamics, and some corresponding numerical examples are given. The result shows that it is a feasible way to solve the problem of the application of Lagrange equation to the mechanics of continuous media by using the variational integral calculus.

    Stability Analysis of Rigid-Liquid-Flexible Coupling Dynamics of Spacecraft Systems by Using the Energy-Casimir Method
    YUE Baozeng, YAN Yulong
    2016, 52(4):  608-618.  DOI: 10.13209/j.0479-8023.2016.095
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    The stability of liquid filled spacecraft with flexible appendage was researched by using energy-Casimir method. Liquid sloshing dynamics was simplified by spring-mass model, and flexible appendage was modeled as a linear shearing beam. Rigid-liquid-flexible coupling dynamics of spacecraft was built. The energy function and the Casimir function were derived by analyzing the energy function of a rigid body, liquid sloshing and a flexible appendage. The nonlinear stability condition of coupled spacecraft system was derived by computing the first and second variation of energy-Casimir function. The stable and unstable regions of the parameter space were given in the final section with numerical computation. Related results show that the inertia matrix, the length of shearing beam, the spacecraft spinning rate, and the filled ratio of liquid fuel tank have strong influence on the stability of coupled spacecraft system.

    Trajectory Tracking Control of 3D Rigid Body Pendulum Attitude Based on Legendre Pseudospectral Method
    GE Xinsheng, ZHU Ning
    2016, 52(4):  619-626.  DOI: 10.13209/j.0479-8023.2016.071
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    The optimal control of the attitude motion of 3D rigid pendulum with initial disturbance is investigated. Combined with the characteristics of the attitude and angular velocity of the 3D rigid pendulum, the closed-loop feedback attitude tracking controller is designed for the external disturbance. Firstly, 3D rigid pendulum attitude trajectory is designed for open loop by use of Legendre pseudospectra method. Then the system’s motion equation is linearized, and the difference between the attitude reference trajectory and actual trajectory motion in 3D rigid pendulum is considered as control variable. Attitude tracking problem is converted to linear time-varying systems attitude regulation problem. Finally, the closed-loop control based on the Legendre pseudospectral method is simulated and analyzed for the optimal control of 3D rigid pendulum, and simulations show that the effectiveness in the case of initial disturbance.

    Modeling and Simulation of Space Robot with Unilateral Contact Based on Complementary Problem
    HU Tianjian, WANG Tianshu, LI Junfeng, QIAN Weiping
    2016, 52(4):  627-633.  DOI: 10.13209/j.0479-8023.2016.072
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    Traditionally, the contact between the end-effector and the target is modeled as a parallel spring-damp model, which requires a time-consumed tuning of values of stiffness and damping factor and an extra force sensor fixed on the end-effector. The above drawbacks inspire the application of complementary problem to uniformly describe the unilateral contact for space robot. A dynamical equation of the space robot with unilateral contact is derived, and a numerical method is developed utilizing the Lemke algorithm. By numerical calculation of a planar 3 degree-of-freedom (DOF) manipulator fastened on a 3 DOF floating base, the effectiveness of the dynamical model is verified.

    Dynamic Model of Power Converters with the Kirchhoff’s Current Constraints
    XIE Yu, FU Hao, CHEN Benyong, FU Jingli
    2016, 52(4):  634-642.  DOI: 10.13209/j.0479-8023.2016.075
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    Based on constraint Euler-Lagrange equations, a general dynamic model is presented for power converters with multiple or single switches. From the research on the energy of power converters, the generalized charges and generalized currents coordinates of dynamic elements are chosen, switching function is introduced, and a set of differential algebraic equations with constraint can be obtained based on Euler-Lagrange equations with the Kirchhoff’s current constraints. The inductor currents and the capacitor voltages are chosen as the state variables, and then the state equations of the converters can be got ultimately. The proposed method is applied to model an ?uk converter circuit with an ideal switch and a three phase PWM rectifier with multiple switches. From the modeling process, it can be seen that this method is unified, clear in physical meaning and strong commonality. Finally, with the obtained state equations, the working process of ?uk converter is simulated in MATLAB, the simulation results coincide with the operation of the power converter, and it verify the effectiveness of the proposed method, which has higher application value to model the more complex power converters.

    Noether Symmetries and Their Inverse Problems of Nonholonomic Systems with Fractional Derivatives
    FU Jingli, FU Liping
    2016, 52(4):  643-652.  DOI: 10.13209/j.0479-8023.2016.074
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    Noether symmetries and their inverse problems of the nonholonomic systems with the fractional derivatives are studied. Based on the quasi-invariance of fractional Hamilton action under the infinitesimal transformations without the time and the general transcoordinates of time-reparametrization, the fractional Noether theorems are established for the nonholonomic constraint systems. Further, the fractional Noether inverse problems are firstly presented for the nonholonomic systems. An example is designed to illustrate the applications of the results.

    Stability and Bifurcation for a Type of Non-autonomous Generalized Birkhoffian Systems
    CAO Qiupeng, CHEN Xiangwei
    2016, 52(4):  653-657.  DOI: 10.13209/j.0479-8023.2016.067
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    Bifurcation for a type of non-autonomous generalized Birkhoffian systems is studied. Gradient representations for this type of non-autonomous generalized Birkhoffian systems are given. The stability of equilibrium point of these systems is discussed by the characteristic of the gradient system. Further the systems which contain some parameter are studied. The stability and the number of equilibrium point will change along with the change of the parameter to produce the bifurcation phenomenon.

    Noether Symmetry and Conserved Quantity for Fractional Birkhoffian Systems in Terms of Riesz Derivatives
    ZHANG Yi, ZHOU Yan
    2016, 52(4):  658-668.  DOI: 10.13209/j.0479-8023.2016.068
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    The Noether symmetry and the conserved quantity for a fractional Birkhoffian system in terms of Riesz fractional derivatives are studied. The fractional Pfaff variational problems are presented and the fractional Birkhoff’s equations are established within Riesz-Riemann-Liouville fractional derivatives and Riesz-Caputo fractional derivatives, respectively. Based on the invariance of the Pfaff action under the infinitesimal transformations, the Noether theorems for the fractional Birkhoffian system are given. The proof of the Noether theorem is done in two steps: first, the Noether theorem under a special one-parameter group of infinitesimal transformations without transforming the time is proved; second, by using a technique of time-reparameterization, the Noether theorem in its general form is obtained. Two examples are given to illustrate the application of the results.

    Precise Exponential Integrator and Its Application in Dynamics of Spacecraft Formation Flying
    DENG Zichen, LI Qingjun
    2016, 52(4):  669-675.  DOI: 10.13209/j.0479-8023.2016.069
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    The dynamic equations of spacecraft formation flying are weakly nonlinear equations since the distance between spacecrafts is quite small compared with the orbital radius of the spacecrafts. To solve weakly nonlinear equations effectively, a precise exponential integrator (PEI) was proposed. Precise integration method (PIM) was applied to calculate exponential function in the formulas of exponential integrators (EI). Firstly, PEI was validated by solving a weakly nonlinear equation compared with Runge-Kutta method. Secondly, the dynamic equations of spacecraft formation flying were obtained through Lagrange equations, and then the equations were tansfered into semi-linear form. Ultimately, PEI and Runge-Kutta method were comparatively used to solve these equations. Through numerical analysis, PEI gave higher precision of the dynamic equations of spacecraft formation flying, indicating that PEI can be applied to other weakly nonlinear problems as well.

    Dynamics Analogy of Thin Elastic Rod and Schrödinger Particle Wave
    WANG Peng, XUE Yun
    2016, 52(4):  676-680.  DOI: 10.13209/j.0479-8023.2016.073
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    The Schrödinger analogy of thin elatic rod is studied. Compared with the Kirchhoff dynamics analogy, the Schrödinger analogy is proposed. By the new analogy, the Kirchhoff equation of elastic rod can be written as curvature equation which is similar to nonlinear Schrödinger equation. Thus, the Jacobi elliptic function solution of Schrödinger equation can be taken into elastic rod equation. The space configurations of the elastic rod described by the solution are given. Schrödinger analogy reveals the relations between the quantum state of wave function and the geometry configuration of elastic rod, and gives a new way to solve the Kirchhoff equation.

    Approximate Lie Symmetries, Approximate Noether Symmetries and Approximate Mei Symmetries of Typical Perturbed Mechanical System
    LOU Zhimei, WANG Yuanbin, XIE Zhikun
    2016, 52(4):  681-686.  DOI: 10.13209/j.0479-8023.2016.080
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    Three methods, which are approximate Lie symmetry method, approximate Noether symmetry method and approximate Mei symmetry method, are adopted to study the first order approximate symmetries and approximate conserved quantities of a typical perturbed mechanical system. Six identical first order approximate symmetries and approximate conserved quantities of the typical perturbed mechanical system are obtained by approximate Lie symmetry method and approximate Noether symmetry method, but only five of them can be obtained by approximate Mei symmetry method.

    Discontinuous Quantities of Kirchhoff Elastic Rod Expressed by Singularity Function
    XUE Yun, WENG Dewei
    2016, 52(4):  687-691.  DOI: 10.13209/j.0479-8023.2016.086
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    Kirchhoff equation of thin elastic rod statics and dynamics must be written in piecewise at section with discontinuous or nonsmooth quantities such as external forces, mass geometry and physical parameters, which leads to inconvenience to the numerical calculation. According to singular function method in calculation of beam bending deformation, these discontinuous or nonsmooth quantities of the rod are expressed by singular function and become continues quantities alone centerline of the rod. Numerical simulations of equilibrium configuration of the rod acted by lateral concentrated load are made by means of Mathematics software. Results explain that introducing singular function to express discontinuous or nonsmooth quantities can avoid complicated calculation and improve the computational efficiency.

    Hamel’s Field Variational Integrator of Geometrically Exact Beam
    WANG Liang, AN Zhipeng, SHI Donghua
    2016, 52(4):  692-698.  DOI: 10.13209/j.0479-8023.2016.079
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    This paper develops a structure-preserving variational integrator for geometrically exact beam in Hamel’s field formalism. A simulation illustrates that the derived algorithm preserves energy, momentum and geometry structure.

    Study on the Vibration Characteristic and Axial-Compressive Stability of the Beam with Simple and Flexible Supports
    XIAO Shifu, CHEN Xueqian, LIU Xinen
    2016, 52(4):  699-707.  DOI: 10.13209/j.0479-8023.2016.085
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    For the beam with simple and flexible supports, a nonlinear dynamic model is established by applying the flexible multi-body dynamic theory. The model can describe both the global rotation and the relative deformation of the beam. The modal and axial-compressive stability of the system are investigated by using analytical and numerical method, and the effect of the movable support stiffness are obtained. The results show that there is great influence on the lower-order frequencies, vibration shape and the buckling mode of the system while the movable support stiffness is smaller than the beam. In this case, they behave to the global rotation characteristic and the lower-order vibration shape in the floating coordinate system is also different to the classical beam, which is affected by the global rotation. However, when the movable support is very stiff, the influence on the lowerorder frequencies, vibration shape and the buckling mode of the system are extremely slight and the uncertainty of the movable support stiffness only lightly affects the higher-order frequencies and vibration shape of the system. The results are important to the constraint boundary design of the beam and the application of the flexible multibody dynamic theory.

    Dynamical Modeling of Multi-Rigid-Body System Based on Gauss Principle of Least Constraint in Generalized Coordinates
    YAO Wenli
    2016, 52(4):  708-712.  DOI: 10.13209/j.0479-8023.2016.087
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     By variational Gauss principle in the form of kinetic energy and explicit express, the meaning of items in Gauss constraints is confirmed. The optimization model of dynamics of multi-rigid-body system in the form of Descartes generalized coordinates is established based on Gauss principle of least constraint in generalized coordinates. The method to write the Gauss constraints in other coordinate system according to the above model is studied. The dynamical problem of multi-rigid-body system can be turned into a problem for a minimum, and provided by the relation between the Descartes generalized coordinates and other coordinates system, the Gauss constraint in this kind of coordinates system is easy to be obtained. The modeling method is more convenient and is of universality. The dynamical models are established for planar motion and fixed axis rotation by Descartes generalized coordinate and Lagrange coordinate, and the validity of this method is proved.

    On Nonholonomic Constraints about the Pure Rolling of Point Contact
    ZHAO Zhen, LIU Caishan, LU Jiandong
    2016, 52(4):  713-716.  DOI: 10.13209/j.0479-8023.2016.084
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    Nonholomonic constraints are involved for 3D point-contact problems. The virtual displacements restricted by the constraints are usually given by Appell-Chetaev’s rule. It has not been very clear of the geometric meaning in configuration space for Appell-Chetaev’s rule of nonholonomic constraints. The authors investigate point contact with pure rolling by two rigid bodies in a multibody system to discover its geometric sense. First, the sufficient and necessary conditions of point contact are given. A ball-plane system is presented to demonstrate the validation of the conditions by deducing the system’s obvious contact constraint originating from them. Two geometric restrictions for pure rolling are obtained by the nonholonomic constraints of pure rolling as well as the contact constraint in velocity level. It proves that the virtual displacements of the two restrictions are same as those of the constraints of point contact with pure rolling obtained by Appell-Chetaev’s rule. So, it is thought that the constraints of pure rolling are constructed by the two geometric restrictions.

    Nonsmooth Dynamical Simulation of Astronautics Separation Device in Guided Stage
    ZHANG Hongjian, ZHUANG Fangfang, QU Zhanlong, JI Baofeng, LIU Guanri, HUANG Cheng
    2016, 52(4):  717-721.  DOI: 10.13209/j.0479-8023.2016.088
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     A new numerical method based on LZB is proposed for nonsmooth dynamics on astronautics separation device in guided stage. Using Euler-Lagrange equations and the LZB method, the general calculating framework describing the whole dynamical motion including contact, collision with friction is established. Compared with FEM simulation time, numerical simulation implies that this method is valid.

    Some Dynamical Problems in Biological Neural Systems
    LU Qishao
    2016, 52(4):  722-726.  DOI: 10.13209/j.0479-8023.2016.083
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    Biological neural systems are extremely complex super-network structures with multiple time scales and spatial levels and carry out the biological functions of sensitivity, cognition , motion control and so on. There are ubiquitous dynamical phenomena in biological neural systems, and some basic problems, including spatiotemporal dynamics, dynamical modeling of networks and intelligence activities, which are surveyed briefly in the implicit connection with mechanics. Prospects of the development of neurodynamics are also concerned.

    Advances in Energetics and Conserved Quantities of Axially Moving Structures
    CHEN Liqun
    2016, 52(4):  727-731.  DOI: 10.13209/j.0479-8023.2016.081
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     Progresses in investigations on energetics and conserved quantities are summarized. The key issue in energetics, the time-rate of the total mechanical energy of axially moving structures, is determined for linear transverse vibration, nonlinear transverse vibration, and coupled planer vibration. The result shows that the mechanical energy is not a constant. Conserved quantities are constructed so that the quantities remain unchanged during those vibrations. The conserved quantities can be used to prove stability of the straight equilibrium configurations and to check the numerical algorithms. Some promising topics are suggested for future research.

    Some Progressions in Study of the Inverse Problem of the Calculus of Variations
    DING Guangtao
    2016, 52(4):  732-740.  DOI: 10.13209/j.0479-8023.2016.070
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    Several aspects of the inverse problem of the calculus of variations and a short overview of the domestic development of this problem are presented. The main topic is a new method to construct Lagrangians from the first integral of mechanical system, and the equivalent Lagrangians and family of Lagrangians can be obtained by the new method. Some examples are given to illustrate the theoretical significance and the application value of this method. Finally, it is pointed out that great attention should be paid to the study of the inverse problem of the calculus of variations.

    Advances in Modeling of Clearance Joints and Dynamics of Mechanical Systems with Clearances
    YAN Shaoze, XIANG Wuweikai, HUANG Tieqiu
    2016, 52(4):  741-755.  DOI: 10.13209/j.0479-8023.2016.094
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    Recent developments in modeling of clearance joints and dynamics of mechanical systems with clearances are reviewed. Different modeling approaches for clearance joints are summarized firstly, which comprise the massless link approach, the non-smooth dynamics approach, the contact force approach and the 3D revolute joint approach. Then, applications of these approaches in the study of the nonlinear dynamics, and performance and reliability evaluation of the mechanical systems with clearances are systematically reviewed. Finally, the key problems and priorities which need to be further studied are proposed, including the modeling of clearace joints considering stick-slip phenomenon, contact surface profile and conformal contact condition, the dynamic analysis and kinematic accuracy evaluation of mechanical systems with both clearances and uncertainties, and the design for clearance joints.

    The Fundamental Equations in Analytical Mechanics for Nonholonomic Systems
    LIU Caishan
    2016, 52(4):  756-766.  DOI: 10.13209/j.0479-8023.2016.082
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    Analytical mechanics is established based on d’Almbert-Lagrange Principle, Gauss principle, Jourdian principle and Hamilton principle, to deal with the dynamics of mechanical systems subject to holonomic or nonholonomic constraints. The governing equation of the systems are derived either by introducing Lagrange’s multipliers to adjoin with the limitation equations for the virtual displacements, or by directly eliminating the constraint equations to achieve minimal formulations. The author presents a survey for the history of analytical mechanics, and explains some basic concepts, such as virtual displacement, ideal constraint, and the correlations between the Lagrange multipliers and the real constraint forces.