Transient three-dimensional contact problems: mortar method. Mixed methods and conserving integration

The present work deals with the development of an energy-momentum conserving method to unilateral contact constraints and is a direct continuation of a previous work (Hesch and Betsch in Comput Mech 2011, doi:) dealing with the NTS method. In this work, we introduce the mortar method and a newly developed segmentation process for the consistent integration of the contact interface. For the application of the energy-momentum approach to mortar constraints, we extend an approach based on a mixed formulation to the segment definition of the mortar constraints. The enhanced numerical stability of the newly proposed discretization method will be shown in several examples.     

Dynamics of multibody systems in planar motion in a central gravitational field

Multibody systems in planar motion are modelled as two or more rigid components that are connected and can move relative to each other. The dynamics of such multibody systems in planar motion in a central gravitational force field is analysed. The equations of motion of the system include the equations for the orbital motion of the bodies, the orientation (attitude) of the assembly, and the relative orientation (shape) of the bodies with respect to each other. Dynamic coupling between these degrees of freedom gives rise to complex dynamical systems that are usually not integrable. Relative equilibria, corresponding to circular orbits of the multibody system, are obtained. The free dynamics has a symmetry due to a cyclic coordinate. Routh reduction is carried out to eliminate this coordinate and obtain the reduced dynamics. The stability of the relative equilibria is analysed using the Routh stability criterion when it is applicable; an expansion of the Hamiltonian in normal form is used otherwise. We apply the general results to a multibody system consisting of two hinged planar bodies, each modelled as a rigid massless link with a point mass at one end with their other ends connected by a hinge joint. We obtain the relative equilibria of this model, and carry out a stability analysis for the relative equilibria. Numerical simulations using a symplectic integrator are carried out for perturbations to these relative equilibria, to confirm their stability properties.     

预防接种情况下非线性饱和接触率SIR流行病模型动力学性态研究

研究了一类预防接种情况下具有一般非线性饱和接触率SIR流行病模型动力学性态。得到决定疾病灭绝和持续生存的基本再生数。当基本再生数小于等于1时，仅存在无病平衡态：当基本再生数大于1时，除存在无病平衡态外，还存在惟一的地方病平衡态。利用Hurwitz判据、Liapunov-Lasalle不变集原理得到各个平衡态局部渐近稳定及无病平衡态全局渐近稳定的条件。特别地。当传染率为双线性时，无病平衡态及地方病平衡态全局渐近稳定。     

Relative equilibria of tethered satellite systems and their stability for very stiff tethers

We study the existence and stability of the relative equilibria of systems of two satellites joined by a tether. Since tethers used in practice are very stiff we consider a stiff tether as a perturbation of an inextensible tether. We show that the equations for relative equilibria and the stability conditions are continuous as stiffness approaches infinity and limit on the equations and conditions relevant to an inextensible tether. We obtain a numerical bifurcation diagram for a class of relative equilibria in the case of an inextensible tether.     

Transient 3d contact problems—NTS method: mixed methods and conserving integration

The present work deals with a new formulation for transient large deformation contact problems. It is well known, that one-step implicit time integration schemes for highly non-linear systems fail to conserve the total energy of the system. To deal with this drawback, a mixed method is newly proposed in conjunction with the concept of a discrete gradient. In particular, we reformulate the well known and widely-used node-to-segment methods and establish an energy-momentum scheme. The advocated approach ensures robustness and enhanced numerical stability, demonstrated in several three-dimensional applications of the proposed algorithm.     

Motions of a rimless spoked wheel: a simple three-dimensional system with impacts

This paper discusses the mechanics of a rigid rimless spoked wheel, or regular polygon, 'rolling' downhill. By 'rolling', we mean motions in which the wheel pivots on one 'support' spoke until another spoke collides with the ground, followed by transfer of support to that spoke, and so on. We carry out three-dimensional (3D) numerical and analytical stability studies of steady motions of this system. At any fixed, large enough slope, the system has a one-parameter family of stable steady rolling motions. We find analytic approximations for the minimum required slope at a given heading for stable rolling in three dimensions, for the case of many spokes and small slope. The rimless wheel shares some qualitative features with passive-dynamic walking machines; it is a passive 3D system with intermittent impacts and periodic motions. In terms of complexity, it lies between one-dimensional impact oscillators and 3D walking machines. In contrast to a rolling disk on a flat surface which has steady rolling motions that are only neutrally stable at best, the rimless wheel can have asymptotic stability. In the limit as the number of spokes approaches infinity, the behavior of the rimless wheel approaches that of a rolling disk in an averaged sense and becomes neutrally stable. Also, in this averaged sense, the piecewise holonomic system (rimless wheel) approaches a non-holonomic system (disk).     

Hard- and soft-modeling approaches for resolution of complex formation of Co2+ and Ni2+ with glycine

Principal component analysis (PCA) was used to obtain information about the number of components in the complex formation equilibria of Co(2+) and Ni(2+) with glycine (Gly). In order to obtain a clearer insight into these complex formation systems, multivariate curve resolution-alternating least squares (MCR-ALS) was used. Using MCR-ALS as a soft-modeling method, well-defined concentration and spectral profiles were obtained under unimodality, non-negativity, and closure constraints. Based on the obtained results, an equilibrium model was proposed and subsequently, a hard-modeling method was used to resolve the complex formation equilibria completely. Due to the presence of multiple equilibria, the resolution of such systems is very difficult. The Co-Gly system was best described by a model consisting of M(GlyH), M(Gly), M(Gly)(2), M(Gly)(2)H, and M(Gly)(3) (M = Co(2+)) with the overall stability constants determined to be 7.10 ± 0.011, 5.14 ± 0.006, 9.28 ± 0.009, 13.75 ± 0.016, and 11.10 ± 0.010, respectively. On the other hand, the system of Ni-Gly was best fitted by a model containing M(GlyH), M(Gly), M(Gly)(2), M(Gly)(3), and M(Gly)(2)(OH) (M = Ni(2+)) with overall stability constants of 10.95 ± 0.04, 6.41 ± 0.03, 11.31 ± 0.03, 15.39 ± 0.06, and 14.32 ± 0.02, respectively.     

具有线性免疫响应的HIV-TB共感染模型

研究HIV与TB发生共感染时的免疫系统—抗原模型。确定了各类平衡点存在的条件以及它们的稳定性，证明了局部稳定性蕴涵全局稳定性。     

An analysis of the equilibria of neural networks with linear interconnections

In this paper, we analyse the equilibria of neural networks which consist of a set of sigmoid nonlinearities with linear interconnections,without assuming that the interconnections are symmetric or that there are no self-interactions. By eliminating these assumptions, we are able to study the effects of imperfect implementation on the behaviour of Hopfield networks. If one views the neural network as evolving on the openn-dimensional hypercubeH = (0, 1) ⁿ , we have the following conclusions as the neural characteristics become steeper and steeper: (i) There is at most one equilibrium in any compact subset ofH, and under mild assumptions this equilibrium is unstable. In fact, the dimension of the stable manifold of this equilibrium is the same as the number of eigenvalues of the interconnection matrix with negative real parts. (ii) There might be some equilibria in the faces ofH, and under mild conditions these are always unstable. Moreover, it is easy to compute the dimension of the stable manifold of each such equilibrium. (iii) A systematic procedure is given for determining which corners of the hypercubeH contain equilibria, and it is shown that all equilibria in the corners ofH are asymptotically stable.     

A finite deformation method for discrete modeling: particle rotation and parameter calibration

We present a finite deformation method for 3-D discrete element modeling. In this method particle rotation is explicitly represented using quaternion and a complete set of interactions is permitted between two bonded particles, i.e., normal and tangent forces, rolling and torsional torques. Relative rotation between two particles is decomposed into two sequence-independent rotations, such that an overall torsional and rolling angle can be distinguished and torques caused by relative rotations are uniquely determined. Forces and torques are calculated in a finite deformation fashion, rather than incrementally. Compared with the incremental methods our algorithm is numerically more stable while it is consistent with the non-commutativity of finite rotations. We study the macroscopic elastic properties of a regularly arranged 2-D and 3-D lattice. Using a micro-to-macro approach based on the existence of a homogeneous displacement field, we study the problem of how to choose the particle-scale parameters (normal, tangent, rolling and torsional stiffness) given the macroscopic elastic parameters and geometry of lattice arrangement. The method is validated by reproducing the wing crack propagation and the fracture patterns under uniaxial compression. This study will provide a theoretical basis for the calibration of the DEM parameters required in engineering applications.     

Bio-Inspired Computational Methods for the Polio Virus Epidemic Model

In 2021, most of the developing countries are fighting polio, and parents are concerned with the disabling of their children. Poliovirus transmits from person to person, which can infect the spinal cord, and paralyzes the parts of the body within a matter of hours. According to the World Health Organization (WHO), 18 million currently healthy people could have been paralyzed by the virus during 1988–2020. Almost all countries but Pakistan, Afghanistan, and a few more have been declared polio-free. The mathematical modeling of poliovirus is studied in the population by categorizing it as susceptible individuals (S), exposed individuals (E), infected individuals (I), and recovered individuals (R). In this study, we study the fundamental properties such as positivity and boundedness of the model. We also rigorously study the model’s stability and equilibria with or without poliovirus. For numerical study, we design the Euler, Runge–Kutta, and nonstandard finite difference method. However, the standard techniques are time-dependent and fail to present the results for an extended period. The nonstandard finite difference method works well to study disease dynamics for a long time without any constraints. Finally, the results of different methods are compared to prove their effectiveness.     

具有三物种的食饵-捕食反应扩散时滞系统的稳定性与行波解

本文研究了在有界区域上带有Neumann边界条件的反应扩散三物种食饵-捕食时滞系统.利用特征值方法和Lyapunov函数找到了该系统平衡点稳定的充分条件,该条件说明时滞限制了稳定性.稳定性中的主要一个结论是当食饵和捕食者间的种内竞争大于种间竞争时正平衡点是全局渐近稳定的.进一步,通过构建上下解证明了当波速相对大时该系统具有连接零平衡点和正平衡点的行波解.     

Role of Friction in Cold Ring Rolling

1. Introduction Cold ring rolling is a main technology used to manu- facture various precise seamless ring shape parts. It has been increasingly used in many industrial fields such as bearing, machine, automobile, petrochemicals, aeronau- tics, astronautics and atomic energy because of its many technical superiorities such as considerable saving in en- ergy and material cost, high quality, high efficiency, and low noise, etc. To research and develop advanced precise cold ring rolling technolog…     

Stable supervisory control of flexible manufacturing systems with fixed supply and demand rates

In this article, we develop a method for computing a stable supervisory control for a class of manufacturing systems (consisting of reconfigurable processors and buffers with limited capacities) with fixed supply and demand rates. It is shown that, with this method, we are able to (i) decide constructively whether a stable supervisory control exists for a given manufacturing system, and (ii) compute the stable supervisory control if it exists. Furthermore, the stable supervisory control thus obtained is complete in the sense that it contains all 'safe' sequences (of possibly concurrent buffer and processor activities) which guarantee the stability of the system. An example is presented to illustrate the proposed method. Implications and computational complexity associated with this method are discussed.     

相关约束Euler-Lagrange方程解的存在唯一性

讨论了存在相关约束的Euler-Lagrange方程解的存在唯一性条件，给出了一个存在唯一性定理。定理中不再要求Фx满秩，从而推广了以前的结论。文中还讨论了定理的应用背景。     

The bipenalty method for arbitrary multipoint constraints

In finite element (FE) analysis, traditional penalty methods impose constraints by adding virtual stiffness to the FE system. In dynamics, this can decrease the critical time step of the system when conditionally stable time integration schemes are used by introducing spurious modes with high eigenfrequencies. Recent studies have shown that using mass penalties alongside traditional stiffness penalties can mitigate this effect for systems with a one single‐point constraint. In the present work, we extend this finding to include systems with an arbitrary set of multipoint constraints. By analysing the generalised eigenvalue problem, we show that the values of spurious eigenfrequencies may be controlled by the choice of stiffness and mass penalty parameters. The method is demonstrated using numerical examples, including a one‐dimensional contact–impact formulation and a two‐dimensional crack propagation analysis. The results show that constraint imposition using the bipenalty method can be employed such that the critical time step of an analysis is unaffected, whereas also displaying superiority over the mass penalty method in terms of accuracy and versatility. Copyright © 2012 John Wiley & Sons, Ltd.     

On functional periodicity as the basis for long-term stability of engineered and natural systems and its relationship to physical laws

In this research note, it is shown that for long-term stability, both engineered systems and natural systems must be either at a stable equilibrium state or have a functional periodicity. Examples are listed in a table. Classical physics such as Newtonian mechanics and thermodynamics are based on the existence of equilibrium states, whereas modern physics such as quantum mechanics and superstring theory suppose the existence of a functional periodicity. The particle/wave duality of matters that forms the basis of quantum mechanics is consistent with the stability argument presented in this paper.     

Oscillations, SE(2)-snakes and motion control: a study of the Roller Racer

This paper is concerned with the problem of motion generation via cyclic variations in selected degrees of freedom (usually referred to as shape variables) in mechanical systems subject to non-holonomic constraints (here the classical system of a disc rolling without sliding on a flat surface). In earlier work, we identified an interesting class of such problems arising in the setting of Lie groups, and investigated these under a hypothesis on constraints, that naturally led to a purely kinematic approach. In the present work, the hypothesis on constraints does not hold, and as a consequence, it is necessary to take into account certain dynamical phenomena. Specifically we concern ourselves with the group SE (2) of rigid motions in the plane and a concrete mechanical realization dubbed the 2-node, 1-module SE (2)snake. In a restricted version, it is also known as the Roller Racer (a patented ride/ toy). Based on the work of Bloch, Krishnaprasad, Marsden and Murray, one recognizes in the example of this paper a balance law called the momentum equation, which is a direct consequence of the interaction of the SE (2)-symmetry of the problem with the constraints. The systematic use of this type of balance law results in certain structures in the example of this paper. We exploit these structures to demonstrate that the single shape freedom in this problem can be cyclically varied to produce a rich variety of motions of the SE (2)-snake. In their study of the snakeboard, a patented modification of the skateboard that also admits the group SE (2) as a symmetry group, Lewis, Ostrowski, Burdick and Murray exploited the same type of balance law as that discussed here to generate motions. A key difference, however, is that, in the present paper, we have only one control variable and thus controllability considerations become somewhat more delicate. In the present paper, we give a self-contained treatment of the geometry, mechanics and motion control of the Roller Racer.     

An improved method for the hot strip mill production scheduling problem

In most research on the hot strip mill production scheduling problem (HSMPSP) arising in the steel industry, it is accepted that a schedule with lower penalty caused by jumps of width, hardness, and gauge will result in lower roller wear, so it is regarded as a better schedule. However, based on the analysis of production processes, it is realised that rolling each coil also cause roller wear. In order to assessing the roller wear associated with production scheduling more precisely, it is necessary to consider it as another factor besides those jumps, especially when complicated constraints are involved. In this paper, an improved method is proposed to quantify the expected wear of the rollers done by those jumps and rolling processes. Then the HSMPSP whose objective is to maximise the total length of all scheduled coils is formulated as a team orienteering problem with time windows and additional production constraints. A heuristic method combining an improved Ant Colony Extended algorithm with local search procedures dedicated to HSMPSP is developed. Finally, computational results on instances generated based on production data from an integrated steel mill in China indicate that the proposed algorithm is a promising solution specific to HSMPSP.     

Stable numerical integration of dynamical systems subject to equality state-space constraints

Numerical simulation of constrained dynamical systems is known to exhibit stability problems even when the unconstrained system can be simulated in a stable manner. We show that not the constraints themselves, but the transformation of the continuous set of equations to a discrete set of equations is the true source of the stability problem. A new theory is presented that allows for stable numerical integration of constrained dynamical systems. The derived numerical methods are robust with respect to errors in the initial conditions and stable with respect to errors made during the integration process. As a consequence, perturbations in the initial conditions are allowed. The new theory is extended to the case of constrained mechanical systems. Some numerical results obtained when implementing the numerical method here developed are shown.