On the Global Geometric Structure of the Dynamics of the Elastic Pendulum

We approach the planar elastic pendulum as a singular perturbation of the pendulum to show that its dynamics are governed by global two-dimensional invariant manifolds of motion. One of the manifolds is nonlinear and carries purely slow periodic oscillations. The other one, on the other hand, is linear and carries purely fast radial oscillations. For sufficiently small coupling between the angular and radial degrees of freedom, both manifolds are global and orbitally stable up to energy levels exceeding that of the unstable equilibrium of the system. For fixed value of coupling, the fast invariant manifold bifurcates transversely to create unstable radial oscillations exhibiting energy transfer. Poincaré sections of iso-energetic manifolds reveal that only motions on and near a separatrix emanating from the unstable region of the fast invariant manifold exhibit energy transfer.     

On application of Newton’s law to mechanical systems with motion constraints

This work is devoted to deriving and investigating conditions for the correct application of Newton’s law to mechanical systems subjected to motion constraints. It utilizes some fundamental concepts of differential geometry and treats both holonomic and nonholonomic constraints. This approach is convenient since it permits one to view the motion of any dynamical system as a path of a point on a manifold. In particular, the main focus is on the establishment of appropriate conditions, so that the form of Newton’s law of motion remains invariant when imposing an additional set of motion constraints on a mechanical system. Based on this requirement, two conditions are derived, specifying the metric and the form of the connection on the new manifold, which results after enforcing the additional constraints. The latter is weaker than a similar condition obtained by imposing a metric compatibility condition holding on Riemannian manifolds and employed frequently in the literature. This is shown to have several practical implications. First, it provides a valuable freedom for selecting the connection on the manifold describing large rigid body rotation, so that the group properties of this manifold are preserved. Moreover, it is used to state clearly the conditions for expressing Newton’s law on the tangent space and not on the dual space of a manifold, which is the natural geometrical space for this. Finally, the Euler–Lagrange operator is examined and issues related to equations of motion for anholonomic and vakonomic systems are investigated and clarified further.     

Metric Description of Singular Defects in Isotropic Materials

Classical elasticity is concerned with bodies that can be modeled as smooth manifolds endowed with a reference metric that represents local equilibrium distances between neighboring material elements. The elastic energy associated with the configuration of a body in classical elasticity is the sum of local contributions that arise from a discrepancy between the actual metric and the reference metric. In contrast, the modeling of defects in solids has traditionally involved extra structure on the material manifold, notably torsion to quantify the density of dislocations and non-metricity to represent the density of point defects. We show that all the classical defects can be described within the framework of classical elasticity using tensor fields that only assume a metric structure. Specifically, bodies with singular defects can be viewed as affine manifolds; both disclinations and dislocations are captured by the monodromy that maps curves that surround the loci of the defects into affine transformations. Finally, we showthat two dimensional defectswith trivial monodromy are purely local in the sense that if we remove from the manifold a compact set that contains the locus of the defect, the punctured manifold can be isometrically embedded in a Euclidean space.     

Approximate inertial manifolds for reaction-diffusion equations in high space dimension

The concept of approximate inertial manifolds was introduced by Foiaset al. (1987) in the case of the two-dimensional Navier-Stokes equations. These manifolds are finite dimensional smooth manifolds such that the orbits enter a very thin neighborhood of the manifold after a transient time; this concept replaces the one of inertial manifold when either an inertial manifold does not exist or its existence is not known. Our aim in this paper is to prove that approximate inertial manifolds exist for reaction-diffusion equations in high space dimension by opposition with exact inertial manifolds whose existence has only been proved in low dimension and for which nonexistence results have been obtained in space dimensionn=4.     

An invariant manifold approach for studying waves in a one-dimensional array of non-linear oscillators

We consider a one-dimensional linear spring-mass array coupled to a one-dimensional array of uncoupled pendula. The principal aim of this study is to investigate the non-linear dynamics of this large-scale system in the limit of weak non-linearities, i.e. when the (fast) non-linear pendulum effects are small compared to the underlying (slow) linear dynamics of the linear spring-mass chain. We approach the dynamics in the context of invariant manifolds of motion. In particular, we prove the existence of an invariant manifold containing the (predominantly) slow dynamics of the system, with the fast pendulum dynamics providing small perturbations to the motions on the invariant manifold. By restricting the motion on the slow invariant manifold and performing asymptotic analysis we prove that the non-linear large-scale system possesses propagation and attenuation zones (PZs and AZs) in the frequency domain, similarly to the corresponding zones of the linearized system. Inside PZs non-linear travelling wave solutions exist, whereas in AZs only attenuating waves are permissible.     

Application of the integral manifold method to the analysis of the spatial motion of a rigid body fixed to a cable

We analyze the spatial motion of a rigid body fixed to a cable about its center of mass when the orbital cable system is unrolling. The analysis is based on the integral manifold method, which permits separating the rigid body motion into the slow and fast components. The motion of the rigid body is studied in the case of slow variations in the cable tension force and under the action of various disturbances.We estimate the influence of the static and dynamic asymmetry of the rigid body on its spatial motion about the cable fixation point. An example of the analysis of the rigid body motion when the orbital cable system is unrolling is given for a special program of variations in the cable tension force. The conditions of applicability of the integral manifold method are analyzed.     

On contact interaction between a moving massive body and a linear elastic half space

We study a class of problems involving the motion of a linear elastic body in frictional contact with a linear elastic half space. The dynamic effects considered are the inertial properties of the body regarded as rigid. We study only those regimes of contact interaction for which the slip velocity with the body taken as absolutely rigid and the time rate of change of the elastic displacements of points of the body and the half space that are on the contact surface are of the same order of magnitude. This work generalizes previous work on similar problems in that we simultaneously consider inertia forces of the body and the convective term in the slip-velocity due to the rigid-body velocity of the slider/indentor. Thus regimes of contact interaction investigated include rolling/sliding and shift-torsion type. We propose a variational formulation of the following two problems: (a) finite contact area and shift-torsion type of contact kinematics, (b) local contact area and general kinematics at the contact surface. Results for an elastic cylinder contacting an elastic half-plane are also given.     

Tracking particles in flows near invariant manifolds via balance functions

Particle moving inside a fluid near, and interacting with, invariant manifolds is a common phenomenon in a wide variety of applications. One elementary question is whether we can determine once a particle has entered a neighbourhood of an invariant manifold, when it leaves again. Here we approach this problem mathematically by introducing balance functions, which relate the entry and exit points of a particle by an integral variational formula. We define, study, and compare different natural choices for balance functions and conclude that an efficient compromise is to employ normal infinitesimal Lyapunov exponents. We apply our results to two different model flows: a regularized solid-body rotational flow and the asymmetric Kuhlmann–Muldoon model developed in the context of liquid bridges. To test the balance function approach, we also compute the motion of a finite size particle in an incompressible liquid near a shear-stress interface (invariant wall), using fully resolved numerical simulation. In conclusion, our theoretically developed framework seems to be applicable to models as well as data to understand particle motion near invariant manifolds.     

Identifying invariant manifold using phase space warping and stochastic interrogation

Advances in the generalization of invariant manifolds to finite time, experimental (or observational) flows have stimulated many recent developments in the approximation of invariant manifolds and Lagrangian coherent structures. This paper explores the identification of invariant manifold like structures in experimental settings, where knowledge of a flow field is absent, but phase space trajectories can be experimentally measured. Several existing methods for the approximation of these structures modified for application when only unstructured trajectory data is available. We find the recently proposed method, based on the concept of phase space warping, to outperform other methods as data becomes limited and show it to extend the finite-time Lyapunov exponent method. This finding is based on a comparison of methods for various data quantities and in the presence of both measurement and dynamic noise.     

二维稳定流形的自适应推进算法

稳定和不稳定流形是研究动力系统全局特性的重要工具. 一般系统的稳定和不稳定流形的曲率在全局范围内会有明显变化,应根据流形曲率的变化采用不同尺寸的网格单元计算全局流形. 然而在现有二维流形算法中,流形网格单元的尺寸在全局范围内是统一的. 为持续有效地计算全局稳定流形,提高计算网格对流形曲率变化的适应性. 本文在偏微分方程算法的基础上提出一种二维稳定流形的自适应推进算法. 该算法的基本思想是根据稳定流形曲率的变化自适应地调整网格单元的尺寸. 该算法首先在系统的稳定特征子空间中确定稳定流形的一个初始估计,该初始估计的网格单元尺寸设置为初始大小. 然后根据稳定流形网格前沿的曲率特点自适应地产生新的备选网格单元,继而根据相切性条件更新备选点的坐标,并将距离平衡点最近的备选点接受为已知点,最后更新稳定流形网格的前沿并自适应地产生新的备选网格单元,通过这个迭代过程使流形网格自适应地向前推进. 本文算法通过引入流形单元尺寸自适应,成功实现了洛伦兹流形和类球面流形的计算,并与偏微分方程算法进行了对比,结果表明自适应推进算法的流形计算单元的尺寸可在全局范围内根据流形曲率自适应地调整. 利用自适应推进算法计算二维稳定流形,可实现稳定流形的自适应推进.      

Riemann–Cartan Geometry of Nonlinear Dislocation Mechanics

We present a geometric theory of nonlinear solids with distributed dislocations. In this theory the material manifold—where the body is stress free—is a Weitzenb?ck manifold, that is, a manifold with a flat affine connection with torsion but vanishing non-metricity. Torsion of the material manifold is identified with the dislocation density tensor of nonlinear dislocation mechanics. Using Cartan’s moving frames we construct the material manifold for several examples of bodies with distributed dislocations. We also present non-trivial examples of zero-stress dislocation distributions. More importantly, in this geometric framework we are able to calculate the residual stress fields, assuming that the nonlinear elastic body is incompressible. We derive the governing equations of nonlinear dislocation mechanics covariantly using balance of energy and its covariance.     

Cauchy's Theorem on Manifolds

A generalization of the Cauchy theory of forces and stresses to the geometry of differentiable manifolds is presented using the language of differential forms. Body forces and surface forces are defined in terms of the power densities they produce when acting on generalized velocity fields. The normal to the boundary is replaced by the tangent space equipped with the outer orientation induced by outward pointing vectors. Assuming that the dimension of the material manifold is m, stresses are modelled as m − 1 covector valued forms. Cauchy's formula is replaced by the restriction of the stress form to the tangent space of the boundary while the outer orientation of the tangent space is taken into account. The special cases of volume manifolds and Riemannian manifolds are discussed. This revised version was published online in August 2006 with corrections to the Cover Date.     

Augmented perpetual manifolds and perpetual mechanical systems-part II: theorem for dissipative mechanical systems behaving as perpetual machines of third kind

Perpetual points in mathematics defined recently, and their significance in nonlinear dynamics and their application in mechanical systems is currently ongoing research. The perpetual points significance relevant to mechanics so far is that they form the perpetual manifolds of rigid body motions of unforced mechanical systems, which lead to the definition of perpetual mechanical systems. The perpetual mechanical systems admit as perpetual points rigid body motions which are forming the perpetual manifolds. The concept of perpetual manifolds extended to the definition of augmented perpetual manifolds that an externally excited multi-degree of freedom mechanical system is moving as a rigid body, and may exhibit particle-wave motion. This article is complementary to the work done so far applied to natural perpetual dissipative mechanical systems with motion defined by the exact augmented perpetual manifolds, whereas the internal forces, and individual energies are examined, to understand further the mechanics of these systems while their motion is in the exact augmented perpetual manifolds. A theorem is proved stating that under conditions when the motion of a perpetual natural dissipative mechanical system is in the exact augmented perpetual manifolds, all the internal forces are zero, which is rather significant in the mechanics of these systems since the operation on augmented perpetual manifolds leads to zero internal degradation. Moreover, the theorem is stating that the potential energy is constant, and there is no dissipation of energy, therefore the process is internally isentropic, and there is no energy loss within the perpetual mechanical system. Also in this theorem is proved that the external work done is equal to the changes of the kinetic energy, therefore the motion in the exact augmented perpetual manifolds is driven only by the changes of the kinetic energy. This is also a significant outcome to understand the mechanics of perpetual mechanical systems while it is in particle-wave motion which is guided by kinetic energy changes. In the final statement of the theorem is stated and proved that the perpetual dissipative mechanical system can behave as a perpetual machine of third kind which is rather significant in mechanical engineering. Noting that the perpetual mechanical system apart of the augmented perpetual manifolds solutions is having other solutions too, e.g., in higher normal modes and in these solutions the theorem is not valid. The developed theory is applied in the only two possible configurations that a mechanical system can have. The first configuration is a perpetual mechanical system without any connection through structural elements with the environment. In the second configuration, the perpetual mechanical system is a subsystem, connected with structural elements with the environment. In both examples, the motion in the exact augmented perpetual manifolds is examined with the view of mechanics defined by the theorem, resulting in excellent agreement between theory and numerical simulations. The outcome of this article is significant in physics to understand the mechanics of the motion of perpetual mechanical systems in the exact augmented perpetual manifolds, which is described through the kinetic energy changes and this gives further insight into the mechanics of particle-wave motions. Also, in mechanical engineering the outcome of this article is significant, because it is shown that the motion of the perpetual mechanical systems in the exact augmented perpetual manifolds is the ultimate, in the sense that there are no internal forces which lead to degradation of the internal structural elements, and there is no energy loss due to dissipation.

     

Solar sail dynamics in the three-body problem: Homoclinic paths of points and orbits

In this paper we consider the orbital dynamics of a solar sail in the Earth-Sun circular restricted three-body problem. The equations of motion of the sail are given by a set of non-linear autonomous ordinary differential equations, which are non-conservative due to the non-central nature of the force on the sail. We consider first the equilibria and linearisation of the system, then examine the non-linear system paying particular attention to its periodic solutions and invariant manifolds. Interestingly, we find there are equilibria admitting homoclinic paths where the stable and unstable invariant manifolds are identical. What is more, we find that periodic orbits about these equilibria also admit homoclinic paths; in fact the entire unstable invariant manifold winds off the periodic orbit, only to wind back onto it in the future. This unexpected result shows that periodic orbits may inherit the homoclinic nature of the point about which they are described.     

Equilibrium configurations of the tethered three-body formation system and their nonlinear dynamics

This paper considers nonlinear dynamics of tethered three-body formation system with their centre of mass staying on a circular orbit around the Earth, and applies the theory of space manifold dynamics to deal with the nonlinear dynamical behaviors of the equilibrium configurations of the system. Compared with the classical circular restricted three body system, sixteen equilibrium configurations are obtained globally from the geometry of pseudo-potential energy surface, four of which were omitted in the previous research. The periodic Lyapunov orbits and their invariant manifolds near the hyperbolic equilibria are presented, and an iteration procedure for identifying Lyapunov orbit is proposed based on the differential correction algorithm. The non-transversal intersections between invariant manifolds are addressed to generate homoclinic and heteroclinic trajectories between the Lyapunov orbits. (3,3)and (2,1)-heteroclinic trajectories from the neighborhood of one collinear equilibrium to that of another one, and (3,6)and (2,1)-homoclinic trajectories from and to the neighborhood of the same equilibrium, are obtained based on the Poincaré mapping technique.     

Inertial Manifolds for the 3D Modified-Leray-$$\alpha $$ Model with Periodic Boundary Conditions

The existence of an inertial manifold for the modified Leray-\(\alpha \) model with periodic boundary conditions in three-dimensional space is proved by using the so-called spatial averaging principle. Moreover, an adaptation of the Perron method for constructing inertial manifolds in the particular case of zero spatial averaging is suggested.     

Structure of the attracting set of a piecewise linear Hénon mapping

Detailed structure of the attracting set of the piecewise linear Hénon mapping (x,y)→(1−a|x|+by,x) with a=8/5 and b=9/25 is described in this paper using the method of dual line mapping. Let A and B denote the fixed saddles in the first quadrant, and in the third quadrant, respectively. It is claimed that (1) the attracting set is the closure of the unstable manifold of saddle B, which includes the unstable manifold of A as its subset, and (2) the basin of attraction is the closure of the stable manifold of A, bounded by the stable manifold of B, which is in the limiting set of the stable manifold of A. Relations of the manifolds of the periodic saddles with the manifolds of the fixed point are given. Symbolic dynamics notations are adopted which renders possible the study of the dynamical behavior of every piece of the manifolds and of every homoclinic or heteroclinic point.     

Hamiltonian structure for a neutrally buoyant rigid body interacting with N vortex rings of arbitrary shape: the case of arbitrary smooth body shape

We present a (noncanonical) Hamiltonian model for the interaction of a neutrally buoyant, arbitrarily shaped smooth rigid body with N thin closed vortex filaments of arbitrary shape in an infinite ideal fluid in Euclidean three-space. The rings are modeled without cores and, as geometrical objects, viewed as N smooth closed curves in space. The velocity field associated with each ring in the absence of the body is given by the Biot–Savart law with the infinite self-induced velocity assumed to be regularized in some appropriate way. In the presence of the moving rigid body, the velocity field of each ring is modified by the addition of potential fields associated with the image vorticity and with the irrotational flow induced by the motion of the body. The equations of motion for this dynamically coupled body-rings model are obtained using conservation of linear and angular momenta. These equations are shown to possess a Hamiltonian structure when written on an appropriately defined Poisson product manifold equipped with a Poisson bracket which is the sum of the Lie–Poisson bracket from rigid body mechanics and the canonical bracket on the phase space of the vortex filaments. The Hamiltonian function is the total kinetic energy of the system with the self-induced kinetic energy regularized. The Hamiltonian structure is independent of the shape of the body, (and hence) the explicit form of the image field, and the method of regularization, provided the self-induced velocity and kinetic energy are regularized in way that satisfies certain reasonable consistency conditions.      

Non-linear normal modes,invariance, and modal dynamics approximations of non-linear systems

Non-linear systems are here tackled in a manner directly inherited from linear ones, that is, by using proper normal modes of motion. These are defined in terms of invariant manifolds in the system's phase space, on which the uncoupled system dynamics can be studied. Two different methodologies which were previously developed to derive the non-linear normal modes of continuous systems — one based on a purely continuous approach, and one based on a discretized approach to which the theory developed for discrete systems can be applied-are simultaneously applied to the same study case-an Euler-Bernoulli beam constrained by a non-linear spring-and compared as regards accuracy and reliability. Numerical simulations of pure non-linear modal motions are performed using these approaches, and compared to simulations of equations obtained by a classical projection onto the linear modes. The invariance properties of the non-linear normal modes are demonstrated, and it is also found that, for a pure non-linear modal motion, the invariant manifold approach achieves the same accuracy as that obtained using several linear normal modes, but with significantly reduced computational cost. This is mainly due to the possibility of obtaining high-order accuracy in the dynamics by solving only one non-linear ordinary differential equation.     

DYNAMIC ANALYSIS OF FLEXIBLE BODY WITH DEFINITE MOVING ATTITUTE

The nonlinear dynamic control equation of a flexible multi-body system with definite moving attitude is discussed.The motion of the aircraft in space is regarded as known and the influence of the flexible structural members in the aircraft on the motion and attitude of the aircraft is analyzed.By means of a hypothetical mode,the defor- mation of flexible members is regarded as composed of the line element vibration in the axial direction of rectangular coordinates in space.According to Kane’s method in dy- namics,a dynamic equation is established,which contains the structural stiffness matrix that represents the elastic deformation and the geometric stiffness matrix that represents the nonlinear deformation of the deformed body.Through simplification the dynamic equation of the influence of the planar flexible body with a windsurfboard structure on the spacecraft motion is obtained.The numerical solution for this kind of equation can be realized by a computer.