On the coupling of non-linear normal modes

The phenomenon of internal resonance is known as the exchange of energy between the modes and the existence of coupled-mode response under a single-mode excitation. This phenomenon is observed whenever a non-linear normal mode loses its stability, called the modal coupling. The details of modal coupling are formulated in the free vibrations of two-degree-of-freedom systems, and compared with internal resonance. The theory is based on the structural change in Poincaré map due to the stability change of normal modes. It is shown that every change in stability of normal modes gives rise to a pitchfork or a period-doubling bifurcation. The functional form is derived to compute the coupled modes by the method of harmonic balance. Examples are given to describe the procedure of stability analysis of non-linear normal modes, to compute the coupled modes, and then to demonstrate that results of internal resonances can be derived by model coupling. Other examples are given to demonstrate that the results of some modal couplings cannot be obtained by internal resonances.     

Stability of Singular Periodic Motions in a Vibro-Impact Oscillator

A single-degree-of-freedom vibro-impact oscillator is considered. Forsome values of parameters, a non-differentiable fixed point of thePoincaré map exists: a local expansion of the Poincaré map around such apoint is given, including a square root term on the impact side. Fromthis approximate map, the stability of the fixed point can beinvestigated, and it is shown that the periodic solution is stable whenthe Floquet multipliers are real.     

On the invariant theory of nonholonomic systems with constraints of non-Chetaev type

This paper deals with the theory of the differential invariant and integral invariant for a nonholonomic system with constraints of non-Chetaev type. It gives the restricted conditions of virtual displacement in velocity space for nonholonomic constraints of non-Chetaev type and extends the Jourdain principle and the canonical equation for the system. It presents and proves generalized Noether theorem, and gives generalized energy integral and cyclic integral for the system. Finally, the basic integral variants for the system are extended and the integral invariant of Poincaré-Cartan and the universal integral invariant of Poincaré for the system are obtained.     

On the closure problem for Darcy's law

In a previous derivation of Darcy's law, the closure problem was presented in terms of an integro-differential equation for a second-order tensor. In this paper, we show that the closure problem can be transformed to a set of Stokes-like equations and we compare solutions of these equations with experimental data. The computational advantages of the transformed closure problem are considerable.Roman Letters A interfacial area of the- interface contained within the macroscopic system, m² - A _e area of entrances and exits for the-phase contained within the macroscopic system, m² - A interfacial area of the- interface contained within the averaging volume, m² - A _e area of entrances and exits for the-phase contained within the averaging volume, m² - B second-order tensor used to respresent the velocity deviation - b vector used to represent the pressure deviation, m^–1 - C second-order tensor related to the permeability tensor, m^–2 - D second-order tensor used to represent the velocity deviation, m² - d vector used to represent the pressure deviation, m - g gravity vector, m/s² - I unit tensor - K C ^–1,–D, Darcy's law permeability tensor, m² - L characteristic length scale for volume averaged quantities, m - characteristic length scale for the-phase, m - l _i i=1, 2, 3, lattice vectors, m - n unit normal vector pointing from the-phase toward the-phase - n _e outwardly directed unit normal vector at the entrances and exits of the-phase - p pressure in the-phase, N/m ² - p intrinsic phase average pressure, N/m² - p –p , spatial deviation of the pressure in the-phase, N/m² - r position vector locating points in the-phase, m - r ₀ radius of the averaging volume, m - t time, s - v velocity vector in the-phase, m/s - v intrinsic phase average velocity in the-phase, m/s - v phase average or Darcy velocity in the \-phase, m/s - v –v , spatial deviation of the velocity in the-phase m/s - V averaging volume, m³ - V volume of the-phase contained in the averaging volume, m³ Greek Letters V /V volume fraction of the-phase - mass density of the-phase, kg/m³ - viscosity of the-phase, Nt/m²     

The Conley Index Over the Circle

We study the Conley index over a base in the case when the base is the circle. Such an index arises in a natural way when the considered flow admits a Poincaré section. In that case the fiberwise pointed spaces over the circle generated by index pairs are semibundles, i.e., admit a special structure similar to locally trivial bundles. We define a homotopy invariant of semibundles, the monodromy class. We use the monodromy class to prove that the Conley index of the Poincaré map may be expressed in terms of the Conley index over the circle.     

The analysis of the spectrum of Lyapunov exponents in a two-degree-of-freedom vibro-impact system

In the study of dynamical systems, the spectrum of Lyapunov exponents has been shown to be an efficient tool for analyzing periodic motions and chaos. So far, different calculating methods of Lyapunov exponents have been proposed. Recently, a new method using local mappings was given to compute the Lyapunov exponents in non-smooth dynamical systems. By the help of this method and the coordinates transformation proposed in this paper, we investigate a two-degree-of-freedom vibro-impact system with two components. For this concrete model, we construct the local mappings and the Poincaré mapping which are used to describe the algorithm for calculating the spectrum of Lyapunov exponents. The spectra of Lyapunov exponents for periodic motions and chaos are computed by the presented method. Moreover, the largest Lyapunov exponents are calculated in a large parameter range for the studied system. Numerical simulations show the success of the improved method in a kind of two-degree-of-freedom vibro-impact systems.     

On the dynamics of cracks in three dimensions

We introduce a three-dimensional dynamic crack propagation law, which is derived from Hamilton's principle. The result is an extension of a previous one obtained, corresponding to the two-dimensional case. As a matter of fact, in an orthogonal plane to the crack front, the geometric condition to be satisfied over the path is the same as in two dimensions. The third mode enters only through the energy release rate. The fact that the physics of the problem is locally two dimensional is a consequence of the virtual motions allowed in the set of admissible crack configurations.     

On the energy conservation during the active deformation in molecular dynamics simulations

In this paper, we examined the energy conservation for the current schemes of applying active deformation in molecular dynamics (MD) simulations. Specifically, two methods are examined. One is scaling the dimension of the simulation box and the atom positions via an affine transformation, suitable for the periodic system. The other is moving the rigid walls that interact with the atoms in the system, suitable for the non-periodic system. Based on the calculation of the external work and the internal energy change, we present that the atom velocities also need to be updated in the first deformation method; otherwise the energy conservation cannot be satisfied. The classic updating scheme is examined, in which any atom crossing the periodic boundary experiences a velocity delta that is equal to the velocity difference between the opposite boundaries. In addition, a new scheme which scales the velocities of all the atoms according to the strain increment is proposed, which is more efficient and realistic than the classic scheme. It is also demonstrated that the Virial stress instead of its interaction part is the correct stress definition that corresponds to Cauchy stress in the continuum mechanics.     

Periodic Solutions of Symmetric Perturbations of Gravitational Potentials

It is shown that for certain symmetric perturbations of gravitational potentials in the space, which admit two first integrals of motion, a circular solution of the unperturbed system with inclination different from 0 and π gives rise to a periodic solution of the reduced dynamics which is defined in the quotient space of the action by the subgroup that fixes the symmetry axis. In the planar case, if we assume that the system admits a first integral of motion which is also symmetric with respect to the origin, then it is shown that each circular solution of the unperturbed problem gives rise to a periodic solution of the perturbed system.     

On general form of Navier-Stokes equations and implicit factored scheme

A general weak conservative form of Navier-Stokes equations expressed with respect to non-orthogonal Curvilinear coordinates and with primitive variables was obtained by using tensor analysis technique, where the contravariant and covariant velocity components were employed. Compared with the current coordinate transformation method, the established equations are concise and forthright, and they are more convenient to be used for solving problems in body-fitted curvilinear coordinate system. An implicit factored scheme for solving the equations is presented with detailed discussions in this paper. For n-dimensional flow the algorithm requires n-steps and for each step only a block tridiagonal matrix equation needs to be solved. It avoids inverting the matrix for large systems of equations and enhances the speed of arithmetic. In this study, the Beam- Warming’s implicit factored schceme is extended and developed in non-orthogonal curvilinear coordinate system.     

Observing Symmetry-Breaking and Chaos in the Normal Form Network

The complex dynamical behaviors of neural networks may deducenew information processing methodology. In this paper, the dynamics of anormal form network with Z ₂ symmetry is studied. Thesecondary Hopf bifurcation of the network is discussed and a two-torusis observed. Examining the phase-locking motions of the two-torus, wepresent the regularity of symmetry-breaking occurring in the system. Ifthe ratio of the two frequencies of the codimension-two Hopf bifurcationis represented by an irreducible fraction, symmetry-breaking occurs wheneither the numerator or the denominator of the fraction is even. Chaoticattractors may be created with sigmoid nonlinearities added to theright-hand side of the normal form equations. The trajectory andsecond-order Poincaré maps of the chaotic attractor are given.The chaotic attractor looks like a butterfly on some of the second-orderPoincaré maps. This is a marvelous example for chaos mimickingnature.     

Dynamics analysis of the Minuteman cover drive

Recursive matrix relations in kinematics and dynamics analysis of a 1-DOF compound planetary gear train are established in the paper. The mechanism of the Minuteman cover drive is a system with four moving links and three gear pairs controlled by one electric motor. Knowing the rotation motion of the output link, the inverse dynamic problem is solved using an approach based on the principle of virtual work, but it has been verified the results in the framework of the Lagrange equations. Finally, some simulation graphs for the input and output angles of rotation, the torque and the power of the actuator are obtained.     

Stable periodic solutions for the hypercycle system

We consider the hypercycle system of ODEs, which models the concentration of a set of polynucleotides in a flow reactor. Under general conditions, we prove the omega-limit set of any orbit is either an equilibrium or a periodic orbit. The existence of an orbitally asymptotic stable periodic orbit is shown for a broad class of such systems.     

Nonlinear Dynamic Stability of Normal and Arthritic Greyhounds

In this paper the dynamic stability of greyhound gait was analyzed within the framework of nonlinear dynamics theory. A video based motion analysis system was utilized to obtain the kinematic data of the hindlimb joints of greyhounds at trot. Phase plane portraits and first return maps for the coxofemoral, femorotibial, and the tarsal joints were calculated from averaged kinematic data for each dog. The analysis was based on the assumption that the steady state dog locomotion could be represented as a nonlinear periodic system. Using the Floquet theory, the dynamic stability of gait was quantified by computing the characteristic multipliers from experimental data. A stability index based on multipliers was used for comparison between normal and arthritic dogs. Phase plane portraits and first return maps of the dogs with transient synovitis were compared with the averaged portraits of the normal dogs. It was observed that the coxofemoral angle exhibited the maximum difference while the femorotibial and the tarsal joints showed little or no difference from those of the normals. Comparison of the Floquet multipliers indicated that the dogs with synovitis had a less stable gait than that of the normal dogs.     

Coulomb law in generalized differential form in problems of dynamics of rigid bodies with combined kinematics

We suggest an exact integral model of sliding and spinning friction constructed under the assumption that the Coulomb law in generalized differential form holds for the surface element in the interior of the contact spot.     

Basic features of the time finite element approach for dynamics

Very general weak forms may be developed for dynamic systems, the most general being analogous to a Hu-Washizu three-field formulation, thus paralleling well-established weak methods of solid mechanics. In this work two different formulations are developed: a pure displacement formulation and a two-field mixed formulation. With the objective of developing a thorough understanding of the peculiar features of finite elements in time, the relevant methodologies associated with this approach for dynamics are extensively discussed. After having laid the theoretical bases, the finite element approximation and the linearization of the resulting forms are developed, together with a method for the treatment of holonomic and nonholonomic constraints, thus widening the horizons of applicability over the vast world of multibody system dynamics. With the purpose of enlightening on the peculiar numerical behavior of the different approaches, simple but meaningful examples are illustrated. To this aim, significant parallels with elastostatics are emphasized. Paper presented at the ‘International Technical Specialists' Meeting on Rotorcraft Basic Research’, March 25–27, 1991, Georgia Institute of Technology, Atlanta, Georgia, USA.     

Persistence of Poincaré mappings in functional differential equations (with application to structural stability of complicated behavior)

A theorem on the dependence of Poincaré mappings for different functional differential equations (FDEs) on the right-hand side of the equation is proved. Together with recent results on hyperbolic sets for noninvertible mappings, this is used to describe how Poincaré mappings and their complicated behavior in the neighborhood of a transversal homoclinic orbit persist under FDE perturbations of the equation. The method is shown to apply to three example equations, where Poincaré mappings with such behavior are known to exist.     

On the developments of Darcy's law to include inertial and slip effects

The empirical Darcy law describing flow in porous media, whose convincing theoretical justification was proposed almost 130 years after its original publication in 1856, has however been extended to account for particular flow conditions. This article reviews historical developments aimed at including inertial and slip effects (respectively, when the Reynolds and Knudsen numbers are not exceedingly small compared to unity). Despite the early empirical extensions to include inertia and slip effects, it is striking to observe that clear formal derivations of physical models to account for these effects were reported only recently.     

Dynamics of a kicked oscillator with a delay in its parametric feedback loop: An analytical study

This paper deals with the behavior of a parametrically-forced analytically-solvable oscillator in the presence of a delay in the feedback loop. In spite of the delay, it is here shown that, when the system is in the strong relaxation regime, a one-dimensional approximation to the Poincaré (stroboscopic) map may be constructed. The influence of both the delay and the amplitude of the impulses on the dynamics is explained in terms of the geometric properties of either the isochrone portrait or the PTC. Interest has recently grown on this kind of excitation because in real-life feedback systems a delay is always present.     

Geometry of mesoscopic dynamics and thermodynamics

Contact geometry provides a natural setting for classical thermo-dynamics. In this paper we use it to derive the structure of mesoscopic dynamics (GENERIC) expressing its compatibility with thermodynamics. In the second part we derive kinematics (Poisson brackets) of a large family of mesoscopic state variables.