One-dimensional Mechanical Systems Research Articles

Euler top and freedom in supersymmetrization of one-dimensional mechanics

In this paper we formulate the Euler top as a system on the complex projective plane, playing the role of phase space, i.e., as a one-dimensional mechanical system. Then we suggest the supersymmetrization scheme of the generic one-dimensional systems with positive Hamiltonian which yields á priori integrable family of N=2k supersymmetric Hamiltonians parameterized by N/2 arbitrary real functions.

A new tunable elastic metamaterial structure for manipulating band gaps/wave propagation

A one-dimensional mechanical lattice system with local resonators is proposed as an elastic metamaterial model, which shows negative mass and negative modulus under specific frequency ranges. The proposed representative units, consisting of accurately arranged rigid components, can generate controllable translational resonance and achieve negative mass and negative modulus by adjusting the local structural parameters. A shape memory polymer is adopted as a spring component, whose Young’s modulus is obviously affected by temperature, and the proposed metamaterials’ tunable ability is achieved by adjusting temperature. The effect of the shape memory polymer’s stiffness variation on the band gaps is investigated detailedly, and the special phenomenon of intersecting dispersion curves is discussed, which can be designed and controlled by adjusting temperature. The dispersion relationship of the continuum metamaterial model affected by temperature is obtained, which shows great tunable ability to manipulate wave propagation.

Jacobi Multipliers in Integrability and the Inverse Problem of Mechanics

We review the general theory of the Jacobi last multipliers in geometric terms and then apply the theory to different problems in integrability and the inverse problem for one-dimensional mechanical systems. Within this unified framework, we derive the explicit form of a Lagrangian obtained by several authors for a given dynamical system in terms of known constants of the motion via a Jacobi multiplier for both autonomous and nonautonomous systems, and some examples are used to illustrate the general theory. Finally, some geometric results on Jacobi multipliers and their use in the study of Hojman symmetry are given.

Disorder-induced topological phase transition in a one-dimensional mechanical system

We numerically investigate the topological phase transition induced purely by disorder in a spring-mass chain. We employ two types of disorders---chiral and random types---to explore the interplay between topology and disorder. By tracking the evolution of real-space topological invariants, we obtain the topological phase diagrams and demonstrate the bilateral capacity of disorder to drive topological transitions, from topologically nontrivial to trivial and vice versa. The corresponding transition is accompanied by the realization of a mechanical topological Anderson insulator. The findings from this study hint that the combination of disorder and topology can serve as an efficient control knob to manipulate the transfer of mechanical energy.

High performance model for buckling of functionally graded sandwich beams using a new semi-analytical method

In this paper, a new semi-analytical approach based on the scaled boundary finite element method (SBFEM) is proposed to solve buckling problem of functionally gradient material (FGM) sandwich beams. Material properties of each individual layer are assumed to be continuously graded along the thickness according to a power law function with respect to the volume fractions. Based on the layer-wise theory, the two-dimensional constitutive model is directly performed in the proposed formulations without any kinematic assumptions of plate theory. The buckling governing equations of FGM sandwich beams based on the SBFEM are derived using the weighted residual method and solved by the means of the eigenvalue analysis. The advantage of the present approach is that only the longitudinal dimension of the beam structure needs to be discretized using one-dimensional higher-order spectral element so that it can be dealt with as a one-dimensional mechanical system while maintaining the analytical characteristics in the thickness direction, which possesses a key feature to make structure modelling more effective and accuracy. To evaluate the validity of proposed formulations, a series of numerical examples involving the element convergence and parametric analysis are carried out and the results are compared with existing solutions available in open literature. The numerical studies confirm the accuracy and adaptability of the presented method for the buckling analysis of FGM sandwich plates.

Synchronization of impacting mechanical systems with a single constraint

This paper addresses the synchronization problem of mechanical systems subjected to a single geometric unilateral constraint. The impacts of the individual systems, induced by the unilateral constraint, generally do not coincide even if the solutions are arbitrarily ‘close’ to each other. The mismatch in the impact time instants demands a careful choice of the distance function to allow for an intuitively correct comparison of the discontinuous solutions resulting from the impacts. We propose a distance function induced by the quotient metric, which is based on an equivalence relation using the impact map. The distance function obtained in this way is continuous in time when evaluated along jumping solutions.The property of maximal monotonicity, which is fulfilled by most commonly used impact laws, is used to significantly reduce the complexity of the distance function. Based on the simplified distance function, a Lyapunov function is constructed to investigate the synchronization problem for two identical one-dimensional mechanical systems. Sufficient conditions for the uncoupled individual systems are provided under which local synchronization is guaranteed. Furthermore, we present an interaction law which ensures global synchronization, also in the presence of grazing trajectories and accumulation points (Zeno behavior). The results are illustrated using numerical examples of a 1-DOF mechanical impact oscillator which serves as stepping stone in the direction of more general systems.

Stable coherent states

We analyze the stability under time evolution of complexifier coherent states (CCS) in one-dimensional mechanical systems. A system of coherent states is called stable if it evolves into another coherent state. It turns out that a system can only poses stable CCS if the classical evolution of the variable for a given complexifier C depends only on z itself and not on its complex conjugate. This condition is very restrictive in general so that only few systems exist that obey this condition. However, it is possible to access a wider class of models that in principle may allow for stable coherent states associated to certain regions in the phase space by introducing action-angle coordinates.

Controlled synchronization of mechanical systems with a unilateral constraint

Abstract: This paper addresses the controlled synchronization problem of mechanical systems subjected to a geometric unilateral constraint as well as the design of a switching coupling law to obtain synchronization. To define the synchronization problem, we propose a distance function induced by the quotient metric, which is based on an equivalence relation using the impact map. A Lyapunov function is constructed to investigate the synchronization problem for two identical one-dimensional mechanical systems. Sufficient conditions for the individual systems and their controlled interaction are provided under which synchronization can be ensured. We present a (coupling) control law which ensures global synchronization, also in the presence of grazing trajectories and accumulation points (Zeno behavior). The results are illustrated using a numerical example.

On quantum and relativistic mechanical analogues in mean-field spin models

Conceptual analogies among statistical mechanics and classical or quantum mechanics have often appeared in the literature. For classical two-body mean-field models, such an analogy is based on the identification between the free energy of Curie–Weiss-type magnetic models and the Hamilton–Jacobi action for a one-dimensional mechanical system. Similarly, the partition function plays the role of the wave function in quantum mechanics and satisfies the heat equation that plays, in this context, the role of the Schrödinger equation. We show that this identification can be remarkably extended to include a wider family of magnetic models that are classified by normal forms of suitable real algebraic dispersion curves . In all these cases, the model turns out to be completely solvable as the free energy as well as the order parameter are obtained as solutions of an integrable nonlinear PDE of Hamilton–Jacobi type. We observe that the mechanical analogue of these models can be viewed as the relativistic analogue of the Curie–Weiss model and this helps to clarify the connection between generalized self-averaging in statistical thermodynamics and the semiclassical dynamics of viscous conservation laws.

Phase transitions in one-dimensional mechanical models of thermodynamics and the physics of the Hall bar system

We study one-dimensional mechanical systems by using a thermodynamic approach. In this framework we discuss the appearance of distinct thermodynamic phases corresponding to different types of trajectories for a charge carrier moving in a Hall bar, i.e. a channel patterned in a two-dimensional electron gas subject to a transverse magnetic field. In this device different kinds of orbits are allowed, cyclotron orbits, skipping orbits (when the particle collides with one channel wall), and traversing trajectories (when the particle collides with both channel walls). The passage from one type of orbits to the other can be characterized as a phase transition; so, in this system two different phase transitions can be identified.

NEW ESTIMATES FOR EVANS' VARIATIONAL APPROACH TO WEAK KAM THEORY

We consider a recent approximate variational principle for weak KAM theory proposed by Evans. As in the case of classical integrability, for one-dimensional mechanical Hamiltonian systems all the computations can be carried out explicitly. In this setting, we illustrate the geometric content of the theory and prove new lower bounds for the estimates related to its dynamic interpretation. These estimates also extend to the case of n degrees of freedom.

On Normal Modes of Vibrating 1-D Mechanical Systems with Discontinuous Properties

A fundamental solution for the problem describing the longitudinal vibration of the rod with viscous damping has been found. A theoretical approach for determination of normal modes of vibrating one-dimensional mechanical systems with 3-stepwise constant properties has been put forward. The transcendental equation has been derived from the solution in the Laplace image space and its roots are natural frequencies. The equation relates thickness of layers and elastic moduli; it makes possible experimental measurement of thin film thickness and elastic moduli. The physical interpretation of the natural frequencies by introducing an “effective” speed of the wave propagation has been proposed.

Generalized Maupertuis’ principle with applications

We give a rigorous proof of the equivalence of Mane’s supercritical potential and the minimal action with respect to an associated Jacobi-Finsler metric. As a consequence, we give an explicit representation of the weak KAM solutions of one-dimensional mechanical systems without the quadratic assumption on the kinetic energy term of the Hamiltonians, and a criterion of the integrability result for such a system of arbitrary degree of freedom by the regularity assumption on Mather’s α-function is discussed.

Fundamental frequencies of mechanical systems with N-piecewise constant properties

This paper presents an analytical derivation of a set of transcendental equations for the normal modes of a one-dimensional mechanical system composed of an arbitrary number of components and provides experimental validation for the case of a two-component system consisting of a fiber partially submerged in liquid. The transcendental equations relate thickness of layers and their elastic moduli and allow possible experimental measurement of thin film thickness and elastic moduli. An effective model of the wave propagation in such systems gives an a- priori qualitative explanation of the natural frequencies change and allows one to predict them quantitatively.

A Remark on Rational Isochronous Potentials

We consider the rational potentials of the one-dimensional mechanical systems, which have a family of periodic solutions with the same period (isochronous potentials). We prove that up to a shift and adding a constant all such potentials have the form or .

On the Helmholtz-Boltzmann thermodynamics of mechanical systems

In an 1884 paper, Boltzmann showed that for a one-dimensional mechanical system with a convex potential energy \(\varphi\) that depends on a parameter V, it is possible to define a temperature T, pressure p, and entropy S that satisfy the Gibbs relation TdS = de + p dV, where \(e = \mathrm{kin. en.} + \varphi\). In the paper we review the extension of the Boltzmann construction to general natural mechanical systems endowed with a fibration over the (possibly multidimensional) space of macroscopic parameters. Moreover, for certain discrete mechanical systems with non-convex potential energies, which are used as models for phase transitions in solids, we compare the thermodynamic pressure p = p(e,V) introduced above with the quasi-static, macroscopic, stress-strain relation.

On Transverse Vibration of Aircraft Landing Gear

Transverse elastic-frictional vibrations of landing gear with respect to a hull of infinite mass during an airplane high-speed motion on the runway are studied theoretically within the framework of the nonlinear shift hypothesis. The authors have derived approximate amplitude-phase equations that describe vibration in one-dimensional mechanical systems with arbitrary analytical velocity-dependent friction characteristics.

On the geometric approach to the motion of inertial mechanical systems

According to the principle of least action, the spatially periodic motions of one-dimensional mechanical systems with no external forces are described in the Lagrangian formalism by geodesics on a manifold-configuration space, the group D of smooth orientation-preserving diffeomorphisms of the circle. The periodic inviscid Burgers equation is the geodesic equation on D with the L 2 right-invariant metric. However, the exponential map for this right-invariant metric is not a C 1 local diffeomorphism and the geometric structure is therefore deficient. On the other hand, the geodesic equation on D for the H 1 rightinvariant metric is also a re-expression of a model in mathematical physics. We show that in this case the exponential map is a C 1 local diffeomorphism and that if two diffeomorphisms are sufficiently close on D ,t heycan be joined by a unique length-minimizing geodesic—a state of the system is transformed to another nearby state by going through a uniquely determined flow that minimizes the energy. We also analyse for both metrics the breakdown of the geodesic flow.

Dual transformations in one-dimensional classical and quantum mechanics

Dual transformations in two-dimensional classical and quantum mechanical systems have been widely studied using conformal mapping techniques but one-dimensional systems have been largely ignored. In this paper we study dual transformations in one-dimensional mechanical systems, both classical and quantum mechanical, using some previously developed methods. A number of examples, mostly involving periodic motion or bound states, are presented. Dual transformations provide interesting connections between hitherto unconnected problems.

THE WZW MODEL AS A DYNAMICAL SYSTEM ON AFFINE LIE GROUPS

Working directly on affine Lie groups, we construct several new formulations of the WZW model. In one formulation WZW is expressed as a one-dimensional mechanical system whose variables are coordinates on the affine Lie group. When written in terms of the affine group element, this formulation exhibits a two-dimensional WZW term. In another formulation WZW is written as a two-dimensional field theory, with a three-dimensional WZW term, whose fields are coordinates on the affine group. On the basis of these equivalent formulations, we develop a translation dictionary in which the new formulations on the affine Lie group are understood as mode formulations of the conventional WZW formulation on the Lie group. Using this dictionary, we also express WZW as a three-dimensional field theory on the Lie group with a four-dimensional WZW term.