Class Of Mechanical Systems Research Articles

Complex nonlinear dynamics of a multidirectional energy harvester with hybrid transduction

Abstract Mechanical energy harvesting has increasing scientific and technological interests due to novel energetic challenges. A critical issue in classical cantilever-based mechanical energy harvesting systems is the lack of multidirectional energy conversion capabilities and, due to that, deviations from the excitation source can drastically reduce their performance. This limitation has led to the development of energy harvesters with attached pendula, serving as a direction coupling mechanism. Nevertheless, the pendulum structure itself can act as an energy absorber, drastically reducing the harvester performance in certain scenarios. In order to overcome this issue, a hybrid transduction multidirectional pendulum-based energy harvester is proposed integrating a piezoelectric element, to capture energy from the principal direction, and an electromagnetic transducer, to harness rotational energy from the pendulum. This paper presents an in-depth analysis of the hybrid multidirectional pendulum-based energy harvester using a nonlinear dynamics perspective to evaluate the energy harvesting performance. A reduced-order model is proposed to represent the essential characteristics of such systems. A parametric analysis using a nonlinear dynamics perspective is carried out to map the system dynamics and performance. The emergence of complex and rich dynamics is observed, including chaos and hyperchaos. Results reveal the most and least effective combinations of structural parameters in terms of energy conversion. Additionally, the dynamical responses and patterns associated with high performance are identified. These responses are often characterized by a blend of irregular complex behaviors, coupled with a mix of oscillatory and rotational patterns of motion, resulting in wider bandwidth systems.

Assembly-induced nonuniform deformation prediction and control method for circular flange connectors with spacers in an optical lens support structure

Widely utilized in engineering, circular flange connections are typically secured by multiple bolts, but always result in nonuniform deformations that significantly impact the system shape and position accuracies. Numerous scholars have studied the deformation characteristics under various structures, external loads, bolt preloads, and tightening sequences in classical rough mechanical systems, but there is a lack of research regarding a high-precision optical system. In optical systems, nonuniform deformation of the support structure can be transmitted to the lens, causing astigmatic aberrations and affecting imaging accuracy. In the optical lens support structure as a circular flange connector, spacers need to be introduced to adjust the gap between two parts for controlling the optical path. These spacers tend to be more flexible than the support structures, thereby significantly affecting assembly-induced nonuniform deformations. However, the influence of spacers has been overlooked in previous works and thus needs to be investigated. In this paper, the effects of spacer parameters and bolt tightening sequences on the assembly-induced nonuniform deformation are investigated by the finite element method, which is verified experimentally, and an empirical formula is proposed to predict the nonuniform assembly-induced deformation based on the FEM results. According to the prediction results, a method for controlling the assembly-induced nonuniform deformation by optimizing the bolt preload distribution is proposed. The influence rules can guide the optimization of spacers and the bolt tightening process. The prediction and control method can effectively predict and reduce the assembly-induced nonuniform deformation of the circular flange with a spacer, reducing the aberration caused by the assembly and improving the accuracy of high-precision optical systems.

On the role of geometric phase in the dynamics of elastic waveguides.

The geometric phase provides important mathematical insights to understand the fundamental nature and evolution of the dynamic response in a wide spectrum of systems ranging from quantum to classical mechanics. While the concept of geometric phase, which is an additional phase factor occurring in dynamical systems, holds the same meaning across different fields of application, its use and interpretation can acquire important nuances specific to the system of interest. In recent years, the development of quantum topological materials and its extension to classical mechanical systems have renewed the interest in the concept of geometric phase. This review revisits the concept of geometric phase and discusses, by means of either established or original results, its critical role in the design and dynamic behaviour of elastic waveguides. Concepts of differential geometry and topology are put forward to provide a theoretical understanding of the geometric phase and its connection to the physical properties of the system. Then, the concept of geometric phase is applied to different types of elastic waveguides to explain how either topologically trivial or non-trivial behaviour can emerge based on the geometric features of the waveguide. This article is part of the theme issue 'Current developments in elastic and acoustic metamaterials science (Part 2)'.

Variational Methods for Evolution

Variational principles for evolutionary systems arise in many settings, both in those describing the physical world and in man-made algorithms for data science and optimization tasks. Variational principles are available for Hamiltonian systems in classical mechanics, gradient flows for dissipative systems, as well as in time-incremental minimization techniques for more general evolutionary problems. Additional challenges arise via the interplay of two or more functionals (e.g. a free energy and a dissipation potential), thus encompassing a large variety of applications in the modeling of materials and fluids, in biology, and in multi-agent systems. Variational principles and associated evolutions are also at the core of the modern approaches to machine learning tasks, since many of them are posed as minimizing an objective functional that models the problem. The discrete and random nature of these problems and the need for accurate computation in high dimension present a set of challenges that require new mathematical insights. Variational methods for evolution allow for the usage of the rich toolbox provided by the calculus of variations, metric-space geometry, partial differential equations, and other branches of applied analysis. The variational methods for evolution have seen a rapid growth over the last two decades. This workshop continued the successful line of meetings (2011, 2014, 2017, and 2020), while evolving and branching into new directions. We have brought together a wide scope of mathematical researchers from calculus of variations, partial differential equations, numerical analysis, and stochastics, as well as researchers from data science and machine learning, to exchange ideas, foster interaction, develop new avenues, and generally bring these communities closer together.

Minkowski Space from Quantum Mechanics

Penrose’s Spin Geometry Theorem is extended further, from SU(2) and E(3) (Euclidean) to E(1, 3) (Poincaré) invariant elementary quantum mechanical systems. The Lorentzian spatial distance between any two non-parallel timelike straight lines of Minkowski space, considered to be the centre-of-mass world lines of E(1, 3)-invariant elementary classical mechanical systems with positive rest mass, is expressed in terms of E(1, 3)-invariant basic observables, viz. the 4-momentum and the angular momentum of the systems. An analogous expression for E(1, 3)-invariant elementary quantum mechanical systems in terms of the basic quantum observables in an abstract, algebraic formulation of quantum mechanics is given, and it is shown that, in the classical limit, it reproduces the Lorentzian spatial distance between the timelike straight lines of Minkowski space with asymptotically vanishing uncertainty. Thus, the metric structure of Minkowski space can be recovered from quantum mechanics in the classical limit using only the observables of abstract quantum mechanical systems.

Elementary examples of renormalization from mechanics and integral calculus

Two classical mechanical systems—block-and-tackle machines and nonlinear oscillators—exhibit renormalization of numerical parameters. A simple one-dimensional integral illustrates several other facets of renormalization (regularization, renormalization group, resummation). These examples provide insight for students and non-specialists into the workings of a subtle, complicated and historically misunderstood subject.

Continuous finite-time terminal sliding mode to solve the tracking problem in a class of mechanical systems

The major contribution of this study is the feedback design of a finite-time convergence sliding mode control to solve the trajectory-tracking problem in a class of mechanical systems. Some advantages are that the controller presents a continuous signal by integration of the high-frequency switching term. Another benefit is the design and implementation of an uncertainty and disturbance estimator (UDE) to robustify the closed-loop system. We use Lyapunov tools to develop the closed-loop stability analysis and to give an expression of the convergence time [Formula: see text] t through this, we can reduce the convergence time by tuning the gains of the controller. We illustrate the performance of the proposed control structure via numerical simulations conducted for a mass-spring-damper system and experiments developed in a pendular system.

A mathematical description of the Weber nucleus as a classical and quantum mechanical system

Wilhelm Weber’s electrodynamics is an action-at-a-distance theory which has the property that equal charges inside a critical radius become attractive. Weber’s electrodynamics inside the critical radius can be interpreted as a classical Hamiltonian system whose kinetic energy is, however, expressed with respect to a Lorentzian metric. In this article we study the Schrödinger equation associated with this Hamiltonian system, and relate it to Weyl’s theory of singular Sturm–Liouville problems.

Modern Theory of Creep of Reinforced Concrete

The important features of the theory of creep of reinforced concrete, identified and published earlier, are explored. The creation and development of the theory of creep of reinforced concrete is based on non-scientific principles take from systems of classical mechanics that do not correspond to this theory. A detailed analysis of the theory used in many countries was performed, while five oversimplifications were identified that reject fundamental experiments, Eurocodes, rules of mathematics and mechanics: listed in the law of creep, oversimplifications that grossly distort the calculation results, not only the deformations themselves, but also subsequent methods for calculating reinforced concrete structures. These include: unnecessarily modified classical Hooke’s law; imposing a property missing from concrete - an algebraic measure of creep; erroneous superposition principle; use of viscoelastic deformations instead of instantaneous nonlinear plastic deformations; replacement of obvious - nonlinear and non-stationary properties of concrete with linear ones, distorting the qualitative side of phenomena inherent only in nonlinear systems. These errors are covered by unreasonable safety factors, which undermines the economic component of the problem, and of the enormous volumes of reinforced concrete used throughout the world, the analyzed unscientific theory of its calculation causes enormous economic damage in global construction.

Experimental Realization of Stable Exceptional Chains Protected by Non-Hermitian Latent Symmetries Unique to Mechanical Systems.

Lines of exceptional points are robust in the three-dimensional non-Hermitian parameter space without requiring any symmetry. However, when more elaborate exceptional structures are considered, the role of symmetry becomes critical. One such case is the exceptional chain (EC), which is formed by the intersection or osculation of multiple exceptional lines (ELs). In this Letter, we investigate a non-Hermitian classical mechanical system and reveal that a symmetry intrinsic to second-order dynamical equations, in combination with the source-free principle of ELs, guarantees the emergence of ECs. This symmetry can be understood as a non-Hermitian generalized latent symmetry, which is absent in prevailing formalisms rooted in first-order Schrödinger-like equations and has largely been overlooked so far. We experimentally confirm and characterize the ECs using an active mechanical oscillator system. Moreover, by measuring eigenvalue braiding around the ELs meeting at a chain point, we demonstrate the source-free principle of directed ELs that underlies the mechanism for EC formation. Our Letter not only enriches the diversity of non-Hermitian exceptional point configurations, but also highlights the new potential for non-Hermitian physics in second-order dynamical systems.

Robust proportional control for trajectory tracking using parameter and perturbation adaptive estimators

The principal contribution of this paper is the feedback design of an output trajectory tracking controller applied to a certain class of mechanical systems. The controller does not need velocity measurements for its implementation and achieves the trajectory tracking control aim. We propose adaptive estimators to determine the plant parameters and constant external perturbations. When the plant is affected by time-variant external perturbations, an observer filter that can be implemented to recover an estimation of the time-variant perturbations can replace the adaptive estimator. The closed-loop stability is analysed throughout Lyapunov tools. We illustrate the performance of the proposed control structure via numerical simulations and experiments, the numerical simulations are carried out for a pendulum coupled to a DC motor and the experiments are accomplished for a mass-spring-damper system.

Riemannian exponential and quantization

This article continues and completes the previous one [18]. First of all, we present two methods of quantization associated with a linear connection given on a differentiable manifold, one of them being the one presented in [18]. The two methods allow quantization of functions that come from covariant tensor fields. The equivalence of both is demonstrated as a consequence of a remarkable property of the Riemannian exponential (Theorem 5.1) that, as far as we know, is new to the literature. In addition, we provide a characterization of the Schrödinger operators as the only ones that by quantization correspond to classical mechanical systems. Finally, it is shown that the extension of the above quantization to functions of a very broad type can be carried out by generalizing the method of [18] in terms of fields of distributions.

A new look into the concept of separable system of Classical Mechanics

The theory of integrable systems of Classical Mechanics has been deeply affected, in the second half of the last century, by the discovery of the integrability of the Korteweg–de Vries equation. New ideas appeared at that time (such as, for instance, the concepts of Lax and bihamiltonian representations, and the notions of Poisson manifold, Poisson pencil, and recursion operator) which have changed our way of seeing the geometry of integrable systems. They also changed our way of seeing the geometry of separable systems. The paper is an attempt to explain why and in what form the idea of recursion operator leads to reshape the definition of separable system, and to discover a new set of separability conditions called “Kowalevski separability conditions”.

Cuplength estimates for periodic solutions of Hamiltonian particle-field systems

We consider a natural class of time-periodic infinite-dimensional nonlinear Hamiltonian systems modelling the interaction of a classical mechanical system of particles with a scalar wave field. When the field is defined on a space torus {mathbb {T}}^d={mathbb {R}}^d/(2pi {mathbb {Z}})^d and the coordinates of the particles are constrained to a submanifold Qsubset {mathbb {T}}^d, we prove that the number of contractible T-periodic solutions of the coupled Hamiltonian particle-field system is bounded from below by the {mathbb {Z}}_2-cuplength of the space Lambda ^{{text {contr}}} Q of contractible loops in Q, provided that the square of the ratio T/2pi of time period T and space period X=2pi is a Diophantine irrational number. The latter condition is necessary since for the infinite-dimensional version of Gromov–Floer compactness as well as for the C^0-bounds we need to deal with small divisors.

Sqrt{Toverline{T}}-deformed oscillator inspired by ModMax

Inspired by a recently proposed Duality and Conformal invariant modification of Maxwell theory (ModMax), we construct a one-parameter family of two-dimensional dynamical systems in classical mechanics that share many features with the ModMax theory. It consists of a couple of sqrt{Toverline{T}}-deformed oscillators that nevertheless preserves duality (q rightarrow p,p rightarrow -q) and depends on a continuous parameter gamma, as in the ModMax case. Despite its nonlinear features, the system is integrable. Remarkably, it can be interpreted as a pair of two coupled oscillators whose frequencies depend on some basic invariants that correspond to the duality symmetry and rotational symmetry. Based on the properties of the model, we can construct a nonlinear map dependent on gamma that maps the oscillator in 2D to the nonlinear one, but with parameter 2gamma. The reason behind the existence of such map can be revealed through a construction of two Lax pairs associated with the system. The dynamics also shows the phenomenon of energy transfer and we calculate a Hannay angle associated to geometric phases and holonomies.

On the reduction of some systems of classical mechanics to the Liouvillian form

On the reduction of some systems of classical mechanics to the Liouvillian form

Constructing span categories from categories without pullbacks

Span categories provide an abstract framework for formalizing mathematical models of certain systems. The mathematical descriptions of some systems, such as classical mechanical systems, require categories that do not have pullbacks, and this limits the utility of span categories as a formal framework. Given categories $\mathscr{C}$ and $\mathscr{C}^\prime$ and a functor $\mathcal F$ from $\mathscr{C}$ to $\mathscr{C}^\prime$, we introduce the notion of an $\mathcal F$ pullback of a cospan in $\mathscr{C}$, as well as the notion of span tightness of $\mathcal F$. If $\mathcal F$ is span tight, then we can form a generalized span category ${\rm Span}(\mathscr{C},\mathcal F)$ and circumvent the technical difficulty of $\mathscr{C}$ failing to have pullbacks. Composition in ${\rm Span}(\mathscr{C},\mathcal F)$ uses $\mathcal F$-pullbacks rather than pullbacks and in this way differs from the category ${\rm Span}(\mathscr{C})$, but reduces to it when both $\mathscr{C}$ has pullbacks and $\mathcal F$ is the identity functor.

Collective dynamics and shattering of disturbed two-dimensional Lennard-Jones crystals.

Elucidating collective dynamics in crystalline systems is a common scientific question in multiple fields. In this work, by combination of high-precision numerical approach and analytical normal mode analysis, we systematically investigate the dynamical response of two-dimensional Lennard-Jones crystal as a purely classical mechanical system under random disturbance of varying strength, and reveal rich microscopic dynamics. Specifically, we observe highly symmetric velocity field composed of sharply divided coherent and disordered regions, and identify the order-disorder dynamical transition of the velocity field. Under stronger disturbance, we reveal the vacancy-driven shattering of the crystal. This featured disruption mode is fundamentally different from the dislocation-unbinding scenario in two-dimensional melting. We also examine the healing dynamics associated with vacancies of varying size. The results in this work advance our understanding about the formation of collective dynamics and crystal disruption, and may have implications in elucidating relevant non-equilibrium behaviors in a host of crystalline systems. Microscopic dynamics and underlying topological defects in the disruption of the Lennard-Jones lattice under random disturbance.

Discrete-time energy-balance passivity-based control

In this paper, new results for passivation and stabilization of discrete-time nonlinear systems via energy balancing are established. When specified on sampled-data systems, the approach is constructive for computing stabilizing digital controllers that assign, at all sampling instants, a target energy profile while stabilizing a target equilibrium. The class of mechanical systems is discussed as an example. Simulations are reported highlighting, for position regulation of a 2R robot, the effect of approximate solutions with respect to standard emulation.

Tunable topological interface states in one-dimensional inerter-based locally resonant lattices with damping

Wave dispersion and topology of phononic crystals in classical mechanical systems are well understood and extensively studied subjects. However, the topological properties of acoustic metamaterials with more complex unit cells having several masses, internal resonators, and inerter elements is an insufficiently investigated topic. In this work, we study a class of locally resonant acoustic systems having diatomic- and triatomic-like mass-in-mass unit cells with inerter elements and different springs connecting outer masses. Winding numbers and signs of band gaps are investigated to assess the topological characteristics of a lattice band structure that support edge/interface modes and whether that property is affected when inerter elements are embedded into the system. The dynamics of finite undamped and damped chains constituted of two connected sub-lattices are investigated to demonstrate the existence of interface modes and their localization in space. We reveal that the presented diatomic-like and triatomic-like mass-in-mass chains can generate several interface modes that reside within both lower and higher frequency band gaps. The concept is illustrated through the investigation of the eigenvalue spectrum for varying and fixed stiffness of outer springs and frequency response function. The effect of arbitrary viscous damping is explored based on steady-state responses of the lattice interface mass points. Numerical analysis reveals that the introduction of inerters in combination with local resonators can significantly shift the band gaps and corresponding interface modes to lower frequency values while keeping the main topological properties of the initial configuration without inerters. The effect of damping is shown to be significant and capable to attenuate both lower and higher frequency interface mode amplitudes. We anticipate that this study will pave the wave for future works on the topic that include more reliable models of inerters in the topological mechanical metamaterial design.