Class Of Mechanical Systems Research Articles

Finite-time Stückelberg interferometry with nanomechanical modes

Stückelberg interferometry describes the interference of two strongly coupled modes during a double passage through an avoided energy level crossing. In this work, we investigate finite-time effects in Stückelberg interferometry and discuss the exact analytical solution of the double passage Stückelberg problem by expanding the finite-time solution of the Landau–Zener problem. Approximating the return probability amplitudes of the double passage in distinct limits reveals uncharted parameter regimes of Stückelberg interferometry where finite-time effects affect the coherent exchange of energy. We find the long-time limit of the exact solution to formally coincide with the well-established adiabatic impulse model which is, to the best of our knowledge, the only regime of Stückelberg interferometry reported so far. Experimentally, we study all predicted regimes using a purely classical, strongly coupled nanomechanical two-mode system of high quality factor. The classical two-mode system consists of the in-plane and out-of-plane fundamental flexural mode of a high stress silicon nitride string resonator, coupled via electric gradient fields. We exploit our experimental and theoretical findings by studying the onset of Stückelberg interference in dependence of the characteristic system control parameters and obtain characteristic excitation oscillations between the two modes even without the explicit need of traversing the avoided crossing. The presented findings are not limited to classical mechanical two-mode systems but can be applied to every strongly coupled (quantum) two-level system, for example a spin-1/2 system or superconducting qubit.

Control of an axially moving viscoelastic Kirchhoff string

The control problem of axially moving strings occurs in a large class of mechanical systems. In addition to the longitudinal displacement, the strings are subject to undesirable transversal vibrations. In this work, in order to suppress these vibrations, we consider a control by a hydraulic touch-roll actuator at the right boundary. We prove uniform stability of the system using a viscoelastic material and an appropriate boundary control force applied to the touch rolls of the actuator. The features of the present work are: taking into account the mass flow entering in and out at the boundaries due to the axial movement of the string and overcoming the difficulty raised by the Kirchhoff coefficient which does not allow us to profit from the dissipativity of the system (as in the existing works so far). We shall make use of an inequality which is new in this theory.

Variational symmetries and pluri-Lagrangian systems in classical mechanics

We analyze the relation of the notion of a pluri-Lagrangian system, which recently emerged in the theory of integrable systems, to the classical notion of variational symmetry, due to E. Noether. We treat classical mechanical systems and show that, for any Lagrangian system with m commuting variational symmetries, one can construct a pluri-Lagrangian 1-form in the (m+1)-dimensional time, whose multi-time Euler-Lagrange equations coincide with the original system supplied with m commuting evolutionary flows corresponding to the variational symmetries. We also give a Hamiltonian counterpart of this construction, leading, for any system of commuting Hamiltonian flows, to a pluri-Lagrangian 1-form with coefficients depending on functions in the phase space.

Derivation and application of quantum Hamilton equations of motion

Non‐relativistic quantum systems are analyzed theoretically or by numerical approaches using the Schrödinger equation. Compared to the options available to treat classical mechanical systems this is limited, both in methods and in scope. However, based on Nelson's stochastic mechanics, the mathematical structure of quantum mechanics has in some aspects been developed into a form analogous to classical analytical mechanics. We show here that finding the Nash equilibrium for a stochastic optimal control problem, which is the quantum equivalent to Hamilton's principle of least action, allows to derive two things: i) the Schrödinger equation as the Hamilton‐Jacobi‐Bellman equation of this optimal control problem and ii) a set of quantum dynamical equations which are the generalization of Hamilton's equations of motion to the quantum world. We derive their general form for the non‐stationary and the stationary case. For the harmonic oscillator, the stationary equations lead to the coherent states, and we establish a numerical procedure to solve for the ground state properties without using the Schrödinger equation. image

Energy Shaping of Mechanical Systems via PID Control and Extension to Constant Speed Tracking

In a recent contribution, it was shown that a class of mechanical systems, which contains many practical examples, can be stabilized via energy shaping without solving partial differential equations. The proposed controller consists of two terms, a partial linearizing state-feedback and a linear PID loop around two new passive outputs. In this brief note we prove that the first, admittedly non-robust, step can be obviated leaving only the linear PID. A second contribution of the note is to propose a slight modification to the controller to go beyond regulation tasks—being able to follow ramp references in the actuated coordinates.

Elastic spin-Hall effect in mechanical graphene

We show that spin–orbit interaction and elastic spin-Hall effect can exist in a classical mechanical system consisting of a two-dimensional honeycomb lattice of masses and springs. The band structure shows the presence of splitting at K point induced by the difference of longitudinal and transverse elastic constant, and this splitting can be regarded as an effective Dresselhaus-type spin–orbit coupling. Interestingly, as an initial displacement away from the equilibrium is applied, the time evolution simulation shows that waves of different spin polarization propagates along different directions at the Γ and K point, which is characteristic of spin-Hall effect. Several cases for spin-Hall effect are also discussed.

The virial theorem and the kinetic energy of particles of a macroscopic system in the general field concept

The virial theorem is considered for a system of randomly moving particles that are tightly bound to each other by the gravitational and electromagnetic fields, acceleration field and pressure field. The kinetic energy of the particles of this system is estimated by three methods, and the ratio of the kinetic energy to the absolute value of the energy of forces, binding the particles, is determined, which is approximately equal to 0.6 . For simple systems in classical mechanics, this ratio equals 0.5 . The difference between these ratios arises by the consideration of the pressure field and acceleration field inside the bodies, which make additional contribution to the acceleration of the particles. It is found that the total time derivative of the system's virial is not equal to zero, as is assumed in classical mechanics for systems with potential fields. This is due to the fact that although the partial time derivative of the virial for stationary systems tends to zero, but in real bodies the virial also depends on the coordinates and the convective derivative of the virial, as part of the total time derivative inside the body, is not equal to zero. It is shown that the convective derivative is also necessary for correct description of the equations of motion of particles.

Quantization due to breaking the commutativity of symmetries. Wobbling oscillator and anharmonic Penning trap

We discuss two examples of classical mechanical systems which can become quantum either because of degeneracy of an integral of motion or because of tuning parameters at resonance. In both examples, the commutativity of the symmetry algebra is breaking, and noncommutative symmetries arise. Over the new noncommutative algebra, the system can reveal its quantum behavior including the tunneling effect. The important role is played by the creation-annihilation regime for the perturbation or anharmonism. Activation of this regime sometimes needs in an additional resonance deformation (Cartan subalgebra breaking).

An approximation theorem in classical mechanics

A theorem by K. Meyer and D. Schmidt says that The reduced three-body problem in two or three dimensions with one small mass is approximately the product of the restricted problem and a harmonic oscillator [7]. This theorem was used to prove dynamical continuation results from the classical restricted circular three-body problem to the three-body problem with one small mass. We state and prove a similar theorem applicable to a larger class of mechanical systems. We present applications to spatial $(N+1)$-body systems with one small mass and gravitationally coupled systems formed by a rigid body and a small point mass.

Poincaré’s Equations for Cosserat Media: Application to Shells

In 1901 Henri Poincare discovered a new set of equations for mechanics. These equations are a generalization of Lagrange's equations for a system whose configuration space is a Lie group which is not necessarily commutative. Since then, this result has been extensively refined through the Lagrangian reduction theory. In the present contribution, we extend these equations from classical mechanical systems to continuous Cosserat media, i.e. media in which the usual point particles are replaced by small rigid bodies, called micro-structures. In particular, we will see how the Shell balance equations used in nonlinear structural dynamics, can be easily derived from this extension of the Poincare's result.

Minimum time control of a pair of two-level quantum systems with opposite drifts**Work supported by ARO MURI under Grant W911NF-11-1-0268.

In this paper we solve two equivalent time optimal control problems. On one hand, we design the control field to implement in minimum time the SWAP (or equivalent) operator on a two-level system, assuming that it interacts with an additional, uncontrollable, two-level system. On the other hand, we synthesize the SWAP operator simultaneously, in minimum time, on a pair of two-level systems subject to opposite drifts. We assume that it is possible to perform three independent control actions, and that the total control strength is bounded. These controls either affect the dynamics of the target system, under the first perspective, or, simultaneously, the dynamics of both systems, in the second view. We obtain our results by using techniques of geometric control theory on Lie groups. In particular, we apply the Pontryagin maximum principle, and provide a complete characterization of singular and nonsingular extremals. Our analysis shows that the problem can be formulated as the motion of a material point in a central force, a well known system in classical mechanics. Although we focus on obtaining the SWAP operator, many of the ideas and techniques developed in this work apply to the time optimal implementation of an arbitrary unitary operator.

Modal self-excitation in a class of mechanical systems by nonlinear displacement feedback

Many devices and processes utilize self-excited oscillation to enhance performance. Recently, much research work has been devoted to the induction of self-excited oscillation in mechanical systems by nonlinear feedback. The present paper investigates the efficacy of a displacement feedback technique in generating self-excited oscillation at the desired mode(s) in a multiple degrees-of-freedom mechanical system. The controller couples the system with a bank of second-order filters and generates the required control force as a nonlinear function of the filter output. The describing function method theoretically explores the dynamics of the system with the control law. The control cost of the controller is studied for the proper choice of the filter parameters. The analytical results are substantiated by the numerical simulation results. The present study reveals that the proposed control laws, if used in an appropriate way, can generate self-excited oscillation in the system at the desired mode(s).

Modal self-excitation by nonlinear acceleration feedback in a class of mechanical systems

The article proposes an acceleration feedback based technique for exciting modal self-oscillation in a class of multi degrees-of-freedom mechanical systems. The controller comprises a bank of second-order filters and the control law is formulated as the nonlinear function of the filter output. A design methodology is developed to excite self-oscillation in any desired mode or combination of modes (mixed-mode oscillation). The choice of control parameters takes into account the control cost and robustness of the controller. The effects of structural damping on the system performance are also studied. Analytical results are confirmed by numerical simulations. An adaptive control is proposed to maintain the oscillation amplitude at the desired level.

Hannay Angle: Yet Another Symmetry-Protected Topological Order Parameter in Classical Mechanics

Topological way of thinking now goes beyond conventional solid materials, and topological characterization of classical mechanical systems governed by Newton's equation of motion begins to attract much attention. To have a deeper insight on physical meaning of topological numbers in mechanical systems, we demonstrate the use of the Hannay angle, a classical counterpart of the Berry phase, as a symmetry protected topological order parameter. We first derive the Hannay angle using a canonical transformation that maps the Newton's equation to the Schr\"{o}dinger type equation. The Hannay angle is then used to characterize a simple spring-mass model topologically with a particular focus on the bulk-edge correspondence and new aspects of the symmetry in a classical system.

Backstepping control for gear transmission servo systems with unknown partially nonsymmetric deadzone nonlinearity

This paper deals with the output tracking control of gear transmission servo (GTS) systems in the presence of deadzone nonlinearity with nonsymmetric beak points and unknown parameters. A novel differentiable deadzone model with nonsymmetric break points is put forward, which greatly facilitates the control design for a class of mechanical systems in the presence of deadzone nonlinearity. A new smooth backstepping controller, based on the newly-developed model, is proposed for the nominal system. Then, guaranteed robust steady-state performance of the closed-loop system with parametric uncertainties is derived by using Lyapunov analysis for the perturbed nonlinear systems. Simulations are carried out to validate the proposed algorithm and analysis in this paper.

Variational approach and deformed derivatives

Recently, we have demonstrated that there exists a possible relationship between q-deformed algebras in two different contexts of Statistical Mechanics, namely, the Tsallis’ framework and the Kaniadakis’ scenario, with a local form of fractional-derivative operators for fractal media, the so-called Hausdorff derivatives, mapped into a continuous medium with a fractal measure. Here, in this paper, we present an extension of the traditional calculus of variations for systems containing deformed-derivatives embedded into the Lagrangian and the Lagrangian densities for classical and field systems. The results extend the classical Euler–Lagrange equations and the Hamiltonian formalism. The resulting dynamical equations seem to be compatible with those found in the literature, specially with mass-dependent and with nonlinear equations for systems in classical and quantum mechanics. Examples are presented to illustrate applications of the formulation. Also, the conserved ​Noether current is worked out.

Threat and Risk Assessment Methodologies in the Automotive Domain

Safety and security are both qualities that concern the overall system. However, these disciplines are traditionally treated independently in the automotive domain. Replacement of classical mechanical systems with safety-critical embedded systems raised the awareness of the safety attribute and caused the introduction of the ISO 26262 standard. In contrast to this, security topics are traditionally seen as attacks of a mechanical nature and as only affecting single vehicles (e.g. door lock and immobilizer related). Due to the increasing interlacing of automotive systems with networks (such as Car2X), new features like autonomous driving, and online software updates, it is no longer acceptable to assume that car fleets are immune to security risks and automated remote attacks. Consequently, future automotive systems development requires appropriate systematic approaches to support cyber security and safety aware development.Therefore, this paper examines threat and risk assessment techniques that are available for the automotive domain and presents an approach to classify cyber-security threats, which can be used to determine the appropriate number of countermeasures that need to be considered. Furthermore, we present a combined approach for safety and security analysis to be applied in early development phases, which is a pre-requisite for consistent engineering throughout the development lifecycle.

Bifurcations and Chaotic Motions of a Class of Mechanical System With Parametric Excitations

Bifurcations and chaotic motions of a class of mechanical system subjected to a superharmonic parametric excitation or a nonlinear periodic parametric excitation are studied, respectively, in this paper. Chaos arising from the transverse intersections of the stable and unstable manifolds of the homoclinic and heteroclincic orbits is analyzed by Melnikov's method. The critical curves separating the chaotic and nonchaotic regions are plotted. Chaotic dynamics are compared for these systems with a periodic parametric excitation or a superharmonic parametric excitation, or a nonlinear periodic parametric excitation. Especially, some new dynamical phenomena are presented for the system with a nonlinear periodic parametric excitation.

Robust Stochastic Sampled-Data H∞ Control for a Class of Mechanical Systems With Uncertainties

This paper considers a class of mechanical systems with uncertainties appearing in all the mass, damping, and stiffness matrices. Two cases, linear fractional and randomly occurring uncertainty formulations, are considered. Since sampled-data controllers have an advantage of implementing with microcontroller or digital computer to lower the implementation cost and time, a robust stochastic sampled-data controller is considered with m sampling intervals whose occurrence probabilities are given constants and satisfy Bernoulli distribution. A discontinuous type Lyapunov functional based on the extended Wirtinger's inequality is constructed with triple integral terms and sufficient conditions that promises the robust mean square asymptotic stability of the concerned system are derived in terms of linear matrix inequalities (LMIs). In an aim to reduce the conservatism, a newly introduced concept called the second-order reciprocally convex approach is employed in deriving the bound for some cross terms that arise while maneuvering the derivative of Lyapunov functional. The obtained LMIs can be easily solved through any of the standard available software. Finally, numerical examples are given to verify the effectiveness of the proposed theoretical results.

Coriolis force induced topological order for classical mechanical vibrations

We show that topological order and vibrational edge modes can exist in a classical mechanical system consisting of a two-dimensional honeycomb lattice of masses and springs. The band structure shows the existence of Dirac cones and unconventional edge states that are similar to the vibrational modes in graphene. Interestingly, as the system is placed on a constantly rotational coordinate system, the Coriolis force resulting from the non-inertial reference frame introduces time-reversal symmetry breaking and leads to topologically nontrivial band gaps. The nontrivial topological orders are further verified by the calculation of Chern numbers for corresponding bands.