Simple Mechanical System Research Articles

The optimal choice of control constraints

The problem of the optimal choice of the limits of a set of possible values of the control during motion for the purpose of obtaining the required form of the attainability set of a linear dynamical system in a specified time interval is considered. Using the method, in which these sets are approximated by ellipsoids, the problem of controlling the parameters of the ellipsoid containing the control vector is solved. Then a functional, which depends on the matrix of the ellipsoid, containing the phase vector, reaches its maximum. The order in which the corresponding formulae are used is illustrated using the example of a simple mechanical system. The results obtained are suitable for systems in which, instead of the control vector, there is an interference vector with controllable boundaries of possible changes and can be extended to stochastic systems.

Topological spectrum of mechanical systems

We consider the method of topological quantization for conservative systems with a finite number of degrees of freedom. Maupertuis' formalism for classical mechanics provides an appropriate scenario which permit us to adapt the method of topological quantization, originally formulated for gravitational field configurations. We show that any conservative system in classical mechanics can be associated with a principal fiber bundle. As an application of topological quantization we derive expressions for the topological spectra of some simple mechanical systems and show that they reproduce the discrete behavior of the corresponding canonical spectra.

A DECOMPOSITION OF A QUASI-OSCILLATOR

A decomposition of a simplest mechanical system – a q-oscillator, is considered. A q-oscillator (quasi-oscillator) is defined as a system consisting of a particle moving on a linear segment with a constant speed and reflecting from the segment's end-points. A theorem, stating that a discrete-time version of q-oscillator can be decomposed into a countable set of discrete-time rotators (a rotator consists of a particle rotating on a circle with a constant angular rate) as well as a metrical theorem, concerning the rate of such decomposition, are proved. Some physics-related examples are discussed. A phase-shifting perturbed rotator and a number-theoretical matrix system modelling the quantum oscillator, are presented.

A bead on a hoop rotating about a horizontal axis: A one-dimensional ponderomotive trap

We describe a simple mechanical system that operates as a ponderomotive particle trap. The system consists of a circular hoop and a frictionless bead, with the hoop rotating about a horizontal axis lying in the plane of the hoop. The system is analyzed in the ideal case of small displacements from the minimum, and the motion of the particle is shown to satisfy the Mathieu equation. If the axis of rotation is tangential to the hoop, the system is an exact analog to the rf Paul ion trap. Various complicating factors such as anharmonic terms, friction, and noise are considered. A working model of the system was constructed using a ball-bearing rolling in a tube along the outside of a bicycle rim. The apparatus demonstrates the operation of an rf Paul trap by reproducing the dynamics of trapped atomic ions and illustrating the manner in which the electric potential varies with time.

Controllability and synthesis of control for nonlinear systems

We solve motion control problems with feedback under the linear approximation of equations of the considered system. Due to the smallness of the nonlinear part and the controllability of the linear approximating system this property is preserved under a nonlinear perturbation. We perform the study by the method of Lyapunov functions and by the comparison method within finite and infinite time intervals. We illustrate the obtained results by an example of controlling a simple mechanical system.

Parameter estimation for mechanical systems via an explicit representation of uncertainty

PurposeThe purpose of this paper is to propose a new computational approach for parameter estimation in the Bayesian framework. A posteriori probability density functions are obtained using the polynomial chaos theory for propagating uncertainties through system dynamics. The new method has the advantage of being able to deal with large parametric uncertainties, non‐Gaussian probability densities and nonlinear dynamics.Design/methodology/approachThe maximum likelihood estimates are obtained by minimizing a cost function derived from the Bayesian theorem. Direct stochastic collocation is used as a less computationally expensive alternative to the traditional Galerkin approach to propagate the uncertainties through the system in the polynomial chaos framework.FindingsThe new approach is explained and is applied to very simple mechanical systems in order to illustrate how the Bayesian cost function can be affected by the noise level in the measurements, by undersampling, non‐identifiablily of the system, non‐observability and by excitation signals that are not rich enough. When the system is non‐identifiable and an a priori knowledge of the parameter uncertainties is available, regularization techniques can still yield most likely values among the possible combinations of uncertain parameters resulting in the same time responses than the ones observed.Originality/valueThe polynomial chaos method has been shown to be considerably more efficient than Monte Carlo in the simulation of systems with a small number of uncertain parameters. This is believed to be the first time the polynomial chaos theory has been applied to Bayesian estimation.

On resonant Rossby-Haurwitz triads

The dynamics of non-divergent flow on a rotating sphere are described by the conservation of absolute vorticity. The analytical study of the non-linear barotropic vorticity equation is greatly facilitated by the expansion of the solution in spherical harmonics and truncation at low order. The normal modes are the well-known Rossby–Haurwitz (RH) waves, which represent the natural oscillations of the system. Triads of RH waves, which satisfy conditions for resonance, are of critical importance for the distribution of energy in the atmosphere. We show how non-linear interactions of resonant RH triads may result in dynamic instability of large-scale components. We also demonstrate a mathematical equivalence between the equations for an orographically forced triad and a simple mechanical system, the forced-damped swinging spring. This equivalence yields insight concerning the bounded response to a constant forcing in the absence of damping. An examination of triad interactions in atmospheric reanalysis data would be of great interest.

On resonant Rossby—Haurwitz triads

The dynamics of non-divergent flow on a rotating sphere are described by the conservation of absolute vorticity. The analytical study of the non-linear barotropic vorticity equation is greatly facilitated by the expansion of the solution in spherical harmonics and truncation at low order. The normal modes are the well-known Rossby—Haurwitz (RH) waves, which represent the natural oscillations of the system. Triads of RH waves, which satisfy conditions for resonance, are of critical importance for the distribution of energy in the atmosphere.We show how non-linear interactions of resonant RH triads may result in dynamic instability of large-scale components. We also demonstrate a mathematical equivalence between the equations for an orographically forced triad and a simple mechanical system, the forced-damped swinging spring. This equivalence yields insight concerning the bounded response to a constant forcing in the absence of damping. An examination of triad interactions in atmospheric reanalysis data would be of great interest.

Modeling and simulation of wear in revolute clearance joints in multibody systems

Abstract The main goal of this work is to develop a methodology for studying and quantifying the wear phenomenon in revolute clearance joints. In the process, a simple model for a revolute joint in the framework of multibody systems formulation is presented. The evaluation of the contact forces developed is based on a continuous contact force model that accounts for the geometrical and materials properties of the colliding bodies. The friction effects due to the contact in the joints are also represented. Then, these contact—impact forces are used to compute the pressure field at the contact zone, which ultimately is employed to quantify the wear developed and caused by the relative sliding motion. In this work, the Archard’s wear model is used. A simple planar multibody mechanical system is used to perform numerical simulations, in order to discuss the assumptions and procedures adopted throughout this work. From the main results obtained, it can be drawn that the wear phenomenon is not uniformly distributed around the joint surface, owing to the fact that the contact between the joint elements is wider and more frequent is some specific regions.

SHILNIKOV BIFURCATION: STATIONARY QUASI-REVERSAL BIFURCATION

A generic stationary instability that arises in quasi-reversible systems is studied. It is characterized by the confluence of three eigenvalues at the origin of complex plane with only one eigenfunction. We characterize the dynamics through the normal form that exhibits in particular, Shilnikov chaos, for which we give an analytical prediction. We construct a simple mechanical system, Shilnikov particle, which exhibits this quasi-reversal instability and displays its chaotic behavior.

Quantitative Structure−Property Relationships for Longitudinal, Transverse, and Molecular Static Polarizabilities in Polyynes

The present work reports for the first time quantitative structure-property relationships, derived at the benchmark CCSD(T)/cc-PVTZ level of theory that estimate the static longitudinal, transverse, and molecular polarizability in polyynes (C2nH2), as a function of their length (L). In the case of independent electron models, regardless of the form of the nuclei potential that the electrons experience, the polarizability increases strongly with system size, scaling as L(4). In contrast, the static longitudinal polarizability in polyynes have a considerably weaker length-dependence (L(1.64)). This is shown to predominantly arise from electron-electron repulsion rather than electron correlation by a systematic study of the polarizability length dependence in several simple quantum mechanical systems (e.g., particle-in-box, simple harmonic oscillator) and other molecular systems (e.g., H2, H2(+), polyynes). Decrease of the electron-electron repulsion term is suggested to be the key factor in enhancing nonlinear polarizability characteristics of linear oligomeric and polymeric materials.

Control of mechanical systems on Lie groups based on vertically transverse functions

The transverse function approach to control provides a unified setting to deal with practical stabilization and tracking of arbitrary trajectories for controllable driftless systems. Controllers derived from that approach offer advantages over those based on more classical techniques for control of nonholonomic systems. Nevertheless, its extension to more general classes, such as critical underactuated mechanical systems, is not immediate. The present paper explores a possible extension by developing a framework that allows one to cast point stabilization problems for (left-invariant) second-order systems on Lie groups, including simple mechanical systems. The approach is based on “vertical transversality,” a property exhibited by derivatives of transverse functions. In this paper, we lay out the theoretical foundations of our approach and present an example to illustrate some of its features.

Wave Energy from the North Sea: Experiences from the Lysekil Research Site

This paper provides a status update on the development of the Swedish wave energy research area located close to Lysekil on the Swedish West coast. The Lysekil project is run by the Centre for Renewable Electric Energy Conversion at Uppsala University. The project was started in 2004 and currently has permission to run until the end of 2013. During this time period 10 grid-connected wave energy converters, 30 buoys for studies on environmental impact, and a surveillance tower for monitoring the interaction between waves and converters will be installed and studied. To date the research area holds one complete wave energy converter connected to a measuring station on shore via a sea cable, a Wave Rider™ buoy for wave measurements, 25 buoys for studies on environmental impact, and a surveillance tower. The wave energy converter is based on a linear synchronous generator which is placed on the sea bed and driven by a heaving point absorber at the ocean surface. The converter is directly driven, i.e. it has no gearbox or other mechanical or hydraulic conversion system. This results in a simple and robust mechanical system, but also in a somewhat more complicated electrical system.

An example of Arnold diffusion for near-integrable Hamiltonians

In this paper, using the ideas of Bessi and Mather, we present a simple mechanical system exhibiting Arnold diffusion. This system of a particle in a small periodic potential can be also interpreted as ray propagation in a periodic optical medium with a near-constant index of refraction. Arnold diffusion in this context manifests itself as an arbitrary finite change of direction for nearly constant index of refraction.

Averaging of zero dynamics for systems controlled by the vertically transverse function approach

In this paper, a method is explored to introduce dissipation in the closed-loop, zerodynamics systems that arise in the vertically transverse function approach (VTFA) recently proposed by the authors. The VTFA is an attempt to extend the transverse function approach (TFA) of Morin and Samson to deal with practical point-stabilization of a class of critical simple mechanical systems on Lie groups. This class comprises systems that are not kinematic reductions and hence fall beyond the scope of application of the TFA as originally formulated. The VTFA gives rise to a nontrivial zero dynamics that ultimately determines the qualitative nature of the trajectories and, in order to constrain the velocity (or “fiber”) coordinates to vanish asymptotically, as required in typical applications, dissipation must be injected into the zero dynamics. Reported below is a possible way to reach that goal based on the adjunction of additional auxiliary inputs, via generalized vertically transverse functions, and the use of nonlinear, high-order averaging theory. Also included is an illustrative example as well as numerical simulations suggesting that the feedback laws herein designed yield promising results.

Chaotic dynamics of simple vibrational systems

This study discusses the development of a technique for analysis of the dynamical regimes of complex mechanical systems consisting of a rotor motor coupled to a system with multi-degrees-of-freedom. To understand the possible qualitatively different dynamical regimes in such systems, a simple mechanical system is considered of the “rotator-oscillator” type with a finite power source. This system has four degrees-of-freedom and is defined in four-dimensional cylindrical phase space with 12 parameters. Near the main resonance the original system is reduced to the Lorenz system with four parameters defined in a three-dimensional Cartesian phase space. This is done with the help of a special change of variables, parameters, and employing an averaging method. Studying the latter system, the existence of one of the chaotic attractors, namely of Lorenz attractor is established. Also established is the Feigenbaum attractor and the alternation. Chaotic limit sets define chaotic behavior of the instantaneous frequency of rotation of the asynchronous motor. The Poincare mappings are presented to show the correspondence of the original 4 dof and averaged 3 dof systems. The qualitative rotational characteristics for different values of the system parameters are obtained. In particular, the system can possess normal Sommerfeld effect, doubled Sommerfeld effect and a so-called scattering of the torque curve. The scattering of the torque curve (which is a known effect in micro-electronics) is likely to be a new effect in mechanics. In contrast to the Sommerfeld effect, when frequency or amplitude jumps occur instantaneously (once the unstable point of the characteristic is reached), the jump to a next stable point may take a certain time, even infinite one. Such chaotic mistuning of the motor frequency would result in random vibrations leading to system wear and damage.

An extension of Hamiltonian systems to the thermodynamic phase space: Towards a geometry of nonreversible processes

It is shown that the intrinsic geometry associated with equilibrium thermodynamics, namely the contact geometry, provides also a suitable framework in order to deal with irreversible thermodynamical processes. Therefore we introduce a class of dynamical systems on contact manifolds, called conservative contact systems, defined as contact vector fields generated by some contact Hamiltonian function satisfying a compatibility condition with some Legendre submanifold of the contact manifold. Considering physical systems' modeling, the Legendre submanifold corresponds to the definition of the thermodynamical properties of the system and the contact Hamiltonian function corresponds to the definition of some irreversible processes taking place in the system. Open thermodynamical systems may also be modeled by augmenting the conservative contact systems with some input and output variables (in the sense of automatic control) and so-called input vector fields and lead to the definition of port contact systems. Finally complex systems consisting of coupled simple thermodynamical or mechanical systems may be represented by the composition of such port contact systems through algebraic relations called interconnection structure. Two examples illustrate this composition of contact systems: a gas under a piston submitted to some external force and the conduction of heat between two media with external thermostat.

Nanoscale quantitative stress mapping with atomic force microscopy

A nanometer scale noninvasive method for quantitatively mapping tensile strain in metals or semiconductors is demonstrated. The technique is based on the Kelvin probe force microscopy detection of changes in the electronic work function of a material resulting from the tensile strain. Measurements are quantified using a simple microlever mechanical system by recording changes in the work function as a function of the applied strain. A linear relationship of the work function on the tensile stress is observed with a stress sensitivity of 1kPa. Finally, the stress distribution in a strained silicon membrane is imaged.

Possible biomechanical origins of the long-range correlations in stride intervals of walking

When humans walk, the time duration of each stride varies from one stride to the next. These temporal fluctuations exhibit long-range correlations. It has been suggested that these correlations stem from higher nervous system centers in the brain that control gait cycle timing. Existing proposed models of this phenomenon have focused on neurophysiological mechanisms that might give rise to these long-range correlations, and generally ignored potential alternative mechanical explanations. We hypothesized that a simple mechanical system could also generate similar long-range correlations in stride times. We modified a very simple passive dynamic model of bipedal walking to incorporate forward propulsion through an impulsive force applied to the trailing leg at each push-off. Push-off forces were varied from step to step by incorporating both “sensory” and “motor” noise terms that were regulated by a simple proportional feedback controller. We generated 400 simulations of walking, with different combinations of sensory noise, motor noise, and feedback gain. The stride time data from each simulation were analyzed using detrended fluctuation analysis to compute a scaling exponent, α . This exponent quantified how each stride interval was correlated with previous and subsequent stride intervals over different time scales. For different variations of the noise terms and feedback gain, we obtained short-range correlations ( α < 0.5 ) , uncorrelated time series ( α = 0.5 ) , long-range correlations ( 0.5 < α < 1.0 ) , or Brownian motion ( α > 1.0 ) . Our results indicate that a simple biomechanical model of walking can generate long-range correlations and thus perhaps these correlations are not a complex result of higher level neuronal control, as has been previously suggested.

Geometry and Control of Human Eye Movements

In this paper, we study the human oculomotor system as a simple mechanical control system. It is a well known physiological fact that all eye movements obey Listing's law, which states that eye orientations form a subset consisting of rotation matrices for which the axes are orthogonal to the normal gaze direction. First, we discuss the geometry of this restricted configuration space (referred to as the Listing space). Then we formulate the system as a simple mechanical control system with a holonomic constraint. We propose a realistic model with musculotendon complexes and address the question of controlling the gaze. As an example, an optimal energy control problem is formulated and numerically solved