Conservative Mechanical Systems Research Articles

Variational Integrators on Manifolds for Constrained Mechanical Systems

Abstract Variational integrators play a pivotal role in the simulation and control of constrained mechanical systems. Current Lagrange multiplier-free variational integrators for such systems are concise but inevitably face issues with parameterization singularities, hindering global integration. To tackle this problem, this paper proposes a novel method for constructing variational integrators on manifolds without introducing Lagrange multipliers, offering the benefit of avoiding singularities. Our approach unfolds in three key steps: (1) the local parameterization of configuration space; (2) the formulation of forced discrete Euler–Lagrange equations on manifolds; and (3) the construction and implementation of high-order variational integrators. Numerical tests are conducted for both conservative and forced mechanical systems, demonstrating the excellent global energy behavior of the proposed variational integrators.

Methods for modeling the dissipative characteristics of layered composites

The main methods for modeling dissipative characteristics of layered composites are considered: the principle of elastic-viscoelastic correspondence and energy method. Mathematical models of damped oscillations are given for layered anisotropic plates. A numerical procedure for solving the problem of damped oscillations of a three-layer monoclinic plate containing a two-stage method for solving a complex eigenvalue problem. It is noted that the energy method for modeling energy dissipation at vibrations of layered composite structures is a special case of the method, based on the principle of elasticviscoelastic correspondence. Shown, that in addition to the classical formulation of the energy method, in which the terms energy balance equations are approximately found by their own forms and vibration frequencies of a conservative mechanical system, there is one more an approximate method for calculating the dissipative characteristics of orthotropic composite structures. The advantages of this method are the possibility of using existing commercial software systems (Ansys et al.), which numerically implement the finite element method, and shortcomings - the need to be limited to the use of classical theories of thinwalled structures (Bernoulli-Euler, Kirchhoff-Love) for descriptions of deformation of composite structures. It has been found that the use the classical formulation of the energy method allows with a high accurately predict the values of mechanical loss coefficients up to ηmax = 0.02. Further increase in damping levels construction is accompanied by a decrease in the accuracy of the forecast.

ТРИСТУПЕНЕВА ЕЛЕКТРИЧНА МАШИНА З ВНУТРІШНІМ ТА ЗОВНІШНІМ РОТОРОМ: ПОРІВНЯЛЬНИЙ АНАЛІЗ ПОКАЗНИКІВ КЕРОВАНОСТІ

Two structures of electric machines with three degrees of freedom of the rotational movement of the rotor are considered - with an external and an internal rotor held on an internal cardan suspension. A three-plane symmetric magnetic system was chosen for the comparative analysis and such a choice was justified. Limitations are formulated that allow considering the rotor as a conservative mechanical system, as well as considering the magnetic field in the electric machine as magnetostatic. Mathematical models of the rotational motion of a body with a fixed center of mass, as well as the force interaction of the magnetic field of a permanent magnet located on the rotor and the control winding located on the stator, are given. The peculiarities of the distribution of electromagnetic forces in both structures are analyzed and the greater efficiency of the formation of these forces in the structure with an internal rotor is shown. Expressions for the components of the electromagnetic moment for use in the "COMSOL Multiphysics" interface are obtained. Forced precession movement for both structures was analyzed in the case of equal radii of the control windings and in the case of the same dimensions of the magnetic systems. It was determined that the structure with an external magnetic circuit is able to provide an order of magnitude smaller angular velocity of precessional movement and, at the same time, an equally smaller range of nutation. References 14, figures 10.

Discovering efficient periodic behaviors in mechanical systems via neural approximators

AbstractIt is well known that conservative mechanical systems exhibit local oscillatory behaviors due to their elastic and gravitational potentials, which completely characterize these periodic motions together with the inertial properties of the system. The classification of these periodic behaviors and their geometric characterization are in an ongoing secular debate, which recently led to the so‐called eigenmanifold theory. The eigenmanifold characterizes nonlinear oscillations as a generalization of linear eigenspaces. With the motivation of performing periodic tasks efficiently, we use tools coming from this theory to construct an optimization problem aimed at inducing desired closed‐loop oscillations through a state feedback law. We solve the constructed optimization problem via gradient‐descent methods involving neural networks. Extensive simulations show the validity of the approach.

Mathematical and Statistical Aspects of Estimating Small Oscillations Parameters in a Conservative Mechanical System Using Inaccurate Observations

This paper selects a set of reference points in the form of an arithmetic progression for planning an experiment to evaluate the parameters of systems of differential equations. This choice makes it possible to construct estimates of the parameters of a system of first-order differential equations based on the reversibility of the observation matrix, as well as estimates of the parameters of a system of second-order differential equations describing vibrations in a mechanical system by switching to a system of first-order differential equations. In turn, the reversibility of the observation matrix used in parameter estimation is established using the Vandermonde formula. A volumetric computational experiment has been carried out showing how, with an increase in the number of observations in the vicinity of reference points and with a decrease in the step of arithmetic progression, the accuracy of estimates of the parameters of the analyzed system increases. Among the estimated parameters, the most important are the oscillation frequencies of a conservative mechanical system, which establish its proximity to resonance, and therefore, determine the stability and reliability of the system.

INTERNAL TIME OF A MECHANICAL SYSTEM

One of the most urgent unsolved problems of modern science is the extension of the second law of thermodynamics to conservative mechanical systems that are in a state far from thermodynamic equilibrium. How does the collective motion of the material points of the system, each of which is described by time-reversible equations, become irreversible? Thus, the increase in entropy of an ideal gas in a state close to equilibrium indicates the irreversible nature of the evolution of a conservative mechanical system, which is definitely an ideal gas. That is, entropy acts as internal time and sets the direction to the future or the "arrow of time". The paper deals with obtaining a characteristic of a conservative mechanical system that would have the properties of entropy. The goal is to extend the second law of thermodynamics to conservative mechanical systems that are in a state far from equilibrium. Analytical methods were used to analyze the differential equations of motion of a phase liquid particle as an image of a mechanical system in a multidimensional space. The evolution of the distribution of the probability of a particle of a phase fluid staying on a hypersurface of equal energy is considered. For a conservative mechanical system, a value is introduced whose properties allow us to apply the term "internal time" to it. Its growth determines the difference between future events and past events, which is an inherent property of subjective time. When approaching the state of equilibrium, internal time slows down, and physical time, accordingly, accelerates relative to internal time. Internal time is as universal as physical time, in the sense that it is determined for each system by a universal formula. The proposed approach made it possible to solve the fundamental problem of replacing the averaging of the probability density in the phase space by averaging in time along the trajectory of the point in the phase space. We used the fact that the equation of motion of a conservative system can be obtained as the equation of motion of a particle of a phase liquid, using the law of conservation of matter. The equations of motion of a particle were transferred to the motion of a point. In contrast, when using the traditional approach, the equation of motion of a point in phase space was taken as the basic law of nature. An understanding of internal time allows us understand the emergence of dissipative structures in the future.

Strict Modes Everywhere – Bringing Order Into Dynamics of Mechanical Systems by a Potential Compatible With the Geodesic Flow

Strict nonlinear normal modes provide very regular families of oscillations within conservative mechanical systems. However, a strict normal mode will generally be an isolated curve within the configuration space of the system. In this letter, we design a potential that will densely fill the configuration space with strict normal modes such that each configuration belongs to one mode and each mode passes through a common point, the equilibrium. As the potential can be realized by (nonlinear) elastic elements it can be used to execute a variety of periodic trajectories very efficiently. Most of the required torques will come from the elastic elements in the system and not from the actuators. We also design a controller stabilizing the system to a desired target mode and a controller performing swing-up and compensating dissipated energy. Finally, we showcase the approach for a two DoF manipulator. The experiments show that the approach performed well for the example system.

How to compute invariant manifolds and their reduced dynamics in high-dimensional finite element models

Invariant manifolds are important constructs for the quantitative and qualitative understanding of nonlinear phenomena in dynamical systems. In nonlinear damped mechanical systems, for instance, spectral submanifolds have emerged as useful tools for the computation of forced response curves, backbone curves, detached resonance curves (isolas) via exact reduced-order models. For conservative nonlinear mechanical systems, Lyapunov subcenter manifolds and their reduced dynamics provide a way to identify nonlinear amplitude–frequency relationships in the form of conservative backbone curves. Despite these powerful predictions offered by invariant manifolds, their use has largely been limited to low-dimensional academic examples. This is because several challenges render their computation unfeasible for realistic engineering structures described by finite element models. In this work, we address these computational challenges and develop methods for computing invariant manifolds and their reduced dynamics in very high-dimensional nonlinear systems arising from spatial discretization of the governing partial differential equations. We illustrate our computational algorithms on finite element models of mechanical structures that range from a simple beam containing tens of degrees of freedom to an aircraft wing containing more than a hundred–thousand degrees of freedom.

Oscillations and stability in the coupled mechanical system

Conservative mechanical systems are considered. Each system is supposed to admit a family of non-degenerate symmetric periodic motions. For the entire set of systems, they form a family of conditionally periodic oscillations, which is assumed to contain a symmetrical periodic motion. Universal couplings for mechanical systems are proposed such that they play the role of controls that ensure the existence, stability, and stabilization of oscillations at the same time for the whole system. Those couplings provide the natural, i.e. without involving any additional controls, stabilization of the oscillation. In addition, mechanical systems perform the synchronization in frequency and phase.

Numerical approach for bifurcation and orbital stability analysis of periodic motions of a 2-DOF autonomous Hamiltonian system

In Spaceflight Dynamics it is often necessary to obtain periodic motions of conservative mechanical systems and analyze their stability and bifurcation. These conservative systems can be described using Hamiltonian equations. We consider bifurcation and orbital stability problem for periodic motions of a 2-DOF autonomous Hamiltonian system. Since it is not possible to obtain analytical solutions to the aforementioned problem for all admissible values of its parameters a two-step numerical approach is proposed. On the first step the so-called base solutions are obtained analytically for particular values of problem’s parameters. The base solutions are then continued to the borders of their existence domains using a numerical algorithm. In course of computation bifurcation points are identified and orbital stability is studied. On the second step new base solutions are identified in the neighborhood of bifurcation points and the continuation process is repeated. Finally, orbital stability and bifurcation diagrams of the resulting families of periodic motions are constructed. Poincare sections are also computed in the neighborhoods of bifurcation points to verify the results. To illustrate this approach, we computed the bifurcation and orbital stability diagrams for families of short-periodic motions originating from Regular precessions of a dynamically-symmetric satellite.

Exponential variational integrators for the dynamics of multibody systems with holonomic constraints

We here explore a geometric integrator scheme that is determined by a discretization of a variational principle using a higher-order Lagrangian that uses exponential type of interpolation functions. The resulting exponential variational integrators are here extended to conservative mechanical systems with constraints. To do so we first present continuous Euler-Lagrangian equations with holonomic constraints and then mimic the process for the discrete case. The resulting schemes are then tested to a typical dynamical multibody system with constraints, i.e the double pendulum showing the good long-time behavior when compared to other traditional methods.

Galerkin‐based mixed time finite elements for structural dynamics

AbstractNumerically stable and accurate time integration methods are important in structural dynamics. We propose such a method based on a mixed variational formulation of the underlying semi‐discrete equations of motion. The newly developed scheme relies on a specific combination of the so‐called continuous Galerkin (cG(k)) method and the discontinuous Galerkin (dG(k)) method. We refer to [2] for an introduction of the original cG(k) and dG(k) methods for ODEs (see also [1]). The mixed approach relies on polynomial approximations of degree k on the level of each time finite element. In particular we make use of hierarchical interpolation polynomials. The present approach is closely related to Hellinger‐Reissner type mixed variational formulations for elastodynamics. The mixed method yields implicit time‐stepping schemes with order of accuracy equal to 2k for the displacements and velocities. In addition to that, the method is capable of conserving the energy in the case of conservative mechanical systems. Numerical examples are presented which highlight the numerical properties of the newly proposed mixed approach.

A general framework for the thermodynamically consistent time integration of open nonlinear thermoelastic systems

AbstractThis work deals with Energy‐Momentum‐Entropy (EME) consistent methods of open thermoelastic systems. While Energy‐Momentum (EM) preserving integrators are well‐known for conservative mechanical systems, Romero introduced in [1] the class of thermodynamically consistent (TC) integrators for coupled thermomechanical systems, which further respect symmetries of the underlying coupled system and are capable of conserving associated momentum maps and can therefore also be termed EME schemes in analogy to EM schemes for conservative systems. As mathematical framework for the geometric structure of the non‐equilibrium thermodynamics the GENERIC (General Equation for Non‐Equilibrium Reversible‐Irreversible Coupling) framework, originally introduced for complex fluids in [2], is used. Since the GENERIC framework does not depend on a specific choice of the thermodynamical state variables [3], we explore the structure of the GENERIC framework using the entropy density (see e.g. [4]), the absolute temperature (see e.g. [5]) and further the internal energy density as thermodynamical state variable. Applying the notion of a discrete gradient in the sense of [6] leads to an EME integrator. As boundary conditions rely on the specific choice of the thermodynamical state variable we extend the GENERIC framework to be suitable for open systems following the procedure in [7].

A temperature‐based GENERIC approach for the thermodynamically consistent integration of thermoelastic solids

AbstractThis work deals with the thermodynamically consistent (TC) time integration of thermoelastic systems with polyconvex density functions using the notion of the tensor‐cross‐product. While energy‐momentum preserving integrators are well‐known for conservative (isothermal) mechanical systems, Romero introduced in [7, 8] the new class of TC integrators. While [8] dealt with the sample application of thermo‐elastodynamics, the scope of application was extended in [2] to coupled thermo‐viscoelastodynamics in temperature form. A first step towards the systematic design of a TC integrator is to cast the evolution equations into the GENERIC (General Equation for Non‐Equilibrium Reversible‐Irreversible Coupling) framework [6] which reveals additional underlying physical structures of the system. Relying on a polyconvex density function and using the notion of the tensor‐cross‐product [1] we arrive at a polyconvex version of the GENERIC framework. Further applying the notion of a discrete gradient leads to a TC integrator.Using the entropy as the thermodynamical state variable as in [5, 8] the GENERIC framework possesses an easy structure. However, this choice of thermodynamical state variable only allows to prescribe entropy Dirichlet boundary conditions directly. This drawback can be compensated by using Lagrange‐multipliers to be able to handle temperature Dirichlet boundary conditions leading to an extended system of algebraic equations to be solved, see [5]. Alternatively, the present work uses the temperature as the thermodynamical state variable, see also [2, 3] and the use of an energy‐based Newton‐Raphson termination criterion. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

The Reaction Mass Biped: Geometric Mechanics and Control

Inverted Pendulum based reduced order models offer many valuable insights into the much harder problem of bipedal locomotion. While they help in understanding leg behavior during walking, they fail to capture the natural balancing ability of humans that stems from the variable rotational inertia on the torso. In an attempt to overcome this limitation, the proposed work introduces a Reaction Mass Biped (RMB). It is a generalization of the previously introduced Reaction Mass Pendulum (RMP), which is a multi-body inverted pendulum model with an extensible leg and a variable rotational inertia torso. The dynamical model for the RMB is hybrid in nature, with the roles of stance leg and swing leg switching after each cycle. It is derived using a variational mechanics approach, and is therefore coordinate-free. The RMB model has thirteen degrees of freedom, all of which are considered to be actuated. A set of desired state trajectories that can enable bipedal walking in straight and curved paths are generated. A control scheme is then designed for asymptotically tracking this set of trajectories with an almost global domain-of-attraction. Numerical simulation results confirm the stability of this tracking control scheme for different walking paths of the RMB. Additionally, a discrete dynamical model is also provided along-with an appropriate Geometric Variational Integrator (GVI). In contrast to non-variational integrators, GVIs can better preserve energy terms for conservative mechanical systems and stability properties (achieved through energy-like lyapunov functions) for actuated systems.

The Routh theorem for mechanical systems with unknown first integrals

In this paper we discuss problems of stability of stationary motions of conservative and dissipative mechanical systems with first integrals. General results are illustrated by the problem of motion of a rotationally symmetric rigid body on a perfectly rough plane.

Thermodynamically consistent integration of coupled thermoelastic systems

AbstractThis work deals with the energy‐momentum‐entropy consistent integration of thermoelastic systems. While energy‐momentum preserving integrators are well‐known for conservative mechanical systems, Romero recently introduced in [6] a thermodynamically consistent (TC) integrator for coupled thermomechanical systems. TC integrators also respect symmetries of the underlying coupled system and are therefore capable of conserving associated momentum maps. A first step towards the systematic design of a TC integrator is to cast the evolution equations into the GENERIC framework. GENERIC stands for General Equation for Non‐Equilibrium Reversible‐Irreversible Coupling and has been originally proposed by Grmela and Öttinger for complex fluids [3]. As a second step applying the notion of a discrete gradient in the sense of Gonzalez [2] leads to a TC integrator. The GENERIC‐based framework reveals additional underlying physical structures of the thermodynamical system due to the separation of irreversible and reversible driving forces. Using the entropy as the thermodynamical state variable as in [4,6] the GENERIC framework yields an easy structure. However, this choice of thermodynamical state variable leads to a restriction in the material model and, more importantly, only allows to prescribe entropy Dirichlet boundary conditions. This drawback can only be compensated by using Lagrange‐multipliers to be able to handle temperature Dirichlet boundary conditions, which unfortunately extends the system of algebraic equations to be solved (see Krüger et al. [5]). Alternatively, the present contribution uses the temperature as the thermodynamical state variable (see also the recent work by Conde Martín et al. [1]). This temperature‐based approach allows to set Dirichlet boundary conditions directly. (© 2016 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Differential invariants of feedback transformations for quasi-harmonic oscillation equations

The goal and the main result of the paper is to provide a complete description of the field of rational differential invariants of one class of second order ordinary differential equations with scalar control parameter with respect to Lie pseudo-group of local feedback transformations. In particular, considered class describes behavior of conservative mechanical systems. We construct the class of rational differential invariants that separate regular orbits. It is well known that differential invariants form algebra with respect to the operation of addition and multiplication (Alekseevskij et al. 1991) [20] . In our case, constructed rational differential operators form a field (in algebraic sense). Rational differential invariants were studied by Rosenlicht (1956, 1963) [25] , [26] , Kruglikov and Lychagin (2011) [24] .

A simplified method of solving Birkhoffian function and Lagrangian

By using the calculus of variations, the conservative mechanical systems can be formulated by Lagrange's equations or Hamilton's equations, which are the basis of establishing, simplifying and integrating the equations of motion. Thus it is important to find the solutions of inverse problems for different dynamical systems so as to construct the most of the Lagrange's equations and Hamilton's equations. However, the Lagrangian or Hamiltonian formulation for a dynamical system, limited by the conditions of self-adjointness, is not directly universal if the physical variables remain without using Darboux transformations. Fortunately, Refs. [7, 11] show that based on the Cauchy-Kovalevsky theorem of the integrability conditions for partial differential equations and the converse of the Poincar lemma, it can be proved that there exists a direct universality of Birkhoff's equation for local Newtonian system by reducing the Newton's equations to a first-order form, which means that all local, analytic, regular, finite-dimensional, unconstrained or holonomic, conservative or non-conservative forms always admit, in a star-shaped neighborhood of a regular point of their variables, a representation in terms of first-order Birkhoff's equations in the coordinate and time variables of the experiment. The systems whose equations of motion are represented by the first-order Birkhoff's equations on a symplectic or a contact manifold spanned by the physical variables are called Birkhoffian systems. At present, one of the most important tasks of Birkhoffian mechanics is to study the method of constructing the Birkhoffian and Birkhoffian functions. However, due to the complexity of Birkhoffian system, there exist only a few of results in the literature. Among them, the most famous main methods in this problem are achieved by Santilli[Santilli R M 1983 Foundations of Theoretical Mechanics II (New York: Springer-Verlag) pp12-15]. But the redundant term in Santilli's second method which is used as the classical construction method, is always neglected. As a result, the calculation procedure is tedious and complicated, and the efficiency is not high. Therefore, it is necessary to simplify the Santilli's second method. In Section 2, we will review the first-order standard form of holonomic system in the frame of Cartesian coordinates, which is the starting point of our studying the Birkhoffian systems. In Section 3, the Birkhoff's equations and the key role of Birkhoffian dynamics functions for deriving Birkhoff's equations are introduced. In Section 4, the redundant items are eliminated by using some mathematical operation skills, and then a more simplified constructing method is put forward. In Section 5, the findings in this study are summarized. Through simplifying the Santilli's second method, we realize that the determining of the Birkhoff's equations by constructing the Birkhoffian functions is equivalent to the determining of its symplectic matrix. This view provides a new perspective for solving the problem of constructing the Birkhoffian functions. Finally, the simplified method is applied to Lagrangian inverse problem, and a simplified method of solving Lagrangian function is obtained.

Nonlinear damping models for linear conservative mechanical systems with preserved eigenspaces: a port-Hamiltonian formulation

This paper introduces linear and nonlinear damping models, which preserve the eigenspaces of conservative linear mechanical problems. After some recalls on the finite dimensional case and on Caughey's linear dampings, an extension to a nonlinear class is introduced. These results are recast in the port-Hamiltonian framework and generalized to infinite dimensional systems. They are applied to an Euler-Bernoulli beam, excited by a distributed force. Simulations yield sounds of xylophone, glockenspiel (etc) and some interpolations for nonlinear dampings.