Integration Of Mechanical Systems Research Articles

Integrable and superintegrable quantum mechanical systems with position dependent masses invariant with respect to one parametric Lie groups. 1. Systems with cylindric symmetry

Cylindrically symmetric quantum mechanical systems with position dependent masses admitting at least one second order integral of motion are classified. It is proved that there exist 68 such systems which are inequivalent. Among them there are thirty superintegrable and twelve maximally superintegrable ones. The arbitrary elements of the corresponding Hamiltonians (i.e.,masses and potentials) are presented explicitly.

Variational integrators and graph-based solvers for multibody dynamics in maximal coordinates

Multibody dynamics simulators are an important tool in many fields, including learning and control in robotics. However, many existing dynamics simulators suffer from inaccuracies when dealing with constrained mechanical systems due to unsuitable integrators with bad energy behavior and problematic constraint violations, for example in contact interactions. Variational integrators are numerical discretization methods that can reduce physical inaccuracies when simulating mechanical systems, and formulating the dynamics in maximal coordinates allows for easy and numerically robust incorporation of constraints such as kinematic loops or contacts. Therefore, this article derives a variational integrator for mechanical systems with equality and inequality constraints in maximal coordinates. Additionally, efficient graph-based sparsity-exploiting algorithms for solving the integrator are provided and implemented as an open-source simulator. The evaluation of the simulator shows improved physical accuracy due to the variational integrator and the advantages of the sparse solvers. Comparisons to minimal-coordinate algorithms show improved numerical robustness, and application examples of a walking robot and an exoskeleton with explicit constraints demonstrate the necessity and capabilities of maximal coordinates.

Lobatto-type variational integrators for mechanical systems with frictional contact

This paper introduces a family of Lobatto IIIA-IIIB methods for simulating mechanical systems with frictional contact. These methods extend the existing schemes by addressing both bilateral and unilateral constraints, as well as set-valued Coulomb friction. The Lobatto IIIA-IIIB methods presented in this paper make a substantial contribution to the ongoing endeavor of developing event-capturing versions of high-order schemes.By stating a generalized version of the principle of virtual action for nonsmooth mechanical systems, the integrators are derived from an appropriate discretization of it. The discrete contact laws are found by discretizing the impenetrabilty condition as well as the Newton-type frictional impact laws in an event-capturing way. The presented discrete contact laws exhibit no contact penetration and satisfy the involved unilateral constraints both on position and on velocity level. This behavior is showcased using benchmark examples.

Feedback Integrators for Mechanical Systems with Holonomic Constraints.

The feedback integrators method is improved, via the celebrated Dirac formula, to integrate the equations of motion for mechanical systems with holonomic constraints so as to produce numerical trajectories that remain in the constraint set and preserve the values of quantities, such as energy, that are theoretically known to be conserved. A feedback integrator is concretely implemented in conjunction with the first-order Euler scheme on the spherical pendulum system and its excellent performance is demonstrated in comparison with the RATTLE method, the Lie–Trotter splitting method, and the Strang splitting method.

Thermodynamics of the classical spin triangle

Abstract The classical spin system consisting of three spins with Heisenberg interaction is an example of a completely integrable mechanical system. In this paper, we explicitly calculate thermodynamic quantities such as density of states, specific heat, susceptibility and spin autocorrelation functions. These calculations are performed (semi-)analytically and shown to agree with corresponding Monte Carlo simulations. It is shown that the thermodynamic functions depend qualitatively on the character of the system in terms of its frustration, especially w. r. t. their low temperature limit. For the long-time autocorrelation function, we find, for certain values of the coupling constants, a decay to constant values in the form of an 1/t damped harmonic oscillation and propose a theoretical explanation.

Advanced methodology for multi-way transmission of ship data treatment from mechanical-navigational technical state sensors with using computational neural network computer systems

The paper presents a methodology for multi-track transmission processing of signals from mechanical-navigation sensors describing the state of the ship's motion vector by using a neural network computer system. The integration of mechanical systems with navigation systems on a sea vessel is aimed at creating a uniform decision-making module for the crew steering the ship and supervising propulsion systems in order to minimize the level of risk in controlling a sea vessel in hazardous situations. The concept of integrating radio devices with navigation and engineering systems will enable effective ship management based on sources coming from external and internal sensors of the ship. The processing of navigational and mechanical signals proposed in the article will allow to build an effective prediction of the developing situation in case of danger according to the proposed algorithm for the officer of the watch in charge of the ship. The paper presents a model of mathematical processing of signals from navigational-mechanical systems and illustrates its operation in an exemplary situation of a threat of collision of a sea vessel

Lie Group integrators for mechanical systems

Since they were introduced in the 1990s, Lie group integrators have become a method of choice in many application areas. These include multibody dynamics, shape analysis, data science, image registration and biophysical simulations. Two important classes of intrinsic Lie group integrators are the Runge–Kutta–Munthe–Kaas methods and the commutator free Lie group integrators. We give a short introduction to these classes of methods. The Hamiltonian framework is attractive for many mechanical problems, and in particular we shall consider Lie group integrators for problems on cotangent bundles of Lie groups where a number of different formulations are possible. There is a natural symplectic structure on such manifolds and through variational principles one may derive symplectic Lie group integrators. We also consider the practical aspects of the implementation of Lie group integrators, such as adaptive time stepping. The theory is illustrated by applying the methods to two nontrivial applications in mechanics. One is the N-fold spherical pendulum where we introduce the restriction of the adjoint action of the group to , the tangent bundle of the two-dimensional sphere. Finally, we show how Lie group integrators can be applied to model the controlled path of a payload being transported by two rotors. This problem is modeled on and put in a format where Lie group integrators can be applied.

An improved implicit method for mechanical systems with set-valued friction

The computational procedures of higher-order implicit integrators for mechanical systems with friction are provided in this paper. The dynamic equations are established using the augmented Lagrangian formulation, and set-valued friction forces are described by projection functions. To reduce the accuracy loss caused by event transitions and to eliminate spurious oscillations in the acceleration, a new robust event-driven scheme, which accurately detects event transitions and corrects the friction forces and accelerations at switching points, is proposed. The numerical performance of the proposed scheme is demonstrated by solving several benchmark problems. Numerical results show that the newly developed scheme can approximately achieve second-order accuracy, and they are more accurate than the classical Moreau time-stepping scheme under close computational efforts. Finally, a slider-crank system is simulated to prove the validity of the developed method for nonlinear mechanical systems with friction.

Integrability and Chaos in Figure Skating

We derive and analyze a three dimensional model of a figure skater. We model the skater as a three-dimensional body moving in space subject to a non-holonomic constraint enforcing movement along the skate's direction and holonomic constraints of continuous contact with ice and pitch constancy of the skate. For a static (non-articulated) skater, we show that the system is integrable if and only if the projection of the center of mass on skate's direction coincides with the contact point with ice and some mild (and realistic) assumptions on the directions of inertia's axes. The integrability is proved by showing the existence of two new constants of motion linear in momenta, providing a new and highly nontrivial example of an integrable non-holonomic mechanical system. We also consider the case when the projection of the center of mass on skate's direction does not coincide with the contact point and show that this non-integrable case exhibits apparent chaotic behavior, by studying the divergence of nearby trajectories We also demonstrate the intricate behavior during the transition from the integrable to chaotic case. Our model shows many features of real-life skating, especially figure skating, and we conjecture that real-life skaters may intuitively use the discovered mechanical properties of the system for the control of the performance on ice.

Symplectic integration with non-canonical quadrature for guiding-center orbits in magnetic confinement devices

We study symplectic numerical integration of mechanical systems with a Hamiltonian specified in non-canonical coordinates and its application to guiding-center motion of charged plasma particles in magnetic confinement devices. The technique combines time-stepping in canonical coordinates with quadrature in non-canonical coordinates and is applicable in systems where a global transformation to canonical coordinates is known explicitly but its inverse is not. A fully implicit class of symplectic Runge-Kutta schemes has recently been introduced and applied to guiding-center motion by Zhang et al. (2014) [9]. Here a generalization of this approach with emphasis on semi-implicit partitioned schemes is described together with methods to enhance performance, in particular avoiding evaluation of non-canonical variables at full time steps. For application in toroidal plasma confinement configurations with nested magnetic flux surfaces a global canonicalization of coordinates for the guiding-center Lagrangian by a spatial transform is presented that allows for pre-computation of the required map in a parallel algorithm in the case of time-independent magnetic field geometry. Guiding-center orbits are studied in stationary magnetic equilibrium fields of an axisymmetric tokamak and a realistic three-dimensional stellarator configuration. Superior long-term properties of symplectic methods are demonstrated in comparison to a conventional adaptive Runge-Kutta scheme. Finally statistics of fast fusion alpha particle losses over their slowing-down time are computed in the stellarator field on a representative sample, reaching a speed-up of the symplectic Euler scheme by more than a factor three compared to usual Runge-Kutta schemes while keeping the same statistical accuracy and linear scaling with the number of computing threads.

Energy-preserving variational integrators for forced Lagrangian systems

The goal of this paper is to develop energy-preserving variational integrators for time-dependent mechanical systems with forcing. We first present the Lagrange-d’Alembert principle in the extended Lagrangian mechanics framework and derive the extended forced Euler-Lagrange equations in continuous-time. We then obtain the extended forced discrete Euler-Lagrange equations using the extended discrete mechanics framework and derive adaptive time step variational integrators for time-dependent Lagrangian systems with forcing. We consider three numerical examples to study the numerical performance of energy-preserving variational integrators. First, we consider the example of a nonlinear conservative system to illustrate the advantages of using adaptive time-stepping in variational integrators. In addition, we demonstrate how the implicit equations become more ill-conditioned as the adaptive time step decreases through a condition number analysis. As a second example, we numerically simulate the time-dependent example of a forced harmonic oscillator to demonstrate the superior energy performance of energy-preserving integrators for mechanical systems with explicit time-dependent forcing. Finally, we consider a damped harmonic oscillator using the adaptive time step variational integrator framework. The adaptive time step increases monotonically for the dissipative system leading to unexpected energy behavior.

Integrated lung tissue mechanics one piece at a time: Computational modeling across the scales of biology

The lung is a delicately balanced and highly integrated mechanical system. Lung tissue is continuously exposed to the environment via the air we breathe, making it susceptible to damage. As a consequence, respiratory diseases present a huge burden on society and their prevalence continues to rise. Emergent function is produced not only by the sum of the function of its individual components but also by the complex feedback and interactions occurring across the biological scales – from genes to proteins, cells, tissue and whole organ - and back again. Computational modeling provides the necessary framework for pulling apart and putting back together the pieces of the body and organ systems so that we can fully understand how they function in both health and disease. In this review, we discuss models of lung tissue mechanics spanning from the protein level (the extracellular matrix) through to the level of cells, tissue and whole organ, many of which have been developed in isolation. This is a vital step in the process but to understand the emergent behavior of the lung, we must work towards integrating these component parts and accounting for feedback across the scales, such as mechanotransduction. These interactions will be key to unlocking the mechanisms occurring in disease and in seeking new pharmacological targets and improving personalized healthcare.

New integrable problems in a rigid body dynamics with cubic integral in velocities

Abstract We introduce a new family of the 2D integrable mechanical system possessing an additional integral of the third degree in velocities. This system contains 20 arbitrary parameters. We also clarify that the majority of the previous systems with a cubic integral can be reconstructed from it as a special version for certain values of those parameters. The applications of this system are extended to include the problem of motion of a particle and rigid body about its fixed point. We announce new integrable problems describing the motion of a particle in the plane, pseudosphere, and surfaces of variable curvature. We also present a new integrable problem in a rigid body dynamics and this problem generalizes some of the previous results for Sokolov-Tsiganov, Yehia, Stretensky, and Goriachev.

Nonstandard separation of variables for the Maxwell–Bloch conservative system

The Maxwell–Bloch conservative system, which originated in laser optic theory, is a nice, well-known example of a completely integrable mechanical system. Here we show how to separate the system into two subsystems: one with a fish-shaped dynamics and the other which is governed by a central force. The behaviour of the whole system is then easily explained with tools that are more elementary than in previous approaches. The separation of the fish-shaped dynamics is new, and it is obtained in an unconventional way, using nonlocal constants of motion.

Polynomial integrals of mechanical systems on a torus with a singular potential

The problem on integrability of the equations of motion of a material point on an n-dimensional Euclidean torus under the action of a force field with the potential energy having singularities at a finite number of points is considered. It is assumed that these singularities contain logarithmic coefficients and, consequently, have a more general form in comparison with power features. The potentials having power-type singularities were considered previously by V.V. Kozlov and D.V. Treshchev. In this work, it is proved that the equations of motion in the problem under consideration admit no nontrivial momentum-polynomial first integral with integrable coefficients on this torus.

Time integration of mechanical systems with elastohydrodynamic lubricated joints using Quasi‐Newton method and projection formulations

SummaryThis work deals with the efficient time integration of mechanical systems with elastohydrodynamic (EHD) lubricated joints. Two novel approaches are presented. First, a projection function is used to formulate the well‐known Swift–Stieber cavitation condition and the mass‐conservative cavitation condition of Elrod as an unconstrained problem. Based on this formulation, the pressure variable from the EHD problem is added to the dynamic equations of a multi‐body system in a monolithic manner so that cavitation is solved within a global iteration. Compared with a partitioned state‐of‐the‐art formulation, where the pressure is solved locally in a nonlinear force element, this global search reduces simulation time.Second, a Quasi‐Newton method of DeGroote is applied during time integration to solve the nonlinear relation between pressure and deformation. Compared with a simplified Newton method, the calculation of the time‐consuming parts of the Jacobian are avoided, and therefore, simulation time is reduced significantly, when the Jacobian is calculated numerically. Solution strategies with the Quasi‐Newton method are presented for the partitioned formulation as well as for the new DAE formulations with projection function. Results are given for a simulation example of a rigid shaft in a flexible bearing. Copyright © 2016 John Wiley & Sons, Ltd.

Integrable 2D Time-Irreversible Systems with a Cubic Second Integral

We construct a very rare integrable 2D mechanical system which admits a complementary integral of motion cubic in the velocities in the presence of conservative potential and velocity-dependent (gyroscopic) forces. Special cases are given interpretation as a motion of a particle on a sphere endowed with a Riemannian metric, a particle in the Euclidean plane, and new generalizations of two cases of motion of a rigid body with a cubic integral, known by names of Goriachev-Chaplygin and Goriachev.

Structured Linearization of Discrete Mechanical Systems for Analysis and Optimal Control

Variational integrators are well-suited for simulation of mechanical systems because they preserve mechanical quantities about a system such as momentum, or its change if external forcing is involved, and holonomic constraints. While they are not energy-preserving they do exhibit long-time stable energy behavior. However, variational integrators often simulate mechanical system dynamics by solving an implicit difference equation at each time step, one that is moreover expressed purely in terms of configurations at different time steps. This paper formulates the first- and second-order linearizations of a variational integrator in a manner that is amenable to control analysis and synthesis, creating a bridge between existing analysis and optimal control tools for discrete dynamic systems and variational integrators for mechanical systems in generalized coordinates with forcing and holonomic constraints. The forced pendulum is used to illustrate the technique. A second example solves the discrete Linear Quadratic Regulator (LQR) problem to find a locally stabilizing controller for a 40 DOF system with six constraints.

An integrated framework for mechatronics based product development in a fuzzy environment

New product development (NPD) is a term used to describe the complete process of bringing a new concept to a state of market readiness. Mechatronics based product requires a multidisciplinary approach for its modeling, design, development and implementation. An integrated and concurrent approach focusing on integrating the mechanical structure with basic three components namely sensors, controllers and actuators is required. This paper aims at developing a framework for a new Mechatronics product development. For conceptual design of Mechatronics system, various tools like Fuzzy Delphi Method (FDM), Fuzzy Interpretive Structural Modeling (FISM), Fuzzy Analytical Network Process (FANP) and Fuzzy Quality Function Deployment (FQFD) are used. Based on the prioritized design requirements, the functional specifications of the required components are developed. Then, Computer Aided Design and control system software are used to develop the detailed system design. Then, a prototype model is developed based on the integration of mechanical system with Sensor, Controller and Electrical units. Performance of the prototype model is monitored and Fuzzy failure mode and effect analysis (FMEA) is then used to rank the potential failures. Based on the results of fuzzy FMEA, the developed model is redesigned. The proposed framework is illustrated with a case study related to developing automatic power loom reed cleaning machine.

Modeling of an Indirect Solar Assisted Heat Pump System for a High Performance Residential House

The combination of solar thermal and heat pump systems as a single solar assisted heat pump (SAHP) system can significantly reduce residential energy consumption in Canada. As a part of Team Ontario's efforts to develop a high performance house for the U.S. Department of Energy's Solar Decathlon 2013 Competition, an integrated mechanical system (IMS) consisting of a SAHP was investigated. The system was designed to provide domestic hot water (DHW), space-heating, space-cooling, and dehumidification. The system included a cold and a hot thermal storage tanks and a heat pump to move energy from the low temperature reservoir to the hot reservoir. Solar thermal collectors supplied heat to the cold storage and operated at a higher efficiency due to the heat pump reducing the temperature of the collector working fluid. The combination of the heat pump and solar thermal collectors allows more heat to be harvested at a lower temperature, and then boosted to a suitable temperature for domestic use via the heat pump. The IMS and the building's energy loads were modeled using the TRNSYS simulation software. A parametric study was conducted to optimize the control, sizing, and configuration of the system. The simulation results suggested that the investigated system can achieve a free energy ratio (FER) of about 0.583 for the high performance house designed for the Ottawa climate.