On integral principles for nonholonomic systems

On the stability of equilibria of nonholonomic systems with nonlinear constraints

Energy integrals for mechanical systems with nonlinear nonholonomic constraints

This paper presents a Hamilton's principle for distributed control of infinite-dimensional systems modeled by a distributed form of the Euler-Lagrange method. The distributed systems are governed by a system of linear partial differential equations in space and time. A generalized potential energy expression is developed that can capture most physical systems including those systems that have no spatial distribution. The Hamilton's principle is applied to derive distributed feedback control methods without resorting to the standard weak-form discretization approach to convert an infinite-dimensional systems to a finite-dimensional systems. It can be shown by the principle of least action that the distributed control synthesized by the Hamilton's principle is a minimum-norm control. A model-reference adaptive control framework is developed for distributed Lagrangian systems in the presence of uncertainty. The theory is demonstrated by an application of adaptive flutter suppression control of a flexible aircraft wing.

A new form of governing equations of fluids arising from Hamilton's principle

The calculus of variations is concerned with finding extremal paths of functionals in analogy with the way that classical calculus seeks to find critical points of functions. Variational calculus plays a central role in classical mechanics, connecting the “Principle of Least Action” and Lagrange’s equations of motion (also called the Euler– Lagrange equations). In that setting, generalized coordinates are introduced to describe the geometric configuration of a mechanical system. In this chapter, classical variational calculus is reviewed and extended to describe systems on Lie groups. Of course, the introduction of coordinates such as Euler angles to describe the orientation of a rigid body can be used to formulate classical variational problems at the expense of introducing singularities. However, it is possible to formulate variational problems on Lie groups without coordinates. This results in the Euler’Poincaré equations.

In the first part of the paper we present a new point of view on the geometry of nonholonomic mechanical systems with linear and affine constraints. The main geometric object of the paper is the nonholonomic connection on the distribution of constraints. By using this connection and adapted frame fields, we obtain the Newton forms of Lagrange–d’Alembert equations for nonholonomic mechanical systems with linear and affine constraints. In the second part of the paper, we show that the Kaluza–Klein theory is best presented and explained by using the framework of nonholonomic mechanical systems. We show that the geodesics of the Kaluza–Klein space, which are tangent to the electromagnetic distribution, coincide with the solutions of Lagrange–d’Alembert equations for a nonholonomic mechanical system with linear constraints, and their projections on the spacetime are the geodesics from general relativity. Any other geodesic of the Kaluza–Klein space that is not tangent to the electromagnetic distribution is also a solution of Lagrange–d’Alembert equations, but for affine constraints. In particular, some of these geodesics project exactly on the solutions of the Lorentz force equations of the spacetime.

In this paper we revive a special, less-common, variational principle in analytical mechanics (Hertz’ principle of least curvature) to develop a novel variational analogue of Euler's equations for the dynamics of an ideal fluid. The new variational formulation is fundamentally different from those formulations based on Hamilton's principle of least action. Using this new variational formulation, we generalize the century-old problem of the flow over a two-dimensional body; we developed a variational closure condition that is, unlike the Kutta condition, derived from first principles. The developed variational principle reduces to the classical Kutta–Zhukovsky condition in the special case of a sharp-edged airfoil, which challenges the accepted wisdom about the Kutta condition being a manifestation of viscous effects. Rather, we found that it represents conservation of momentum. Moreover, the developed variational principle provides, for the first time, a theoretical model for lift over smooth shapes without sharp edges where the Kutta condition is not applicable. We discuss how this fundamental divergence from current theory can explain discrepancies in computational studies and experiments with superfluids.

The paper presents a framework for analysis of beams interacting with nonlinear‐poroelastic, layered continuums (e.g., clayey soils) when subjected to time‐dependent loads. The poroelastic layered continuum is characterized by a nonlinear‐elastic constitutive relationship that relates the secant shear modulus to the induced shear strain. The Biot's theory of consolidation is combined with a dynamic beam‐continuum interaction model to develop the analysis. The vertical consolidation settlement of the beam and the excess pore pressure in the porous continuum are assumed to be products of separable functions, and the extended Hamilton's principle of least action is applied to obtain the differential equations governing the inertial consolidation motion of the beam‐continuum system and the dissipation of excess pore pressure. An iterative numerical algorithm is used to solve these coupled differential equations following one‐dimensional finite element analysis in which the implicit Wilson‐Θ time integration scheme is used to obtain the time history of beam and continuum responses. The novelty of the framework is that it rigorously takes into account the nonlinear poroelastic soil‐structure interaction within a dynamic time‐integration framework with minimal computational resources. The characteristics of this newly developed model are illustrated through examples.

The Hamilton action principle, also known as the principle of least action, and Lagrange equations are an integral part of intermediate and advanced undergraduate mechanics. Although the Hamilton principle is oft stated as ‘the action for any nearby trial world-line is greater than the action for the classical world-line’, the landscape of action in the space of world-lines is rarely explored. Here, for three common problems from introductory physics—a free particle, a uniformly accelerating particle, and a simple harmonic oscillator—we present families of trial world-lines, characterized by a few parameters, that evolve continuously from their respective classical world-lines. With explicit analytical expressions available for the action, they permit a graphical visualization of the action landscape in the space of nearby world-lines. Although these trial world-lines form only a subset of the space of all nearby world-lines, they provide a pedagogical tool that complements the traditional Lagrange equation approach. This tutorial is aimed at advanced undergraduate physics students.

In this article, from the integral variational principles for embedded variation identity of high-order nonholonomic constrained systems, three kinds of dynamics for high-order nonholonomic constrained systems are obtained, including the vakonomic dynamical model, Lagrange-d'Alembert model and a new one if utilizing respectively three kinds of conditional variation to them. And the integral variational principles for embedded variation identity of high-order nonholonomic constrained systems is also fitted for the general nonholonomic systems when the constrained equation is reduced to a first-order one. Then, the vakonomic dynamic, Chetaev dynamics and a new model of general nonholonomic systems can also be obtained. Finally, two illustrated examples are used to verify the validity of the theory.

The Lagrange Problem with Nonholonomic Constraints

The d’Alembert–Lagrange principle (DLP) is designed primarily for dynamical systems under ideal geometric constraints. Although it can also cover linear-velocity constraints, its application to nonlinear kinematic constraints has so far remained elusive, mainly because there is no clear method whereby the set of linear conditions that restrict the virtual displacements can be easily extracted from the equations of constraint. On recognition that the commutation rule traditionally accepted for velocity displacements in Lagrangian dynamics implies displaced states that do not satisfy the kinematic constraints, we show how the property of possible displaced states can be utilized ab initio so as to provide an appropriate set of linear auxiliary conditions on the displacements, which can be adjoined via Lagrange's multipliers to the d’Alembert–Lagrange equation to yield the equations of state, and also new transpositional relations for nonholonomic systems. The equations of state so obtained for systems under general nonlinear velocity and acceleration constraints are shown to be identical with those derived (in Appendix A) from the quite different Gauss principle. The present advance therefore solves a long outstanding problem on the application of DLP to ideal nonholonomic systems and, as an aside, provides validity to axioms as the Chetaev rule, previously left theoretically unjustified. A more general transpositional form of the Boltzmann–Hamel equation is also obtained.

Multi-product, multi-venders inventory models with different cases of the rational function under linear and non-linear constraints via geometric programming approach

Discrete Lagrange problems with constraints valued in a Lie group

This paper focuses on the study of complete dynamic modeling and maximum dynamic load carrying capacity computation of N-flexible links and N-flexible joints mobile manipulator undergoing large deformation. Nonlinear dynamic analysis relies on the Timoshenko theory of beams. In order to model the system completely and precisely, structural and joint flexibility, nonlinear strain-displacement relationship, payload, and non-holonomic constraints will be considered to. A finite element solution method based on mixed method is applied to model the shear deformation. This procedure is considerably more involved than displacement based element and shear deformation can be readily included without inducing the shear locking in the element. Another goal of this paper is to present a computational procedure for determination of the maximum dynamic load of geometrically nonlinear manipulators with structural and joint flexibility. An effective measure named as Moment-Height Stability (MHS) measure is applied to consider the dynamic stability of a wheeled mobile manipulator. Simulations are performed for mobile base manipulator with two flexible links and joints. The results represent that dynamic stability constraint is sensitive when calculating the maximum carrying load. Furthermore, by changing the trajectory of end effector, allowable load also changes. The effect of torsional spring parameter on the joint deformation is investigated in a parametric sensitivity study. The findings show that, by the increase of torsional stiffness, the behavior of system approaches to a system with rigid joints and allowable load of robot is also enhanced. A comparison is also made between the results obtained from small and large deformation models. Fluctuation range in obtained figures for angular displacement of links and end effector path is bigger for large deformation model. Experimental results are also provided to validate the theoretical model and these have good agreement with the simulated results.