On Hamel’s equations

Zenkov, V Dmitry

doi:10.2298/tam160612011z

Cited by 10 publications

(6 citation statements)

References 21 publications

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“…Note that all these variables are introduced relative to the body-fixed coordinate frame. Similar representations of the dynamics of a pendulum have been proposed and can be found in [18,24,25]. We use this dynamic model and control theory (Pontryagin minimum principle) to develop optimal control strategies for building maintenance units.…”

Section: System Dynamicsmentioning

confidence: 95%

“…The dynamics of the BMU (system) can be derived also from the Euler-Lagrange equations in spherical coordinates. However, that would introduce singularities when the vector ρ is collinear with z axis because the equations describing the physical movement of the BMU in spherical coordinates have denominators vanishing under such situation [24]. Thus the dynamic model expressed in spherical coordinates does not fully represent the physics of the BMU, and is therefore not advised for computational purpose.…”

Section: System Dynamicsmentioning

confidence: 99%

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Optimal control and stabilization of building maintenance units based on minimum principle

Wang¹,

Ahmed²

2021

JIMO

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In this paper we present a mathematical model describing the physical dynamics of a building maintenance unit (BMU) equipped with reaction jets. The momentum provided by reaction jets is considered as the control variable. We introduce an objective functional based on the deviation of the BMU from its equilibrium state due to external high-wind forces. Pontryagin minimum principle is then used to determine the optimal control policy so as to minimize possible deviation from the rest state thereby increasing the stability of the BMU and reducing the risk to the workers as well as the public. We present a series of numerical results corresponding to three different scenarios for the formulated optimal control problem. These results show that, under high-wind conditions the BMU can be stabilized and brought to its equilibrium state with appropriate controls in a short period of time. Therefore, it is believed that the dynamic model presented here would be potentially useful for stabilizing building maintenance units thereby reducing the risk to the workers and the general public.2010 Mathematics Subject Classification. Primary: 49J15; Secondary: 93C95.

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Section: System Dynamicsmentioning

confidence: 95%

Section: System Dynamicsmentioning

confidence: 99%

Optimal control and stabilization of building maintenance units based on minimum principle

Wang¹,

Ahmed²

2021

JIMO

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“…De Leon also referred to the Voronec equations the same year [56]. Along with the Chaplygin equations and the equations of the nonholonomic systems written in terms of quasi-velocities, known as the Euler-Poincaré-Chetayev-Hamel equations, the Voronec equations form core tools in the study of nonholonomic systems (see [12,34,35,61,71]).…”

Section: The Voronec Principlementioning

confidence: 99%

Gyroscopic Chaplygin systems and integrable magnetic flows on spheres

Dragović¹,

Gajić²,

Jovanović³

2021

Preprint

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We introduce and study the Chaplygin systems with gyroscopic forces with a special emphases on the important subclass of such systems with magnetic forces. This class of nonholonomic systems is natural and has not been treated before. We describe the gyroscopic Chaplygin systems on fiber spaces and the reduction procedure for the corresponding G-Chaplygin systems. The existence of an invariant measure and the problem of Hamiltonization are studied, both within the Lagrangian and the almost-Hamiltonian framework. In addition, we introduce problems of rolling of a ball with the gyroscope without slipping and twisting over a plane and a sphere in R n as examples of gyroscopic SO(n)-Chaplygin systems. We describe an invariant measure and provide a class of examples, which allow the Chaplygin Hamiltonization, and prove the integrability of the obtained magnetic geodesic flows on the sphere S n−1 . In particular, we introduce the generalized Demchenko case in R n , when the inertia operator of the system (ball + gyroscope) is proportional to the identity operator. The reduced system is automatically Hamiltonian and represents the magnetic geodesic flow on the sphere S n−1 endowed with the round-sphere metric, under the influence of the homogeneous magnetic field. The magnetic geodesic flow problem on the two-dimensional sphere is well known, but for n > 3 was not studied before. We perform explicit integrations in elliptic functions of the systems for n = 3 and n = 4, and provide the case study of the solutions in both dimensions.

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“…The same year de Leon also referred to the Voronec equations in [76]. Now we can say that the Voronec equations, together with the Chaplygin equations and the equations of nonholonomic systems written in terms of quasi-velocities, known as the Euler-Poincaré-Chetayev-Hamel equations, form the central tools in the study of nonholonomic systems (e.g., see [15,45,46,82,102]).…”

Section: Demchenko's Phd Thesis and Contemporary Nonholonomic Mechanicsmentioning

confidence: 99%

Demchenko’s nonholonomic case of a gyroscopic ball rolling without sliding over a sphere after his 1923 Belgrade doctoral thesis

Dragović

Gajić

Jovanović

2020

Theor appl mech (Belgr)

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We present an integrable nonholonomic case of rolling without sliding of a gyroscopic ball over a sphere. This case was introduced and studied in detail by Vasilije Demchenko in his 1923 doctoral dissertation defended at the University of Belgrade, with Anton Bilimovic as the advisor. These results are absolutely unknown to modern researchers. The study is based on the C. Neumann coordinates and the Voronec principle. By using the involved technique of elliptic functions, a detailed study of motion is performed. Several special classes of trajectories are distinguished, including regular and pseudoregular precessions. The so-called remarkable trajectories, introduced by Paul Painlev?e and Anton Bilimovic, are described in the present case. The historical context of the results as well as their place in contemporary mechanics are outlined.

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On Hamel’s equations

Abstract: This paper reviews recent results on the extension of Hame?s formalism to infinite-dimensional mechanical systems and to variational integrators. Of a particular interest are applications to the dynamics and numerical integration of systems with velocity constraints.

Cited by 10 publications

References 21 publications

Optimal control and stabilization of building maintenance units based on minimum principle

Optimal control and stabilization of building maintenance units based on minimum principle

Gyroscopic Chaplygin systems and integrable magnetic flows on spheres

Demchenko’s nonholonomic case of a gyroscopic ball rolling without sliding over a sphere after his 1923 Belgrade doctoral thesis

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