On the Geometry of Nonholonomic Dynamics

1994 ◽  
Vol 61 (3) ◽  
pp. 689-694 ◽  
Author(s):  
H. Esse´n
Keyword(s):  
Equations Of Motion ◽  
Nonholonomic Systems ◽  
Vector Equation ◽  
Finite Degree ◽  
Abstract Form ◽  
Nonholonomic Dynamics ◽  
Starting Point ◽  
Projection Approach ◽  
Local Tangent ◽  
Lagrange’S Method

The formulation and derivation of equations of motion for finite degree-of-freedom nonholonomic systems, is discussed. The starting point is Newton’s equation of motion in the 3K-dimensional unconstrained configuration space of K particles. Constraints represent knowledge that motion is only possible along some directions in the local tangent spaces. Only projections of the 3K-dimensional vector equation onto these allowed directions are of interest. The formalism is essentially that of Kane-Appell cast into an abstract form. It is shown to give the same equations as Hamel’s generalization of Lagrange’s method. The algorithmic advantage of the Kane-Appell projection approach is stressed.

scholarly journals An explicit construction of the dimension-9 operator basis in the standard model effective field theory

2020 ◽  
Vol 2020 (11) ◽  
Author(s):  
Yi Liao ◽  
Xiao-Dong Ma
Keyword(s):  
Field Theory ◽  
Standard Model ◽  
Effective Field Theory ◽  
Equations Of Motion ◽  
Baryon Number ◽  
Explicit Construction ◽  
Effective Field ◽  
Starting Point ◽  
The Standard Model ◽  
Symmetry Relations

Abstract We investigate systematically dimension-9 operators in the standard model effective field theory which contains only standard model fields and respects its gauge symmetry. With the help of the Hilbert series approach to classifying operators according to their lepton and baryon numbers and their field contents, we construct the basis of operators explicitly. We remove redundant operators by employing various kinematic and algebraic relations including integration by parts, equations of motion, Schouten identities, Dirac matrix and Fierz identities, and Bianchi identities. We confirm counting of independent operators by analyzing their flavor symmetry relations. All operators violate lepton or baryon number or both, and are thus non-Hermitian. Including Hermitian conjugated operators there are $$ {\left.384\right|}_{\Delta B=0}^{\Delta L=\pm 2}+{\left.10\right|}_{\Delta B=\pm 2}^{\Delta L=0}+{\left.4\right|}_{\Delta B=\pm 1}^{\Delta L=\pm 3}+{\left.236\right|}_{\Delta B=\pm 1}^{\Delta L=\mp 1} $$ 384 Δ B = 0 Δ L = ± 2 + 10 Δ B = ± 2 Δ L = 0 + 4 Δ B = ± 1 Δ L = ± 3 + 236 Δ B = ± 1 Δ L = ∓ 1 operators without referring to fermion generations, and $$ {\left.44874\right|}_{\Delta B=0}^{\Delta L=\pm 2}+{\left.2862\right|}_{\Delta B=\pm 2}^{\Delta L=0}+{\left.486\right|}_{\Delta B=\pm 1}^{\Delta L=\pm 3}+{\left.42234\right|}_{\Delta B=\mp 1}^{\Delta L=\pm 1} $$ 44874 Δ B = 0 Δ L = ± 2 + 2862 Δ B = ± 2 Δ L = 0 + 486 Δ B = ± 1 Δ L = ± 3 + 42234 Δ B = ∓ 1 Δ L = ± 1 operators when three generations of fermions are referred to, where ∆L, ∆B denote the net lepton and baryon numbers of the operators. Our result provides a starting point for consistent phenomenological studies associated with dimension-9 operators.

Constrained optimal terrain following/threat avoidance trajectory planning using network flow

2014 ◽  
Vol 118 (1203) ◽  
pp. 523-539 ◽  
Author(s):  
R. Zardashti ◽  
A. A. Nikkhah ◽  
M. J. Yazdanpanah
Keyword(s):  
Cost Function ◽  
Trajectory Planning ◽  
Network Flow ◽  
Equations Of Motion ◽  
Minimum Cost ◽  
Masking Effect ◽  
Search Problem ◽  
Starting Point ◽  
Terrain Following ◽  
Threat Avoidance

AbstractThis paper focuses on the trajectory planning for a UAV on a low altitude terrain following/threat avoidance (TF/TA) mission. Using a grid-based approximated discretisation scheme, the continuous constrained optimisation problem into a search problem is transformed over a finite network. A variant of the Minimum Cost Network Flow (MCNF) to this problem is then applied. Based on using the Digital Terrain Elevation Data (DTED) and discrete dynamic equations of motion, the four-dimensional (4D) trajectory (three spatial and one time dimensions) from a starting point to an end point is obtained by minimising a cost function subject to dynamic and mission constraints of the UAV. For each arc in the grid, a cost function is considered as the combination of the arc length, fuel consumption and flight time. The proposed algorithm which considers dynamic and altitude constraints of the UAV explicitly is then used to obtain the feasible trajectory. The resultant trajectory can increase the survivability of the UAV using the threat region avoidance and the terrain masking effect. After obtaining the feasible trajectory, an improved algorithm is proposed to smooth the trajectory. The numeric results are presented to verify the capability of the proposed approach to generate admissible trajectory in minimum possible time in comparison to the previous works.

The Anchor-Last Deployment Problem for Inextensible Mooring Lines

1975 ◽  
Vol 97 (3) ◽  
pp. 1046-1052 ◽  
Author(s):  
Robert C. Rupe ◽  
Robert W. Thresher
Keyword(s):  
Numerical Model ◽  
Equations Of Motion ◽  
Simultaneous Equations ◽  
Mooring Line ◽  
Integration Scheme ◽  
Lumped Mass ◽  
Mooring Lines ◽  
Concentrated Masses ◽  
Predictor Corrector ◽  
Lagrange’S Method

A lumped mass numerical model was developed which predicts the dynamic response of an inextensible mooring line during anchor-last deployment. The mooring line was modeled as a series of concentrated masses connected by massless inextensible links. A set of angles was used for displacement coordinates, and Lagrange’s Method was used to derive the equations of motion. The resulting formulation exhibited inertia coupling, which, for the predictor-corrector integration scheme used, required the solution of a set of linear simultaneous equations to determine the acceleration of each lumped mass. For the selected cases studied the results show that the maximum tension in the cable during deployment will not exceed twice the weight of the cable and anchor in water.

Hamilton’s Principle, Lagrange’s Method, and Ship Motion Theory

1984 ◽  
Vol 28 (04) ◽  
pp. 229-237 ◽  
Author(s):  
Touvia Miloh
Keyword(s):  
Lagrangian Density ◽  
Transfer Functions ◽  
Equations Of Motion ◽  
Steady Motion ◽  
Harmonic Oscillation ◽  
Wave Radiation ◽  
Forward Speed ◽  
Forward Motion ◽  
Radiation Problem ◽  
Lagrange’S Method

Lagrange's equations of motion, describing the motion of several bodies on or below a free surface, are here derived from Hamilton's variational principle. The Lagrangian density is obtained by extending Luke's principle to the wave-radiation problem, and the hydrodynamical loads on the bodies are expressed in terms of the Lagrangian density and its derivatives with respect to the generalized coordinates of the bodies. First we consider a forced harmonic oscillation without a forward speed and then we discuss the case of the same oscillatory motion superimposed on arbitrary steady motion. In both cases we employ Lagrange's method to derive the transfer functions between the generalized forces and the amplitudes of the harmonic motions, in terms of added mass, damping, and the hydrostatic restoring coefficients. The case of a steady forward motion, for which the transfer function is already known, is obtained as a particular case of the general solution.

Conservation Principles for Systems With a Varying Number of Degrees-of-Freedom

1998 ◽  
Vol 65 (3) ◽  
pp. 719-726 ◽  
Author(s):  
S. Djerassi
Keyword(s):  
Angular Momentum ◽  
Degrees Of Freedom ◽  
Mechanical Energy ◽  
Equations Of Motion ◽  
Nonholonomic Systems ◽  
Linear Momentum ◽  
Dynamical Equations ◽  
The Third ◽  
Conservation Principles

This paper is the third in a trilogy dealing with simple, nonholonomic systems which, while in motion, change their number of degrees-of-freedom (defined as the number of independent generalized speeds required to describe the motion in question). The first of the trilogy introduced the theory underlying the dynamical equations of motion of such systems. The second dealt with the evaluation of noncontributing forces and of noncontributing impulses during such motion. This paper deals with the linear momentum, angular momentum, and mechanical energy of these systems. Specifically, expressions for changes in these quantities during imposition and removal of constraints are formulated in terms of the associated changes in the generalized speeds.

Dynamics of the Planetary Roller Screw Mechanism

2015 ◽  
Vol 8 (1) ◽  
Author(s):  
Matthew H. Jones ◽  
Steven A. Velinsky ◽  
Ty A. Lasky
Keyword(s):  
Steady State ◽  
Slip Velocity ◽  
Equations Of Motion ◽  
Viscous Friction ◽  
Contact Angles ◽  
Dynamic Equations ◽  
Roller Screw ◽  
Angular Velocities ◽  
Screw Mechanism ◽  
Lagrange’S Method

This paper develops the dynamic equations of motion for the planetary roller screw mechanism (PRSM) accounting for the screw, rollers, and nut bodies. First, the linear and angular velocities and accelerations of the components are derived. Then, their angular momentums are presented. Next, the slip velocities at the contacts are derived in order to determine the direction of the forces of friction. The equations of motion are derived through the use of Lagrange's Method with viscous friction. The steady-state angular velocities and screw/roller slip velocities are also derived. An example demonstrates the magnitude of the slip velocity of the PRSM as a function of both the screw lead and the screw and nut contact angles. By allowing full dynamic simulation, the developed analysis can be used for much improved PRSM system design.

Modeling, Validation and Prototype Development of Electric Sail Tether Deployment Systems for CubeSats

2019 ◽  
Author(s):  
Benjamin E. Hargis ◽  
Benjamin F. Brandt ◽  
Stephen L. Canfield ◽  
Michael Tinker
Keyword(s):  
Control Strategies ◽  
Equations Of Motion ◽  
Satellite System ◽  
Prototype System ◽  
Prototype Development ◽  
Damping Control ◽  
Flight Operations ◽  
Development And Validation ◽  
Thrust Forces ◽  
Lagrange’S Method

Abstract The Electric sail concept is based on a distributed tether satellite system with tether lengths on the order of thousands-of meters. The system must deploy from stowed arrangement into a selected flight configuration in which thrust forces are transmitted through the tether to the satellite body. The system must be stable through deployment procedure and maintain stable, desired configuration during flight operations. Understanding the dynamic behavior of the satellite bodies and distributed, conductive tether are critical to long-range design and development of the Electric Sail concept. This paper’s contribution is the presentation, development and validation of a mathematical model for simulating E-Sail deployment of a prototype system for testing on the MSFC Robotic Flat Floor Facility. A massed tether model is developed using the bead and string concept with equations of motion derived from Lagrange’s Method. The model is validated using infrared motion capture data produced by controlled experiments of a representative tether portion outfitted with IR targets. Further, a prototype is presented which will be used to investigate an E-Sail deployment approach and associated control. The design of this system will allow for deployment on specially designed flat floor facilities at MSFC. The prototype will be used to: 1) gather data for validation of system dynamic model, 2) evaluate alternative deployment strategies, 3) evaluate tether reel-out and damping control strategies.

A geometric approach to the transpositional relations in dynamics of nonholonomic mechanical systems

2018 ◽  
Vol 15 (07) ◽  
pp. 1850112 ◽  
Author(s):  
Mahdi Khajeh Salehani
Keyword(s):  
Equations Of Motion ◽  
Mechanical Systems ◽  
Nonholonomic Constraints ◽  
Geometric Approach ◽  
Nonholonomic Mechanical Systems ◽  
Nonholonomic Dynamics ◽  
Geometric Objects

Exploring the geometry of mechanical systems subject to nonholonomic constraints and using various bundle and variational structures intrinsically present in the nonholonomic setting, we study the structure of the equations of motion in a way that aids the analysis and helps to isolate the important geometric objects that govern the motion of such systems. Furthermore, we show that considering different sets of transpositional relations corresponding to different transitivity choices provides different variational structures associated with nonholonomic dynamics, but the derived equations (being referred to as the generalized Hamel–Voronets equations) are equivalent to the Lagrange–d’Alembert equations. To illustrate results of this work and as some applications of the generalized Hamel–Voronets formalisms discussed in this paper, we conclude with considering the balanced Tennessee racer, as well as its modification being referred to as the generalized nonholonomic cart, and an [Formula: see text]-snake with three wheeled planar platforms whose snake-like motion is induced by shape variations of the system.

NONHOLONOMIC DYNAMICS OF SECOND ORDER AND THE HEISENBERG SPINNING PARTICLE

2012 ◽  
Vol 09 (07) ◽  
pp. 1220010
Author(s):  
MIRCEA CRASMAREANU ◽  
IULIAN STOLERIU
Keyword(s):  
Equations Of Motion ◽  
Second Order ◽  
Spinning Particle ◽  
Nonholonomic Dynamics

The equations of motion for the associated constrained Lagrangian to a nonholonomic Lagrangian of second order are computed. The spinning particle subject to the Heisenberg constraint is treated as example and its dynamics is completely described.

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