scholarly journals Ehresmann connection in the geometry of nonholonomic systems

2012 ◽  
Vol 91 (105) ◽  
pp. 19-24
Author(s):  
Aleksandar Baksa
Keyword(s):  
Dynamic System ◽  
Equations Of Motion ◽  
Nonholonomic Constraints ◽  
Nonholonomic Systems ◽  
Geometrical Properties ◽  
Ehresmann Connection ◽  
Affine Constraints

This article deals with a dynamic system whose motion is constrained by nonholonomic, reonomic, affine constraints. The article analyses the geometrical properties of the ?reactions" of nonholonomic constraints in Voronets?s equations of motion. The analysis shows their link with the torsion of the Ehresmann connection, which is defined by the nonholonomic constraints.

D’Alembert’s Principle and the Equations of Motion for Nonholonomic Systems

2006 ◽  
Author(s):  
B. F. Feeny
Keyword(s):  
Variational Methods ◽  
Equations Of Motion ◽  
Nonholonomic Constraints ◽  
Nonholonomic Systems ◽  
Rigid Bodies ◽  
Virtual Power ◽  
Lagrangian Equations ◽  
Conservative Systems ◽  
Principle Of Virtual Power ◽  
D'alembert's Principle

D'Alembert's principle is manipulated in the presence of nonholonomic constraints to derive the principle of virtual power in nonholonomic form, and Lagrange's equations for nonholonomic systems. The Lagrangian equations had been expressed previously for conservative systems, derived by variational methods. The D'Alembert derivation confirms the roles of constrained and unconstrained Lagrangians directly by the presence of constrained and unconstrained velocities in D'Alembert's principle. The constrained form of nonconservative generalized forces is also determined for both particles and rigid bodies. An example is a rolling disk.

scholarly journals Numerical Integration of Multibody Dynamic Systems Involving Nonholonomic Equality Constraints

2021 ◽  
Author(s):  
Sotirios Natsiavas ◽  
Panagiotis Passas ◽  
Elias Paraskevopoulos
Keyword(s):  
Dynamic Systems ◽  
Equations Of Motion ◽  
Time Integration ◽  
Nonholonomic Constraints ◽  
Coupled System ◽  
Weak Form ◽  
Integration Scheme ◽  
Motion Constraints ◽  
Multibody Dynamic ◽  
Time Integration Scheme

Abstract This work considers a class of multibody dynamic systems involving bilateral nonholonomic constraints. An appropriate set of equations of motion is employed first. This set is derived by application of Newton’s second law and appears as a coupled system of strongly nonlinear second order ordinary differential equations in both the generalized coordinates and the Lagrange multipliers associated to the motion constraints. Next, these equations are manipulated properly and converted to a weak form. Furthermore, the position, velocity and momentum type quantities are subsequently treated as independent. This yields a three-field set of equations of motion, which is then used as a basis for performing a suitable temporal discretization, leading to a complete time integration scheme. In order to test and validate its accuracy and numerical efficiency, this scheme is applied next to challenging mechanical examples, exhibiting rich dynamics. In all cases, the emphasis is put on highlighting the advantages of the new method by direct comparison with existing analytical solutions as well as with results of current state of the art numerical methods. Finally, a comparison is also performed with results available for a benchmark problem.

Sliding Mode Control in Ziegler’s Pendulum With Tracking Force: Novel Modeling Considerations

2013 ◽  
Author(s):  
Ehsan Sarshari ◽  
Nastaran Vasegh ◽  
Mehran Khaghani ◽  
Saeid Dousti
Keyword(s):  
Stability Analysis ◽  
Sliding Mode Control ◽  
Dynamic System ◽  
Linear Stability Analysis ◽  
Sliding Mode ◽  
Chaotic Behavior ◽  
Equations Of Motion ◽  
Sliding Mode Controller ◽  
Gravity Effect ◽  
Tracking Force

Ziegler’s pendulum is an appropriate model of a non-conservative dynamic system. By considering gravity effect, new equations of motion are extracted from Newton’s motion laws. The instability of equilibriums is determined by linear stability analysis. Chaotic behavior of the model is shown by numerical simulations. Sliding mode controller is used for eliminating chaos and for stabilizing the equilibriums.

Toward a Minimal Dynamic Model of a 6-DOF Parallel Robot

1997 ◽  
Author(s):  
L. Beji ◽  
M. Pascal ◽  
P. Joli
Keyword(s):  
Dynamic Model ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Computational Cost ◽  
Closed Form Solution ◽  
Parallel Robot ◽  
Form Solution ◽  
Lagrangian Formalism ◽  
Inverse Kinematic Problem ◽  
Geometrical Properties

Abstract In this paper, an architecture of a six degrees of freedom (dof) parallel robot and three limbs is described. The robot is called Space Manipulator (SM). In a first step, the inverse kinematic problem for the robot is solved in closed form solution. Further, we need to inverse only a 3 × 3 passive jacobian matrix to solve the direct kinematic problem. In a second step, the dynamic equations are derived by using the Lagrangian formalism where the coordinates are the passive and active joint coordinates. Based on geometrical properties of the robot, the equations of motion are derived in terms of only 9 coordinates related by 3 kinematic constraints. The computational cost of the obtained dynamic model is reduced by using a minimum set of base inertial parameters.

Towards an Energy-Based Linear Analysis of Nonholonomic Systems

1999 ◽  
Author(s):  
Marwan Bikdash ◽  
Richard A. Layton
Keyword(s):  
Steady State ◽  
Singular Systems ◽  
Nonholonomic Constraints ◽  
Nonholonomic Systems ◽  
Linear Analysis ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Indirect Approach ◽  
Physical Systems ◽  
Future Work

Abstract Guidelines toward an energy-based, linear analysis of discrete physical systems are presented, based on previous work in systematic modeling using Lagrangian differential-algebraic equations (DAEs). Recent work in this area is extended by accommodating nonholonomic constraints and explicit inputs. An equilibrium postulate is proposed and equilibrium is characterized for static and steady-state conditions. Lagrangian DAEs are linearized using a local, indirect approach. Alternate descriptor formulations leading to different linear singular systems are compared and one formulation is determined to be a good foundation for future work in linear analysis using Lagrangian DAEs.

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.

Concurrent Multiple-Time-Scale Simulations With Improved Numerical Dissipation for Structural Dynamics

2013 ◽  
Author(s):  
Tejas Ruparel ◽  
Azim Eskandarian ◽  
James Lee
Keyword(s):  
Time Scale ◽  
Equations Of Motion ◽  
Time Integration ◽  
Nonholonomic Constraints ◽  
Numerical Dissipation ◽  
Multiple Time ◽  
Multiple Time Scale ◽  
Constraint Forces ◽  
Conservation Of Energy ◽  
Coupled Equations

Work presented in this paper describes the formulation for implementation of a concurrent multiple-time-scale integration method with improved numerical dissipation capabilities. This approach generalizes the previous Multiple Grid and Multiple Time-Scale (MGMT) Method [1] implemented for the Newmark family of algorithms. The framework is largely based upon the fundamental principles of Lagrange multipliers used to enforce workless nonholonomic constraints and Domain Decomposition Methods (DDM) to obtain coupled equations of motion for distinct regions of a continuous domain. These methods when combined together systematically yield constraint forces that not only ensure conservation of energy but also enforce continuity of velocities across the interfaces. Multiple grid connections between (non-conforming) sub-domains are handled using Mortar elements whereas coupled multiple-time-scale equations are derived for the Generalized-α Method [2]. We show that MGMT Method can be easily extended to incorporate the Generalized-α family of time integration algorithms, hence allowing selective discretization in space and time along with controlled numerical dissipation for distinct grids. We also show that interface energy across connecting sub-domains is identically zero, further assuring global energy balance and continuity of velocities across connecting sub-domains.

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.

Explicit Poincaré equations of motion for general constrained systems. Part I. Analytical results

2007 ◽  
Vol 463 (2082) ◽  
pp. 1421-1434 ◽  
Author(s):  
Firdaus E Udwadia ◽  
Phailaung Phohomsiri
Keyword(s):  
Equations Of Motion ◽  
Mechanical Systems ◽  
Nonholonomic Constraints ◽  
Constrained Systems ◽  
Analytical Results ◽  
Multi Body ◽  
D'alembert's Principle

This paper gives the general constrained Poincaré equations of motion for mechanical systems subjected to holonomic and/or nonholonomic constraints that may or may not satisfy d'Alembert's principle at each instant of time. It also extends Gauss's principle of least constraint to include quasi-accelerations when the constraints are ideal, thereby expanding the compass of this principle considerably. The new equations provide deeper insights into the dynamics of multi-body systems and point to new ways for controlling them.

Sign in / Sign up

Export Citation Format

Share Document