THE LAGRANGIAN DERIVATION OF KANE’S EQUATIONS

2007 ◽  
Vol 31 (4) ◽  
pp. 407-420 ◽  
Author(s):  
Kourosh Parsa
Keyword(s):  
Kinetic Energy ◽  
Partial Derivative ◽  
Equations Of Motion ◽  
Forward Kinematics ◽  
Body System ◽  
Generalized Coordinates ◽  
Kane’S Equations ◽  
Kane's Equations ◽  
Multi Body ◽  
Velocity Level

The Lagrangian approach to the development of dynamics equations for a multi-body system, constrained or otherwise, requires solving the forward kinematics of the system at velocity level in order to derive the kinetic energy of the system. The kinetic-energy expression should then be differentiated multiple times to derive the equations of motion of the system. Among these differentiations, the partial derivative of kinetic energy with respect to the system generalized coordinates is specially cumbersome. In this paper, we will derive this partial derivative using a novel kinematic relation for the partial derivative of angular velocity with respect to the system generalized coordinates. It will be shown that, as a result of the use of this relation, the equations of motion of the system are directly derived in the form of Kane’s equations.

scholarly journals Classical n-body system in geometrical and volume variables: I. Three-body case

2021 ◽  
pp. 2150140
Author(s):  
A. M. Escobar-Ruiz ◽  
R. Linares ◽  
Alexander V. Turbiner ◽  
Willard Miller
Keyword(s):  
Kinetic Energy ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Center Of Mass ◽  
Symmetric Polynomial ◽  
Body System ◽  
Generalized Coordinates ◽  
New Variables ◽  
Three Body ◽  
Mass Dependent

We consider the classical three-body system with [Formula: see text] degrees of freedom [Formula: see text] at zero total angular momentum. The study is restricted to potentials [Formula: see text] that depend solely on relative (mutual) distances [Formula: see text] between bodies. Following the proposal by J. L. Lagrange, in the center-of-mass frame we introduce the relative distances (complemented by angles) as generalized coordinates and show that the kinetic energy does not depend on [Formula: see text], confirming results by Murnaghan (1936) at [Formula: see text] and van Kampen–Wintner (1937) at [Formula: see text], where it corresponds to a 3D solid body. Realizing [Formula: see text]-symmetry [Formula: see text], we introduce new variables [Formula: see text], which allows us to make the tensor of inertia nonsingular for binary collisions. In these variables the kinetic energy is a polynomial function in the [Formula: see text]-phase space. The three-body positions form a triangle (of interaction) and the kinetic energy is [Formula: see text]-permutationally invariant with respect to interchange of body positions and masses (as well as with respect to interchange of edges of the triangle and masses). For equal masses, we use lowest order symmetric polynomial invariants of [Formula: see text] to define new generalized coordinates, they are called the geometrical variables. Two of them of the lowest order (sum of squares of sides of triangle and square of the area) are called volume variables. It is shown that for potentials which depend on geometrical variables only (i) and those which depend on mass-dependent volume variables alone (ii), the Hamilton’s equations of motion can be considered as being relatively simple. We study three examples in some detail: (I) three-body Newton gravity in [Formula: see text], (II) three-body choreography in [Formula: see text] on the algebraic lemniscate by Fujiwara et al., where the problem becomes one-dimensional in the geometrical variables and (III) the (an)harmonic oscillator.

A fresh insight into Kane's equations of motion

Robotica ◽  
2015 ◽  
Vol 35 (3) ◽  
pp. 498-510 ◽  
Author(s):  
H. Nejat Pishkenari ◽  
S. A. Yousefsani ◽  
A. L. Gaskarimahalle ◽  
S. B. G. Oskouei
Keyword(s):  
Dynamic Systems ◽  
Rapid Development ◽  
Equations Of Motion ◽  
Nonholonomic Constraints ◽  
Kane’S Equations ◽  
Kane's Equations ◽  
The Matrix ◽  
Impulsive Forces ◽  
Multi Body ◽  
Fresh Insight

SUMMARYWith rapid development of methods for dynamic systems modeling, those with less computation effort are becoming increasingly attractive for different applications. This paper introduces a new form of Kane's equations expressed in the matrix notation. The proposed form can efficiently lead to equations of motion of multi-body dynamic systems particularly those exposed to large number of nonholonomic constraints. This approach can be used in a recursive manner resulting in governing equations with considerably less computational operations. In addition to classic equations of motion, an efficient matrix form of impulse Kane formulations is derived for systems exposed to impulsive forces.

A Large Displacement Formulation Using Euler’s Angles

1999 ◽  
Author(s):  
G. Biakeu ◽  
F. Thouverez ◽  
J. P. Laine ◽  
L. Jezequel
Keyword(s):  
Kinetic Energy ◽  
Equations Of Motion ◽  
Static Solution ◽  
Element Formulation ◽  
First Order ◽  
Nonlinear Relation ◽  
Newton Raphson ◽  
Body Formulation ◽  
Multi Body ◽  
Euler’S Angles

Abstract The goal of this paper is to present a flexible multi-body formulation involving large displacements. This method is based on a separate discretisation of the kinetic and the internal energies. To introduce flexibility, we discretize the structure in elements (of two nodes): on each element of the beam discretisation, the local frame is defined using Euler’s angles. A finite element formulation is then applied to describe the evolution of these angles along the beam neutral fibre. For the kinetic energy, each element is cut into two rigid bars whose characteristics are given by a first order Taylor factorisation on the general kinetic energy expression. These bars are linked by a nonlinear relation. We obtain the equations of motion by applying the Lagrange’s equations to the system. These equations are solved using the Newmark method in dynamic and a Newton-Raphson technique while looking for a static solution. The method is then applied to very classic problems such as the curved beam problem proposed by authors such as Simo [6, 9], Lee [4] or the rotational rod presented by Avello [1] and Simo [7, 8] etc...

An Applied Form of Kane’s Equations of Motions

2005 ◽  
Author(s):  
Hossein Nejat Pishkenari ◽  
Amir Lotfi Gaskarimahalle ◽  
Seyed Babak Ghaemi Oskouei ◽  
Ali Meghdari
Keyword(s):  
Degrees Of Freedom ◽  
Control Unit ◽  
Generalized Coordinates ◽  
Kane’S Equations ◽  
Kane's Equations ◽  
The Matrix ◽  
Angular Velocities ◽  
New Form ◽  
Derivatives Of ◽  
Equations Of Motions

In this paper we have presented a new form of Kane’s equations. This new form is expressed in the matrix form with the components of partial derivatives of linear and angular velocities relative to the generalized speeds and generalized coordinates. The number of obtained equations is equal to the number of degrees of freedom represented in a closed form. Also the equations can be rearranged to appear only one of the time derivatives of generalized speeds in each equation. This form is appropriate especially when one intends to derive equations recursively. Hence in addition to the simplicity, the amount of calculations is noticeably reduced and also can be used in a control unit.

Dynamics Modeling of “CEDRA” Rescue Robot on Uneven Terrains

2004 ◽  
Author(s):  
A. Meghdari ◽  
S. H. Mahboobi ◽  
A. L. Gaskarimahalle
Keyword(s):  
Inverse Kinematics ◽  
High Mobility ◽  
Dynamical Analysis ◽  
Forward Kinematics ◽  
Rough Terrain ◽  
Body System ◽  
Dynamics Modeling ◽  
Wheeled Mobile Robots ◽  
Rescue Robot ◽  
Multi Body

In this paper an effective approach for kinematic and dynamic modeling of high mobility wheeled mobile robots (WMR) has been presented. As an example of these robots, the method has been applied on CEDRA rescue robot which is a complex, multibody mechanism. The model is derived for 6-DOF motions enabling movement in x, y, z directions, as well as pitch, roll and yaw rotations. Forward kinematics equations are derived using Denavit-Hartenberg method and the wheels Jacobian matrices. Moreover the inverse kinematics of the robot is obtained and solved for the wheel velocities and steering commands in terms of desired velocity, heading and measured link angles. Finally dynamical analysis of the rover has been thoroughly studied. Due to the complexity of this multi-body system especially on rough terrain, Kane’s method of dynamics has been used to model this problem. The approach has been developed in such a way that it can easily be extended to other mechanisms and rovers.

Extended Kane’s Equations for Nonholonomic Variable Mass System

1982 ◽  
Vol 49 (2) ◽  
pp. 429-431 ◽  
Author(s):  
Z.-M. Ge ◽  
Y.-H. Cheng
Keyword(s):  
Equations Of Motion ◽  
Variable Mass ◽  
Mass System ◽  
Variable Mass System ◽  
Kane’S Equations ◽  
Kane's Equations ◽  
Variable Mass Systems

An extension of Kane’s equations of motion for nonholonomic variable mass systems is presented. As an illustrative example, equations of motion are formulated for a rocket car.

Dynamic Analysis of Abrasive Filaments in Contact with Different Workpiece Geometries

2018 ◽  
Vol 12 (6) ◽  
pp. 892-900
Author(s):  
Eckart Uhlmann ◽  
◽  
Christian Sommerfeld
Keyword(s):  
Equations Of Motion ◽  
Machining Process ◽  
Surface Finishing ◽  
Body System ◽  
Single Filament ◽  
Contact Behavior ◽  
Motion System ◽  
Contact Behaviour ◽  
Plastic Matrix ◽  
Multi Body

Abrasive brushes are often used for surface finishing and deburring and consist of a brush body with fixed, highly flexible abrasive filaments. During the brushing process the highly flexible abrasive filaments deform tangentially and axially and adapt to the shape of the workpiece. The contact behaviour of abrasive brushes in the machining process is very complex and has been insufficiently investigated so far. Abrasive brushes consist of a brush body with fixed, highly flexible abrasive filaments and are often used for surface finishing and deburring. During the brushing process, the highly flexible abrasive filaments deform tangential and axial and adapt to the shape of the workpiece. The mentioned contact behavior of the abrasive brush during the machining process is complex, and has not yet been sufficiently investigated. To better understand the contact behavior and, thus, the brushing process, a model of an abrasive filament is proposed in this study. The model describes the dynamic behavior of a single filament in contact with different workpiece geometries. The filament is discretized into a multi-body system of rigid links connected with rotational springs and rotational dampers, and the workpiece is approximated by using a polynomial. The contact of the multi-body system representing the filament with the surface of the workpiece is described by using Hertz’s theory of elastic contact and Coulomb’s law of friction. Based on this, a system of equations of motion for the multi-body system is obtained by using Lagrangian mechanics. A numerical solution of the equation of motion system was calculated by using experimentally determined material and contact properties of the filament as a composite of a plastic matrix and abrasive grains. A comparison of the calculated results with experimental data yielded satisfactory agreement for the contact between the filament and different workpiece geometries.

scholarly journals Corrigendum: New Form of Kane's Equations of Motion for Constrained Systems

2007 ◽  
Vol 30 (1) ◽  
pp. 286-288 ◽  
Author(s):  
Carlos M. Roithmayr ◽  
Abdulrahman H. Bajodah ◽  
Dewey H. Hodges ◽  
Ye-Hwa Chen
Keyword(s):  
Equations Of Motion ◽  
Constrained Systems ◽  
Kane’S Equations ◽  
Kane's Equations ◽  
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