Lagrange’s Equations, Hamilton’s Equations, and Kane’s Equations: Interrelations, Energy Integrals, and a Variational Principle

1995 ◽  
Vol 62 (2) ◽  
pp. 505-510 ◽  
Author(s):  
D. L. Mingori
Keyword(s):  
Analytical Mechanics ◽  
Energy Integral ◽  
Governing Equations ◽  
Kane’S Equations ◽  
Kane's Equations ◽  
Lagrange’S Equations ◽  
Energy Integrals ◽  
Hamilton's Equations ◽  
Lagrange's Equations ◽  
Hamilton’S Equations

A new viewpoint is suggested for expressing the governing equations of analytical mechanics. This viewpoint establishes a convenient framework for examining the relationships among Lagrange’s equations, Hamilton’s equations, and Kane’s equations. The conditions which must be satisfied for the existence of an energy integral in the context of Kane’s equations are clarified, and a generalized form of Hamilton’s Principle is presented. Generalized speeds replace generalized velocities as the velocity variables in the formulation. The development considers holonomic systems in which the generalized forces are derivable from a potential function.

Kane’s Equations With Undetermined Multipliers—Application to Constrained Multibody Systems

1987 ◽  
Vol 54 (2) ◽  
pp. 424-429 ◽  
Author(s):  
J. T. Wang ◽  
R. L. Huston
Keyword(s):  
Multibody Systems ◽  
Automated Analysis ◽  
Numerical Analyses ◽  
Constrained Multibody Systems ◽  
Controlled Systems ◽  
Kane’S Equations ◽  
Kane's Equations ◽  
Lagrange’S Equations ◽  
Lagrange's Equations

A procedure for automated analysis of constrained multibody systems is presented. The procedure is based upon Kane’s equations and the concept of undetermined multipliers. It is applicable with both free and controlled systems. As with Lagrange’s equations, the multipliers are identified as scalar components of constraining forces and moments. The advantage of using Kane’s equations is that they are ideally suited for development of algorithms for numerical analyses. Also, generalized speeds and quasi-coordinates are readily accommodated. A simple example illustrating the concepts is presented.

Dynamics of Large Constrained Flexible Structures

1988 ◽  
Vol 110 (1) ◽  
pp. 78-83 ◽  
Author(s):  
F. M. L. Amirouche ◽  
R. L. Huston
Keyword(s):  
Structure Analysis ◽  
Flexible Structures ◽  
Orthogonal Complement ◽  
Space Structures ◽  
Governing Equations ◽  
Finite Segment ◽  
Closed Loops ◽  
Constraint Equations ◽  
Kane’S Equations ◽  
Kane's Equations

This paper presents an automated procedure useful in the study of large constrained flexible structures, undergoing large specified motions. The structure is looked upon as a “partially open tree” system, containing closed loops in some of the branches. The governing equations are developed using Kane’s equations as formulated by Huston et al. The accommodation of the constraint equations is based on the use of orthogonal complement arrays. The flexibility and oscillations of the bodies is modelled using finite segment modelling, structure analysis, and scaling techniques. The procedures developed are expected to be useful in applications including robotics, space structures, and biosystems.

A geometrical interpretation of Kane’s Equations

1992 ◽  
Vol 436 (1896) ◽  
pp. 69-87 ◽  
Keyword(s):  
Tangent Space ◽  
Equations Of Motion ◽  
Rigid Bodies ◽  
Analytical Mechanics ◽  
Kane’S Method ◽  
Kane's Method ◽  
Kane’S Equations ◽  
Kane's Equations ◽  
Spanning Set ◽  
Geometric Picture

The method for the development of the equations of motion for systems of constrained particles and rigid bodies, developed by T. R. Kane and called Kane’s Equations, is discussed from a geometric viewpoint. It is shown that what Kane calls partial velocities and partial angular velocities may be interpreted as components of tangent vectors to the system’s configuration manifold. The geometric picture, when attached to Kane’s formalism shows that Kane’s Equations are projections of the Newton-Euler equations of motion onto a spanning set of the configuration manifold’s tangent space. One advantage of Kane’s method, is that both non-holonomic and non-conservative systems are easily included in the same formalism. This easily follows from the geometry. It is also shown that by transformation to an orthogonal spanning set, the equations can be diagonalized in terms of what Kane calls the generalized speeds. A further advantage of the geometric picture lies in the treatment of constraint forces which can be expanded in terms of a spanning set for the orthogonal complement of the configuration tangent space. In all these developments, explicit use is made of a concrete realization of the multidimensional vectors which are called K -vectors for a K -component system. It is argued that the current presentation also provides a clear tutorial route to Kane’s method for those schooled in classical analytical mechanics.

scholarly journals Numerical Modeling of the Motion of the Flat Textile Structures

2011 ◽  
Vol 2011 ◽  
pp. 1-8
Author(s):  
Piotr Szablewski
Keyword(s):  
Numerical Modeling ◽  
Boundary Conditions ◽  
Discrete Model ◽  
Gravity Force ◽  
Dynamic Conditions ◽  
Textile Structure ◽  
Sewing Thread ◽  
Lagrange’S Equations ◽  
Lagrange's Equations ◽  
Time Boundary

In many problems from the field of textile engineering (e.g., fabric folding, motion of the sewing thread) it is necessary to investigate the motion of the objects in dynamic conditions, taking into consideration the influence of the forces of inertia and changing in the time boundary conditions. This paper deals with the model analysis of the motion of the flat textile structure using Lagrange's equations in two variants: without constraints and with constraints. The motion of the objects is under the influence of the gravity force. Lagrange's equations have been used for discrete model of the structure.

scholarly journals IV. On some applications of dynamical principles to physical phenomena

1885 ◽  
Vol 176 ◽  
pp. 307-342 ◽  
Keyword(s):  
Physical Phenomenon ◽  
Good Deal ◽  
Rigid Bodies ◽  
Great Success ◽  
Material System ◽  
Electric Currents ◽  
Conservation Of Energy ◽  
Lagrange’S Equations ◽  
Physical Phenomena ◽  
Lagrange's Equations

1. The tendency to apply dynamical principles and methods to explain physical phenomena has steadily increased ever since the discovery of the principle of the Conservation of Energy. This discovery called attention to the ready conversion of the energy of visible motion into such apparently dissimilar things as heat and electric currents, and led almost irresistibly to the conclusion that these too are forms of kinetic energy, though the moving bodies must be infinitesimally small in comparison with the bodies which form the moving pieces of any of the structures or machines with which we are acquainted. As soon as this conception of heat and electricity was reached mathematicians began to apply to them the dynamical method of the Con­servation of Energy, and many physical phenomena were shown to be related to each other, and others predicted by the use of this principle; thus, to take an example, the induction of electric currents by a moving magnet was shown by von Helmholtz to be a necessary consequence of the fact that an electric current produces a magnetic field. Of late years things have been carried still further; thus Sir William Thomson in many of his later papers, and especially in his address to the British Association at Montreal on “Steps towards a Kinetic Theory of Matter,” has devoted a good deal of attention to the description of machines capable of producing effects analogous to some physical phenomenon, such, for example, as the rotation of the plane of polarisation of light by quartz and other crystals. For these reasons the view (which we owe to the principle of the Conservation of Energy) that every physical phenomenon admits of a dynamical explanation is one that will hardly be questioned at the present time. We may look on the matter (including, if necessary, the ether) which plays a part in any physical phenomenon as forming a material system and study the dynamics of this system by means of any of the methods which we apply to the ordinary systems in the Dynamics of Rigid Bodies. As we do not know much about the structure of the systems we can only hope to obtain useful results by using methods which do not require an exact knowledge of the mechanism of the system. The method of the Conservation of Energy is such a method, but there are others which hardly require a greater knowledge of the structure of the system and yet are capable of giving us more definite information than that principle when used in the ordinary way. Lagrange's equations and Hamilton's method of Varying Action are methods of this kind, and it is the object of this paper to apply these methods to study the transformations of some of the forms of energy, and to show how useful they are for coordinating results of very different kinds as well as for suggesting new phenomena. A good many of the results which we shall get have been or can be got by the use of the ordinary principle of Thermodynamics, and it is obvious that this principle must have close relations with any method based on considerations about energy. Lagrange’s equations were used with great success by Maxwell in his ‘Treatise on Electricity and Magnetism,’ vol. ii., chaps. 6, 7, 8, to find the equations of the electromagnetic field.

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