Crack dynamics via Lagrange's equations and generalized coordinates

2001 ◽  
Vol 148 (1-4) ◽  
pp. 79-92 ◽  
Author(s):  
J. K. Dienes
Keyword(s):  
Generalized Coordinates ◽  
Lagrange’S Equations ◽  
Crack Dynamics ◽  
Lagrange's Equations

A New Method for Dynamic Modeling of Flexible-Link Flexible-Joint Manipulators

2011 ◽  
Vol 134 (1) ◽  
Author(s):  
M. Vakil ◽  
R. Fotouhi ◽  
P. N. Nikiforuk
Keyword(s):  
Closed Form ◽  
Dynamic Model ◽  
Lagrange Multipliers ◽  
New Method ◽  
Flexible Link ◽  
Flexible Joint ◽  
Generalized Coordinates ◽  
Lagrange’S Equations ◽  
Mass Moment ◽  
Lagrange's Equations

In this article, by combining the assumed mode shape method and the Lagrange’s equations, a new and efficient method is introduced to obtain a closed-form finite dimensional dynamic model for planar Flexible-link Flexible-joint Manipulators (FFs). To derive the dynamic model, this new method separates (disassembles) a FF into two subsystems. The first subsystem is the counterpart of the FF but without joints’ flexibilities and rotors’ mass moment of inertias; this subsystem is referred to as a Flexible-link Rigid-joint manipulator (FR). The second subsystem has the joints’ flexibilities and rotors’ mass moment of inertias, which are excluded from the FR; this subsystem is called Flexible-Inertia entities (FI). While the method proposed here employs the Lagrange’s equations, it neither requires the derivation of the lengthy Lagrangian function nor its complex derivative calculations. This new method only requires the Lagrangain function evaluation and its derivative calculations for a Single Flexible link manipulator on a Moving base (SFM). By using the dynamic model of a SFM and the Lagrange multipliers, the dynamic model of the FR is first obtained in terms of the dependent generalized coordinates. This dynamic model is then projected into the tangent space of the constraint manifold by the use of the natural orthogonal complement of the Jacobian constraint matrix. Therefore, the dynamic model of the FR is obtained in terms of the independent generalized coordinates and without the Lagrange multipliers. Finally, the joints’ flexibilities and rotors’ mass moment of inertias, which are included in the FI, are added to the dynamic model of the FR and a closed-form dynamic model for the FF is derived. To verify this new method, the results of simulation examples, which are obtained from the proposed method, are compared with those of a full-nonlinear finite element analysis, where the comparisons indicate sound agreement

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.

scholarly journals Growing bubbles rising in line

2001 ◽  
Vol 5 (2) ◽  
pp. 65-73 ◽  
Author(s):  
John F. Harper
Keyword(s):  
Reynolds Numbers ◽  
Dissipation Function ◽  
Supersaturated Solution ◽  
Liquid Theory ◽  
High Reynolds Numbers ◽  
Lagrange’S Equations ◽  
Spherical Bubbles ◽  
High Reynolds ◽  
Lagrange's Equations

Over many years the author and others have given theories for bubbles rising in line in a liquid. Theory has usually suggested that the bubbles will tend towards a stable distance apart, but experiments have often showed them pairing off and sometimes coalescing. However, existing theory seems not to deal adequately with the case of bubbles growing as they rise, which they do if the liquid is boiling, or is a supersaturated solution of a gas, or simply because the pressure decreases with height. That omission is now addressed, for spherical bubbles rising at high Reynolds numbers. As the flow is then nearly irrotational, Lagrange's equations can be used with Rayleigh's dissipation function. The theory also works for bubbles shrinking as they rise because they dissolve.

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