scholarly journals Proof of Lagrange's Equations of Motion, &c.

Nature ◽  
1903 ◽  
Vol 67 (1740) ◽  
pp. 415-415
Author(s):  
R. F. W.
Keyword(s):  
Equations Of Motion ◽  
Lagrange’S Equations ◽  
Lagrange's Equations

On the Kinematics and Kinetics of Mechanical Seals, Rotors, and Wobbling Bodies

2005 ◽  
Author(s):  
Itzhak Green
Keyword(s):  
Equations Of Motion ◽  
Mechanical Seals ◽  
Kinematical Constraint ◽  
Lagrange’S Equations ◽  
Dynamic Analyses ◽  
Kinematical Model ◽  
Kinetics Of ◽  
Kinematics And Kinetics ◽  
Lagrange's Equations

Mechanical seals, rotors, and wobbling bodies are characterized by a kinematical constraint that prevents them from having integral motion with respect to their own frame. A valid kinematical model is a prerequisite to subsequent dynamic analyses. Three previous works have suggested distinctly different kinematical models to the same problem. The analysis herein presents yet another kinematical model that preserves (actually enforces) the proper kinematical constraint. The outcome reaffirms one of the previous models. The equations of motion are derived using Lagrange’s equations to complement results obtained previously by Newton-Euler mechanics.

scholarly journals Proof of Lagrange's Equations of Motion, &c.

Nature ◽  
1903 ◽  
Vol 67 (1740) ◽  
pp. 415-415
Author(s):  
W. MCF. ORR
Keyword(s):  
Equations Of Motion ◽  
Lagrange’S Equations ◽  
Lagrange's Equations

scholarly journals Finite Element Method-Based Elastic Analysis of Multibody Systems. A Review

Mathematics ◽  
2022 ◽  
Vol 10 (2) ◽  
pp. 257
Author(s):  
Sorin Vlase ◽  
Marin Marin ◽  
Negrean Iuliu
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Mechanical System ◽  
Multibody Systems ◽  
Equations Of Motion ◽  
Alternative Methods ◽  
Engineering Applications ◽  
Lagrange’S Equations ◽  
Lagrange's Equations ◽  
Element Method

This paper presents the main analytical methods, in the context of current developments in the study of complex multibody systems, to obtain evolution equations for a multibody system with deformable elements. The method used for analysis is the finite element method. To write the equations of motion, the most used methods are presented, namely the Lagrange equations method, the Gibbs–Appell equations, Maggi’s formalism and Hamilton’s equations. While the method of Lagrange’s equations is well documented, other methods have only begun to show their potential in recent times, when complex technical applications have revealed some of their advantages. This paper aims to present, in parallel, all these methods, which are more often used together with some of their engineering applications. The main advantages and disadvantages are comparatively presented. For a mechanical system that has certain peculiarities, it is possible that the alternative methods offered by analytical mechanics such as Lagrange’s equations have some advantages. These advantages can lead to computer time savings for concrete engineering applications. All these methods are alternative ways to obtain the equations of motion and response time of the studied systems. The difference between them consists only in the way of describing the systems and the application of the fundamental theorems of mechanics. However, this difference can be used to save time in modeling and analyzing systems, which is important in designing current engineering complex systems. The specifics of the analyzed mechanical system can guide us to use one of the methods presented in order to benefit from the advantages offered.

A Large Deformation Finite Element for Pipes Conveying Fluid Based on the Absolute Nodal Coordinate Formulation

2007 ◽  
Author(s):  
Michael Stangl ◽  
Johannes Gerstmayr ◽  
Hans Irschik
Keyword(s):  
Finite Element ◽  
Large Deformation ◽  
Absolute Nodal Coordinate Formulation ◽  
Equations Of Motion ◽  
Directional Derivatives ◽  
Absolute Nodal Coordinate ◽  
Lagrange’S Equations ◽  
The Absolute ◽  
Lagrange's Equations ◽  
Conveying Fluid

A novel pipe finite element conveying fluid, suitable for modeling large deformations in the framework of Bernoulli Euler beam theory, is presented. The element is based on a third order planar beam finite element, introduced by Berzeri and Shabana, on basis of the absolute nodal coordinate formulation. The equations of motion for the pipe-element are derived using an extended version of Lagrange’s equations of the second kind for taking into account the flow of fluids, in contrast to the literature, where most derivations are based on Hamilton’s Principle or Newtonian approaches. The advantage of this element in comparison to classical large deformation beam elements, which are based on rotations, is the direct interpolation of position and directional derivatives, which simplifies the equations of motion considerably. As an advantage Lagrange’s equations of the second kind offer a convenient connection for introducing fluids into multibody dynamic systems. Standard numerical examples show the convergence of the deformation for increasing number of elements. For a cantilever pipe, the critical flow velocities for increasing number of pipe elements are compared to existing works, based on Euler elastica beams and moving discrete masses. The results show good agreements with the reference solutions applying only a small number of pipe finite elements.

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