Instantaneous impulses in multibody dynamics

2015 ◽  
Vol 22 (4) ◽  
pp. 581-635
Author(s):  
P Lidström
Keyword(s):  
Multibody Dynamics ◽  
Virtual Work ◽  
Mass Matrix ◽  
Full Rank ◽  
Rigid Bodies ◽  
Unilateral Constraints ◽  
Lagrange’S Equations ◽  
Bilateral Constraints ◽  
The Impact ◽  
Lagrange's Equations

This paper considers instantaneous impulses in multibody dynamics. Instantaneous impulses may act on the multibody from its exterior or they may appear in its interior as a consequence of two of its parts interacting by an impact imposed by a unilateral constraint. The theory is based on the Euler laws of instantaneous impulses, which may be seen as a complement to the Euler laws for regular motions. Based on these laws, and specific continuum properties of the quantities involved, local balance laws for momentum and moment of momentum, involving instantaneous impulses and introducing the Cauchy impulse tensor, are derived. Thermodynamical restrictions on the impulse tensor are formulated based on the dissipation inequality. By stating a principle of virtual work for instantaneous impulses, and demonstrating its equivalence to Euler’s laws, Lagrange’s equations are derived. Lagrange’s equations are convenient to use in the case of multibody dynamics containing rigid as well as flexible parts. A central theme of this paper is the discussion of the interaction between parts of the multibody and their relation to geometrical and kinematical constraints. This interaction is severely affected by the presence of friction, which is notoriously difficult to handle. In a preparation for this discussion we first consider the one-point impact between two rigid bodies. The importance of the so-called impact tensor for this problem is demonstrated. In order to be able to handle the impact laws of Poisson and Stonge, an impact process, governed by a system of ordinary differential equations, is defined. Within this model phenomena, such as slip stop, slip start and slip direction reversal, may be handled. For a multibody with an arbitrary number of parts and multiple impacts, the situation is much more complicated and certain simplifications have to be introduced. Equations of motion for a multibody, consisting of rigid parts and in the presence of ideal bilateral constraints and unilateral constraints involving friction, are formulated. Unique solutions are obtained, granted that the mass matrix of the multibody system is non-singular, the constraint matrices satisfy specific full rank conditions and that the friction is not too high.

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.

Equivalence of Lagrange’s equations for non-material volume and the principle of virtual work in ALE formulation

2019 ◽  
Vol 231 (3) ◽  
pp. 1141-1157 ◽  
Author(s):  
Kai-Dong Chen ◽  
Jia-Peng Liu ◽  
Jia-Qi Chen ◽  
Xiao-Yu Zhong ◽  
Aki Mikkola ◽  
...  
Keyword(s):  
Virtual Work ◽  
Principle Of Virtual Work ◽  
Lagrange’S Equations ◽  
Ale Formulation ◽  
Material Volume ◽  
Lagrange's Equations

Steps Towards Non-Smooth Multibody Dynamics

2021 ◽  
Author(s):  
Friedrich Pfeiffer
Keyword(s):  
System Theory ◽  
Multibody Dynamics ◽  
Multibody System ◽  
Local Development ◽  
Impact Behavior ◽  
Unilateral Constraints ◽  
The Past ◽  
Bilateral Constraints ◽  
Large Systems ◽  
Contact And Impact

Abstract Before the background of many thoughts about contact and impact behavior with and without friction in the past centuries a comprehensive theory appeared not before the second half of the last century, mainly connected with the names of Moreau in Montpellier and Panagiotopoulos in Thessaloniki. My former Institute has been part of this evolution focusing on non-smooth multibody dynamics and on large systems. The local development from simple impact to complex contact systems including all possible contact details will be subject of the paper, considering also the necessary mathematical evolution from classical multibody system theory with bilateral constraints and single-valued forces to non-smooth multibody system theory with unilateral constraints and set-valued forces. Paper will be illustrated by practical examples.

scholarly journals On Euler-Lagrange's Equations: A New Approach

2020 ◽  
Vol 21 (2) ◽  
pp. 359
Author(s):  
G. E. O. Giacaglia ◽  
W. Q. Lamas
Keyword(s):  
Differential Geometry ◽  
Dynamical Systems ◽  
Linear Algebra ◽  
Mechanical Systems ◽  
Rigid Bodies ◽  
New Approach ◽  
New Formalism ◽  
Separate Part ◽  
Lagrange’S Equations ◽  
Lagrange's Equations

A new formalism is proposed to study the dynamics of mechanical systems composed of N connected rigid bodies, by introducing the concept of $6N$-dimensional composed vectors. The approach is based on previous works by the authors where a complete formalism was developed by means of differential geometry, linear algebra, and dynamical systems usual concepts. This new formalism is a method for the description of mechanical systems as a whole and not as each separate part. Euler-Lagrange's Equations are easily obtained by means of this formalism.

A New O(n) Method for Inverting the Mass Matrix for Serial Chains Composed of Rigid Bodies

2005 ◽  
Author(s):  
Yunfeng Wang ◽  
Gregory S. Chirikjian
Keyword(s):  
Multibody Dynamics ◽  
Inverse Dynamics ◽  
Mass Matrix ◽  
Rotation Group ◽  
Rigid Bodies ◽  
Euclidean Group ◽  
The Past ◽  
Lumped Masses ◽  
Serial Chains ◽  
And Robotics

Over the past several decades a number of O(n) methods for forward and inverse dynamics computations have been developed in the multibody dynamics and robotics literature. In this paper, a method developed in 1973 by Fixman for O(n) computation of the mass-matrix determinant for a polymer chain consisting of point masses is adapted and modified. In other recent papers, we and our collaborators recently extended this method in order for Fixman’s results to be applicable to robotic manipulator models with lumped masses. In the present paper we extend these ideas further to the case of serial chains composed of rigid-bodies. This requires the use of relatively deep mathematics associated with the rotation group, SO(3), and the special Euclidean group, SE(3), and how to differentiate functions of group-valued argument.

scholarly journals Derivation of Equations for Flexible Multibody Systems in Terms of Quasi-Coordinates from the Extended Hamilton’s Principle

1993 ◽  
Vol 1 (2) ◽  
pp. 107-119 ◽  
Author(s):  
L. Meirovitch
Keyword(s):  
Multibody Systems ◽  
Equations Of Motion ◽  
Rigid Bodies ◽  
Flexible Multibody Systems ◽  
Hamilton’S Principle ◽  
Hamilton's Principle ◽  
Lagrange’S Equations ◽  
Extended Hamilton’S Principle ◽  
Flexible Multibody ◽  
Lagrange's Equations

Early derivations of the equations of motion for single rigid bodies, single flexible bodies, and flexible multibody systems in terms of quasi-coordinates have been carried out in two stages. The first consists of the use of the extended Hamilton’s principle to derive standard Lagrange’s equations in terms of generalized coordinates and the second represents a transformation of the Lagrange’s equations to equations in terms of quasi-coordinates. In this article, hybrid (ordinary and partial) differential equations for flexible multibody systems are derived in terms of quasi-coordinates directly from the extended Hamilton's principle. The approach has beneficial implications in an eventual spatial discretization of the problem.

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