Note on an apparently forgotten theorem about solid rigid dynamics

Chica, Francisco Javier Gil; Polo, Manuel Pérez; Molina, Manuel Pérez

doi:10.1088/0143-0807/35/4/045003

Cited by 6 publications

(5 citation statements)

References 3 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…They also derived the equation of motion based on the screw theory of a rigid body in space. Similarly, in Chica et al [9], the same result is derived in a more current language. The authors state that this important property seems to be forgotten, and up to the publication date of this paper, they had not found an appropriate demonstration.…”

Section: Introductionsupporting

confidence: 66%

Equimomental systems representations of point-masses of planar rigid-bodies

Romero

Vieira

Martins

2023

Preprint

View full text Add to dashboard Cite

Equimomental point mass systems facilitate the definition of the inertial properties of a rigid body and simplify the dynamic analysis of mechanisms and machines. In this work, the equimomental systems of three point-masses of planar rigid bodies are investigated using the concept of pseudo-inertia matrix. It is found that given a planar rigid body, it is always possible to determine an equimomental system of three equal masses located at the vertices of an isosceles triangle. A procedure is presented to determine equimomental systems with different masses, guaranteeing that the masses are positive. It is shown that it is always possible to choose an equimomental system of three point-masses located at the vertices of an isosceles triangle with a prescribed position of one mass. The conditions for prescribing the position of two and three point-masses are also investigated. A first idealized example shows the step-by-step procedure for determining an equimomental system of three point-masses of a planar rigid body. The proposed model is applied to a symmetric connecting rod in a second example due to its wide use in combustion engines. A third example shows an equimomental system of an asymmetric bucket of an excavator.

show abstract

Section: Introductionsupporting

confidence: 66%

Equimomental systems representations of point-masses of planar rigid-bodies

Romero

Vieira

Martins

2023

Preprint

View full text Add to dashboard Cite

show abstract

“…The center of mass enters only into the first term of the Lagrangian. So, variation of the action with respect to y 0 gives the equation (16), whose solution we already know, see equation (17). It is said that three functions y 0 i describe the translational degrees of freedom of a rigid body.…”

Section: Ab Ab B a A A A A B Ab B Amentioning

confidence: 99%

“…The theory of a rigid body, including the convenient equations of motion for the independent degrees of freedom, was formulated by Euler, Lagrange and Poisson already at the dawn of the development mechanics [1][2][3], and enters now as a chapter in the standard books on classical mechanics [4][5][6][7][8][9][10][11]. However, a didactically systematic formulation of the equations of motion is regarded not as an easy task [12][13][14][15][16][17][18]. For instance, J E Marsden, D D Holm and T S Ratiu in their work [19] dated 1998 write: 'It was already clear in the last century that certain mechanical systems resist the usual canonical formalism, either Hamiltonian or Lagrangian, outlined in the first paragraph.…”

Section: Introductionmentioning

confidence: 99%

Lagrangian and Hamiltonian formulations of asymmetric rigid body, considered as a constrained system

Дериглазов

2023

Eur. J. Phys.

View full text Add to dashboard Cite

This work is devoted to a systematic exposition of the dynamics of a rigid body, considered as a system with kinematic constraints. Having accepted the variational problem in accordance with this, we no longer need any additional postulates or assumptions about the behavior of the rigid body. All the basic quantities and characteristics of a rigid body, as well as the equations of motion and integrals of motion, are obtained from the variational problem by direct and unequivocal calculations within the framework of standard methods of classical mechanics. Several equivalent forms for the equations of motion of rotational degrees of freedom are deduced and discussed on this basis. Using the resulting formulation, we revise some cases of integrability, and discuss a number of peculiar properties, that are not always taken into account when formulating the laws of motion of a rigid body.

show abstract

“…This can be repeated for the other three extended vectors showing that the set of all four vectors q 1 ,q 2 ,q 3 , andq 4 comprises a set of mutually orthogonal unit vectors, sometimes called an orthonormal frame. Finally, since the group O(4) acts transitively on orthonormal frames, this set of four point-masses can be transformed into any other orthonormal frame by some matrix U ∈ O (4). In particular, they may be transformed into the vertices of a regular tetrahedron given in the previous Section.…”

Section: Theorem 1 Let ξ Be the Inertia Matrix Of A General Rigid Bodmentioning

confidence: 99%

“…They claimed, "This important property seems to have been forgotten, as we have not found any proper demonstration at all." [4].…”

Section: Introductionmentioning

confidence: 99%

Rigid body dynamics using equimomental systems of point-masses

Laus

Selig

2019

Acta Mech

View full text Add to dashboard Cite

The inertia matrix of any rigid body is the same as the inertia matrix of some system of four pointmasses. In this work, the possible disposition of these point-masses is investigated. It is found that every system of possible point-masses with the same inertia matrix can be parameterised by the elements of the orthogonal group in four-dimensional modulo-permutation of the points. It is shown that given a fixed inertia matrix, it is possible to find a system of point-masses with the same inertia matrix but where one of the points is located at some arbitrary point. It is also possible to place two point-masses on an arbitrary line or three of the points on an arbitrary plane. The possibility of placing some of the point-masses at infinity is also investigated. Applications of these ideas to rigid body dynamics are considered. The equation of motion for a rigid body is derived in terms of a system of four point-masses. These turn out to be very simple when written in a 6-vector notation.

show abstract

Note on an apparently forgotten theorem about solid rigid dynamics

Abstract: We re-derive a general procedure to substitute any rigid body by an equivalent system of exactly four masses, located at vertices of an irregular tetrahedron.

Cited by 6 publications

References 3 publications

Equimomental systems representations of point-masses of planar rigid-bodies

Equimomental systems representations of point-masses of planar rigid-bodies

Lagrangian and Hamiltonian formulations of asymmetric rigid body, considered as a constrained system

Rigid body dynamics using equimomental systems of point-masses

Contact Info

Product

Resources

About