Abstract
We recall that a body B is a collection of material points P: = B = {P}. In general, all of the particles of a body may change their relative positions in B. and in this case the body is said to be deformable. At the other extreme, a rigid body is a body with the property that the straight line distance between every pair of its particles is constant in time. This idealization of a body that cannot be deformed, however great may be the forces and torques that act upon it, is so intuitively natural that it is often used without mention. The reader surely will recognize that our basic definition of a reference frame embodied the concept of rigidity. In fact, there were several occasions in Chapter 1 where the concept was quietly invoked. In particular, it was tacitly supposed for the mechanical device shown in Fig. 1.3 that the radius R of the wheel and the length L of the hinged rod did not vary with time. These are typical examples of rigid bodies whose motions will be investigated in this chapter. The kinematics of a rigid body in general motion in space will be studied. The main objective will be to learn how the velocity and acceleration of the particles of a rigid body are related to the translational and rotational parts of its motion.
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References
1. Classical Sources
Chasles, M, Note sur les propriétés générales du système de deux corps semblables entr’eux et placés d’une manière quelconque dans lèespace; et sur le déplacement fini ou infiniment petit d’un corps solide libre. Bulletin des Sciences Mathématiques par Férussac 14, 321–326 (1830). It is interesting that roughly four years later, L. Poinsot published his famous geometrical theory of the rotation of a rigid body about a fixed point, in which he states the screw theorem without reference to Chasles. This preliminary version, published in 1834, was expanded in two subsequent Papers that appcared much later: Poinsot, L., Théorie nouvelle de la rotation des corps, Liouville’s Journal de, Mathématiques 16, 9–129, 289–336 (1851). See also Ball, R. S., The Theory of Screws. Cambridge University Press, Cambridge, 1900. This is an advanced treatise on the mathematical theory of screw motion. The thorough annotated bibliography and Ballés parablc on the dynamical motions of a rigid body make useful and enjoyable reading Tor anyone interested in tbe history of mechanics. Geometrical constructions of the kinematical theorems on rigid body motion may be found in the book by Routh, E. J., Dynamics of a System of Rigid Budies, Dover, New York, 1960, Chapter 5, 184–229.
Euler, L., Mémoires de léAcadémie de Berlin, 1750; and Novi Commentaires de Saint-Pétersbourg 20, 189 (1775). See the advanced treatise by Whittaker, E. T., Analytical Dynamics, Cambridge University Press, Cambridge, 1927, Chapter 1.
2. Sources for Finite Rotations
Beatty, M., Vector representation of rigid body rotation, American Journal of Physics 31, 134–135 (1963).
Grubin, C., Vector representation of rigid body rotation, American Journal of Physics 30, 416–417 (1962). Equation (2.7) is derived by solving a simple vector differential equation as described in Problem 2.5.
Schwartz, H., Derivation of the matrix of rotation about a given direction as a simple exercise in matrix algebra, American Journal of Physics 31, 730–731 (1963). Equation (2.7) is derived by matrix methods.
3. Engineering Applications Sources
Long, R., Engineering Science Mechanics, Prentice-Hall, Englewood Cliffs, New Jersey, 1963. Chapter 6 treats the theory of rigid body motion.
Meriam, J. L., Dynamics, 2nd Edition, Wiley, New York, 1975. Many excellent collateral problems and additional examples of engineering interest may be found in Chapters 5 and 7.
Shames, I., Engineering Mechanics, Vol. 2: Dynamics, 2nd Edition, Prentice-Hall, Englewood Cliffs, New Jersey, 1966. Chapter 15 is a good source for similar problems and examples useful for parallel study.
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Beatty, M.F. (1986). Kinematics of Rigid Body Motion. In: Principles of Engineering Mechanics. Mathematical Concepts and Methods in Science and Engineering, vol 32. Springer, Boston, MA. https://doi.org/10.1007/978-1-4899-7285-9_2
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