Abstract
Newton’s Principia (1687) stands among the world’s greatest scientific and intellectual achievements, and justly so, but it does not address all of the general principles of mechanics. In Newton’s book there is no theory of general dynamical systems nor of rigid bodies, and nothing on the mechanics of deformable solid and fluid continua. Newton’s theory for mass points is just insufficiently general to deliver a unifying method for their study. More than half a century of research and struggle with solutions of special problems would pass before the first of the general principles of mechanics applicable to all bodies was discovered by Euler in 1750, and thereafter.
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References
BEATTY, M. F., On the motion of lineal bodies subject to linear damping, Journal of Applied Mechanics 64, 227–9 (1997). It is shown that the three special problems studied by Wilms (1995), all of which are described by precisely the same equation of motion, belong to a more general class of problems for which the equation of motion is exactly the same for every lineal rigid body regardless of its geometry. This result is presented in the text. The extension of these results to homogeneous, rigid plates and shells is given by Beatty, M.F., On the motion of thin plates and shells subject to Stokes damping, Mechanics Research Communications 29, 321–6 (2002).
CARROLL, M. M., Singular constraints in rigid-body dynamics, American Journal of Physics 52, 1010–12 (1984). Sometimes a seemingly simple problem of rigid body dynamics has a more subtle solution than one anticipates. This paper presents an interesting example. Here the ends of a rigid rod of length l = r2−r1 are smoothly constrained to move on concentric circles of radii r1 and r2 in the vertical plane, and the rod is released from rest in its horizontal position. It becomes quickly apparent in the problem formulation that the idea of smooth (frictionless) constraint reactions is inconsistent with Euler’s laws of motion. Carroll shows that this is due to a singularity in the constraint forces.
DUGAS, R., A History of Mechanics, Central Book Co., New York, 1955. Euler’s mechanics of a particle and of solid bodies are discussed in Chapters 3 (4 pages) and 6 (3 pages). The last describes Euler’s (1760) paper in which the important role of a moving body reference frame is emphasized. Euler thus derives his famous differential equations for the general motion of a rigid body referred to principal body axes, and also characterizes the general spatial motion of a rigid body by a decomposition into the motion of the center of mass and the motion relative to the center of mass. This book covers a great body of the history of mechanics from Aristotle (384–322 B.C.) through Einstein and modem physical theories of mechanics and quantum mechanics, which accounts for its terse and incomplete description of Euler’s contributions to the foundations of mechanics. In this regard, however, all reports pale in comparison with Truesdell’s studies of Euler’s classical works on the mechanics of all bodies, cited below.
FOX, E. A., Mechanics, Harper and Row, New York, 1967. This is a unified treatment of the principles of mechanics of a particle and both rigid and deformable bodies, including an introduction to continuum mechanics. Equilibrium and motion of a rigid body are discussed in Chapters 6 and 10.
GREENWOOD, D. T., Principles of Dynamics, Prentice-Hall, Englewood Cliffs, New Jersey, 1965. Euler’s dynamical equations for a rigid body are developed in Chapter 8. Poinsot’s interesting geometrical construction characterizing the motion of rolling cones, the free motion of general rigid bodies, infinitesimal stability, and the motion of a spinning top are presented as well.
LONG, R. R., Engineering Science Mechanics, Prentice-Hall, Englewood Cliffs, New Jersey, 1963. Vector and tensor methods are integrated to study topics in engineering mechanics. Chapters 6 and 7 cover rigid body motion. Overall, this is an excellent resource for collateral reading and for additional problems and examples. Several problems below are modeled after those presented here.
MACMILLAN, W. D., Dynamics of Rigid Bodies, Dover, New York, 1960. The rolling, pivoting, and slipping motion of a sphere on a surface, possibly curved, is studied in Chapter 8. The billiard ball problem is a special case.
MERIAM, J. L., and KRAIGE, L. G., Engineering Mechanics. Vol. 2, Dynamics, 3rd Edition, Wiley, New York, 1992. Plane motion of a rigid body, including work-energy, impulse-momentum, and impact topics, is discussed in Chapter 6. Three dimensional problems that include gyroscopic motion are introduced in Chapter 7. There are many problems and examples useful for collateral study.
McGILL, D. J., and KING, W. W., Engineering Mechanics. An Introduction to Dynamics, 2nd Edition, PWS-Kent, Boston, 1989. This is a well-illustrated text with numerous engineering applications and exercises. Euler’s equations in two and three dimensions are studied in Chapters 4, 5, and 7.
NOLL, W., On the Foundation of the Mechanics of Continuous Media, Technical Report No. 17, Department of Mathematics, Carnegie Institute of Technology (now Carnegie-Mellon University), Pittsburgh, June 1957. The principle of mutual action is here derived within a formal mathematical framework for nonlinear continua. Much of this material, however, has appeared differently in subsequent papers: (1) The foundations of classical mechanics in the light of recent advances in continuum mechanics, Proceedings of the Berkeley Symposium on the Axiomatic Method, 226–81, Amsterdam, 1959; (2) La méchanique classique, basée sur un axiome d’objectivité, Colloque International sur la Méthode Axiomatique dans les Méchaniques Classiques et Nouvelles, 47–56, Paris, 1959; (3) Lectures on the foundations of continuum mechanics and thermodynamics, Archive for Rational Mechanics and Analysis 52, 62–92 (1973). These works are reprinted in a volume of selected papers by Noll: The Foundations of Mechanics and Thermodynamics, Springer-Verlag, New York, 1974. See also Noll’s more recent reports cited in the References of Chapter 5.
SHAMES, I. H., Engineering Mechanics. Vol. 2, Dynamics, 2nd Edition, Prentice-Hall, Englewood Cliffs, New Jersey, 1966. This is good resource for collateral study and additional examples. Plane dynamics of rigid bodies is presented in Chapter 16, work—energy and impulse—momentum methods in Chapter 17, and Euler’s equations for general rigid body motion are treated in Chapter 18.
SYNGE, J. L., and GRIFFITH, B. A., Principles of Mechanics, 3rd Edition, McGraw-Hill, New York, 1959. Chapters 7, 8, 12, and 14 cover plane and spatial motion of a rigid body. Applications to mechanical devices, impulsive motion, the spinning top, and the gyroscope are included. Several examples and problems in the current chapter are modeled after those presented in this classical text.
TRUESDELL, C, Essays in the History of Mechanics, Springer-Verlag, Berlin, Heidelberg, New York, 1968. The development of the foundation principles of classical mechanics of the 17th and 18th centuries due Newton (1687), Euler (1750), Lagrange (1788) and others is traced in Chapters 2, 3 and 5. The reader will discover here a fascinating survey of principal results of the nearly 60 years of research following new paths set by Newton’s Principia and culminating in Euler’s discovery of the precise mathematical formulation of the general principles of momentum and moment of momentum for all bodies, be they solid or fluid, and from which Euler derives his famous general equations of motion for a rigid body. See also Truesdell’s The Rational Mechanics of Flexible or Elastic Bodies 1638–1788. Introduction to Leonhardi Euleri Opera Omnia, Vols 10 and 11, 2nd Series, Orell Füssli Turici, Switzerland, pp. 222–9, 250–4, 1960. These works on the history of development of the mechanics of deformable solid and fluid bodies, and rigid bodies are recommended for reading by all advanced students of engineering and applied mathematics. They are the principal source and inspiration for historical remarks in the introduction to this chapter.
WILMS, E. V, Three single degree of freedom systems with linear damping, Mechanics Research Communications 22, 233–7 (1995). Wilms observes that the single degree of freedom, plane motion of three specific lineal rigid bodies subject to Stokes resistance and certain additional systems of forces is governed by exactly the same equation of motion for each of three very different systems. (See Problems 10.64 through 10.68.) Beatty (1997) generalized this result for all lineal bodies. See also Wilms, E. V., and Cohen, H., The motion of a thin circular ring with linear damping, American Journal of Physics 55, 394, 469 (1987). Additional applications of linear damping may be found in the references to these articles.
YEH, H., and ABRAMS, J. I., Principles of Mechanics of Solids and Fluids, Vol. 1, Particle and Rigid Body Mechanics, McGraw-Hill, New York, 1960. Chapter 12 treats Euler’s equations for rigid body motion, including study of the gyrocompass and the unbalanced rotor problem. Some examples and problems in the current text are patterned after those presented in this book.
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Beatty, M.F. (2006). Dynamics of a Rigid Body. In: Principles of Engineering Mechanics. Mathematical Concepts and Methods in Science and Engineering, vol 33. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-31255-2_6
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