Rigid body mechanics

For the model of free point particles the Newtonian equations present by far the simplest and most efficient analytical fonnalism. In contrast, for chains of rigid bodies, there are several different, but equally applicable, analytical methods in mechanics, with their spe-... 

From this kind of continuum mechanics one can move further towards the domain of almost pure mathematics until one reaches the field of rational mechanics, which harks back to Joseph Lagrange s (1736-1813) mechanics of rigid bodies and to earlier mathematicians such as Leonhard Euler (1707-1783) and later ones such as Augustin Cauchy (1789-1857), who developed the mechanics of deformable bodies. The preeminent exponent of this kind of continuum mechanics was probably Clifford Truesdell in Baltimore. An example of his extensive writings is A First Course in... 

Mechanics is the physical science that deals with the effects of forces on the state of motion or rest of solid, liquid, or gaseous bodies. The field may he divided into the mechanics of rigid bodies, the mechanics of deformable bodies, and the mechanics of fluids. 

A rigid body is one that does not deform. True rigid bodies do not exist in nature however, the assumption of rigid body behavior is usually an acceptable accurate simplification for examining the state of motion or rest of structures and elements of structures. The rigid body assumption is not useful in the study of structural failure. Rigid body mechanics is further subdivided into the study of bodies at rest, stalks, and the study of bodies in motion, dynamics. 

Dynamics is the study of the mechanics of rigid bodies in motion. It is usually subdivided into khiematics, the study of the motion of bodies without reference to the forces causing that motion or to the mass of bodies, and kinetics, the study of the relationship between the forces acting on a body, the mass and geometry of the body, and the resulting motion of the body. 

E. T. Whittaker, A Treatise on the Analytical Mechanics of Particles and Rigid Bodies, 4th edn., Cambridge University Press, Cambridge, 1937. 

Plants use osmotic pressure to achieve mechanical rigidity. The very high solute concentration in the plant cell vacuole draws water into the cell (Fig. 2-13). The resulting osmotic pressure against the cell wall (turgor pressure) stiffens the cell, the tissue, and the plant body. When the lettuce in your salad wilts, it is because loss of water has reduced turgor pressure. Sudden alterations in turgor pressure produce the movement of plant... 

To consider the quantum mechanics of rotation of a polyatomic molecule, we first need the classical-mechanical expression for the rotational energy. We are considering the molecule to be a rigid rotor, with dimensions obtained by averaging over the vibrational motions. The classical mechanics of rotation of a rigid body in three dimensions is involved, and we shall simply summarize the results.2... 

This is the classical-mechanical Hamiltonian for the rotation of a rigid body. [Note the similarity of (5.20), (5.22), and (5.23) to the corresponding equations for linear motion we get the equations for rotational motion by replacing the velocity v with the angular velocity to, the linear momentum p with the angular momentum P, and the mass with the principal moments of inertia.)... 

The energy levels from the quantum mechanical solution of the rotation of a rigid body have the characteristic feature of increasing separation with angular momentum. The energy levels are given by the expression ... 

The principal source of, 3C relaxation is intramolecular dipole-dipole (DD) interaction between a l3C nucleus and neighboring protons. A simplified picture of this mechanism is shown in Fig. 1. A, 3C nucleus, situated in a rigid body which is... 

The determination of accurate molecular structure from molecular rotational resonance (MRR) spectra has always been a great challenge to this branch of spectroscopy [/]. There are three basic facts which make this task feasible (1) the free rotation of a rigid body is described in classical as well as in quantum mechanics by only three parameters, the principal inertial moments of the body, Ig, g = x, v, z ... 

In mechanics, the inertial properties of a rotating rigid body are fully described by its inertial moment tensor I. We can simplify the subsequent equations if we employ in place of I the closely related planar moment tensor P, apparently first used by Kraitchman [4], At any stage of the calculations, however, an equivalent equation could be given which involves I instead of P. The principal planar moments P (g = x, y, z) are the three eigenvalues of the planar moment tensor P and the principal inertial moments Ig the eigenvalues of the inertial moment tensor I. Pg, Ig, and the rotational constants Bg = f/Ig are equivalent inertial parameters of the problem investigated (/conversion factor). 

A branch of mechanics dealing with the motion of rigid bodies without reference to their masses or the forces acting on the bodies, kite... 

Solution of crystal structures can be aided by rigid body refinement of a molecular mechanics optimized structure. A recent example of this is the work of Boeyens and Oosthuizen. The crystal structures of (15-ane N5)Cu(II) and (16-ane N5)Ni(II) were refined with the aid of calculated models from a force field described earlier. This method, however, does not refine the internal molecular parameters. 

Broadly speaking, our description of continuum mechanics will be divided along a few traditional lines. First, we will consider the kinematic description of deformation without searching for the attributes of the forces that lead to a particular state of deformation. Here it will be shown that the displacement fields themselves do not cast a fine enough net to sufficiently distinguish between rigid body motions, which are often of little interest, and the more important relative motions that result in internal stresses. These observations call for the introduction of other kinematic measures of deformation such as the various strain tensors. Once we have settled these kinematic preliminaries, we turn to the analysis of the forces in continua that lead to such deformation, and culminate in the Cauchy stress principle. 

In addition to the importance that attaches to rigid body motions, shearing deformations occupy a central position in the mechanics of solids. In particular, permanent deformation by either dislocation motion or twinning can be thought of as a shearing motion that can be captured kinematically in terms of a shear in a direction s on a plane with normal n. 

H. B. G. Casimir, The Rotation of a Rigid Body in Quantum Mechanics, Leyden thesis, J. B. Wolters, the Hague, Netherlands (1931). 

There are two proper explanations, one based on physical intuition and the other based on the principle of material objectivity. The latter is discussed in many books on continuum mechanics.19 Here, we content ourselves with the intuitive physical explanation. The basis of this is that contributions to the deviatoric stress cannot arise from rigid-body motions -whether solid-body translation or rotation. Only if adjacent fluid elements are in relative (nonrigid-body) motion can random molecular motions lead to a net transport of momentum. We shall see in the next paragraph that the rate-of-strain tensor relates to the rate of change of the length of a line element connecting two material points of the fluid (that is, to relative displacements of the material points), whereas the antisymmetric part of Vu, known as the vorticity tensor 12, is related to its rate of (rigid-body) rotation. Thus it follows that t must depend explicitly on E, but not on 12 ... 

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