Rigid body mechanics, dynamics

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. 

As demonstrated by the foregoing two formulations, some problems taken from mechanics can be formulated by using only Newton s laws of motion these are called mechanically determined problems. The dynamics of rigid bodies in the absence of friction, statically determined problems of rigid bodies, and mechanics of ideal fluids provide examples of this class. Some other mechanics problems, however, require knowledge beyond Newton s laws of motion. These are called mechanically undetermined problems. The dynamics of rigid bodies with friction and the mechanics of deformable bodies provide examples of this class. 

In dynamics, it is well known that planar motion of a rigid body can be decomposed into translation with and rotation about a reference point. Similarly, in continuum mechanics, strain is decomposed into normal strain (stretch or translation) and shear strain (rotation). It is thus inferred that the deformation gradient could be decomposed into stretch and rotation. Before we proceed, let us discuss the rigid body motion induced deformation gradient. 

In Chapter 2, the general notational and arithmetic concepts necessary for modelling robotic mechanisms and formulating their kinematic and dynamic equations are presented. These include a modified system of spatial notation and arithmetic, the kinematic and dynamic parameters used to describe mechanisms, and the general joint model which will be used throughout this book to describe the interactions between rigid bodies of a system. 

The reader is assumed to have some knowledge of vector mechanics and matrix algebra. A basic knowledge of rigid-body kinematics and dynamics would be helpful, but it is not necessary. 

Ross, D.F. Klingenberg, D.J. 1997. Dynamic Simulation of Flexible Fibers composed of Linked Rigid Bodies. /. Chem. Phys. 106 (7) 2949-2960 Schmid, C.F. Klingenberg, D.J. 2000. Mechanical Flocculation of Flowing Fiber Suspensions. Phys. Rev. Lett. 84 (2) 290-293... 

The determination of dense fluid properties from ab initio quantum mechanical calculations still appears to be some time from practical completion. Molecular dynamics and Monte Carlo calculations on rigid body motions with simple interacting forces have qualitatively produced all of the essential features of fluid systems and quantitative agreement for the thermodynamic properties of simple pure fluids and their mixtures. These calculations form the basis upon which perturbation methods can be used to obtain properties for polyatomic and polar fluid systems. All this work has provided insight for the development of the principle of corresponding state methods that describe the properties of larger molecules. 

The inclusion of the all-purpose rigid-body dynamic model in the design and control improves the mechanical system behaviour and performance fidelity. However, friction imposes additional nonlinearity in the dynamic equations of connected moving bodies. The discontinuous nature of all kinds of friction causes vibrations and stick-slip effects which limit the accuracy of end-effector position or path. 

Finally, in Sect. 4.3, we turn our attention to the kinematic constraint instability. This section begins by the introduction of the Painleve s paradoxes which play an important role in the kinematic constraint instability mechanism. The self-locking property which is another consequence of MctiOTi in the rigid body dynamics is also discussed in this section. This effect has a prominent presence in the study of the lead screws in Chaps. 7 and 8. The concepts presented here form the basis for the study of the kinematic constraint instability in the lead screws in Chap. 8. 

In Sects. 4.3.1 and 4.3.2, we study the classic Painleve s example and derive the conditions for the occurrence of the paradoxes. In Sect. 4.3.3, the concept of self-locking is introduced which is closely related to the kinematic constraint instability mechanism. In the rigid body systems, this phenomenon is sometimes known as jamming or wedging [97]. As we will see later on, the self-locking is an important aspect of the study of the dynamics of the lead screws. In Sect. 4.3.4, a simple model of a vibratory system is analyzed where the kinematic constraint mechanism leads to instability. In the study of disc brake systems, similar instability mechanism is sometimes referred to as sprag-slip vibration [7]. Some further references are given in Sect. 3.3.5. 

The area of rigid body dynamical systems with contact and friction belongs to the study of nonsmooth systems. See, for example, [52, 104-108] for the theory of nonsmooth mechanics. Mathematical concepts, such as Filippov systems, measure differential inclusions, and linear complimentarity problems (LCP) are used to describe and analyze these systems. The book by Brogliato [96] is an excellent reference on these subjects and discusses a great number of relevant works. 

The scope of the series covers the entire spectrum of solid mechanics. Thus it includes the foundation of mechanics variational formulations computational mechanics statics, kinematics and dynamics of rigid and elastic bodies vibrations of solids and structures dynamical systems and chaos the theories of elasticity, plasticity and viscoelasticity composite materials rods, beams, shells and membranes structural control and stability soils, rocks and geomechanics fracture tribology experimental mechanics biomechanics and machine design. 

Regulation of Calcium Homeostasis Calcium homeostasis necessitates the maintenance of a dynamic equilibrium of calcium fluxes between three different compartments which harbor the mineral ion in vastly different concentrations. Thus, homeostatic control mechanisms ought to modulate calcium fluxes between different body compartments in a way which allows the generation and maintenance of steep concentration gradients between the skeletal tissue, the extracellular fluid and the intracellular - that is, the cytoplasmic compartment. Of particular importance thereby is the rigid control of plasma free Ca " ", because even small deviations from the normal level induce profound changes in both intracellular free Ca, as well as in the amount of calcium deposited at skeletal sites, inevitably causing adverse effects on bone health (cf. Whedon 1980). 

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