Rigid body dynamics

Hayward et al. 1994] Hayward, S., Kitao, A., Go, N. Harmonic and anharmonic aspects in the dynamics of BPTI A normal mode analysis and principal component analysis. Prot. Sci. 3 (1994) 936-943 [Head-Gordon and Brooks 1991] Head-Gordon, T., Brooks, C.L. Virtual rigid body dynamics. Biopol. 31 (1991) 77-100... 

The key to these more efficient treatments is a natural canonical formulation of the rigid body dynamics in terms of rotation matrices. The orientational term of the Lagrangian in these variables can be written simply as... 

These difficulties have led to a revival of work on internal coordinate approaches, and to date several such techniques have been reported based on methods of rigid-body dynamics [8,19,34-37] and the Lagrange-Hamilton formalism [38-42]. These methods often have little in common in their analytical formulations, but they all may be reasonably referred to as internal coordinate molecular dynamics (ICMD) to underline their main distinction from conventional MD They all consider molecular motion in the space of generalized internal coordinates rather than in the usual Cartesian coordinate space. Their main goal is to compute long-duration macromolecular trajectories with acceptable accuracy but at a lower cost than Cartesian coordinate MD with bond length constraints. This task mrned out to be more complicated than it seemed initially. 

When we evaluate the Green-Kubo relations for the transport coefficients we solve the equations of motion for the molecules. They are often modelled as rigid bodies. Therefore we review some of basic definition of rigid body dynamics [10]. The centres of mass of the molecules evolve according to the ordinary Newtonian equations of motion. The motion in angular space is more complicated. Three independent coordinates a, =(a,a,-2, ,3), i = 1, 2,. . N where N is the number of molecules, are needed to describe the orientation of a rigid body. (Note that a, is not a vector because it does not transform like a vector when the coordinate system is rotated.) The rate of change of a, is... 

Although rigid body methods can be effective for simulations of rigid models, we focus here on methods that apply effectively also to flexible models. Therefore, this chapter is not concerned with the methods of rigid body dynamics, either in the form of the older and more problematic Euler angles method, or as represented by the more recent and more effective quaternion methods. 

Essentially, the problem of adiabatic leak is mitigated by the presence of constraints, whether in the form of steric hindrance, external forces, or in reduction of dimensionality (for example, simulations in torsion space). In simulations of proposed nanomachines, for example, rigid body dynamics is sometimes applied to components or substructures in order to make the entire nanostructure more rigid. All of this suggests that the prediction of failure modes in some nanomachine designs due strictly to vibrational motion is overly pessimistic. However, this does not by any means minimize concerns such as the difficulty of building desired nanostructures, chemical reactivity. 

A fundamental approach, now routinely employed in essentially all quantal treatments of molecular collisions (both inelastic and reactive), is to use rotating coordinate systems which are generalizations of those commonly used to describe rigid-body dynamics. Additionally, the most accurate and widely used quantal approximations for rotationally inelastic molecular collisions are those based on the so-called sudden assumption (essentially a time-scale criterion in which an internal degree of freedom is assumed to be slow compared to the time scale or suddenness of the collision). We give a brief summary of these ideas, focussing on Curtiss role both in his research and as a mentor. We conclude with a summary of subsequent developments which show the success of Curtiss research and mentoring. 

Gorr, G. V., Kudryashova, L. V., and Stepanova, L. A. Classical Problems of Rigid-Body Dynamics. Naukova Kumka, Kiev (1978). 

Kozlov, V. V. Methods of Qualitative Analysis on Rigid-Body Dynamics, Iz-datePstvo Mosk. Gos. Univers., Moscow (1980). 

Arkhangelsky, Yu. A. Analytic Rigid-Body Dynamics. Nauka, Moscow (1977). Bogoyavlensky, 0. I. Integrable Euler equations on 50(4) and their physical applications. Comm, in Math. Phys. 93 (1984), 417-436. 

Bogoyavlensky, O. I. Rigid-body dynamics with n ellipsoid cavities filled with magnetic liquid. Dokl. Akad. Nauk SSSR. 272 (1983), No. 6, 1364-1367. Bogoyavlensky, 0. I. "Periodic solutions in the pulsar rotation model. Dokl. Akad. Nauk SSSR. 276 (1984), No. 2, 343-347. 

Kharlamov, M. P. Topologic analysis of classical integrable systems in rigid-body dynamics. Dokl. Akad. Nauk SSSR. 278 (1983), No. 6, 1322-1325. Pogosyan, T. I. Critical Integral Surfaces in the Clebsch Problem. Mekhanika Tverdogo Tela 16, 19-24. Kiev. 

Pogosyan, T. I. Construction of Bifurcation Sets in One Problem of Rigid-Body Dynamics. Mekhanika Tverdogo Tela, Kiev, issue 12 (1980), 9-16. 

In the simplest case, that of rigid body dynamics, U3 and U2 are both zero. Then... 

The forces F with some additional damping are included in the rigid body dynamics equations of motion. Using this methodology, a finite element model of the lumbar spine was incorporated in the occupant model, and the loads acting on the spine were calculated. Prelimin results show that the new spinal model can work successfully as compared to experimental data. Further investigations are being conducted in order to construct a layered model with alternate discs and vertebrae. 

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. 

The ATB model is based on the rigid body dynamics which uses Euler s equations of motion with constraint relations of the type employed in the Lagrange method. The model has been successfully used to study the articulated human body motion under various types of body segment and joint loads. The technology of robotic telepresence will provide remote, closed-loop, human control of mobile robots. 

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 Sect. 3.2, we have seen that a dynamical system with unilateral or bilateral frictional contact can possess a peculiar characteristic, namely the inertia matrix may be asymmetric and nonpositive definite. Painleve was the first to point out the difficulties that may arise in such cases [53, 95]. As we will see in this section through examples, the presence of a kinematic constraint with friction could lead to situations where the equations of motion of the system do not have a bounded solution (inconsistency) or the solution is not unique (indeterminacy). These situations where the existence and uniqueness properties of the solution of the equations of motion are violated are known as the Painleve s paradoxes. There is a vast literature on the general theory of the rigid body dynamics with frictional constraints... 

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. 

Dupont PE, Yamajako SP (1994) Jamming and wedging in constrained rigid-body dynamics. In Proceedings of the 1994 IEEE international conference on robotics and automation, vol 3, pp 2349-2354... 

Stewart DE (20(X)) Rigid-body dynamics with friction and impact. Siam Rev C Soc Ind Appl... 

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