Rigid Body Systems

More generally, consider a collection of N atoms (masses trii, positions qt, i = 1,2. N) subject to an external potential U. The kinetic energy is 

The atomic positions may be referred to a coordinate system placed at the center of mass of the collection 

By choosing an appropriate coordinate frame, it is possible to insure that the symmetric matrix If is a diagonal matrix. 

An alternative form for the rotational kinetic energy may be obtained by writing 

We now insert the expression for S, into the rotational kinetic energy, to obtain 1 N j N 

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We consider the rigid body system consisting of one solute molecule with an arbitrary shape and solvent molecules with a spherical shape. The solute molecule is modeled as fused hard spheres with different radii and the solvent molecules ate modeled as hard spheres with the same radius. This represents aqueous soludon of organic solute at infinite diludon. 

R. Featherstone. The Dynamics of Rigid Body Systems with Multiple Concurrent Contacts. In Robotics Research The Third Internatiotud Symposium, pages 189-196. MIT Press, 1986. 

Glocker, C. 2001. On frictionless impact models in rigid-body systems. Philosophical Transactions of the Royal Society of London, A359 2385-2404. 

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. 

Our study of the lead screw drives entails systems with a single bilateral contact with friction (between lead screw threads and nut threads). As demonstrated by the example in Sect. 4.3.4, in order to study the behavior of a system in the paradoxical regions of parameters, a compliant approximation to rigid contact may be used. In Chap. 8, the limit process approach presented in [51] is utilized to determine the true motion of a 1-DOF lead screw drive model under similar paradoxical conditions. In the limit process approach, the behavior of the rigid body system is taken as that of a similar system with compliant contacts when the contact stiffness tends to infinity. Related to this topic, a discussion of the method of penalizing function can be found in Brogliato ([96], Chap. 2). Other examples include [101-103]. 

In Sect. A3 A A we studied an example where an approximate solution was obtained in the region of paradoxes by adding compliance to the two bodies in contact. In Sect. 8.6.1 below, we will present numerical results of a similar approach apphed to the lead screw and nut (i.e. using the 2-DOF model of Sect 5.5). But first, we will take a closer look at the behavior of the rigid body system under the cmiditimis of the paradoxes. The approach adopted here is based on the limiting process described in [51] where the law of motion of the rigid body system is taken as that of the system with compliant ccmtact when the contact stiffness tends to infinity. 

It has been observed by [27, 24] that the equations of motion of a free rigid body are subject to reduction. (For a detailed discussion of this interesting topic, see [23].) This leads to an unconstrained Lie-Poisson system which is directly solvable by splitting, i.e. the Euler equations in the angular momenta ... 

S. Reich, Symplectic integrators for systems of rigid bodies. Fields Institute Communications, 10, 181-191 (1996). 

Modification of inertia of hydrogen-only rigid bodies is a simple and safe way to balance different frequencies in the system, and it usually allows one to raise to 10 fs. Unfortunately, the further increase appears problematic because of various anharmonic effects produced by collisions between non-hydrogen atoms [48]. 

Routh, E.J. (1905) Dynamics of a System of Rigid Bodies, Macmillan Co., London. 

When dealing with the motions of rigid bodies or systems of rigid bodies, it is sometimes quite difficult to directly write out the equations of motion of the point in question as was done in Examples 2-6 and 2-7. It is sometimes more practical to analyze such a problem by relative motion. That is, first find the motion with respect to a nonaccelerating reference frame of some point on the body, typically the center of mass or axis of rotation, and vectorally add to this the motion of the point in question with respect to the reference point. 

Vector spaces which occur in physical applications are often direct products of smaller vector spaces that correspond to different degrees of freedom of the physical system (e.g. translations and rotations of a rigid body, or orbital and spin motion of a particle such as an electron). The characterization of such a situation depends on the relationship between the representations of a symmetry group realized on the product space and those defined on the component spaces. 

Kellman, M. E., Amar, F., and Berry, R. S. (1980), Correlation Diagrams for Rigid and Nonrigid Three-Body Systems, J. Chem. Phys. 73, 2387. 

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