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Concept# Rigid body

Summary

In physics, a rigid body, also known as a rigid object, is a solid body in which deformation is zero or negligible. The distance between any two given points on a rigid body remains constant in time regardless of external forces or moments exerted on it. A rigid body is usually considered as a continuous distribution of mass.
In the study of special relativity, a perfectly rigid body does not exist; and objects can only be assumed to be rigid if they are not moving near the speed of light. In quantum mechanics, a rigid body is usually thought of as a collection of point masses. For instance, molecules (consisting of the point masses: electrons and nuclei) are often seen as rigid bodies (see classification of molecules as rigid rotors).
Kinematics
Linear and angular position
The position of a rigid body is the position of all the particles of which it is composed. To simplify the description of this position, we exploit the property that the body is rigid, na

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Johannes Richard Cornelis Geerit Kruis

Kinematic couplings are used when two rigid bodies need to be repeatedly and accurately positioned with respect to each other. They allow for sub-micron positioning repeatability by suppressing play and reducing strains in the bodies. Typical applications are lens mounts, work piece mounts and docking interfaces for astrophysics, semiconductor and metrology applications. This thesis generalizes the well-known concept of two-body kinematic couplings to three-body kinematic mounts. The goal of the thesis is: To pave the way for high precision assembly using kinematic mounts by providing an exhaustive catalogue of all twobody and three-body kinematic mounts and to test key configurations experimentally. The main contributions of this thesis are: - State of the art survey of essential knowledge in the field of kinematic couplings. - Rigorous problem statement for the design of two-body and three-body kinematic mounts. - Rigorous limitation of the scope of research to three-body kinematic mounts whose contact points lie exclusively on three convergent orthogonal lines and whose constraint lines are parallel to these lines. - An exhaustive catalogue of three-body kinematic mounts consisting of seven configurations in 3D and nine configurations in 2D. - An exhaustive set of four conditions satisfied by three-body 3-dimensional kinematic mounts. - An exhaustive set of seven conditions satisfied by three-body 2-dimensional kinematic mounts. - Realization of a two-body kinematic mount and a three-body kinematic mount in metal, and precise measurement of their positioning accuracy on a 3D coordinate measurement machine at the Swiss Federal Institute of Metrology. Positioning error of 0.2 microns and 5 microradian achieved with two-body kinematic mounts. Positioning error of 1 micron and 50 microradian achieved with three-body kinematic mounts. - Realization of three-body kinematic mounts in Silicon by Deep Reactive Ion Etching processes (DRIE) and experimental measurement of their positioning error. - Physical implementation of nesting forces and assembly methods allowing for the physical construction of kinematic mounts. - Physical realizations in robotics, optics and aerospace using our new kinematic mounts.

This thesis aims to find, for the first time, a direct relation between the size and morphology of small metallic nanostructures (gold in this case) supported on a metal-oxide surface to their catalytic activity. In this perspective, three main topics have been treated during this thesis. The first part concerns the design and realization of a Scanning Tunneling Microscope (STM) supposed to work over a wide temperature range (4K < T ≤ 300K). This new device replaces an existing solution. The coarse approach of the scanning tip towards the sample is realized by an axial motor based on a sapphire prism gliding on shear piezos in the stick&slip mode. The new STM is very rigid moving the resonance frequencies to higher values with respect to the existing solution. Operation of the motor down to T = 8K has been proven, however topographic imaging has been performed only in the temperature range 77K < T ≤ 300K. The second part of this work focuses on a study of the evolution of the morphology of gold nanoparticles on a TiO2(110) surface. Size-selected clusters Aun+ (n = 5, 7) are deposited at a well defined kinetic energy on the surface held at room temperature. Subsequent annealing of the sample has been performed stepwise. After each temperature increase, the morphology has been determined by STM. The evolution as a function of surface temperature has been studied for two different surface reconstructions, TiO2(110)-(1×1) and TiO2(110)-(2×1). The deposition process leads only to small fragmentation and the morphology is stable up to T = 400K. Further increase of the surface temperature leads to sintering of the particles by Ostwald ripening as shown by an exponential decrease of the island density with temperature. The main topic of this thesis, the correlation between morphology and catalytic activity, is described in the last part of this manuscript. For the first time we are able to relate the onset of CO2 production from CO and O2 to a clear change in the morphology. The catalytic activity of the particles strongly depends on their size and dimensionality. The relative activity per particle has been determined and we find a clear maximum for clusters containing 60 atoms and are 3 to 4 monolayers high. These results are discussed in contrast of literature data on the same but also on different metal-oxide surfaces.

The stability for all generic equilibria of the Lie-Poisson dynamics of the so(4) rigid body dynamics is completely determined. It is shown that for the generalized rigid body certain Cartan subalgebras (called of coordinate type) of so(n) are equilibrium points for the rigid body dynamics. In the case of so(4) there are three coordinate type Cartan subalgebras whose intersection with a regular adjoint orbit gives three Weyl group orbits of equilibria. These coordinate type Cartan subalgebras are the analogues of the three axes of equilibria for the classical rigid body in so(3). In addition to these coordinate type Cartan equilibria there are others that come in curves.

2012