Degree of freedom, motional

Translation is completely described by three degrees of freedom—motion of the entire system in the x-, y-, or z- direction (the three motions on the top of Fig-... 

As we showed in Section 6.2, the stationary states for a particle in a box have specific values for kinetic (translational) energy. For a one-dimensional box, we showed that En = n2h2/%mL2. For a three-dimensional box (side length L, volume V = l ) each of the three degrees of freedom (motion in the x-, y-, and z-directions) has its own quantum number, usually called nx,ny, and nz respectively. The energy is given by... 

Kee, D., Measurement on Range of Two Degrees of Freedom Motion for Analytic Generation of Workspace, Journal of the Ergonomics Society of Korea, Vol. 15, 1997, pp. 15-24. 

A body in space has six degrees of freedom. Motion can therefore be resolved into six axes, namely three linear referred to as X, Y and Z and three corresponding rotational axes referred to as A, B and C as shown in Fig. 12.1. 

For example motion, turn over motion of a starfish was selected to study motion control of deformable robots, because it is a dynamic motion, which is driven by synchronized neuron ring[208]. One of the problems of deformable robots is that motion control is difficult because whose bodies have virtually infinite degrees of freedom. Motion generation of gel robots is not a simple problem if we formulate it in an ordinary maimer. Learning fi om real starfishes and other works, there should exist an alternative methodology. In this section, biologically inspired method to drive deformable robots is proposed. The assumption is that only one or a few points are controlled by whole parts of the robot, which work cooperatively. 

Figure 12 represents a human body model with 39 bodies and 45 degrees of freedom. The degrees of freedom are distributed as follows 3 at the neck, 2 at each collar bone, 3 at each shoulder, 2 at each elbow, 2 at each wrist, 1 for finger motion on each hand, 2 at the waist, 3 at each hip, 1 at each knee and 3 at each foot. In addition, there are 6 degrees of freedom of the base body, which is taken to be the hips. The 3, 2 and 1 degrees of freedom motions at the joints have been represented with spherical, universal and revolute joints, respectively. For convenience, the model used for computation used only revolute joints. Universal joints were represented as two perpendicular revolute joints and spherical joints as three mutually perpendicular revolute joints. 

The two expressions for bo may be brought into formal identity as follows. On adsorption, the three degrees of translational freedom can be supposed to appear as two degrees of translational motion within the confines of a two-... 

In an ideal molecular gas, each molecule typically has translational, rotational and vibrational degrees of freedom. The example of one free particle in a box is appropriate for the translational motion. The next example of oscillators can be used for the vibrational motion of molecules. 

Consider a gas of N non-interacting diatomic molecules moving in a tln-ee-dimensional system of volume V. Classically, the motion of a diatomic molecule has six degrees of freedom—tln-ee translational degrees corresponding to the centre of mass motion, two more for the rotational motion about the centre of mass and one additional degree for the vibrational motion about the centre of mass. The equipartition law gives (... 

These results do not agree with experimental results. At room temperature, while the translational motion of diatomic molecules may be treated classically, the rotation and vibration have quantum attributes. In addition, quantum mechanically one should also consider the electronic degrees of freedom. However, typical electronic excitation energies are very large compared to k T (they are of the order of a few electronvolts, and 1 eV corresponds to 10 000 K). Such internal degrees of freedom are considered frozen, and an electronic cloud in a diatomic molecule is assumed to be in its ground state f with degeneracy g. The two nuclei A and... 

Although in principle the microscopic Hamiltonian contains the infonnation necessary to describe the phase separation kinetics, in practice the large number of degrees of freedom in the system makes it necessary to construct a reduced description. Generally, a subset of slowly varying macrovariables, such as the hydrodynamic modes, is a usefiil starting point. The equation of motion of the macrovariables can, in principle, be derived from the microscopic... 

For very fast reactions, as they are accessible to investigation by pico- and femtosecond laser spectroscopy, the separation of time scales into slow motion along the reaction path and fast relaxation of other degrees of freedom in most cases is no longer possible and it is necessary to consider dynamical models, which are not the topic of this section. But often the temperature, solvent or pressure dependence of reaction rate... 

Consider the collision of an atom (denoted A) with a diatomic molecule (denoted BC), with motion of the atoms constrained to occur along a line. In this case there are two important degrees of freedom, the distance R between the atom and the centre of mass of the diatomic, and the diatomic intemuclear distance r. The Flamiltonian in tenns of these coordinates is given by ... 

All the theory developed up to this point has been limited in the sense that translational motion (the continuum degree of freedom) has been restricted to one dimension. In this section we discuss the generalization of this to three dimensions for collision processes where space is isotropic (i.e., collisions in homogeneous phases, such as in a... 

To generalize what we have just done to reactive and inelastic scattering, one needs to calculate numerically integrated trajectories for motions in many degrees of freedom. This is most convenient to develop in space-fixed Cartesian coordinates. In this case, the classical equations of motion (Hamilton s equations) are given... 

In addition to affecting the number of active degrees of freedom, the fixed n also affects the iinimolecular tln-eshold E in). Since the total angular momentum j is a constant of motion and quantized according to... 

Variational RRKM theory is particularly important for imimolecular dissociation reactions, in which vibrational modes of the reactant molecule become translations and rotations in the products [22]. For CH —> CHg+H dissociation there are tlnee vibrational modes of this type, i.e. the C—H stretch which is the reaction coordinate and the two degenerate H—CH bends, which first transfomi from high-frequency to low-frequency vibrations and then hindered rotors as the H—C bond ruptures. These latter two degrees of freedom are called transitional modes [24,25]. C2Hg 2CH3 dissociation has five transitional modes, i.e. two pairs of degenerate CH rocking/rotational motions and the CH torsion. 

Molecular dynamics tracks tire temporal evolution of a microscopic model system tlirough numerical integration of tire equations of motion for tire degrees of freedom considered. The main asset of molecular dynamics is tliat it provides directly a wealtli of detailed infonnation on dynamical processes. 

Figure C3.2.12. Experimentally observed electron transfer time in psec (squares) and theoretical electron transfer times (survival times, Tau a and Tau b) predicted by an extended Sumi-Marcus model. For fast solvents tire survival times are a strong Emction of tire characteristic solvent relaxation dynamics. For slower solvents tire electron transfer occurs tlirough tire motion of intramolecular degrees of freedom. From [451.

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