Six degrees of freedom

Figure 6 AIR-1 articulated robot arm with six degrees of freedom Robot performs ultrasonic inspection of a large nozzle weld on a BWR main circulation pipe.

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 (... 

A ship has six degrees-of-freedom, i.e. it is free to move in six directions simuitan-eousiy, nameiy three iinear motions - surge (forward), sway (iaterai) and heave... 

An object is generally a three dimensional constmct whose position is dehned by its location (3 degrees of freedom- x, y, z) and by its orientation (3 rotations). Thus an object is constrained if six degrees of freedom of the object are constrained. If less than six degrees of freedom are constrained, the object is under constrained and can be viewed as a mechanism. It is also called under-determined. If the object is only considered in two dimensions, then three constraints are needed to dehne the object (x, y, rotation). When an object is just constrained it is called determinate or statically-determinate. 

Three-dimensional structures Three-dimensional structures are also quite interesting. A determinate structure must satisfy S = dN — 6. This is a result of 3 constraints needed for each node and the object as a whole can move in six degrees of freedom. As in the case of 2-d, this is a necessary but not sufficient condition. 

Eor seven replicates (six degrees of freedom) of the fortified control samples,... 

In the adiabatic bend approximation (ABA) for the same reaction,18 the three radial coordinates are explicitly treated while an adiabatic approximation was used for the three angles. These reduced dimensional studies are dynamically approximate in nature, but nevertheless can provide important information characterizing polyatomic reactions, and they have been reviewed extensively by Clary,19 and Bowman and Schatz.20 However, quantitative determination of reaction probabilities, cross-sections and thermal reaction rates, and their relation to the internal states of the reactants would require explicit treatment of five or the full six degrees-of-freedom in these four-atom reactions, which TI methods could not handle. Other approximate quantum approaches such as the negative imaginary potential method16,21 and mixed classical and quantum time-dependent method have also been used.22... 

An overview of the time-dependent wavepacket propagation approach for four-atom reactions together with the construction of ab initio potential energy surfaces sufficiently accurate for quantum dynamics calculations has been presented. Today, we are able to perform the full-dimensional (six degrees-of-freedom) quantum dynamics calculations for four-atom reactions. With the most accurate YZCL2 surface for the benchmark four-atom reaction H2 + OH <-> H+H2O and its isotopic analogs, we were able to show the following ... 

The number of fundamental vibrational modes of a molecule is equal to the number of degrees of vibrational freedom. For a nonlinear molecule of N atoms, 3N - 6 degrees of vibrational freedom exist. Hence, 3N - 6 fundamental vibrational modes. Six degrees of freedom are subtracted from a nonlinear molecule since (1) three coordinates are required to locate the molecule in space, and (2) an additional three coordinates are required to describe the orientation of the molecule based upon the three coordinates defining the position of the molecule in space. For a linear molecule, 3N - 5 fundamental vibrational modes are possible since only two degrees of rotational freedom exist. Thus, in a total vibrational analysis of a molecule by complementary IR and Raman techniques, 31V - 6 or 3N - 5 vibrational frequencies should be observed. It must be kept in mind that the fundamental modes of vibration of a molecule are described as transitions from one vibration state (energy level) to another (n = 1 in Eq. (2), Fig. 2). Sometimes, additional vibrational frequencies are detected in an IR and/or Raman spectrum. These additional absorption bands are due to forbidden transitions that occur and are described in the section on near-IR theory. Additionally, not all vibrational bands may be observed since some fundamental vibrations may be too weak to observe or give rise to overtone and/or combination bands (discussed later in the chapter). 

Figure 7.2 Definition of the molecular replacement problem and the six degrees of freedom needed to describe it.

Suppose that we consider a gas like nitrogen that can form a diatomic molecule. There are still three degrees of freedom for each atom, so each nitrogen molecule would have six degrees of freedom, and it would take six numbers to specify the positions of the atoms composing the molecule. There are 3 at degrees of freedom per molecule, where nat is the number of atoms per molecule. 

The rans-M(A-A)2B2 matrix M4 has six degrees of freedom and five spectroscopically independent elements. It is certainly within the realm of the... 

The unit cell is 2(7 ). The two La atoms sit on a C3h site, and the six chlorine atoms are on a Cs site (see Appendix 4). Since the Hermann-Mauguin nomenclature cites that the unit cell is primitive (Pb6 /m) we need not reduce it. For the two La atoms there are six degrees of freedom (3n,Z ) = 3 x 1 x 2 = 6. The six Cl atoms possess 18 degrees of freedom (3 , Z ) = 3x3x2=18. Since all vibrational modes can be considered external modes, we need only correlate the site group to factor group. For the... 

Thus of the six degrees of freedom, five have been accounted for. The sixth will describe the movement of the two atoms relative to each other without changing the center of the mass. This is the vibration of the molecule ... 

Samples the six degrees of freedom involved in the relative placement of two three-dimensional rigid structures and scores their fit. Companion programs SPHGEN, DISTMAP, and CHEMGRID. Silicon Graphics. 

The tabulated t-value ( double-sided test) for a 95 percent significance level and six degrees of freedom is 2.45. Thus, there is greater than a 95 percent probability that the two analyses are not the same, i.e., the hypothesis may be rejected with a likelihood of less than 5 percent error. Note that this test has not indicated whether one analysis was better than the other, but rather whether or not there was any significant difference between the two. 

Let us consider first the center-of-mass motions for each of the two I2 molecules in a unit cell. These types of motion account for six degrees of freedom and give rise to two kinds of lattice vibration. When both I2 molecules in a given cell move in phase with each other (say for example both are displaced in the +x direction at the same time), there are... 

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