Displacement, vibrational analysis

A continuing periodic change in a displacement with respect to a fixed reference. The motion will repeat after a certain interval of time. Vibration analysis monitors the noise or vibrations generated by plant machinery or systems to determine their actual operating condition. The normal monitoring range for vibration analysis is from less than 1 to 20,000 Hertz. 

In VFF the molecular vibrations are considered in terms of internal coordinates qs (s = 1..3N — 6, where N is the number of atoms), which describe the deformation of the molecule with respect to its equilibrium geometry. The advantage of using internal coordinates instead of Cartesian displacements is that the translational and rotational motions of the molecule are excluded explicitly from the very beginning of the vibrational analysis. The set of internal coordinates q = qs is related to the set of Cartesian atomic displacements x = Wi by means of the Wilson s B-matrix [1] q = Bx. In the harmonic approximation the B-matrix depends only on the equilibrium geometry of the molecule. 

So far, we have discussed only the vibrations whose displacements occur along the molecular axis. There are, however, two other normal vibrations in which the displacements occur in the direction perpendicular to the molecular axis. They are not treated here, since the calculation is not simple. It is clear that the method described above will become more complicated as a molecule becomes larger. In this respect, the GF matrix method described in Sec. 1.12 is important in the vibrational analysis of complex molecules. 

It is important to note that the linearised displacement and velocity coefficients, commonly used in lateral vibration analysis, only facilitate the estimation of change of oil film force components from an equilbrium condition. In contrast with this, the oil film force components given by equations [5] and [6] are the total values. The estimation of the oil film force components at any location of the journal within the bearing... 

If a vibrational analysis using 3N displacement vectors is being carried out, then the representations that correspond to simple molecular translation or rotation must now be removed. In a character table, the irreducible representations for these degrees of freedom are indicated in the rightmost columns with the symbols x, y, z and R, Ry, Rz-... 

The simplest description of vibrational degrees of lieedom of a molecule with N atoms is in terms of 3N — 6 or 3N — 5 (for linear molecules) normal vibrational modes. Vibrational analysis concerns the study of these normal vibrational modes. It is possible to define mass-weighted normal mode coordinates which provide an equivalent description of the molecular vibrations. Normal mode coordinate Qk a given normal mode k (k = 1, 3N — 6) corresponds to a specific vibrational pattern (displacements from equilibrium) on the molecule, for which all atoms oscillate at the same frequency k-... 

Before considering other concepts and group-theoretical machinery, it should once again be stressed that these same tools can be used in symmetry analysis of the translational, vibrational and rotational motions of a molecule. The twelve motions of NH3 (three translations, three rotations, six vibrations) can be described in terms of combinations of displacements of each of the four atoms in each of three (x,y,z) directions. Hence, unit vectors placed on each atom directed in the x, y, and z directions form a basis for action by the operations S of the point group. In the case of NH3, the characters of the resultant 12x12 representation matrices form a reducible representation... 

If the displacements of the atoms are given in terms of the harmonic normal modes of vibration for the crystal, the coherent one-phonon inelastic neutron scattering cross section can be analytically expressed in terms of the eigenvectors and eigenvalues of the hannonic analysis, as described in Ref. 1. 

Time-domain plots must be used for all linear and reciprocating motion machinery. They are useful in the overall analysis of machine-trains to study changes in operating conditions. However, time-domain data are difficult to use. Because all the vibration data in this type of plot are added together to represent the total displacement at any given time, it is difficult to directly see the contribution of any particular vibration source. 

Mathematical techniques allow us to quantify total displacement caused by all vibrations, to convert the displacement measurements to velocity or acceleration, to separate this data into its components using FFT analysis, and to determine the amplitudes and phases of these functions. Such quantification is necessary if we are to isolate and correct abnormal vibrations in machinery. 

The atomic temperature factor, or B factor, measures the dynamic disorder caused by the temperature-dependent vibration of the atom, as well as the static disorder resulting from subtle structural differences in different unit cells throughout the crystal. For a B factor of 15 A2, displacement of an atom from its equilibrium position is approximately 0.44 A, and it is as much as 0.87 A for a B factor of 60 A2. It is very important to inspect the B factors during any structural analysis a B factor of less than 30 A2 for a particular atom usually indicates confidence in its atomic position, but a B factor of higher than 60 A2 likely indicates that the atom is disordered. 

The relative displacements of the masses in the two normal modes of this coupled oscillator are shown to the right in Fig. 2. This method the form of the normal modes is particularly useful in the analysis of molec vibrations (see Chapter 9). 

The actual calculation consists of minimizing the intramolecular potential energy, or steric energy, as a function of the nuclear coordinates. The potential-energy expressions derive from the force-field concept that features in vibrational spectroscopic analysis according to the G-F-matrix formalism [111]. The G-matrix contains as elements atomic masses suitably reduced to match the internal displacement coordinates (matrix D) in defining the vibrational kinetic energy T of a molecule ... 

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