Periodic motion, vibrational analysis

The boundary conditions established by the machine design determine the freedom of movement permitted within the machine-train. A basic understanding of this concept is essential for vibration analysis. Free vibration refers to the vibration of a damped (as well as undamped) system of masses with motion entirely influenced by their potential energy. Forced vibration occurs when motion is sustained or driven by an applied periodic force in either damped or undamped systems. The following sections discuss free and forced vibration for both damped and undamped systems. 

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. 

Abstract. The development of modern spectroscopic techniques and efficient computational methods have allowed a detailed investigation of highly excited vibrational states of small polyatomic molecules. As excitation energy increases, molecular motion becomes chaotic and nonlinear techniques can be applied to their analysis. The corresponding spectra get also complicated, but some interesting low resolution features can be understood simply in terms of classical periodic motions. In this chapter we describe some techniques to systematically construct quantum wave functions localized on specific periodic orbits, and analyze their main characteristics. 

For an understanding of the vibrations of a polyatomic molecule, should be first a preliminary analysis of the oscillations of a molecule composed of two atoms linked by covalent binding. Such a molecule, with N= 2 atoms, shows N = 3x2 -5 = 1 modes of vibration. The steering as defined by the covalent binding of the two atoms is the only special steering, it is ordinary to accept that atoms will move (in a periodic motion) after direction of the covalent connection. Assembly oscillation may be considered in relation to several systems of reference. It may choose as origin of the system of reference the center of gravity of the diatomic assembly. 

Among the main theoretical methods of investigation of the dynamic properties of macromolecules are molecular dynamics (MD) simulations and harmonic analysis. MD simulation is a technique in which the classical equation of motion for all atoms of a molecule is integrated over a finite period of time. Harmonic analysis is a direct way of analyzing vibrational motions. Harmonicity of the potential function is a basic assumption in the normal mode approximation used in harmonic analysis. This is known to be inadequate in the case of biological macromolecules, such as proteins, because anharmonic effects, which MD has shown to be important in protein motion, are neglected [1, 2, 3]. 

Molecular dynamics examines the temporal evolution of a collection of atoms on the basis of an explicit integration of the equations of motion. From the point of view of diffusion, this poses grave problems. The time step demanded in the consideration of atomic motions in solids is dictated by the periods associated with lattice vibrations. Recall our analysis from chap. 5 in which we found that a typical period for such vibrations is smaller than a picosecond. Hence, without recourse to clever acceleration schemes, explicit integration of the equations of motion demands time steps yet smaller than these vibrational periods. 

The dielectric spectroscopy is an essential probe for nondestructive studies required for biomolecule analysis. The study of internal motion/dynamics related to the dielectric relaxation in biomolcules require the coverage of periodic vibration along with other mechanisms, such as diffusion, molecular orientations, and relaxation processes. 

Vibrational Motion in Polyatomic Molecules. Normal-mode analysis of vibrational motion in polyatomic molecules is the method of choice when there are several vibrational degrees of freedom. The actual vibrations of a polyatomic molecule are completely disordered, or aperiodic. However, these complicated vibrations can be simplified by expressing them as linear combinations of a set of vibrations (i.e., normal modes) in which all atoms move periodically in straight lines and in phase. In other words, all atoms pass through their equilibrium positions at the same time. Each normal mode can be modeled as a harmonic oscillator. The following rules are useful to determine the number of normal modes of vibration that a molecule possesses ... 

The integration step is repeated until the time of interest is reached. To maintain numerical stability the time step At must be smaller than the fastest motion in the system. Bond vibrations and hard-core atomic collisions have periods of femtoseconds and picoseconds. A typical time step that is used to integrate the equations of motion is therefore a femtosecond (10 s). The number of steps accessible to a typical computational setup is about 10, which makes it possible to reach timescales of hundreds of nanoseconds or 10" s in biological molecules. A considerable body of research in numerical analysis aims at increasing this time step and extending the overall timescale. So far, these efforts were not able to increase the computational efficiency by more than a factor of 2 or 3. 

The simulation of structures using pair potential methods gives important information, including unit cell dimensions, atomic positions and details of atomic motion including lattice vibrations (phonon modes). Further analysis permits the calculation of heat capacities, the dependence of volume with temperature and the prediction of vibrational spectra, such as IR and neutron spectroscopies. Codes that perform such periodic structure energy minimisation using pair potential models include METAPOCS, THBREL and GULP (Table 4.1). All have been used successfully to model framework structures. 

The above analysis, shared by many spectroscopists in the field of small molecules, can be further expanded when vibrational spectroscopy is considered in the field of polymers and macromolecules in general. The wiggling of polymers adds new flavor to physics and chemistry. The translational periodicity of infinite polymers with perfect stmcture generates phonons and collective vibrations which give rise to absorption or Raman scattering bands that escape the interpretation based on the traditional spectroscopic correlations. The concept of collective motions forms the basis for the understanding of the vibrations of finite chain molecules which form a nonnegligible part of industrially relevant materials. On the other hand, real polymer samples never show perfect chemical, strereochemical, and conformational structure. Symmetry is broken and new bands appear which become characteristic of specific types of disorder. 

Most solids have ordered arrangements of particles with a very restricted range of motion. Particles in the solid state cannot move freely past one another so they only vibrate about fixed positions. Consequently, solids have definite shapes and volumes. Because the particles are so close together, solids are nearly incompressible and are very dense relative to gases. Solid particles do not diffuse readily into other solids. However, analysis of two blocks of different solids, such as copper and lead, that have been pressed together for a period of years shows that each block contains some atoms of the other element. This demonstrates that solids do diffuse, but very slowly (Figure 13-2). 

It is now 10 years since Carney, Sprandel and Kern (1) published their much cited review of variational ro-vibrational calculations on triatomic systems. It is therefore interesting to consider how the subject has progressed in the intervening period and in particular to focus on the new areas of theoretical spectroscopy that can now be explored with modern supercomputers. At the time of the review in 1978, it was taken for granted that the Eckart Hamiltonian was the one to choose for studying the nuclear motions of polyatomic systems. It was further widely assumed that the role of electronic structure calculations in the solution of the nuclear motion problem was to obtain force constants and rotational constants to be used in perturbation-theoretic analysis. 

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