The vibrating rotor

If there was no interaction between vibration and rotation, the energy levels would be given by the simple sum of the expression giving the vibrational levels for the anharmonic oscillator, equation (6.188), and that describing the rotational levels of the rigid rotor, equation (6.162). There is an interaction, however during a vibration the moment of inertia of the molecule changes, and therefore so also does the rotational constant. We may therefore use a mean value of Bv for the rotational constant of the vibrational level considered, i.e. 

Consequently for the rotational levels in a given vibrational level, the term values are given by 

The full expression for the term values of the vibrating rotor is therefore 

Relationships connecting ae, //., ye with coe, coexe and Be have been given by Pekeris [67] and Dunham [65], In a very detailed study of vibration-rotation interactions, Dunham [65] has shown that the term values for a vibrating rotor should actually be expressed as a double power series, given by 

The coefficients in (6.202) are very closely related to the standard spectroscopic parameters according to the following  

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Zhang J Z H 1999 The semirigid vibrating rotor target model for quantum polyatomic reaction dynamics J. 

The Seetion entitled The BasiC ToolS Of Quantum Mechanics treats the fundamental postulates of quantum meehanies and several applieations to exaetly soluble model problems. These problems inelude the eonventional partiele-in-a-box (in one and more dimensions), rigid-rotor, harmonie oseillator, and one-eleetron hydrogenie atomie orbitals. The eoneept of the Bom-Oppenheimer separation of eleetronie and vibration-rotation motions is introdueed here. Moreover, the vibrational and rotational energies, states, and wavefunetions of diatomie, linear polyatomie and non-linear polyatomie moleeules are diseussed here at an introduetory level. This seetion also introduees the variational method and perturbation theory as tools that are used to deal with problems that ean not be solved exaetly. 

For a symmetric rotor molecule such as methyl fluoride, a prolate symmetric rotor belonging to the C3 point group, in the zero-point level the vibrational selection mle in Equation (6.56) and the character table (Table A. 12 in Appendix A) show that only... 

For a spherical rotor belonging to the octahedral Of, point group, Table A.43 in Appendix A, in conjunction with the vibrational selection rules of Equation (6.56), show that the only allowed transitions are... 

As in Section 5.2.4 on rotational spectra of asymmetric rotors, we do not treat this important group of molecules in any detail, so far as their rotational motion is concerned, because of the great complexity of their rotational energy levels. Nevertheless, however complex the stack associated with the v = 0 level, there is a very similar stack associated with each excited vibrational level. The selection mles for transitions between the rotational stacks of the vibrational levels are also complex but include... 

In a molecule such as the asymmetric rotor formaldehyde, shown in Figure 5.1(f), the a, b and c inertial axes, of lowest, medium and highest moments of inertia, respectively, are defined by symmetry, the a axis being the C2 axis, the b axis being in the yz plane and the c axis being perpendicular to the yz plane. Vibrational transition moments are confined to the a, b or c axis and the rotational selection mles are characteristic. We call them... 

This general behaviour is characteristic of type A, B and C bands and is further illustrated in Figure 6.34. This shows part of the infrared spectrum of fluorobenzene, a prolate asymmetric rotor. The bands at about 1156 cm, 1067 cm and 893 cm are type A, B and C bands, respectively. They show less resolved rotational stmcture than those of ethylene. The reason for this is that the molecule is much larger, resulting in far greater congestion of rotational transitions. Nevertheless, it is clear that observation of such rotational contours, and the consequent identification of the direction of the vibrational transition moment, is very useful in fhe assignmenf of vibrational modes. 

Misalignment causes vibrations, which may eventually lead to failure. The vibrations may also cause cracks between the rotor bars and the end rings. 

The maximum permissible vibration, in mils (0.001 in.), during shop test at rated speed shall be equal to the square root of 20,000 divided by the sum of rated speed plus meeh-anieal and eleetrieal runout for the overhung rotor design. Only the aetual total measured runout may be subtraeted from the unfiltered peak-to-peak amplitude measured during testing to attain the shaft vibration. The meehanieal-plus-eleetrieal runout subtraeted from the unfiltered peak-to-peak amplitude shall not exeeed 0.5 mils regardless of total runout. 

After the eonelusion of the 4 hr hot test, the expander rotor speed shall be smoothly deereased and inereased from rated speed to approximately 500 rpm to obtain a reeord of the vibration levels. 

Figure 4-6 is an amplitude-speed eurve showing the loeation of the running speed to the eritieal speed, and the amplitude inerease near the eritieal speed. When the rotor amplifieation faetor, as measured at the vibration probe, is greater than or equal to 2.5, that frequeney is ealled eritieal and the eorresponding shaft rotational frequeney is ealled a eritieal speed. For the purposes of this standard, a eritieally damped system is one in whieh the amplifieation faetor is less than 2.5. 

Flexible rotors are designed to operate at speeds above those corresponding to their first natural frequencies of transverse vibrations. The phase relation of the maximum amplitude of vibration experiences a significant shift as the rotor operates above a different critical speed. Hence, the unbalance in a flexible rotor cannot simply be considered in terms of a force and moment when the response of the vibration system is in-line (or in-phase) with the generating force (the unbalance). Consequently, the two-plane dynamic balancing usually applied to a rigid rotor is inadequate to assure the rotor is balanced in its flexible mode. 

This procedure is based on the observation of the orbital movement of the shaft eenterline. Three signal piekups are employed, of whieh two probes measure the vibration amplitudes of the rotor in two mutually perpendieular direetions. These two signals trace the orbit of the shaft centerline. The third probe is used to register the once-per-revolution reference point and is called the keyphazor. A schematic arrangement of these probes is shown in Figure 17-6. 

Calculating the electronic barrier with an accuracy of 0.1 kcal/mol is only possible for very simple systems. An accuracy of 1 kcal/mol is usually considered a good, but hard to get, level of accuracy. The situation is slightly better for relative energies of stable species, but a 1 kcal/mol accuracy still requires a significant computational effort. Thermodynamic corrections beyond the rigid rotor/harmonic vibrations approximation are therefore rarely performed. 

Rotating machines subject to imbalance caused by turbulent or unbalanced media flow include pumps, fans, and compressors. A good machine design for these units incorporates the dynamic forces of the gas or liquid in stabilizing the rotating element. The combination of these forces and the stiffness of the rotor-support system (i.e., bearing and bearing pedestals) determine the vibration level. Rotor-support stiffness is important... 

Lift/gravity dijferential Lift, which is designed into a machine-train s rotating elements to compensate for the effects of gravity acting on the rotor, is another source of imbalance. Because lift does not always equal gravity, there is always some imbalance in machine-trains. The vibration component due to the lift/gravity differential effect appears at the fundamental or 1 x frequency. 

If the operator removes the rotor from the balancing shaft without marking the point of bore and shaft contact, it may not be in the same position when reassembled. This often shifts the rotor by several mils as compared to the axis on which it was balanced, thus causing an imbalance to be introduced. The vibrations that result are usually enough to spoil what should have been a precision balance and produce a barely acceptable vibration level. In addition, if the resultant vibration is resonant with some part of the machine or structure, a more serious vibration could result. 

It is true that the extra weight of non-rotating parts (i.e., frame and foundation) reduces the vibration somewhat from the free-in-space amplitude. However, it is possible to reach precision balancing levels in only two or three additional runs, providing the smoothest running rotor. The extra effort to the balance operator is minimal because he already has the feel of the rotor and has the proper setup and tools in hand. In addition, there is a large financial pay off for this minimal extra effort due to decreased bearing and seal wear. 

If there is no resonant condition to modify the resultant vibration phase, then the phase for both vertical and horizontal readings are essentially the same even though the vertical and horizontal amplitudes do not necessarily correspond. In actual practice, this may be slightly off due to other vibration sources such as misalignment. In performing the analysis, what counts is that when the source of the vibration is primarily from imbalance, then the vertical reading phase differences between one end of the rotor and the other will be very similar to the phase differences when measured horizontally. For example, vibrations 60° out of phase vertically would show 60° out of phase horizontally within 20 per cent. 

Molecules in the gas phase have rotational freedom, and the vibrational transitions are accompanied by rotational transitions. For a rigid rotor that vibrates as a harmonic oscillator the expression for the available energy levels is ... 

It is possible that the complexes benzene- -HX can be described in a similar way, but in the absence of any observed non-rigid-rotor behaviour or a vibrational satellite spectrum, it is not possible to distinguish between a strictly C6v equilibrium geometry and one of the type observed for benzene- ClF. In either case, the vibrational wavefunctions will have C6v symmetry, however. 

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