Resonant zone

De-Leon N, Davis M J and Heller E J 1984 Quantum manifestations of classical resonance zones J. Chem. Phys. 80 794... 

At very low frequencies the movement of the panel will be controlled by the stiffness, as inertia is a dynamic force and cannot come into effect until the panel has measurable velocity. Stiffness controls the performance of the panel at low frequencies until resonance occurs. As the driving frequency increases, the resonance zone is passed and we enter the mass-controlled area. The increase in the sound-reduction index with frequency is approximately linear at this point, and can be represented by Figure 42.8. 

Confirmation analysis In most cases, the occurrence of dynamic resonance can be quickly confirmed. When monitoring phase and amplitude, resonance is indicated by a 180° phase shift as the rotor passes through the resonant zone. Figure 44.44 illustrates a dynamic resonance at 500 rpm, which shows a dramatic amplitude increase in the frequency-domain display. This is confirmed by the 180° phase shift in the time-domain plot. Note that the peak at 1200 rpm is not resonance. The absence of a phase shift, coupled with the apparent modulations in the FFT, discount the possibility that this peak is resonance-related. 

Function of speed The high amplitudes at the rotor s natural frequency are strictly speed dependent. If the energy source, in this case speed, changes to a frequency outside the resonant zone, the abnormal vibration will disappear. 

Higher-speed machines may be designed to operate between the first and second, or second and third, critical speeds of the rotor assembly. As these machines accelerate through the resonant zones or critical speeds, their natural frequency is momentarily excited. As long as the ramp rate limits the duration of excitation, this mode of operation is acceptable. However, care must be taken to ensure that the transient time through the resonant zone is as short as possible. 

Hall, G. R. 1984 Resonance zones in two-parameter families of circle homeomorphisms. SIAM Jl Math. Anal. 15(6), 1075-1081. 

Figure 17. The Poincare surface of section for T-shaped Hel2 with the initial vibrational state of I2 given by v = 10. Two bottlenecks to intramolecular energy transfer are shown, together with a 5 1 resonance zone and the dissociation dividing surface. From top to bottom the figures show how trajectories escape the first and then the second intramolecular botdenecks. The bottom panel shows trajectories passing the separatrix for dissociation. [From M. J. Davis and S. K. Gray, J. Chem. Phys. 84, 5389 (1986).]...

There are a number of open issues associated with statistical descriptions of unimolecular reactions, particularly in many-dimensional systems. One fundamental issue is to find a qualitative criterion for predicting if a reaction in a many-dimensional system is statistical or nonstatistic al. In a recent review article, Toda [17] discussed different aspects of the Arnold web — that is, the network of nonlinear resonances in many-dimensional systems. Toda pointed out the importance of analyzing the qualitative features of the Arnold web— for example, how different resonance zones intersect and how the intersections further overlap with one another. However, as pointed out earlier, even in the case of fully developed global chaos it remains challenging to define a nonlocal reaction separatrix and to calculate the flux crossing the separatrix in a manydimensional phase-space. 

If the resonant tori, which are the invariant tori whose rotational numbers are rational, are broken under perturbations, the pairs of elliptic and hyperbolic cycles are created in the resonance zone. This fact is known as a result of the Poincare-Birkhoff theorem [4], which holds only if the twist condition, Eq. (2), is satisfied. Around elliptic cycles thus created, new types of tori, which are... 

One can Immediately ask, what If there Is not such an EBK torus for the coupled system The empirical result Is that the method may well work anyway Table 111 shows results (26) obtained by adiabatic quantization of the Hase ( ) HCC two-degrees-of-free-dom problem. Away from the 5 2 resonances (see Fig. 5) the adiabatic and Hase results are In accord, but the adiabatic method also successfully quantizes the resonance zones. Another Illustration Is given In Table IV where a two-degrees-of-freedom model of HOD Is adlabatlcally quantized above the classical dissociation threshold... 

The normal mode representation of the phase space trajectories contains the same information as the local mode representation. However, the resonance region on the normal mode phase space map contains the local mode trajectories (la, lb 2a, 2b) and the stable fixed points Ca and C t,. The trajectories contained within the resonance zone are not free to explore the entire 0 < tp < n range whereas the trajectories outside the resonance zone do explore the 0 < ip < n range and are therefore classified as normal mode trajectories. The fixed point B (Iz = I = +2) is unstable, because it lies on a separatrix, and is located at the north pole of the normal mode polyad phase sphere. The stable fixed point A (7Z = — I = —2) is located at the south pole. 

Figure 9.15 Evolution of the polyad phase sphere from the local mode to the normal mode limit as the strength of the 1 1 coupling term (antithetical to the local mode limit) is increased from 0 (part a) to oo (part f). As the coupling term, <5 in Eq. (9.4.174) increases from 0, first in part (c) one trajectory (level 4, at highest E) falls through the unstable fixed point into the normal mode region (antisymmetric stretch) eventually, in part (e), the resonance zone fills the entire phase sphere finally, in part (f), the normal mode limit is reached (from Xiao and Kellman, 1989).

Figure 4-87. General schematic of high-profile electron-cyclotron resonance (ECR) diseharge system (a) configuration of the discharge system (b) axial variation of magnetic field, showing one or more resonance zones.

Figure 10 Poincare map for two noniinearly coupled oscillators at fixed energy. The energetic periphery is labeled X. (A) The elliptic fixed point of an orbit with a),/o)2 = /i. (B, C) Elliptic fixed points of two distinct orbits with mjuiz - Vj. (D) Sep-aratrix surrounding the Wj/wj = Vi resonance zone. The separatrix pinches off at four hyperbolic fixed points, three of which are more or less clearly visible. (E) One of four elliptic fixed points of an orbit with V (the other three are not visi-...

Figures 7-17 show hoist trip (half cycle) recordings with and without speed reduction zones, SRZ, with enlarged views of the resonance zones.

At the higher speed of a large skip hoist, the resonance zone is passed faster. 

The behavior of molecules within the intermolecular bottleneck, that is to say intramolecular dynamics, is also of interest. For example, trajeaories can be trapped for signficant periods of time within resonance zones inside the intermolecular bottleneck. A classical resonance is a region of phase space where, locally, the condition... 

Let us consider in detail what occurs inside the resonance zone when fx is small enough. Let us choose some uq = 2nM/N and reduce the map to the normal form up to terms of order iV - 1. Assuming that the values of /i are small and u is close to a o, we can derive an expression analogous to formula (10.4.19) ... 

Proof. Let us suppose, for definiteness, that the first Lyapunov value Li is negative. Then, the invariant curve exists when /x > 0. The resonance zone adjoining at the point /jL = = a o) corresponds to periodic orbits of period-... 

Fig 11.7.3. An example of an unstable torus with five pairs of periodic orbits in a resonant zone (a) and on its boundary. 

Note that both the saddles and nodes appearing inside the resonant wedge (called Arnold tongue, sometimes) lie on the invariant curve (stable if Li < 0 or unstable if L >0). Since the only stable invariant curve that can go through a saddle is its unstable manifold, and since the only unstable curve that can also go through a saddle is its stable manifold, it follows that inside the resonance zone the invariant curve is the union of the separatrices of saddles (imstable separatrices if Li < 0, or stable separatrices if L > 0) that terminate at the nodes. 

As /i increases within a resonant zone other periodic orbits with the same rotation number M/N may appear. In some cases, the boundary of the resonant zone can lose its smoothness at some points, like in the example shown in Fig. 11.7.4 here, the resonant zone consists of the union of two regions D and Z>2 corresponding to the existence of, respectively, one and two pairs of periodic orbits on the torus. The points C and C2 in Fig. 11.7.4 correspond to a cusp-bifurcation. At the point S corresponding to the existence of a pair of saddle-node periodic orbits the boundary of the resonant zone is non-smooth. 

Fig. 11.7.4. Sketch of the local structure of stability regions of periodic orbits near the boundary of a resonant zone far beyond the critical threshold.

In fact, no common upper bound exists on the number of the periodic orbits which can be generated from a fixed point of a smooth map through the given bifurcation. If the smoothness r of the map is finite, the absence of this upper estimate is obvious because it follows from the proof of the last theorem that to estimate the number of the periodic orbits within the resonant zone 1/ = M/N the map must be brought to the normal form containing terms up to order (AT — 1). In this case the smoothness of the map must not be less than (iV — 1). Hence, we can estimate only a finite number of resonant zones if the smoothness is finite. 

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