Motional Branch

Passive oscillator mode Impedance analysis of the forced oscillation of the quartz plate provides valuable information about the coating even if the active mode is not applicable anymore. For impedance analysis, a frequency generator is used to excite the crystal to a constraint vibration near resonance while monitoring the complex electrical impedance and admittance, respectively, dependent on the applied frequency (Figure 2B). For low load situations near resonance, an equivalent circuit with lumped elements - the so-called Butterworth—van-Dyke (BVD) circuit — can be applied to model the impedance data. The BVD circuit combines a parallel and series (motional branch) resonance circuit. The motional branch consists of an inductance Lq, a capacitance Cq, and a resistance Rq. An additional parallel capacitance Co arises primarily from the presence of the dielectric quartz material between the two surface electrodes (parallel plate capacitor) also containing parasitic contributions of the wiring and the crystal holder (Figure 2B). 

Choosing a thickness-polarized, thickness-vibrating piezoelectric element as an example, we can define the applied voltage V, current i, dimensions b, h, and /, and the output force and velocity F and the cross-sectional area bounded by b and / can be defined by A here it is assumed this area is electroded on the top and bottom faces of the element and that b and / are much greater than h. Treating it as a collection of discrete circuit elements as shown in Fig. lb, the Van Dyke circuit, allows the analysis of one resonance within the isolated element. Most piezoelectric materials are capacitive insulators, and the shunt capacitance Cs = bl/p h is the constant capacitance present across the element. The additional branch in the circuit represents the specific resonance being analyzed, the motional branch with inductance L, resistance R, and capacitance C. Many impedance analyzers provide this circuit as a means to model the electrical behavior of the piezoelectric element. The coupling coefficient for this... 

It turns out that there is another branch of mathematics, closely related to tire calculus of variations, although historically the two fields grew up somewhat separately, known as optimal control theory (OCT). Although the boundary between these two fields is somewhat blurred, in practice one may view optimal control theory as the application of the calculus of variations to problems with differential equation constraints. OCT is used in chemical, electrical, and aeronautical engineering where the differential equation constraints may be chemical kinetic equations, electrical circuit equations, the Navier-Stokes equations for air flow, or Newton s equations. In our case, the differential equation constraint is the TDSE in the presence of the control, which is the electric field interacting with the dipole (pemianent or transition dipole moment) of the molecule [53, 54, 55 and 56]. From the point of view of control theory, this application presents many new features relative to conventional applications perhaps most interesting mathematically is the admission of a complex state variable and a complex control conceptually, the application of control teclmiques to steer the microscopic equations of motion is both a novel and potentially very important new direction. 

The Debye model is more appropriate for the acoustic branches of tire elastic modes of a hanuonic solid. For molecular solids one has in addition optical branches in the elastic wave dispersion, and the Einstein model is more appropriate to describe the contribution to U and Cj from the optical branch. The above discussion for phonons is suitable for non-metallic solids. In metals, one has, in addition, the contribution from the electronic motion to Uand Cy. This is discussed later, in section (A2.2.5.6T... 

Seiten-. side, lateral, -achse, /. lateral axis, -axunerkung, /. marginal note, -ansatz, m. side attachment, side arm, side tube, etc. -ansicht, /. side view, profile, -arm, m. side arm side tube side branch, -bewegung, /. lateral motion. -destUlat, n. side stream, side cut. -druck, m. lateral pressure, -eck, n., -ecke,/. lateral summit, -fldche, /. lateral face flat side, facet. 

The most celebrated textual embodiment of the science of energy was Thomson and Tait s Treatise on Natural Philosophy (1867). Originally intending to treat all branches of natural philosophy, Thomson and Tait in fact produced only the first volume of the Treatise. Taking statics to be derivative from dynamics, they reinterpreted Newton s third law (action-reaction) as conservation of energy, with action viewed as rate of working. Fundamental to the new energy physics was the move to make extremum (maximum or minimum) conditions, rather than point forces, the theoretical foundation of dynamics. The tendency of an entire system to move from one place to another in the most economical way would determine the forces and motions of the various parts of the system. Variational principles (especially least action) thus played a central role in the new dynamics. 

Sequence valves control the sequence of operation between two branches in a hydraulic circuit. In other words, they enable one component within the system to automatically set another component into motion. An example of the use of a sequence valve is in an aircraft landing gear actuating system. 

Nesterovich NI (1979) Equations of turbulent motion of heterogeneous mixtures (in Russian). Prepr of Inst of Theor and Appl Mech USSR Academ of Science, Siberian Branch 8 28... 

As the tangent plane rolls on the primitive surface, it may happen that the two branches of the connodal curve traced out by its motion ultimately coincide. The point of ultimate coincidence is called a plait point, and the corresponding homogeneous state, the critical state. 

It should be noted that there is a considerable difference between rotational structure narrowing caused by pressure and that caused by motional averaging of an adiabatically broadened spectrum [158, 159]. In the limiting case of fast motion, both of them are described by perturbation theory, thus, both widths in Eq. (3.16) and Eq (3.17) are expressed as a product of the frequency dispersion and the correlation time. However, the dispersion of the rotational structure (3.7) defined by intramolecular interaction is independent of the medium density, while the dispersion of the vibrational frequency shift (5 12) in (3.21) is linear in gas density. In principle, correlation times of the frequency modulation are also different. In the first case, it is the free rotation time te that is reduced as the medium density increases, and in the second case, it is the time of collision tc p/ v) that remains unchanged. As the density increases, the rotational contribution to the width decreases due to the reduction of t , while the vibrational contribution increases due to the dispersion growth. In nitrogen, they are of comparable magnitude after the initial (static) spectrum has become ten times narrower. At 77 K the rotational relaxation contribution is no less than 20% of the observed Q-branch width. If the rest of the contribution is entirely determined by... 

Fig. 3.13. Density-dependence of the Qo, branch line width y of methane (the dashed line is for pure vibrational dephasing, supposed to be Unear in density), (o) experimental data (with error bars) [162] Top part rotational contribution yR and its theoretical estimation in motional narrowing limit [162] (solid line) the points were obtained by subtraction of dephasing contribution y

The "M"-flame case shows a different kind of flame interaction illustrated in Figure 5.2.4 (MF). The "M"-shape comprises two reactive sheets separated by fresh reactants. This gives rise to flame-flame interactions between neighboring branches of the "M"-shape [41]. The case presented corresponds to an equivalence ratio O = 1.13, a mixture flow velocity v/v = 1.13 m/s, a modulation level fixed to v = 0.50m/s, and a modulation frequency/= 150 Hz. The description of the flame motion over a cycle of excitation starts as in the flame-plate interaction. A velocity perturbation is generated at... 

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