Z-plane

Let us consider the scheme showed in Fig. I to calculate the field scattered by a rough cylindrical surface (i.e. a wire). The wire is illuminated by a monochromatic, linearly polarized plane wave at an angle of incidence a with its axis of symmetry. The surface is described, in a system fixed to the wire, by p = h (cylindrical coordinates. We shall denote the incident wave vector lying on the x-z plane as kj and the emergent wave vector simply as k. 

Fig.4 Cross-sections in the (y z) plane of the cracks used in the computations. The dimensions are given in mm.

The physical situation of interest m a scattering problem is pictured in figure A3.11.3. We assume that the initial particle velocity v is comcident with the z axis and that the particle starts at z = -co, witli x = b = impact parameter, andy = 0. In this case, L = pvh. Subsequently, the particle moves in the v, z plane in a trajectory that might be as pictured in figure A3.11.4 (liere shown for a hard sphere potential). There is a point of closest approach, i.e., r = (iimer turning point for r motions) where... 

M continually decreases under the influence of spin-spin relaxation which destroys the initial phase coherence of the spin motion within they z-plane. In solid-state TREPR, where large inliomogeneous EPR linewidths due to anisotropic magnetic interactions persist, the long-time behaviour of the spectrometer output, S(t), is given by... 

Using the described algorithm the flow domain inside the cone-and-plate viscometer is simulated. Tn Figure 5.17 the predicted velocity field in the (r, z) plane (secondary flow regime) established inside a bi-conical rheometer for a non-Newtonian fluid is shown. 

Stability in the z-plane 7.6.1 Mapping from the s-plane into the z-plane... 

Just as transient analysis of continuous systems may be undertaken in the. v-plane, stability and transient analysis on discrete systems may be conducted in the z-plane. It is possible to map from the. v to the z-plane using the relationship... 

Equation (7.63) results in a polar diagram in the z-plane as shown in Figure 7.16. Figure 7.17 shows mapping of lines of constant a (i.e. constant settling time) from the. V to the z-plane. From Figure 7.17 it can be seen that the left-hand side (stable) of the. v-plane corresponds to a region within a circle of unity radius (the unit circle) in the z-plane. 

Fig. 7.19 Corresponding pole locations on both sand z-planes.

Figure 7.19 shows corresponding pole locations on both the. v-plane and z-plane. 

In sections 5.4 and 6.6, compensator design in the. v-plane and the frequency domain were discussed for continuous systems. In the same manner, digital compensators may be designed in the z-plane for discrete systems. 

The seript file examp75.m simulates the Jury stability test undertaken in Example 7.5. With the eontroller gain K in Example 7.5 (Figure 7.14) set to 9.58 for marginal stability see equation (7.75), the roots of the denominator of the elosed-loop pulse transfer funetion are ealeulated, and found to lie on the unit eirele in the z-plane. 

Example 7.6 constructs the root-locus in the z-plane for the digital control system in Example 7.4 (Figure 7.14). 

The n orbital is symmetric with respect to both the jc-z plane and the y-z plane. It is antisymmetric with respect to the molecular (x-y) plane. On the other hand, ti is antisymmetric with respect to the y-z plane. 

The implications of the Kirchhoff hypothesis on the laminate displacements u, V, and w in the x-, y-, and z-directions are derived by use of the laminate cross section in the x-z plane shown in Figure 4-4. The displacement in the x-direction of point B from the undeformed middle surface to the deformed middle surface is Uo (the symbol nought (°) is used to designate middle-surface values of a variable). Because line ABCD remains straight under deformation of the laminate, the displacement at point C is... 

These coupled second-order partial differential equations do not have a closed-form solution. Accordingly, the approximate numerical technique of finite differences is employed. First, however, the boundary conditions must be prescribed in order to complete the formulation of the problem. Symmetry of the laminate about several planes permits reduction of the region of consideration to a quarter of the laminate cross section in the y-z plane at any value of x as shown in Figure 4-52. There, along the stress-free upper surface. 

Thus, the plate is in a state of generalized plane strain in the x-z plane. 

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