Stacking model

The initial design is analysed using CA at a component level for their combined ability to achieve the important customer requirement, this being the tolerance of 0.2 mm for the plunger displacement. Only those characteristics involved in the tolerance stack are analysed. The worst case tolerance stack model is used as directed by the customer. This model assumes that each component tolerance is at its maximum or minimum limit and that the sum of these equals the assembly tolerance, given by equation 2.16 (see Chapter 3 for a detailed discussion on tolerance stack models) ... 

In general, tolerance stack models are based on either the wor.st case or statistical approaches, including those given in the references above. The worst case model (see equation 3.1) assumes that each component dimension is at its maximum or minimum limit and that the sum of these equals the assembly tolerance (initially this model was presented in Chapter 2). The tolerance stack equations are given in terms of bilateral tolerances on each component dimension, which is a common format when analysing tolerances in practice. The worst case model is ... 

The bilateral tolerance stack model including a factor for shifted component distributions is given below. It is derived by substituting equations 3.11 and 3.18 into equation 3.2. This equation is similar to that derived in Harry and Stewart (1988), but using the estimates for Cp and a target Cp for the assembly tolerance... 

In addition to understanding the statistical tolerance stack models and the FMEA process in developing a process capable solution, the designer should also address the physical assembly aspects of the tolerance stack problem. Any additional failure costs determined using CA are independent of whether the tolerances assigned to the assembly stack are capable or not. As presented in Chapter 2, the Component... 

The inadequacy of the worst case model is evident and the statistical nature of the tolerance stack is more realistic, especially when including the effects of shifted distributions. This has also been the conclusion of some of the literature discussing tolerance stack models (Chase and Parkinson, 1991 Harry and Stewart, 1988 Wu et al., 1988). Shifting and drifting of component distributions has been said to be the chief reason for the apparent disenchantment with statistical tolerancing in manufacturing (Evans, 1975). Modern equipment is frequently composed of thousands of components, all of which interact within various tolerances. Failures often arise from a combination of drift conditions rather than the failure of a specific component. These are more difficult to predict and are therefore less likely to be foreseen by the designer (Smith, 1993). 

The inadequacy of the worst case approach to tolerance stack design compared to the statistical approach is evident, although it still appears to be popular with designers. The worst case tolerance stack model is inadequate and wasteful when the capability of each dimensional tolerance is high > 1.33). Some summarizing comments on the two main approaches are given below. 

Fig. 8. Diffraction space according to the "disordered stacking model" (a) achiral (zigzag) tube (b) chiral tube. The parallel circles represent the inner rims of diffuse coronae, generated by streaked reflexions. The oo.l nodes generate sharp circles. In (a) two symmetry related 10.0 type nodes generate one circle. In the chiral case (b) each node generates a separate corona [9].

Fig. 13. Simulated diffraction space of a 10-layer monochiral MWCNT with Hamada indices (40+8/ , 5+k) with / =0,...,9. In (a), (a ) and (02) the initial stacking at ( q was ABAB. whereas in (b), (b[) and (b2) the initial stacking was random, (a) The normal incidence pattern has a centre of symmetry only. (3 )(a2) The cusps are of two different types. The arc length separating the cusps is c (b) The normal incidence pattern now exhibits 2mm symmetry. (b )(b2) The cusps are distributed at random along the generating circles of the evolutes. These sections represent the diffuse coronae referred to in the "disordered stacking model" [17].

In practice, the observed distortion is frequently strong. Thus, the correlation-function minimum is not flat. This is demonstrated in most of the dashed and solid curves in Fig. 8.21. They show model correlation functions of the paracrys-talline stacking model with varying amount of disorder. Computation82 is based on Eq. (8.104), p. 180. 

Figure 8.42. ID structural models with inherent loss of long-range order, (a) Paracrystalline lattice after HOSEMANN. The lattice constants (white rods) are decorated by centered placement of crystalline domains (black rods), (b) Lattice model with left-justified decoration, (c) Stacking model with formal equivalence of both phases (no decoration principle)... 

The stacking model (Fig. 8.42c) does not carry this inconsistency [128,229], It cannot be discriminated from the lattice models if the polydispersity is strong. For small polydispersity even the lattice models make physical sense, because then the mutual penetration is negligible. Computation and fitting of stacking and lattice models are described in Sects. 8.7.3.4 and 8.133. 

Properties and Application. The two independent statistical distributions of the two-phase stacking model are the distributions of amorphous and crystalline thicknesses, h (x) and ii2 x). Both distributions are homologous. The stacking model is commutative and consistent. If the structural entity (i.e., the stack as a whole) is found to show medium or even long-ranging order, the lattice model and its variants should be tested, in addition. As a result the structure and its evolution mechanism may more clearly be discriminated. 

Model Construction. In the stacking model alternating amorphous and crystalline layers are stacked. Likewise the combined thicknesses in the convolution polynomial are generated by alternating convolution from the independent distributions hi =h h2, h4 = hi hi, andh = hi h2- In general it follows... 

In analogy to the treatment of the stacking model Jo (s) = 0 is valid, if the structural entities are embedded in matrix material. Compact material, again, may require a correction because of the merging of particles from abutting structural entities... 

As has already been mentioned in the discussion of the stacking model, such equations are particularly useful for the analysis of nanostructured material with weak disorder in order both to assess the perfection of the material and to discriminate among lattice and stacking models (cf. Sect. 8.8.3). 

Figure 8.48. Best fits of stacking model and lattice model to the data from Fig. 8.48. The lattice model fits much better. Data sets are shifted for clarity...

A. Type I—Direct-Stacking Models Transthyretin and Superoxide Dismutase... 

This direct-stacking model (Olofsson et al., 2004 Serag et al., 2002) therefore proposes that TTR maintains much of its native structure, including the native dimer interface, in the fibrillar state. A new interaction interface is gained with the shifting of /(-strands at the ends of two sheets, driving fibril formation. 

D) Cartoon representation of the direct-stacking model of h/]2m fibrils, from Benya-mini et al. (2003). Rectangles represent the intact strands, where /(-strands B, E, and D mosdy obscure the view of strands F and C. /(-Strands A and G of the monomer have separated from the core (dotted lines), allowing stacking of the remaining strands to form a continuous /(-sheet. The gained interactions are indicated by the closed circles. 

In summary, two different Gain-of-Interaction models have been proposed for the fibrillar structure of /12m. The cross-(3 spine model (Ivanova et al., 2004) proposes a core composed of C-terminal /1-hairpins, and the direct-stacking model (Benyamini et al., 2003) proposes a core of native-like /12m molecules with their N- and C-terminal strands displaced. 

Fig. 2. Stacking model.69 Phenyl rings form the regular stacking structure between L and D molecules. Bold lines show the a-helical axes. Reproduced with permissions from Elsevier Science.

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