Dimensional variability

Non-dimensionalization of the stress is achieved via the components of the rate of deformation tensor which depend on the defined non-dimensional velocity and length variables. The selected scaling for the pressure is such that the pressure gradient balances the viscous shear stre.ss. After substitution of the non-dimensional variables into the equation of continuity it can be divided through by ieLr U). Note that in the following for simplicity of writing the broken over bar on tire non-dimensional variables is dropped. 

Simila.rityAna.Iysis, Similarity analysis starts from the equation describing a system and proceeds by expressing all of the dimensional variables and boundary conditions in the equation in reduced or normalized form. Velocities, for example, are expressed in terms of some reference velocity in the system, eg, the average velocity. When the equation is rewritten in this manner certain dimensionless groupings of the reference variables appear as coefficients, and the dimensional variables are replaced by their normalized relatives. If another physical system can be described by the same equation with the same numerical values of the coefficients, then the solutions to the two equations (normalized variables) are identical and either system is an accurate model of the other. 

Component reliability will vary as a function of the power of a dimensional variable in a stress function. Powers of dimensional variables greater than unity magnify the effect. For example, the equation for the polar moment of area for a circular shaft varies as the fourth power of the diameter. Other similar cases liable to dimensional variation effects include the radius of gyration, cross-sectional area and moment of inertia properties. Such variations affect stability, deflection, strains and angular twists as well as stresses levels (Haugen, 1980). It can be seen that variations in tolerance may be of importance for critical components which need to be designed to a high reliability (Bury, 1974). 

The measures of dimensional variability from Conformability Analysis (CA) (as described in Chapters 2 and 3), specifically the Component Manufacturing Variability Risk, q, is useful in the allocation of tolerances and subsequent analysis of their distributions in probabilistic design. The value is determined from process capability maps for the manufacturing process and knowledge of the component s material and geometry compatibility with the process. In the specific case to the th component bilateral tolerance, it was shown in Chapter 3 that the standard deviation estimates were ... 

In experimental load studies, the measurable variables are often surface strain, acceleration, weight, pressure or temperature (Haugen, 1980). A discussion of the techniques on how to measure the different types of load parameters can be found in Figliola and Beasley (1995). The measurement of stress directly would be advantageous, you would assume, for use in subsequent calculations to predict reliability. However, no translation of the dimensional variability of the part could then be accounted for in the probabilistic model to give the stress distribution. A better test would be to output the load directly as shown and then use the appropriate probabilistic model to determine the stress distribution. 

Figure 4.28 shows the derivation of equation 4.38 from the algebra of random variables. (Note, this is exaetly the same approaeh deseribed in Appendix VIII to find the probability of interferenee of two-dimensional variables.)... 

We know that the eoeffieient of variation, C, of the applied torque is approximately 0.1, and that the final loading stress variable will have a similar level of variation beeause the dimensional variables have a very small varianee eontribution in eompar-ison. We also know the ultimate shear strength parameters of the weak link material, therefore substituting in equation 4.91 and rearranging to set the right-hand side to zero gives ... 

The varianee eontribution in pereent of the dimensional variable, a, to the varianee of the stress then beeomes ... 

Plotting this data as a Pareto chart gives Figure 3. It shows that the load is the dominant variable in the problem and so the stress is very sensitive to changes in the load, but the dimensional variables have little impact on the problem. Under conditions where the standard deviation of the dimensional variables increased for whatever reason, their impact on the stress distribution would increase to the detriment of the contribution made by the load if its standard deviation remained the same. 

Among the dimensional variables of the problem, five parameters have independent dimensions and Eq. (6.41) may be written in dimensionless form. Choosing parameters Pl, CpL, U, ATs, and taking into account r-theorem (Sedov 1993), Eq. 

In view of the complexity of the material, it is difficult to unequivocally assign a particular mechanism for electronic conduction in the polymer, although some evidence exists to suggest that it involves three-dimensional variable-range hopping as found for other polymers [213], It has also been suggested that the conductivity of polyaniline is a combination of both ionic and electronic conductivity [207], and is... 

The law of conservation of dimensions can be applied to arrange the variables or parameters that are important in a given problem into a set of dimensionless groups. The original set of (dimensional) variables can then... 

It is important to realize that the process of dimensional analysis only replaces the set of original (dimensional) variables with an equivalent (smaller) set of dimensionless variables (i.e., the dimensionless groups). It does not tell how these variables are related—the relationship must be determined either theoretically by application of basic scientific principles or empirically by measurements and data analysis. However, dimensional analysis is a very powerful tool in that it can rovide a direct guide for... 

Figure 16.3 shows the dependence of the QY from the non-dimensional variables y and w. The increase of both acceptor and donor concentration, until back reactions become important (outside the simple model hypotheses), or the decrease of the absorbed light [Pg.359]

Also nonlinear methods can be applied to represent the high-dimensional variable space in a smaller dimensional space (eventually in a two-dimensional plane) in general such data transformation is called a mapping. Widely used in chemometrics are Kohonen maps (Section 3.8.3) as well as latent variables based on artificial neural networks (Section 4.8.3.4). These methods may be necessary if linear methods fail, however, are more delicate to use properly and are less strictly defined than linear methods. 

FIGURE 2.17 Projection of the object points from a two-dimensional variable space on to a direction b45 giving a latent variable with a high variance of the scores, and therefore a good preservation of the distances in the two-dimensional space. 

For univariate data, only one variable is measured at a set of objects (samples) or is measured on one object a number of times. For multivariate data, several variables are under consideration. The resulting numbers are usually stored in a data matrix X of size ii x in where the n objects are arranged in the rows and the m variables in the columns. In a geometric interpretation, each object can be considered as a point in an m-dimensional variable space. Additionally, a property of the objects can be stored in a vector y (nx 1) or several properties in a matrix Y nxq) (Figure 2.19). 

Principal component analysis (PCA) can be considered as the mother of all methods in multivariate data analysis. The aim of PCA is dimension reduction and PCA is the most frequently applied method for computing linear latent variables (components). PCA can be seen as a method to compute a new coordinate system formed by the latent variables, which is orthogonal, and where only the most informative dimensions are used. Latent variables from PCA optimally represent the distances between the objects in the high-dimensional variable space—remember, the distance of objects is considered as an inverse similarity of the objects. PCA considers all variables and accommodates the total data structure it is a method for exploratory data analysis (unsupervised learning) and can be applied to practical any A-matrix no y-data (properties) are considered and therefore not necessary. 

FIGURE 3.4 Different distributions of object points in a three-dimensional variable space. 

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