Dimensioning constraints

As defined in Chapter 3, constraints are limitations that restrict the position or relationship between features of the geometric shape. Some examples of constraints are making two lines parallel or perpendicular to each other. We call these geometric constraints. We call the process of dimensioning the length of a line or the diameter of circle dimensioning constraints (see Figure 7-4). 

Bi-directional associativity Geometry and Dimensioning Constraints in 3D-CAD-product models ... 

The most commonly used method for applying constraints, particularly in molecula dynamics, is the SHAKE procedure of Ryckaert, Ciccotti and Berendsen [Ryckaert et a 1977]. In constraint dynamics the equations of motion are solved while simultaneous satisfying the imposed constraints. Constrained systems have been much studied in classics mechanics we shall illustrate the general principles using a simple system comprising a bo sliding down a frictionless slope in two dimensions (Figure 7.8). The box is constrained t remain on the slope and so the box s x and y coordinates must always satisfy the equatio of the slope (which we shall write as y = + c). If the slope were not present then the bo... 

In terms of the derived general relationships (3-1) and (3-2), x, y, and h are independent variables—cost and volume, dependent variables. That is, the cost and volume become fixed with the specification of dimensions. However, corresponding to the given restriedion of the problem, relative to volume, the function g(x, y, z) =xyh becomes a constraint funedion. In place of three independent and two dependent variables the problem reduces to two independent (volume has been constrained) and two dependent as in functions (3-3) and (3-4). Further, the requirement of minimum cost reduces the problem to three dependent variables x, y, h) and no degrees of freedom, that is, freedom of independent selection. 

Thus, for a given stiffness (F/8), panel dimensions I, b) and edge-constraints (C) the lightest panel is the one with the smallest value of p/E ... 

A similar anomalous behavior has been detected also in 3d polymer melts but only for rather short chains [41] for longer chains, several regimes occur because of the onset of entanglement (reptation ) effects. In two dimensions, of course, the topological constraints experienced by a chain from... 

The requirement of dimensional consistency places a number of constraints on the form of the functional relation between variables in a problem and forms the basis of the technique of dimensional analysis which enables the variables in a problem to be grouped into the form of dimensionless groups. Since the dimensions of the physical quantities may be expressed in terms of a number of fundamentals, usually mass, length, and time, and sometimes temperature and thermal energy, the requirement of dimensional consistency must be satisfied in respect of each of the fundamentals. Dimensional analysis gives no information about the form of the functions, nor does it provide any means of evaluating numerical proportionality constants. 

An object is generally a three dimensional constmct whose position is dehned by its location (3 degrees of freedom- x, y, z) and by its orientation (3 rotations). Thus an object is constrained if six degrees of freedom of the object are constrained. If less than six degrees of freedom are constrained, the object is under constrained and can be viewed as a mechanism. It is also called under-determined. If the object is only considered in two dimensions, then three constraints are needed to dehne the object (x, y, rotation). When an object is just constrained it is called determinate or statically-determinate. 

If we consider the size of a polymer molecule, assuming that it consists of a freely rotating chain, with no constraints on either angle or rotation or of which regions of space may be occupied, we arrive at the so-called unperturbed dimension, written (r)o Such an approach fails to take account of... 

Ad(ii) On catalysts with pores and cavities of molecular dimensions, exemplified by mordenite and ZSM-5, shape selectivity provides constraints of the transition state on the S 2 path in either preventing axial attack as that of methyl oxonium by isobutanol in mordenite that has to "turn the comer" when switching the direction of fli t through the main channel to the perpendicular attack of methyl oxonium in the side-pocket, or singling out a selective approach from several possible ones as in the chiral inversion in ethanol/2-pentanol coupling in HZSM-5 (14). Both of these types of spatial constraints result in superior selectivities to similar reactions in solutions. 

Conclusive evidence has been presented that surface-catalyzed coupling of alcohols to ethers proceeds predominantly the S 2 pathway, in which product composition, oxygen retention, and chiral inversion is controlled 1 "competitive double parkir of reactant alcohols or by transition state shape selectivity. These two features afforded by the use of solid add catalysts result in selectivities that are superior to solution reactions. High resolution XPS data demonstrate that Brpnsted add centers activate the alcohols for ether synthesis over sulfonic add resins, and the reaction conditions in zeolites indicate that Brpnsted adds are active centers therein, too. Two different shape-selectivity effects on the alcohol coupling pathway were observed herein transition-state constraint in HZSM-5 and reactant approach constraint in H-mordenite. None of these effects is a molecular sieving of the reactant molecules in the main zeolite channels, as both methanol and isobutanol have dimensions smaller than the main channel diameters in ZSM-S and mordenite. 

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