Stiffness ratios

Therefore, the lower bound on the stiffness ratio at the beginning of batch distillation is given approximately by... 

At the other extreme of Distefano s sample problems, for the largest initial charge, the maximum-stiffness ratio is of the order of 1500, which is considered to be a relatively large value. In this case, more than 10,000 time steps are required to distih 90 percent of the initial change, and the problem is better handled by a stiff integrator. 

The stiffness ratios (i.e. stiffness of the foam sandwich beam relative to the original solid beam) are also given in Fig. 2.21. In both cases the values given are independent of the original solid material or its dimensions, so this provides a good design chart. The design of solid/foam sandwich structures is also considered in Chapter 3 in the laminate analysis. 

If this is then divided into the previous enhancement ratio, q, it is possible to observe the way in which one stiffness increases at the expense of the other. Fig. 2.32 shows this transverse/axial stiffness ratio as a function of the depth of the corrugations. It may be seen that when the depth is less than four times the wall thickness then the axial stiffness ratio is better than the transverse stiffness ratio. However, when the depth is greater than four times the wall thickness then the transverse stiffness ratio dominates. 

For both odd- and even-layered special cross-ply laminates, the extensional stiffnesses, Aj., are independent of N, the number of layers (although the N individual lamina thickneWs can be summed to get the total laminate thickness t, so N is implicit im quations (4.78) and (4.82)). However, A., and A22 depend on M, the cross-ply ratio, and on F, the lamina stiffness ratio, as shown in Figures 4-22 and 4-23. For a typical glass-fiber-reinforced lamina, F =. 3, so A, varies from. 65Q.nt to... 

Table 5-1 Equivalent Bending Stiffness Ratios (After Ashton [5-11])...

For a specially orthotropic square boron-epoxy plate with stiffness ratios 0 /022= 10 and (Di2-t-2D66) = 1, the four lowest frequencies are displayed in Table 5-3 along with the four lowest frequencies of an isotropic plate. There, the factor k is defined as... 

Ko is the slip stiffness component of the control tire (p is the ratio between the tread and carcass stiffness p is the stiffness ratio of the experimental to the control tread... 

Seader and Henley (1998) provided a detailed account of the merits and demerits of different integration methods with typical conventional batch distillation examples. They have noted three particular issues with the integration methods applied in batch distillation calculations that require attention. These are the truncation error, stability and stiffness ratio. These will be briefly discussed in the following. 

E+17 Stiffness Ratios . 1915E+13. 4215E+10. 6458E+08. 1502E+08. 8294E+07... 

Another reason for the stiffness of the system is the wide range of thermal transients as indicated before. The time constant for the thermal transient depends on the square of the particle size. Therefore for a system where the ratio of largest to smallest size is 100 the stiffness ratio becomes 10. ... 

Such systems are potentially stiff or ill-conditioned. The width of the domain, in which the local time constants may be found, provides a measure of the stiffness or ill-conditioning of the system. The local stiffness ratio, S, is thus given by the expression... 

S is nothing other than the stiffness ratio of the system. The number of steps for Euler s explicit algorithm increases with the stiffness of the system. This conclusion is quite general or, in other words, explicit algorithms are not suitable for integrating stiff systems. 

To examine the extent of this problem, an eigenvalue analysis was done for many different reactor transient and steady state time profiles. The range of stiffness ratios (absolute value of ratio of largest to smallest eigenvalue, real parts only) observed for the different reactor zones was as follows ... 

This indicates that the equations are extremely stiff with a constantly changing stiffness ratio throughout the reactor. Thus, the equations might be very stiff for a time, then moderately stiff for a while and then mildly stiff at various other locations within the reactor. 

In the mode I problem a crack of length 2a is located along xj = 0, -o < X2 < a, and figure 6 shows the influence of the shear parameter kq and the inhomogeneity parameter a on the stress intensity factor k. In this case fci turns out to be independent of 6 or the stiffness ratio. More generally, from the formulation of the problem it can be shown that <7n(0,X2) is invariant with respect to a 90-degree material rotation. In the figure kq = 1 corresponds to the isotropic material [17]. 

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