Force propagator

Syndioselective polymerizations of propene are somewhat less regioselective than the isoselective reactions, with the typical highly syndiotactic polymer showing a few percent of the monomer units in head-to-head placement [Doi, 1979a,b Doi et al., 1984a,b, 1985 Zambelli et al., 1974, 1987]. The mode of insertion is secondary, contrary to what is expected for a carbanion propagating center. Apparently, steric requirements imposed by the counterion derived from the initiator force propagation to proceed by secondary insertion. 

Polymerization of the bulky monomer chloral yields an optically active product when one uses a chiral initiator, e.g., lithium salts of methyl (+)- or (—)-mandelate or (R)- or (S)-octanoate [Corley et al., 1988 Jaycox and Vogl, 1990 Qin et al., 1995 Vogl, 2000], The chiral initiator forces propagation to proceed to form an excess of one of the two enantiomeric helices. The same driving force has been observed in the polymerization of triphenyl-methyl methacrylate at —78°C in toluene by initiating polymerization with a chiral complex formed from an achiral initiator such as n-butyllithium and an optically active amine such as (+)-l-(2-pyrrolidinylmethyl)pyrrolidine [Isobe et al., 2001b Nakano and Okamoto, 2000 Nakano et al., 2001]. Such polymerizations that proceed in an unsymmetrical manner to form an excess of one enantiomer are referred to as asymmetric polymerizations [Hatada et al., 2002]. Asymmetric polymerization has also been observed in the radical... 

Figure 10.5 Transverse force propagation in the ideal string.

On removal of the electric field, the ionic movement decreases dramatically and the initial alignment is recreated by elastic forces propagated from the surface boundary layers as well as by conduction. Therefore, switch-off times, /off, are an order of magnitude longer ( 20-100 ms) due to the high degree of disorder caused by the flow of liquid crystal and the absence of a restoring field effect. 

Three or more molecules may assemble in the solid state to form a finite assembly with connecting forces propagated in 3D. The components of such an assembly will typically form a polyhedral shell. The shell may accommodate chemical species as guests. The polyhedron may be based on a prism or antiprism, as well as one of the five Platonic (e.g. cube, tetrahedron) or 13 Archimedean (e.g. truncated tetrahedron) solids.4... 

High Speed Cutting, Fig. 13 Cutting force propagation for a wide range of cutting speeds (Neugebauer et al. 2011)... 

FIGURE 3 (a) Force propagation during pullout process (b) Free body diagram of forces. 

The force propagator across joint i is simply the transpose of the accelmtion propa tor across joint i. Thus, we may rewrite Equation 4.62 as follows ... 

The spatial acceleration of link i is now a function of the spatial acceleration of the preceding link, a, i, and the spatial contact force propagated back to the tip of link i. Note that, given f i, this recursion progresses from the base of the chain to the last link. 

The scalar ( rations (multiplications, additions) required to compute A and A using the Force Propagation Method are shown in Table 4.6. These scalar operations are given for ap AT degree-of-freedom manipulator with simple Involute and/or prismatic joints. Note that 1)1, K, and L)), may all be computed off-line, and that the initial condition, (Ag) = 0, allows some simplification in the first iteration of the Forward Recursion. The computational complexity of the complete algorithm is 0(N), an improvement over the previous two algorithms. The efficient coordinate tiansfcMmations described in Chapter 3 are utilized in every case. 

Table 4.S Algorithm for the Force Propagation Method (Method III)...

Table 4.7 Additional Equations fa fl, Force Propagation Method...

The efficient computation of fl and A was discussed in detail in Chapt 4. The most efficient method known for the computation of both fl and A for iV < 21 is the Unit Force Method (Method II), which is O(AT ) for an A/ degree-of-freedom manipulator with revolute and/or prismatic joints. For N > 21, the 0(N) Force Propagation Method (Method III) is the most efficient. The use of these two methods will be discussed further in Section 5.1. 

The two tables differ only in the algorithm used to compute the inverse operational space inertia matrix, A and the coefficient fl. In Chapter 4, the efficient computation of these two quantities was discussed in some detail. It was detomined that the Unit Force Method (Method II) is the most efficient algorithm for these two matrices together for N < 21. The Force Propagation Method (Method ni) is the best solution for and fl for AT > 21. The scalar opmtions required for Method II are used in Table 5.1, while those required for Method III are used in Table 5.2. 

Note that the number of operations listed for fl and A in Table 5.2 is less than the total given for these two quantities in the 0 N) Force Propagation Method in Chapter 4. This reduction was achieved through a little insight First we note that the first recursion in the open-chain Direct Dynamics algorithm of... 

Brandi, Johanni, and Otter [3] computes the articulated-body inertia of each link in the chain, starting at the tip and moving back to the base. This same recursion is the first recursion in the Force Propagation Method for computing A. That is, there is an overlap of computations between the solution for the q)en-chain acceleration terms, tjopen and Xopen. and the calculation of the inverse ( rational space inertia matrix, A for this case. This fact was taken into account when the operations were tabulated. The ( rations listed for SI and A in Table 5.2 include only the second recursion for A and the additional opoations needed for SI. The recursion which computes the articulated-body inertias is included in the computatimis for open and x<,pe . 

Obviously, the PFR dependence on Da increases faster than the CSTR dependence and the conversion in the PFR is larger than that in the CSTR forced propagation is more efficient than loeal mixing only. In particular, our rigorous analysis shows that the TAP dependence lies in between those of the CSTR and the PFR. 

Figure 2. As sUder and substrate are brought into contact along the normal direction (left), asperities from both suifrices meet to ft)rm areas of real contact (middle) and an attractive molecular force field (adheaon forces) propagates aoross the interfiice (right).

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