Torsional relation

Irrespective of the functional fonn of the van der Waals interaction, some force fields reduce the energy computed for 1,4-related atoms (i.e., torsionally related) by a constant scale factor. 

This has thorough consequences for all torsion-related entries of the beam constitutive matrix P, since it eliminates the dependence on the membrane properties stemming from the shell constitutive sub-matrix A(s). The torsional stiffness P44 of Eq. (8.8c) is only governed by the twisting stiffness of the wall ... 

Conformational free energy simulations are being widely used in modeling of complex molecular systems [1]. Recent examples of applications include study of torsions in n-butane [2] and peptide sidechains [3, 4], as well as aggregation of methane [5] and a helix bundle protein in water [6]. Calculating free energy differences between molecular states is valuable because they are observable thermodynamic quantities, related to equilibrium constants and... 

It is generally recognized that the flexibility of a bulk polymer is related to the flexibility of the chains. Chain flexibility is primarily due to torsional motion (changing conformers). Two aspects of chain flexibility are typically examined. One is the barrier involved in determining the lowest-energy conformer from other conformers. The second is the range of conformational motion around the lowest-energy conformation that can be accessed with little or no barrier. There is not yet a clear consensus as to which of these aspects of conformational flexibility is most closely related to bulk flexibility. Researchers are advised to first examine some representative compounds for which the bulk flexibility is known. 

With appropriate caUbration the complex characteristic impedance at each resonance frequency can be calculated and related to the complex shear modulus, G, of the solution. Extrapolations to 2ero concentration yield the intrinsic storage and loss moduH [G ] and [G"], respectively, which are molecular properties. In the viscosity range of 0.5-50 mPa-s, the instmment provides valuable experimental data on dilute solutions of random coil (291), branched (292), and rod-like (293) polymers. The upper limit for shearing frequency for the MLR is 800 H2. High frequency (20 to 500 K H2) viscoelastic properties can be measured with another instmment, the high frequency torsional rod apparatus (HFTRA) (294). 

T and are the glass-transition temperatures in K of the homopolymers and are the weight fractions of the comonomers (49). Because the glass-transition temperature is directly related to many other material properties, changes in T by copolymerization cause changes in other properties too. Polymer properties that depend on the glass-transition temperature include physical state, rate of thermal expansion, thermal properties, torsional modulus, refractive index, dissipation factor, brittle impact resistance, flow and heat distortion properties, and minimum film-forming temperature of polymer latex... 

From an FMEA of the system design, a Severity Rating S) = 1 was allocated, relating to a safety critical failure in service. It is required to find the optimum unequal angle section size from the standard sizes available. It is assumed that the load is carried at the section s centre of gravity, G, and only stresses due to bending of the section are considered, that is, the torsional effects are minimal. The combined weight of the beam and tie rod are not to be taken into account. 

The parameter redundancy is also the reason that care should be exercised when trying to decompose energy differences into individual terms. Although it may be possible to rationalize the preference of one conformation over another by for example increased steric repulsion between certain atom pairs, this is intimately related to the chosen functional form for the non-bonded energy, and the balance between this and the angle bend/torsional terms. The rotational banier in ethane, for example, may be reproduced solely by an HCCH torsional energy term, solely by an H-H van der Waals repulsion or solely by H-H electrostatic repulsion. Different force fields will have (slightly) different balances of these terms, and while one force field may contribute a conformational difference primarily to steric interactions, another may have the... 

Figure 59.15 is an example of a keyed shaft that shows the key size versus the shaft diameter. Because of standardization and interchangeability, keys are generally proportioned with relation to shaft diameter instead of torsional load. 

The constant G, called the shear modulus, the modulus of rigidity, or the torsion modulus, is directly comparable to the modulus of elasticity used in direct-stress applications. Only two material constants are required to characterize a material if one assumes the material to be linearly elastic, homogeneous, and isotropic. However, three material constants exist the tensile modulus of elasticity (E), Poisson s ratio (v), and the shear modulus (G). An equation relating these three constants, based on engineering s elasticity principles, follows ... 

Robeson et al. studied the secondary loss transitions of a series of poly(arylene ether)s using a torsion pendulum.15 They found that the secondary loss transitions are closely related to the segmental motion of the aryl ether bonds. The secondary... 

Fig. 5 —The relation between the rotated angle of reflection beam (a) and the torsional angle of cantilever ( /), (1-rotated angle of the reflection beam, a 2-incident angle, y 3-torsional angle of the cantilever, / 4-reflection surface before torsion of the cantilever 5-reflection surface after torsion of the cantilever).

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