Rigid-body displacement

Gray JJ, Moughon S, Wang C, Schueler-Furman O, Kuhlman B, Rohl CA, Baker D. Protein-protein docking with simultaneous optimization of rigid-body displacement and side chain conformations. J. Mol. Biol. 2354 331 281-299. 

The anisotropic rigid-body displacements of a molecule can be described in terms of translation (T, vibration along a straight-line path), libration (L, vibration along an arc), and screw (S, a combination of vibration and translation that may be regarded as vibration along a helical path). The mean-square amplitude of translational vibration is usually referred to as a system of Cartesian coordinates and unit vectors. S is the mean correlation between libration about an axis and translation parallel to this axis. Each of these three components can be expressed as... 

Flexural strength of core plates and sandwich plates were determined by three point bend test as shown in Fig. 2. The span was 150 mm. Deflection of the center of the span was measured with a laser displacement censor. Vertical movements of the plate on the supporting points were also measured with displacement transducers which were used to delete the rigid body displacement from the measured deflection. The applied load was measured with a load cell. Shear strength of core plates were determined by double shearing test as shown in Fig. 3. 

X is the 6 X 1 rigid body displacement subvector of x for block i Hij is the 12 x 12 decomposed Hessian matrix for an isolated system of blocks i and ... 

In this method, the degrees of freedom of the structure are reduced to the 6 rigid body displacements plus the lowest natural frequency modeshapes. The great advantage of this method is that the reduced mass and stiffness matrices are diagonal (Hurty and Rubinstein 1964). 

Consider a material s body in Figure 4.1 before deformation. After internal or external loading (force, thermal, electromagnetic field, gravity, etc.), the body is deformed, and may also have rigid body displacement, into the deformed spatial body shown in Figure 4.1. A material s point is defined in the Cartesian coordinate system (ei, 62, 63) by a position vector X in the undeformed body and position vector x in the deformed body u is the displacement vector. Throughout this chapter, bold face letters will represent tensors. Obviously,... 

Johnson et al. (3) have shown how the elastic energy can be computed in the case of a spherical punch by adding a rigid body displacement to the classical elastic solution. In fact, this method is quite general for punches of any shape and can be proved by the general theory of elasticity (4) stresses and discontinuities of displacement are those of fracture mechanics ... 

In order to perform a FE analysis it is necessary to constraint the model so as to prevent rigid body displacements and rotations. Anatomically, the LV movement is restrained by the right ventricle and the atria, by the pericardial sac, and by the aorta, none of which constitutes an absolute clamp. The common approach is to hinge the model at the base constraining it from vertical movement. This approach, due to the St. Venant principle, seems to have only a local effect and thus it will not affect the stress distribution in most of the LV. 

There is a rigorous theory for the extraction of rigid-body displacement tensors from ADPs (Schomaker, V. Trueblood, K. N. Acta Cryst. 1968, B24,63-76, as well as for the calculation of ADPs from lattice dynamics (Gramaccioli, C. M. Filippini, G. Lattice-dynamical evaluation of temperature factors in non-rigid molecular crystals a first application to aromatic hydrocarbons, Acta Cryst. 1983, A39, 784-791). 

Under consideration of the independence of internal forces and rigid body displacements and the independence of material damping and rigid body velocities, it yields... 

Simple models of spline tooth contact have been presented by Buckingham [5] and Neale [6], These are based effectively upon the complete slip theory of contact [10] in wdiich the elastic deformations are considered to be small compared to the slip displacements. The externally imposed di lacements due to misalignment of the male spline with respect to the female are used to find local (rigid-body) displacements within the contact. Components of the relative displacement in the plane of the tooth contact provide a first approximation to m nitude and direction of the local slip. In addition it is expected that the contact will occur within the region in which there is an interference (i.e. a negative value of the relative... 

The dimensionless rigid body displacement Ho is coupled to the force balance equation, which, for an elliptical contact reads ... 

J. J. Gray, S. Moughon, C. Wang, O. Schueler-Forman, B. Kuhlman, C. A. Rohl, and D. Baker, /. Mol. Biol., 331, 281-299 (2003). Protein-Protein Docking with Simultaneous Optimization of Rigid-Body Displacement and Side-Chain Conformations. 

The data supplied by TTCI for tbe P3 loads provides six degree-of-freedom accelerations at a single node located at tbe interface between the cradle and rail car deck. As was done for the PI load cases, modifications were made to each load case acceleration time-history to remove large rigid body displacements. This modification was performed in a manner analogous to that performed for the PI load cases. AH of the load cases were taken from the Track 9 snapshot at 62 mph. This snapshot was assumed to represent the most conservative of those provided. 

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