Ellipsoid joint

The ellipsoid Joint modifies the spheroid joint by a head shaped as an ellipsoid (football-shaped). It has greater motion than the condylar joint but less than the spheroid joint. An example of that is the radiocarpal articrrlatioa... 

Figure 8 A joint principal coordinate projection of the occupied regions in the conformational spaces of linear (Ala) (triangles) and its conformational constraint counterpart, cyclic-CAla) (squares), onto the optimal 3D principal axes. The symbols indicate the projected conformations, and the ellipsoids engulf the volume occupied by the projected points. This projection shows that the conformational volume accessible to the cyclic analog is only a small subset of the conformational volume accessible to the linear peptide, (Adapted from Ref. 41.)...

For linear models the joint confidence region is an Alp-dimensional ellipsoid. All parameters encapsulated within this hyperellipsoid do not differ significantly from the optimal estimates at the probability level of 1-a. 

Determination of confidence limits for non-linear models is much more complex. Linearization of non-linear models by Taylor expansion and application of linear theory to the truncated series is usually utilized. The approximate measure of uncertainty in parameter estimates are the confidence limits as defined above for linear models. They are not rigorously valid but they provide some idea about reliability of estimates. The joint confidence region for non-linear models is exactly given by Eqn. (B-34). Contrary to ellipsoidal contours for linear models it is generally banana-shaped. 

Given all the above it can be shown that the (1 -a)lOO%> joint confidence region for the parameter vector k is an ellipsoid given by the equation ... 

In certain occasions the volume criterion is not appropriate. Fn particular when we have an ill-conditioned problem, use of the volume criterion results in an elongated ellipsoid (like a cucumber) for the joint confidence region that has a small volume however, the variance of the individual parameters can be very high. We can determine the shape of the joint confidence region by examining the cond( ) which is equal to and represents the ratio of the principal axes of... 

The confidence intervals defined for a single random variable become confidence regions for jointly distributed random variables. In the case of a multivariate normal distribution, the equation of the surface limiting the confidence region of the mean vector will now be shown to be an n-dimensional ellipsoid. Let us assume that X is a vector of n normally distributed variables with mean n-column vector p and covariance matrix Ex. A sample of m observations has a mean vector x and an n x n covariance matrix S. 

Finally, the dotted curve in Fig. 13 traces the relation between v and vs for rigid prolate ellipsoids of revolution [see Peterlin (16) or Frisch and Simha (6 )] with axial ratio proportional to molecular weight. This curve lies very far from those for flexible molecules except for very low values of the axial ratio p. This seems to exhaust the available information of the type represented by Fig. 13. In connection with the behavior of DNA and perhaps other naturally occurring macromolecules, it would be interesting to have calculations for rods with one or two or at most a small number of flexible joints, such as might correspond to almost completely helical structures [see Section III D]. In spite of the absence of theories for this and possibly other relevant molecular models, it is often possible to arrive at useful indications of conformation by comparing the experimental data with Fig. 13. 

Table 6.1 Weld Joint Efficiencies for Ellipsoidal and Torispherical Heads (Source Adapted from Ref. 10). 

The ASME design formula from UG-32 for ellipsoidal heads having a major-to-minor axis ratio of 2 1 and being subjected to pressure on the concave side is t = PD/ 2SE — 0.2P), where t is the minimum thickness of the head (in.), P is the MAWP (psi), D is the internal diameter of the major diameter of the ellipsoid (in.) (and equal to the inside diameter of the vessel), S is the allowable stress of the material, as listed in ASME Section II, and E is the weld joint efficiency. ... 

Joint confidence regions With two model parameters the confidence limits are defined by elliptic contours. With three parameters these limits are defined by ellipsoidic shells. With many parameters, these limits are defined by hypereUipsoids. 

Approximate inference regions for nonlinear models are defined by analogy to the linear models. In particular, the (l-a)100% joint confidence region for the parameter vector k is described by the ellipsoid. 

Figure 8.2 (a) Individual confidence limits, (b) Joint confidence limits or confidence ellipse (ellipsoid). 

The ellipsoidal dished head with a major to minor axis ratio of 2 1 is popular for economic reasons, even though the theory for thin-walled vessels predicts that the head of this shape should have twice the thickness of a hemispherical head where the major and minor axes are equal. Such an ellipsoidal head used for vessels under internal pressure has the same thickness as the cylindrical shell if the same allowable stresses and joint efficiencies are applied to both parts. The 1962 ASME Code Section VIII, Division 1 gives the following equation for the thin-walled ellipsoidal dished heads with a 2 1 major to minor axis ratio ... 

Systematic location errors could occur due to high deformation of the rock specimen. To minimize the travel-time residuals, systematic location errors associated with picking errors and the velocity variations due to microcracking were removed by the application of the joint-hypocenter determination (JHD) method (Frohlich 1979). Using the JHD method, "station corrections" can be determined that account for consistent inaccuracies of the wave velocity along the travel path especially near sensor positions. To delineate structures inside a clouded AE event distribution the collapsing method, which was first reported by Jones and Steward [1997], can be applied. This method describes how the location of an AE event can be moved within its error ellipsoid in order that the distribution of movements for every event of a cloud approximates that of normally distributed location uncertainties. This does not make the location uncertainties in the dataset smaller but it highlights structures already inherent within the unfocussed dataset. 

The structure of the articular sitrfaces further classify joints into plane, spheroid, concfylar, ellipsoid, trochoid, sellar, wAginglymusioiris. 

Example 11.6. Determine the reinforcement requirements of an 8 in. ID nozzle that is centrally located in a 2 1 ellipsoidal head. The inside diameter of the head skirt is 41.75 in. The allowable stress of both the head and nozzle material is 17.5 ksi. The design pressure is 700 psi and the design temperature is 500 F. There is no corrosion and the weld joint efficiency is = 1.0. See Fig. 11.13.1 for details of a nozzle. 

The dummy is modelled by 13 rigid parts (illustrated by the skeleton on the sketch) which are connected by 12 joints. Each joint is spherical and has complex angular loading and unloading characteristics. A certain number of ellipsoids are added to the members, on which contact may occur. The model allows contact between ellipsoids and ellipsoids or planes, typically between pelvis and seat, foot and foot planes, chin and upper torso, arms and legs, head and knees, etc. 

In a crystal, displacements of atomic nuclei from equilibrium occur under the joint influence of the intramolecular and intermolecular force fields. X-ray structure analysis encodes this thermal motion information in the so-called anisotropic atomic displacement parameters (ADPs), a refinement of the simple isotropic Debye-Waller treatment (equation 5.33), whereby the isotropic parameter B is substituted by six parameters that describe a libration ellipsoid for each atom. When these ellipsoids are plotted [5], a nice representation of atomic and molecular motion is obtained at a glance (Fig. 11.3), and a collective examination sometimes suggests the characteristics of rigid-body molecular motion in the crystal, like rotation in the molecular plane for flat molecules. Lattice vibrations can be simulated by the static simulation methods of harmonic lattice dynamics described in Section 6.3, and, from them, ADPs can also be estimated [6]. 

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