Concave Spherical Surface Research Articles

Monte Carlo calculation of free molecule flow over concave sphericalsurfaces.

Then the relation (13) can be used to express the moments mx(r niy(r and mxy(r^ in terms of normal moments of the triangular element. Since the normal moments m?, Wg, and mg are in no way coupled to unknowns of an adjacent element (see Fig. lb), it is possible to eliminate these unknowns per element using Gauss elimination. This reduces the final number of degrees of freedom from 15 to 12, which is identical to the number of degrees of freedom in the completely compatible, displacement type element of Clough and Felippa.1 3. Discussion of Results In Fig. 2, four different plate bending problems are drawn. The square plate is either simply supported (ss) or clamped (c) and it is loaded by a uniformly distributed load (u) or by a concentrated central force (c). For these four cases, the central deflections and four moments are determined. The results obtained with the Herrmann element and with the refined element are compared with results given by Timoshenko.4 Only one octal of the square plate is needed to derive the results for these problems. The errors (in per cents) are given in Table 1; here n denotes the number of elements in which half a side of the square is divided. An error of zero is an error of less than 0.5%; a plus or minus stands for overor underestimation of a certain quantity. The average errors in the four deflections and four moments show that a division of half a side in eight Herrmann elements leads to about the same accuracy as a division in two parabolic elements. Particularly, it can be observed that the Herrmann element gives accurate values of the moments in the center of the plate where dM fbx = 0, but the errors in the moments along the edge are large. This phenomenon, that the accuracy of displacements obtained with the Herrmann element depends on the value of dTkf/dx, and not on M, can also be observed in simple beam bending problems. The accuracy of the central deflection of the plates using the refined element is about the same as the accuracy that can be obtained with the same

A Laser Unequal Path Interferometer for the Optical Shop

The application of laser technology has been extended to optical shop testing by incorporating a cw, helium-neon gas laser in a package that houses a modified Twyman-Green interferometer. This modification provides for optical testing over large path differences with an auxiliary set of lenses used in the long path and a small reference flat used in the short path of the interferometer. With this technique, f/0.7 spherical mirrors have been tested (at the center of curvature) to an accuracy of 1/10 wavelength at the surface, and various other optical systems have been tested in both double pass and single pass. Two of the advantages of this testing method are (1) the capability of testing spherical concave surfaces without physically contacting the surface and (2) the ability to use small reference surfaces for large optical components or systems. The device known as a laser unequal path interferometer can be used with a set of null lenses to qualify aspheric surfaces. The unit is portable and capable of testing in any orientation under various environmental conditions. Several applications of this device are presented to illustrate its versatility.

Concave Spherical Crystals of Barium-Copper-Stearate for Use in Long Wave-Length X-Ray Spectrometers

To perform the double function of diffracting and focusing a monochromatic beam of x-rays, multiple monomolecular layers of barium-copper-stearate were deposited upon an accurately ground concave spherical surface of glass. With a spherical crystal of radius of 218 cm, grating space of about 50.5A, preliminary photographs of the Al Kα,β region, component lines not resolved, were obtained in 30 minutes at 3 kv and 15 ma. It is believed that the intensity and resolving power from such a spherical grating are adequate for measurements with an ionization chamber of absorption coefficients at wave-lengths above 8A.