Summary Equations 4.7 through 4.13 conveniently describe the model at Station i, j and are statically correct since the summation of forces at any time during the solution will equal zero. There are two such sets of equations, one for the x-system and one for the y-system, at each mesh point, i, j. The number of stations in each direction is equal to the number of increments plus 4. As an example, a problem divided into eight increments in the x direction and eight increments in the y direction would require equations at 12 stations in each direction. Thus, the number of equations required to describe the system would be 288, 144 for the y-beams and 144 for the x-beams. This readily explains the need to resort to digital computers to perform the mathematical manipulations. Details of Solutions For solving the large number of simultaneous equations that result in each half-cycle of the alternating-direction iterative method, Matlock and Haliburton used an efficient twopass method to solve linearly elastic beam-columns. The method involves the elimination of four unknowns, two each in two passes. The first pass from top to bottom eliminates deflections w2 and w1 from each equation. (See equation 4.7.) The second pass, in i-2 reverse order, eliminates deflections w+2 and w+1 from each equation and thus results in X X X X One of the valuable assets of this method is that boundary conditions as normally discussed are automatically provided with two dummy stations specified at each end of each beam in the system. These dummy stations in reality have no bending stiffness; therefore, a bending stiffness equal to zero is input for them. Equation 4.7 is then formulated for every station in the beam plus two dummy stations on each end. X i, j To solve for wț ̧; then, we consider the plate to be two systems of orthogonal beams interconnected at Station i,j by Sf, the fictitious closure-spring constant. Figure B-5 shows a view of a grid-beam system with closure springs acting during solution. A comparable view of the slab model with torsion bars present is shown in figure B-6. With the beam-column as a basic tool, we obtain the solution of the system of equations for plates and slabs as follows: 1. Solve each x-beam successively through the system; consider all the y-beams to be held fixed in space. At any particular solution of any x-beam, the fictitious closure spring acts as restraint on the x-beam of interest. 2. After all x-beams have been solved and their new deflection pattern is known, alternate or change directions; and fix the x-beams in this new pattern. Figure B-5. Grid beam system during closure process with fictitious spring acting between x-bear y-beam at Station ij. 3. Solve for the deflected shape of each y-beam in turn. The fictitious springs no as loads or restraints on the y-beams, serving to transfer the load that has been stor them from the deflected x-beams. 4. Repeat this procedure alternately until all of the load is properly distrib throughout the system. At this point, the summation of static forces at each joint in system will equal zero within the specified tolerance, and the deflection of the x-b j system, wj, at any point will equal the deflection of the y-beam system, w i,j same point within the specified tolerance so that the term S (w;-w;) vanishes. X at Qi,j 4 Figure B-6. Plate represented in the closure process as two orthogonal systems with closure spring acting between them at Station i,j. Introduction Lightweight Mirror Structures * Eric Y. Loytty Since 1925, different approaches have been followed in the production of lightweight structures for astronomical mirrors (refs. 1 and 2); however, with the increasing use of optics in sophisticated airborne applications, weight reduction has become a severe problem. Along with the lightweight requirement, thermal stability has become supercritical for the implementation of a successful optical system. The trend in critical airborne or orbiting optical systems is to satisfy the following requirements: 1. A large primary mirror capable of detecting faint sources and increasing the angular precision of the data gathered (ref. 3) 2. Lightweight mirrors capable of withstanding inertial and gravitational forces 3. Dimensional stability in the presence of temperature changes and thermal gradients. The primary mirror of the 200-inch Hale telescope, located on Palomar Mountain in southern California, is a good example of the design problems involved in making large lightweight mirrors because its design Hale Telescope Design The material chosen to manufacture the 200-inch Hale mirror blank was developed in the Corning Glass Works research laboratory (ref. 4). This special borosilicate glass has a *This paper is an adaptation of the paper "Ultralightweight Mirror Blanks," by E.Y.Loytty and C.F. De Voe, which appeared in the IEEE Transactions on Aerospace and Electronic Systems, pp. 300-305, Vol. AES-5, No. 2; March 1969. embodies all of the technical reasoning used today in lightweight mirror designs. linear coefficient of thermal expansion of 2.5 x 106 °C1. Although the expansivity of this glass was one-third that of a conventional soda-lime composition, a conventional solid design would still have significant thermal distortions. These thermal problems were reduced by designing the mirror with a ribbed back (fig. 1) and the thickest section about 4 inches (ref. 5); therefore, no point in the disk has more than 2 inches for a surface. This reduction in section, along with the increased surface area in contact with the surrounding |