Elimination of degrees of freedom by structural splining.

Abstract

5 10 15 REDESIGN CYCLES Fig. 1 Design cycles for 5-element panel; (n = 5, Y\ = 0.8, A u = 1 0 + 2.0,work the derivatives dkjdpi were evaluated numerically by incrementing successive values of p t . However, other techniques 7 for evaluating derivatives will be used in future studies in order to reduce solution times.Since the constraint equation is not linear in the design parameters, the direction of VA cr , the gradient of /l cr , is a function of the p .'s. In carrying out computations in the weight-minimization mode using Eq. (14) it was found that the value of k cr for the new design was not exactly A 0 . Hence, the following procedure was employed: the weight-minimization mode was employed until A cr (p) fell below a specified value. At this point a "lambdamodification mode" was entered in which the weight was held constant at the last value attained by weight minimization, while the value of X cr was increased along the direction of steepest ascent subject to the weight constraint. The following equation governs the lambda-modification mode 5 :where c 2 is a positive constant. Numerical ResultsA computer program was written in which the element stiffness, aerodynamic, and mass matrices were generated according to Eq. (9) for a specified structural mass fraction, 77, and a specified set of design parameters, p.. These were assembled to form an eigenvalue problem of the form given by Eq. (10). The value of a was systematically varied until the critical value for the given design was obtained. Equations (14) and (15) were employed to determine design changes for weight-minimization steps and for lambda-modification steps, respectively.The results shown in Figs. 1 and 2 represent improved designs and not optimal designs and represent only preliminary studies of the use of the weight-minimization, lambda-modification procedure. Figure 1 is for a panel with structural mass fraction r\ -0.8 and having five finite elements, i.e., n = 5. The value of A 0 , as determined for n = 5, was A 0 =. 342.901.In Fig. 1 weight-minimization proceeded until A cr fell below a limit set at A L = ^0 -2.0. Lambda-modification was allowed to raise A cr above A 0 with an upper limit of A u = A 0 + 2.0. It is seen that a weight reduction of 3.6% of the total weight was possible with A cr = 343.190. The best design gives a total weight reduction of 4.0% with A cr = 341.315. Reference 5 contains similar figures for a 5-element panel with A u = A 0 and for a 9-element panel. The 9-element problem was terminated after four weightminimization steps with no lambda-modification. Figure 2 shows the shape of the best designs for the 5-element and 9-element panels.Although it might be possible to obtain better designs than the best design in Fig. 1, it is significant to note that the percent weight saved is considerably more than was obtained by Turner, 3 and the shape of the best designs shown in Fig. 2 indicate that Turner's Use of n = 3 did not allow the design to assume a form resembling an optimum design.Weisshaar 8 recently published resu...