Fiber-Optic Sensors for the Study of Spacecraft-Thruster Interactions: Ion Sputtering

0.01 0.1 0.3700 0.2889 0.1758 0.0998 1.0 0.2885 0.2556 0.1722 0.0996 10.0 0.1755 0.1720 0.1487 0.0975 100.0 0.0996 0.0994 0.0974 0.0839 0.1 0.1 0.6570 0.5121 0.3106 0.1762 1.0 0.5059 0.4500 0.3040 0.1758 10.0 0.3058 0.3001 0.2612 0.1720 100.0 0.1734 0.1731 0.1698 0.1474 1.0 0.1 1.1216 0.8729 0.5202 0.2941 1.0 0.8009 0.7320 0.5062 0.2933 10.0 0.4704 0.4647 0.4194 0.2854 100.0 0.2658 0.2654 0.2622 0.2363 10.0 0.1 1.4679 1.1505 0.6885 0.3895 1.0 0.9546 0.9003 0.6605 0.3878 10.0 0.5478 0.5442 0.5116 0.3721 100.0 0.3087 0.3085 0.3064 0.2880 100.0 0.1 1.5093 1.1924 0.7269 0.4127 1.0 0.9787 0.9284 0.6940 0.4106 10.0 0.5599 0.5567 0.5274 0.3917 100.0 0.3154 0.3152 0.3134 0.2969 of a cantilever beam (for which ° is very large ) A, B, and C can be derived from the analysis presented here in by excluding the terms with °. Numerical Results When Eq. (7) derived here is used, the fundamental frequencies of spring-hinged cantilever beams with concentrated mass and ro- tary inertia at the free end can be evaluated for several cases of interest. The numerical results for the frequency parameter ¸ 1=4 f are presented in Table 1 for different values ofM, ¸J, and rota- tional spring stiffness parameter °. For a cantilever beam with very large values of ° and without concentrated mass and rotary iner- tia, Eq. (7) yields 1.879 for the frequency parameter, which is in very good agreement (<1% difference ) with the published result ¸ 1=4 f D1:875 ) (Ref. 3). The present results given in Table 1, when compared to corresponding results obtained by very accurate e nite element method, 4 are in excellent agreement with a four signie cant e gures accuracy for most of the cases. Somewhat less agreement was found for very low values of combined °, ¸M, andJ, where the accuracy is up to three signie cant e gures, even though the dif- ference is less than 1%. The e nite element results were compared with results of Lee, 5 and excellentagreement was reported for those presented. The present results are also in excellent agreement with those of Lee, except for a few combinations of °, ¸J, andM. (For example, for ° D10, ¸J DM D0:01, the difference is about 9.5%, and for ° D1:0, ¸J DM D0:01, the difference is about 5%. ) Inas- muchas allother results arein very good agreement, theauthors feel the present results are more accurate. Conclusions Asimpleformulaforthefreevibrationbehavioroftheupperstage of a rocket or missile has been presented. This problem is treated mathematically as a spring-hinged cantilever beam with a concen- trated mass and rotary inertia at the free end. The Rayleigh -Ritz method was used to obtain the fundamental frequency from suit- ably derived admissible functions that represent the spring-hinged boundarycondition.Notethatthefundamentalfrequenciesobtained in the present study agree very well with those obtained by using a very accurate e nite element method. Thus, the present formula can be reliably used to evaluate the fundamental frequency, which is an importantdesignparameter,ofaspring-hingedcantileverbeamwith end concentrated mass and rotary inertia, forming a mathematical model of the upper stage of a rocket/missile. The formula presented should be very useful for design engineers, who need simple, but accurate, closed-form solutions during the design phase.