132 R. N. Coppolino direction, and locally normal to the shell wall, respectively (applied boundary accelerations are not included in this exercise), as depicted below in Fig. 13.9. In addition, the following time history force, {Fe}, along with its normalized shock spectrum, shown below in Fig. 13.10, is imposed separately on the applied load distributions, Fig. 13.9 Applied Load Distributions 1 0.5 F(t) Normalized SRS 0 –0.5 –1 0 0.1 0.2 0.3 0.4 0.5 t (sec) Applied Force Time History and Normalized SRS Freq (Hz) 0.6 0.7 0.8 0.9 1 4 3 2 1 0 0 10 20 30 40 50 60 70 80 0.9 100 Fig. 13.10 Applied Force Time History and Normalized Shock Spectrum The normalized shock spectrum associated with the applied force time history suggests a full-scale value for f*, the dynamic response cut-off frequency, on the order of 70–80 Hz. The 150th model of the shell finite element model is about 73 Hz (corresponding to the 1/20th sub-scale value of 1452 Hz). For the present example, the response of individual (real) modes is described by, {¨qn}+[2ξnωn] {˙qn}+ ω 2 n {qn}= T n e {Fe}. (13.9) The modal gains for each of the four applied load distributions are T n e . Regarding the first three applied load distributions described in Fig. 13.9, the dynamic loads, {σ}, of interest are the cumulative “body” loads at a series of 26 axial stations, defined by the following particular mode acceleration equations: {σ}={σ}D+{σ}QS = LTM¨q {¨qn}+ LTMFe {Fe}, LTM¨q = T b [M] [ ] , LTMFe = T b [ e] . (13.10)
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