Issue 49

S.A. Bochkarev et alii, Frattura ed Integrità Strutturale, 49 (2019) 814-830; DOI: 10.3221/IGF-ESIS.49.15 823  ( ) i D are generated in the known fashion using Young's modulus s E and Poisson’s ratio  . Geometrical and physicomechanical parameters of the problem are summarized in Tab. 5, and a comparison of the obtained results is given in Tab. 6. Without fluid With fluid Wave numbers Short circuited Open circuited Wave numbers Short circuited Open circuited j m [21] Present [21] Present j m [21] Present [21] Present 1 1 88.2310 88.1261 93.7900 93.7266 1 1 44.7600 44.6100 48.2710 48.1062 2 216.260 217.239 228.227 229.386 2 107.390 107.797 115.614 115.968 0 275.956 280.586 275.956 280.586 3 154.131 154.925 166.065 166.705 3 311.647 313.207 330.649 332.755 4 189.327 190.099 204.211 204.721 4 366.968 368.502 392.134 393.952 5 217.190 217.515 234.469 234.471 2 1 37.3160 36.8135 40.1090 39.6043 2 1 19.9160 19.5781 21.4640 21.3157 2 114.210 114.107 122.677 122.616 2 62.5970 62.3712 67.6230 67.4905 3 193.791 194.308 207.916 208.565 3 108.575 108.650 117.248 117.433 4 259.346 260.087 278.434 279.316 4 149.022 149.255 160.912 161.282 5 308.464 308.553 331.551 331.658 5 182.817 182.814 197.400 197.590 Table 4 : A comparison of the natural vibration frequencies  (Hz) of a simply supported shell with or without fluid at different combinations of electrical boundary conditions. Geometric data Material properties Fluid properties Parameter Value Parameter Value Parameter Value L , m 0.3   (1) (2) s s s E E E , GPa 69 f  , km/m 3 1000 (1) R , m 0.1   (1) (2)    0.3 c , m/s 1483 (2) R , m 0.15   (1) (2) s s s    , kg/m 3 2700   (1) ( 2) h h h , m 0.002 Table 5 : Physicomechanical and geometrical parameters of the system, consisting of elastic coaxial shells and a fluid. j m [36] Phase mode Present j m [36] Phase mode Present 1 1 391.1 out-of-phase 390.3 2 1 435.6 out-of-phase 434.9 2 846.7 out-of-phase 846.3 2 907.1 out-of-phase 905.2 3 1397.5 out-of-phase 1396.5 1 996.8 in-phase 995.2 1 1736.6 in-phase 1732.5 3 1401.3 out-of-phase 1399.7 4 1908.5 out-of-phase 1907.3 * 1822.2 mixed phase 1819.1 5 2317.2 out-of-phase 2312.5 * 1892.6 mixed phase 1891.2 * 2623.4 mixed phase 2579.5 * 2265.3 mixed phase 2262.2 3 1 403.0 out-of-phase 403.2 4 1 382.5 out-of-phase 383.3 1 671.3 in-phase 671.4 1 561.9 in-phase 562.9 2 858.30 out-of-phase 857.7 2 791.0 out-of-phase 792.0 2 1344.8 in-phase 1343.8 2 1075.5 in-phase 1076.7 3 1352.4 out-of-phase 1351.5 3 1267.5 out-of-phase 1268.8 4 1810.7 out-of-phase 1811.1 3 1676.9 in-phase 1678.0 3 2010.6 in-phase 2008.6 4 1729.2 out-of-phase 1731.3 Table 6 : A comparison of the natural vibration frequencies  (Hz) of clamped-clamped coaxial shells containing an ideal compressible fluid in the annular gap between them. Note that in [36], it was first shown that for two elastic coaxial shells, along with the in-phase (the direction and number of meridional half-waves m coincide) and out-of-phase (opposite directions) vibration modes there also exist mixed modes (the number of half-waves varies). From the data presented in Tab. 6, we may conclude that the results obtained in the framework of the model used in this paper agree fairly well with the data of the analytical solution presented in [36].