PSI - Issue 24

Lorenzo Beretta et al. / Procedia Structural Integrity 24 (2019) 267–278 L.Beretta, E.Marotta,P.Salvini / Structural Integrity Procedia 00 (2019) 000 – 000

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The mesh cannot be considered as a simple two-dimensional object since its structure comprehends full areas and empty areas, repeated in space with particular shapes and dimensions. For that reason, it is necessary to provide adjustments, Soedel (2004).

Table 1 . Free Vibration Tests.

Table 2. Forced Vibration Tests.

Free Vibrations

Tension y (N)

Tension x (N)

Frequency (Hz)

Forced Vibrations

Tension y (N)

Tension x (N)

Frequency (Hz)

TEST 1 TEST 2 TEST 3 TEST 4 TEST 5 TEST 6 TEST 7 TEST 8 TEST 9 TEST 10 TEST 11 TEST 12 TEST 13 TEST 14 TEST 15

TEST 1 TEST 2 TEST 3 TEST 4 TEST 5 TEST 6 TEST 7 TEST 8 TEST 9 TEST 10 TEST 11 TEST 12 TEST 13 TEST 14 TEST 15

0,5949 0,6615 0,8739 0,9664 1,2098 1,4430 1,5089 1,6566 1,9489 2,1936 2,7205 2,7662 3,0973 3,5070 3,7163

0,7574 0,9285 1,1082 1,3080 1,5213 1,9424 2,1910 2,4470 2,8765 3,1873 3,7222 3,6379 4,1613 4,8904 4,9212

215,6 253,1 274,2 293,0 321,1 360,9 377,3 400,8 435,9 459,4 503,9 501,6 539,1 574,2 581,3

0,4841 0,6160 0,6508 0,9021 1,1177 1,2396 1,3680 1,6630 1,8008 2,1906 2,2984 2,3596 3,0112 3,0694 3,2649

0,5875 0,8112 0,9557 1,1435 1,5428 1,6420 1,9045 2,3700 2,4993 2,9680 3,1790 3,2773 4,0710 4,2602 4,5612

189,8 201,6 236,7 271,9 316,4 330,5 349,2 389,1 405,5 440,6 461,7 459,4 520,3 522,7 550,8

On the other hand, it cannot even be considered as an ordinary one-dimensional object, because some information is inevitably lost. Even if reducing the problem to one dimension is possible, it encounters the necessity to overcome the obstacle of proportions between width and length, Weaver et al. (1990). The approximation of the mesh to a one-dimensional object (vibrating wire) fits well to the measured data if a golden rectangle, inscribed in the edge circumference and under the influence of the highest tension alone, is considered. A golden rectangle is a rectangle whose sides are in the golden ratio ( ) 1 5 2 r = + between them. Moreover, only the highest tension is considered because it represents the load in the stiffest direction, i.e. the one that contributes the most to the variation of potential (deformation) energy during vibrations. With these clarifications and making calculations, the first natural frequency fits the following (One Dimensional Approximation model): being the diameter D and the mass per unit area known data. When considering the mesh as a two-dimensional object, the approximation of the circular net fits very well, if some adjustments are made. As previously said, analytical formulas in Paragraph 2 and 3 do not take into account neither the mesh strong anisotropy nor its complex geometry. So, two coefficients c 1 and c 2 , that consider such aspects, have been introduced in the circular net formula: ( ) 1 2 , , 2 x y m n m n c a c T T r      +  = (7) 1  3 1 0.8107 1.7013 0.5257 x D x T T D D   (6)

These two coefficients are estimated with the least-squares method directly on the experimental data of free