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MODELLING OF MECHANICAL SYSTEM VOLUME 2 Episode 9

Tham khảo tài liệu 'modelling of mechanical system volume 2 episode 9', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 298 Structural elements In the particular case of a square plate they are immediately obtained as in-phase modes fn m n2 m 822 m 2X2 anm Pn m 2L 2 n 2 n2 m2 out-of-phase modes fnm Jj m2 - .ffir 2L 2 n 2 n2 m2 2 an m ftn m f 1 1 783 Hz Figure 5.17 shows the mode shapes 1 1 1 2 and 2 2 of a steel plate though assuming V 0 Lx 4.04 m Ly 4 m as computed by the finite f 1 2 out-of-phase 1037Hz f 1 2 in-phase 1172 Hz f 2 2 1444 Hz Figure 5.17. First modes of a plate fixed at the four edges Plates in-plane motion 299 element method. The computed natural frequencies are found to be less than the Rayleigh-Ritz values f 1 1 789 Hz f 1 2 out-of phase 1048 Hz f 1 2 in-phase 1180Hz f 2 2 1578Hz as expected from the Rayleigh minimum principle. The unsatisfactory result for the mode 2 2 concerning both the value of the natural frequency and the mode shape which is clearly marked by a coupling between the x and the y directions could be corrected by selecting more complicated trial functions such as X 2nx 2ny nx 3ny 2 2 a2 2 sin j sin J 1 3 sin sin L . Y n . 2nx X Í 2ny X . 3nx X ny 2 2 2 2 sin L sin L A 3 sin L sin L which would produce non-vanishing coupling terms between 2 2 and 1 3 as well as between 2 2 and 1 3. 5.3.6.4 Plate loaded by a concentrated in-plane force spatial attenuation of the local response A rectangular plate Lx Ly h is loaded by an in-plane force concentrated at P x0 y0 see Figure 5.18. The four edges are on sliding supports. We are interested in studying the field of the normal longitudinal stress Nxx x y . Colour plate 8 illustrates the results of a finite element computation referring to a longitudinal load applied to the plate centre. It is worth noticing that magnitude of displacement and stress fields are sharply peaked in the close vicinity of the loaded point in agreement with Saint-Venant s principle. It is also noted that the support reactions are essentially normal to the edges and not distributed uniformly. As expected longitudinal reactions are .