Mixed forms of free oscillations of a rectangular CFCF-plate
https://doi.org/10.17586/2226-1494-2023-23-2-413-421
Abstract
Mixed (symmetrically/antisymmetric) forms of natural oscillations of a thin rectangular plate of constant thickness, in which two parallel sides are rigidly pinched, and the other two are free (CFCF plate, C — clamped, F — free), are studied. When all the conditions of the boundary value problem are satisfied, a resolving infinite homogeneous systemof linear algebraic equations with respect to unknown coefficients of the series is obtained using two hyperbolo- trigonometric series of the coordinate deflection function. Even functions on one coordinate and odd functions on another coordinate were used to obtain symmetric-antisymmetric waveforms. As a parameter, the resulting system contains the relative frequency of free oscillations. Nontrivial solutions of the reduced system were found by the method of successive approximations in combination with a search of the frequency parameter. Numerical results are obtained for the spectrum of the first six mixed (symmetric/antisymmetric — S-A and A-S) forms of free oscillations of a thin square CFCF plate of constant thickness. The natural frequencies were compared with the results of other authors and with known experimental values. The influence on the accuracy of the results of the number of members held in rows (the size of the reduced system) and the number of iterations is investigated. 3-D images of the found waveforms are presented. The results obtained can be used in the design of various sensors and sensors using the resonance phenomenon.
About the Authors
M. V. Sukhoterin
Admiral Makarov State University of Maritime and Inland Shipping
Russian Federation
Russian Federation
Mikhail V. Sukhoterin — D.Sc., Associate Professor, Head of
Department
Saint-Petersburg, 198035
sc 16496923700
E. I. Rasputina
Admiral Makarov State University of Maritime and Inland Shipping
Russian Federation
Russian Federation
Elena I. Rasputina — PhD (Physics & Mathematics), Associate
Professor
Saint-Petersburg, 198035
sc 57204360630
N. F. Pizhurina
Admiral Makarov State University of Maritime and Inland Shipping
Russian Federation
Russian Federation
Natalya F. Pizhurina — PhD, Associate Professor, Professor, Saint-Petersburg, 198035
sc 57213917458
References
1. Leissa A.W., Qatu M.S. Vibrations of Continuous Systems. McGraw-Hill Companies, 2011, 507 p.
2. Ilanko S., Monterrubio L., Mochida Y. The Rayleigh-Ritz Method for Structural Analysis. Hoboken, NJ, Wiley & Sons, 2014, 240 p. https://doi.org/10.1002/9781118984444
3. Monterrubio L.E., Ilanko S. Proof of convergence for a set of admissible functions for the Rayleigh–Ritz analysis of beams and plates and shells of rectangular planform. Computers and Structures, 2015, vol. 147, pp. 236–243. https://doi.org/10.1016/j.compstruc.2014.09.008
4. Narita Y. Polya counting theory applied to combination of edge conditions for generally shaped isotropic plates. EPI International Journal of Engineering, 2019, vol. 2, no. 2, pp. 194–202. https://doi.org/10.25042/epi-ije.082019.16
5. Narita Y., Innami M. Identifying all combinations of boundary conditions for in-plane vibration of isotropic and anisotropic rectangular plates. Thin-Walled Structures, 2021, vol. 164, pp. 107320. https://doi.org/10.1016/j.tws.2020.107320
6. Lopatin A.V., Morozov E.V. Fundamental frequency and design of the CFCF composite sandwich plate. Composite Structures, 2011, vol. 93, no. 2, pp. 983–991. https://doi.org/10.1016/j.compstruct.2010.06.023
7. Nagino H., Mikami T., Mizusawa T. Three-dimensional free vibration analysis of isotropic rectangular plates using the B-spline Ritz method. Journal of Sound and Vibration, 2008, vol. 317, no. 1-2, pp. 329–353. https://doi.org/10.1016/j.jsv.2008.03.021
8. Liu X., Banerjee J.R. A spectral dynamic stiffness method for free vibration analysis of plane elastodynamic problems. Mechanical Systems and Signal Processing, 2017, vol. 87, part A, pp. 136–160. https://doi.org/10.1016/j.ymssp.2016.10.017
9. Liu X., Banerjee J.R. Free vibration analysis for plates with arbitrary boundary conditions using a novel spectral-dynamic stiffness method. Computers & Structures, 2016, vol. 164, pp. 108–126. https://doi.org/10.1016/j.compstruc.2015.11.005
10. Kshirsagar S., Bhaskar K. Accurate and elegant free vibration and buckling studies of orthotropic rectangular plates using untruncated infinite series. Journal of Sound and Vibration, 2008, vol. 314, no. 3–5, pp. 837–850. https://doi.org/10.1016/j.jsv.2008.01.013
11. Du J., Li W.L., Jin G., Yang T., Liu Z. An analytical method for the in-plane vibration analysis of rectangular plates with elastically restrained edges. Journal of Sound and Vibration, 2007, vol. 306, no. 3-5, pp. 908–927. https://doi.org/10.1016/j.jsv.2007.06.011
12. Li R., Zheng X., Wang P., Wang B., Wu H., Cao Y., Zhu Z. New analytic free vibration solutions of orthotropic rectangular plates by a novel symplectic approach. Acta Mechanica, 2019, vol. 230, no. 9, pp. 3087–3101. https://doi.org/10.1007/s00707-019-02448-1
13. Eisenberger M., Deutsch A. Solution of thin rectangular plate vibrations for all combinations of boundary conditions. Journal of Sound and Vibration, 2019, vol. 452, pp. 1–12. https://doi.org/10.1016/j.jsv.2019.03.024
14. Deutsch A., Tenenbaum J., Eisenberger M. Benchmark vibration frequencies of square thin plates with all possible combinations of classical boundary conditions. International Journal of Structural Stability and Dynamics, 2019, vol. 19, no. 11, pp. 1950131. https://doi.org/10.1142/S0219455419501311
15. Sukhoterin M., Baryshnikov S., Knysh T., Rasputina E. Stability of rectangular cantilever plates with high elasticity. E3S Web of Conferences, 2021, vol. 244, pp. 04004. https://doi.org/10.1051/e3sconf/202124404004
16. Lekhnitskii S.G. Anisotropic Plates. New York, Gordon & Breach, 1968, 534 p.
17. Singal R.K., Gorman D.J., Forques S.A. A comprehensive analytical solution for free vibration of rectangular plates with classical edge coditions: Experimental verification. Experimental Mechanics, 1992, vol. 32, no. 1, pp. 21–23. https://doi.org/10.1007/BF02317979
18.
Review
For citations:
Sukhoterin M.V., Rasputina E.I., Pizhurina N.F. Mixed forms of free oscillations of a rectangular CFCF-plate. Scientific and Technical Journal of Information Technologies, Mechanics and Optics. 2023;23(2):413-421. (In Russ.) https://doi.org/10.17586/2226-1494-2023-23-2-413-421
Views:
5
Email this article 