- PII
- S3034575825010071-1
- DOI
- 10.7868/S3034575825010071
- Publication type
- Article
- Status
- Published
- Authors
- Volume/ Edition
- Volume 89 / Issue number 1
- Pages
- 90-105
- Abstract
- The problem of maximizing the value of the first natural frequency for a functionally graded material depending on the variation law of Young's modulus is considered. It is assumed that there is a limitation on the average integral value of Young's modulus. The effect of variable material properties on the value of the first natural frequency is shown using the finite element method for the numerical solution of a two-dimensional axisymmetric problem of free oscillations of a cylinder. The optimality condition is obtained using the methods of variational calculus based on the general formulation of the problem for an inhomogeneous elastic isotropic body. It is noted that the left-hand side of this condition has a quadratic form. The problem of finding the optimal variation law of Young's modulus is essentially nonlinear in the general case and special numerical methods must be used to solve it. Three special cases are considered using the obtained optimality condition: bending vibrations of a circular solid plate, longitudinal vibrations of a rod and radial vibrations of a solid thin disk, taking into account the corresponding hypotheses. The optimal variation laws of Young's modulus and the displacement function are obtained in analytical form for all problems. Particularly, in the problem for the disk, a representation is proposed for the radial component of the displacement field, which is described by a linear law. It is shown that in this case the corresponding radial and tangential components of the stress tensor are equal. The sought-for function of the change in Young's modulus along the radial coordinate is found in analytical form from the equation of motion and the boundary condition on the outer boundary. An analytical expression is obtained for determining the value of the natural frequency, corresponding to the found solution. The accuracy of this formula is estimated by comparing it with the numerical solution obtained using the finite element method in the FlexPDE package. A comparison of the values of the natural frequency for homogeneous and inhomogeneous disks is made.
- Keywords
- цилиндр стержень пластина диск функционально-градиентный материал метод конечных элементов модуль Юнга оптимизация собственная частота
- Date of publication
- 03.02.2025
- Year of publication
- 2025
- Number of purchasers
- 0
- Views
- 60
References
- 1. Suresh S., Mortensen A. Fundamentals of Functionally Graded Materials. London: IOM Commun. Ltd., 1998. 165 p.
- 2. Birman V., Byrd L. Modeling and analysis of functionally graded materials and structures // Appl. Mech. Rev. 2007. V. 60. № 5. P. 195-216. https://www.doi.org/10.1115/1.2777164
- 3. Kieback B., Neubrand A., Riedel H. Processing techniques for functionally graded materials // Mater. Sci. Eng., A. 2003. V. 362. № 1-2. https://www.doi.org/10.1016/S0921-5093 (03)00578-1
- 4. Naebe M., Shirvanimoghaddam K. Functionally graded materials: A review of fabrication and properties // Appl. Materials Today. 2016. V. 5. P. 223-245. https://www.doi.org/10.1016/j.apmt.2016.10.001
- 5. Селяев В.П., Карташов В.А., Клементьев В.Д., Лазарев А.Л. Функционально-градиентные композиционные строительные материалы и конструкции. Саранск: Мордовский государственный ун-т им. Н.П. Огарева, 2005. 160 с.
- 6. Saleh B., Jiang J., Fathi R. et al. 30 Years of functionally graded materials: An overview of manufacturing methods, applications and future challenges // Composites, Pt. B. 2020. V. 201. Art. No. 108376. https://www.doi.org/10.1016/j.compositesb.2020.108376
- 7. Boggarapu V., Gujjala R., Ojha S. et al. State of the art in functionally graded materials // Compos. Struct. 2021. V. 262. Art. No. 113596. https://www.doi.org/10.1016/j.compstruct.2021.113596
- 8. Ольхофф Н. Оптимальное проектирование конструкций: Вопросы вибрации и потери устойчивости. Механика. Новое в зарубежной науке: Сб. статей. Вып. 27. М.: Мир, 1981. 277 с.
- 9. Баничук Н.В. Оптимизация форм упругих тел. М.: Наука, 1980. 256 с.
- 10. Баничук Н.В. Введение в оптимизацию конструкций. М.: Наука, 1986. 304 с.
- 11. Yang S.T., Liang Y.J. Stacking sequence optimization of composite laminates for maximum fundamental frequency using Bayesian optimization computational framework // Results in Engng. 2024. V. 23. Art. No. 102586. https://www.doi.org/10.1016/j.rineng.2024.102586
- 12. Narita Y., Robinson P. Maximizing the fundamental frequency of laminated cylindrical panels using layerwise optimization // Int. J. of Mech. Sci. 2006. V. 48. № 12. P. 1516-1524. https://www.doi.org/10.1016/j.ijmecsci.2006.06.008
- 13. Trias D., Maimi P., Blanco N. Maximization of the fundamental frequency of plates and cylinders // Compos. Struct. 2016. V. 156. P. 375-384. https://www.doi.org/10.1016/j.compstruct.2015.08.034
- 14. Marzok A., Waisman H. Topology optimization of extruded beams modeled with the XFEM for maximizing their natural frequencies // Mech. Res. Commun. 2024. V. 135. Art. No. 104234. https://www.doi.org/10.1016/j.mechrescom.2023.104234
- 15. Kamgar R., Rahmani F., Rahgozar R. Geometrical and material optimization of the functionally graded doubly-curved shell by metaheuristic optimization algorithms // Structures. 2024. V. 62. Art. No. 106254. https://www.doi.org/10.1016/j.istruc.2024.106254
- 16. Jensen J.S., Pedersen N.L. On maximal eigenfrequency separation in two-material structures: the 1D and 2D scalar cases // J. of Sound&Vibr. 2006. V. 289. № 4-5. P. 967-986. https://www.doi.org/10.1016/j.jsv.2005.03.028
- 17. Du J., Olhoff N. Topological design of freely vibrating continuum structures for maximum values of simple and multiple eigenfrequencies and frequency gaps // Struct.&Multidisc. Optimiz. 2007. V. 34. P. 91-110. https://www.doi.org/10.1007/s00158-007-0101-y
- 18. Sun J., Tian Q., Hu H., Pedersen N.L. Topology optimization for eigenfrequencies of a rotating thin plate via moving morphable components // J. of Sound&Vibr. 2019. V. 448. P. 83-107. https://www.doi.org/10.1016/j.jsv.2019.01.054
- 19. Li Q., Wu Q., Dou S. et al. Nonlinear eigenvalue topology optimization for structures with frequency-dependent material properties // Mech. Syst.&Signal Proc. 2022. V. 170. Art. No. 108835. https://www.doi.org/10.1016/j.ymssp.2022.108835
- 20. Meng Z., Yang G., Wu Q. et al. Reliability-based topology optimization for fundamental frequency maximization with frequency band constraints // Mech. Syst.&Signal Proc. 2023. V. 195. Art. No. 110295. https://www.doi.org/10.1016/j.ymssp.2023.110295
- 21. Bachour R.S., Nicoletti R. Natural frequencies and band gaps of periodically corrugated beams // J. of Vibr.&Acoust. 2021. V. 143. № 4. Art. No. 044502. https://www.doi.org/10.1115/1.4048889
- 22. Shi J., Wang W., Fan Y. et al. Creating absolute band gap based on frequency locking of three wave modes in a wavy plate // J. of Sound&Vibr. 2024. V. 592. Art. No. 118623. https://www.doi.org/10.1016/j.jsv.2024.118623
- 23. Zhou L., Han W., Wan S. Low frequency band gap for box girder attached IDVAs // ThinWalled Struct. 2022. V. 174. Art. No. 109088. https://www.doi.org/10.1016/j.tws.2022.109088
- 24. Niordson F. Optimal disks in vibration // Int. J. of Solids&Struct. 1997. V. 34. № 23. P. 2957-2968. https://www.doi.org/10.1016/S0020-7683 (97)00186-8
- 25. Blom A.W., Setoodeh S., Hol J.M.A.M., Gurdal Z. Design of variable-stiffness conical shells for maximum fundamental eigenfrequency // Comput.&Struct. 2008. V. 86. № 9. P. 870-878. https://www.doi.org/10.1016/j.compstruc.2007.04.020
- 26. Blom A.W., Stickler P.B., Gurdal Z. Optimization of a composite cylinder under bending by tailoring stiffness properties in circumferential direction // Compos. Pt. B: Engng. 2010. V. 41. № 2. P. 157-165. https://www.doi.org/10.1016/j.compositesb.2009.10.004
- 27. Ватульян А.О., Недин Р.Д. Об одной задаче оптимизации для преднапряженной пластины с переменной жесткостью // Пробл. прочн. и пластич. 2024. Т. 86. № 2. С. 202-214. https://www.doi.org/10.32326/1814-9146-2024-86-2-202-214
- 28. Lurie A.I., Belyaev A. Theory of Elasticity. Berlin; Heidelberg: Springer, 2005. 1050 p. https://www.doi.org/10.1007/978-3-540-26455-2
- 29. Ломакин В.А. Теория упругости неоднородных тел. М.: Ленанд, 2014. 376 с.
- 30. Калинчук В.В., Белянкова Т.И. Динамика поверхности неоднородных сред. М.: Физматлит, 2009. 316 с.
- 31. Ватульян А.О. Коэффициентные обратные задачи механики. М.: Физматлит, 2019. 272 с.
- 32. Vatulyan A.O., Dudarev V.V., Mnukhin R.M. Functionally graded cylinders: Vibration analysis // ZAMM. J. of Appl. Math.&Mech. 2023. V. 103. № 11. Art. No. e202200430. https://www.doi.org/10.1002/zamm.202200430
- 33. Vatulyan A.O., Dudarev V.V., Mnukhin R.M. Identification of characteristics of a functionally graded isotropic cylinder // Int. J. of Mech.&Mater. in Design. 2021. V. 17. № 2. P. 321-332. https://www.doi.org/10.1007/s10999-020-09527-5
- 34. Абовский Н.П., Андреев Н.П., Деруга А.П. Вариационные принципы теории упругости и теории оболочек. М.: Наука, 1978. 288 с.
- 35. Gupta V.K., Murthy P.N. Optimal design of uniform non-homogeneous vibrating beams // J. of Sound Vibr. 1978. V. 59. № 4. P. 521-531. https://www.doi.org/10.1016/S0022-460X (78)80132-1
