On the Optimality Conditions in the Weight Minimization Problem for a Shell of Revolution at a Given Vibration Frequency

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Abstract

We consider shallow elastic shells with a given circular boundary and seek an axisymmetric shell shape minimizing the weight at a given fundamental frequency of shell vibrations. Using the resulting formula for the linear part of the increment of the frequency functional, the multiplicity of the minimum natural frequency of vibrations of the shell is estimated. The Fréchet differentiability of the frequency functional is also established, and optimal conditions for minimizing the weight of the shell at a given fundamental vibration frequency are obtained.

About the authors

M. O. Arabyan

Yerevan State University, Yerevan, 0025, Armenia

Author for correspondence.
Email: arabyan.mariam@ysu.am
Ереван, Армения

References

  1. Lagrange J.L. Sur la figure des colonnes // Miscellanea Taurinensia. 1770-1773. V. 5.
  2. Arabyan M.H. Boundary-value problems and associated eigen-value problems for systems describing vibrations of a rotation shell // New York J. of Math. 2019. V. 25. P. 1350-1367.
  3. Plaut R.H., Johnson L.W., Parbery R. Optimal forms of shallow shells with circular boundary. Part 1. Maximum fundamental frequency // ASME J. of Appl. Mech. 1984. V. 51. № 3. P. 526-530.
  4. Plaut R.H., Johnson L.W. Optimal forms of shallow shells with circular boundary. Part 2. Maximum buckling load // ASME J. of Appl. Mech. 1984. V. 51. № 3. P. 531-535.
  5. Plaut R.H., Johnson L.W. Optimal forms of shallow shells with circular boundary. Part 3. Maximum enclosed volume // ASME J. of Appl. Mech. 1984. V. 51. № 3. P. 536-539.
  6. Abdulla U.G., Cosgrove E., Goldfarb J. On the Frechet differentiability in optimal control of coeffients in parabolic free boundary problems // Evolution Equat. and Control Theory. 2017. V. 6. № 3. P. 319-344.
  7. Bucur D., Buttazzo G. Variational Methods in Shape Optimization Problems. Boston, 2005.
  8. He Y., Guo B.Z. The existence of optimal solution for a shape optimization problem on starlike domain // J. Optim. Theory and Appl. 2012. V. 152. P. 21-30.
  9. Hinton E., Sienz J., Ozakca M. Analysis and Optimization of Shells of Revolution and Prismatic Shells. London, 2003.
  10. Krivoshapko S. Optimal shells of revolution and main optimization // Строительная механика инженерных конструкций и сооружений. 2019. Т. 15. № 3. С. 201-209.
  11. Lellep J., Hein H. Optimization of clamped plastic shallow shells subjected to initial impulsive loading // Eng. Optim. 2002. V. 34. № 5. P. 545-556.
  12. Neittaanmaki P., Sprekels J., Tiba D. Optimization of Elliptic Systems. New York, 2006.
  13. Olhoff N., Plaut R.H. Bimodal optimization of vibrating shallow arches // Int. J. of Solids and Struct. 1983. V. 19. № 6. P. 553-570.
  14. Stupishin L.Yu., Kolesnikov A.G., Nikitin K.E. Optimal design of flexible shallow shells on elastic foundation // J. of Appl. Eng. Sci. 2017. V. 15. № 3. P. 345-349.
  15. Velichkov B. Existence and Regularity Results for Some Shape Optimization Problems. Springer, 2015.
  16. Wang G., Wang L., Yang D. Shape optimization of an elliptic equation in an exterior domain // SIAM J. Control Optim. 2006. V. 45. № 2. P. 532-547.
  17. Ozakca M., Gogus M.T. Structural analysis and optimization of bells using finite elements // J. New Music Res. 2004. V. 33. № 1. P. 61-69.
  18. Григолюк Э.И. Нелинейные колебания и устойчивость пологих стержней и оболочек // Изв. АН СССР. Отдел техн. наук. 1955. № 3. С. 33-68.
  19. Timoshenko S., Woinorowsky-Krieger. S. Theory of Plates and Shells. New York, 1959.
  20. Timoshenko S. Strength of Materials. Part 2. Advanced Theory and Problems. New York, 1976.
  21. Колмогоров А.Н., Фомин С.В. Элементы теории функций и функционального анализа. М., 1976.
  22. Bramble J.H., Osborn J.E. Rate of convergence estimates for nonselfadjoint approximations // Math. Comput. 1973. V. 27. P. 525-549.
  23. Аrabyan М.H. On the existence of solutions of two optimization problems // J. of Optim. Theory and Appl. 2018. V. 177. P. 291-315.
  24. Алекееев В.M., Tихомиров В.M., Фомин С.В. Оптимальное управление. М., 1979.

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