حل دقیق برای معادلات فرکانسی ارتعاشات آزاد شعاعی و عرضی یک ورق دایره‌ای با شرایط مرزی مختلف

نوع مقاله: پژوهشی

نویسندگان

1 دانشگاه اصفهان

2 دانشگاه کاشان

چکیده

در این تحقیق، معادله‌های فرکانسی ارتعاشات آزاد شعاعی و عرضی یک ورق دایره‌ای با سه نوع تکیه گاه آزاد، ساده و گیردار به صورت حل دقیق بسته استخراج شده است. برای به دست آوردن معادلات حاکم بر ارتعاشات شعاعی و عرضی ورق دایره‌ای از اصل همیلتون استفاده شده است. در حالت ارتعاشات شعاعی، دو معادله دیفرانسیل وابسته به هم حاصل می‌شود که با استفاده از روش تجزیه هلمهولتز، آن دو معادله دیفرانسیل از یکدیگر مستقل می‌شوند و سپس با روش جداسازی متغیرها به صورت تحلیلی قابل حل خواهند بود. در حالت ارتعاشات عرضی ورق دایره‌ای، معادلات حاکم نیز با استفاده از روش جداسازی متغیرها حل شده و یک رابطه دقیق بسته برای معادله فرکانسی در هر شرط مرزی به دست می‌آید. در نهایت فرکانس‌های طبیعی حاصل از معادله‌های فرکانسی به دست آمده با نتایج المان محدود مقایسه گردیده است.

کلیدواژه‌ها


عنوان مقاله [English]

Exact solution for frequency equations of radial and transverse vibration of a circular plate with various boundary conditions

نویسندگان [English]

  • Mohammad Heidari-Rarani 1
  • Shahram Hosseini-Chaleshtori 2
  • Keivan Torabi 2
چکیده [English]

In this study, closed-form relations are obtained for the frequency equations of radial (in-plane) and transverse (out-of–plane) free vibration of a circular plate with free, simply-support, and clamped boundary conditions. Governing equations are obtained using Hamilton's principal. In the case of radial vibration, governing equations are coupled to each other. So, they are decoupled using Helmholtz decomposition technique and solved using separation of variables method. In the case of transverse vibration, governing equations are analytically solved using separation of variables method. Finally, natural frequencies obtained from the frequency equations are compared with finite element results for an isotropic homogeneous circular plate with various boundary conditions.

کلیدواژه‌ها [English]

  • Radial vibration
  • Transverse vibration
  • Circular plate
  • Exact solution
[1] Leissa, A. (1993). “Vibration of Plates”. Acoustical Society of America, Woodbury, NY.
[2] Wah, T. (1961). “Vibration of circular plate”. Journal of the Acoustical Society of America, Vol. 34, pp. 275–281.
[3] Jomezadeh, E., Saidi, A.R. (2009). “Analytical solution for free vibration of transversely isotropic sector plates using a boundary layer function”. Thin-Walled Structures, Vol. 47, pp. 82–88.
[4] Farag, N.H., Pan J. (1998). “Free and forced in-plane vibration of rectangular plates”. Journal of the Acoustical Society of America, Vol. 103, pp. 408–413.
[5] Farag, N.H., Pan J. (1998). “Free and forced in-plane vibration of rectangular plates”. Journal of the Acoustical Society of America, Vol. 103, pp. 408–413.
[6] Bardell, N.S., Langley J.M., Dunsdon. (1996). “On the free in-plane vibration of isotropic rectangular plates”. Journal of Sound and Vibration, Vol. 191, pp. 459–467.
[7] Rizzi, S.A., Doyle J.F., Doyle. (1992). “Spectral analysis of wave motion in plane solids with boundaries, Transactions of the American Society of Mechanical Engineers”. Journal of Vibration and Acoustics, Vol. 114, pp. 133–140.
[8] Love, A.E.H. (1944). “A Treatise on the Mathematical Theory of Elasticity”. fourth ed., Dover, New York.
[9] Hosseini-Hashemi, Sh., Es’haghi M., Rokni Damavandi Taher, H. (2010). “An exact analytical solution for freely vibrating piezoelectric coupled circular/annular thick plates using Reddy plate theory”. Composite Structures, Vol. 92, pp. 1333–1351.
[10] Bisadi, H., Es’haghi, M., Rokni, H., Ilkhani, M. (2012). “Benchmark solution for transverse vibration of annular Reddy plates”. International Journal of Mechanical Sciences, Vol. 56, pp. 35–49.
[11] McGee III, O.G., Kim J.W., Kim, Y.S. (2010). “Influence of boundary stress singularities on the vibration of clamped and simply-supported sectorial plates with arbitrary radial edge conditions”. Journal of Sound and Vibration, Vol. 329, pp. 5563-5583.
[12] Chen, S.S.H., Liu T.M. (1975). “Extensional vibration of thin plates of various shapes”. Journal of the Acoustical Society of America, Vol. 58.
[13] Zhou, Z.H., Wong K.W., Xu, X.S., Leung, A.Y.T. (2011). “Natural vibration of circular and annular thin plates by Hamiltonian approach”. Journal of Sound and Vibration, Vol. 330, pp. 1005-1017.
[14] Helmut F. Bauer., Werner Eidel. (2010). “Transverse vibration and stability of spinning circular plates of constant thickness and different boundary conditions”. Journal of Sound and Vibration, Vol. 300, pp. 877–895.
[15] Hosseini-Hashemi, Sh., Rezaee V., Atashipour, S.R., Girhammar, U.A. (2012). “Accurate free vibration analysis of thick laminated circular plates with attached rigid core”. Journal of Sound and Vibration, Vol. 331, pp. 5581-5596.
[16] Hosseini-Hashemi, Sh., Es’haghi M., Rokni Damavandi Taher, H, Fadaie, M. (2010). “Exact closed-form frequency equations for thick circular plates using a third-order shear deformation theory”. Journal of Sound and Vibration, Vol. 329, pp. 3382-3396.
[17] Shanqing, Li., Hong Yuan (2012). “Green quasifunction method for free vibration of clamped thin plates”. Acta Mechanica Solida Sinica, Vol. 25, pp. 37–45.
[18] Shariyat, M., Jafari A. A., Alipour, M. M. (2013). “Investigation of the thickness variability and material heterogeneity effects on free vibration of the viscoelastic circularplates”. Acta Mechanica solida sinica, Vol. 26, pp. 83–98.
[19] Hasheminejad, Seyyed M., Ghaheri Ali., Rezaei, Shahed. (2012). “Semi-analytic solutions for the free in-plane vibrations of confocal annular elliptic plates with elastically restrained edges”. Journal of Sound and Vibration, Vol. 331, pp. 434-456.
[20] Shaban, M., Alipour M. M. (2011). “Semi-analytical solution for free vibrationof thick functionally graded plates rested onelastic foundation with elastically restrained edge”. Acta Mechanica Solida Sinica, Vol. 24, pp. 340–354.
[21] Chakraverty, S., Pradhan K.K. (2014). “Free vibration of exponential functionally graded rectangular plates in thermal environment with general boundary conditions”. Aerospace Science and Technology, Vol. 36, pp. 132–156.
[22] Alipour, M.M., Shariyat M. (2014). “An analytical global–local taylor transformation-based vibration solution for annular FGM sandwich plates supported by nonuniform elastic foundations”. Archives of Civil and Mechanical Engineering, Vol. 14, pp. 6-24.
[23] Doyle, J.F. (1997). “Wave Propagation in Structures”. Springer, New York.
[24] Achenbach, J.D. (1973). “Wave Propagation in Elastic Solid”. North-Holland Publishing, Amsterdam.
[25] Soedel, W. (1981). “Vibrations of Shells and Plates”. Marcel Dekker, New York.
[26] Szilard, R. (1974). “Theory and Analysis of Plates: Classical and Numerical Methods”. Prentice-Hall, Englewood Cliffs, NJ.