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Free and forced vibrations of an elastically interconnected annular plates system

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Abstract

An analytical solution is presented for determining the natural frequencies, mode shapes, and forced responses of a system of elastically connected annular plates with general boundary conditions. By applying the Kirchhoff’s plate theory, the motion of \(n\) elastically connected plates is described through a set of \(n\) differential equations. These equations are coupled, thus, hard to solve. A new variable change is presented to uncouple the equations and obtain one decoupled equation for each plate. These equations are solved separately and analytically, and the natural frequencies, mode shapes, and forced responses of the separated plates are obtained. The frequencies of the original system are those calculated analytically for the decoupled system, and the mode shapes and forced responses are obtained by applying the inverse transform. A three plates system with clamped edges is solved to demonstrate this approach. The effects of stiffness coefficients of elastic layers, inner edge radius of the plates, and the frequency of the external harmonic force on the answers are assessed. Applying ABAQUS software, the analytical solution is validated, where a good agreement is observed.

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References

  1. Oniszczuk, Z.: Free transverse vibrations of elastically connected simply supported double-beam complex system. J. Sound Vib. 232(2), 387–403 (2000). https://doi.org/10.1006/jsvi.1999.2744

    Article  Google Scholar 

  2. Oniszczuk, Z.: Forced transverse vibrations of an elastically connected complex simply supported double-beam system. J. Sound Vib. 264(2), 273–286 (2003). https://doi.org/10.1016/S0022-460X(02)01166-5

    Article  MATH  Google Scholar 

  3. Zhang, Y.Q., Lu, Y., Wang, S.L., Liu, X.: Vibration and buckling of a double-beam system under compressive axial loading. J. Sound Vib. 318(1), 341–352 (2008). https://doi.org/10.1016/j.jsv.2008.03.055

    Article  Google Scholar 

  4. Ariaei, A., Ziaei-Rad, S., Ghayour, M.: Transverse vibration of a multiple-Timoshenko beam system with intermediate elastic connections due to a moving load. Arch. Appl. Mech. 81(3), 263–281 (2011). https://doi.org/10.1007/s00419-010-0410-2

    Article  MATH  Google Scholar 

  5. Ghafarian, M., Ariaei, A.: Free vibration analysis of a system of elastically interconnected rotating tapered Timoshenko beams using differential transform method. Int. J. Mech. Sci. 107, 93–109 (2016). https://doi.org/10.1016/j.ijmecsci.2015.12.027

    Article  Google Scholar 

  6. Hao, Q., Zhai, W., Chen, Z.: Free vibration of connected double-beam system with general boundary conditions by a modified Fourier-Ritz method. Arch. Appl. Mech. 88(5), 741–754 (2018). https://doi.org/10.1007/s00419-017-1339-5

    Article  Google Scholar 

  7. Foroozandeh, S., Ariaei, A.: Vibration and buckling of a multiple-Timoshenko beam system joined by intermediate elastic connections under compressive axial loading. Arch. Appl. Mech. 88(7), 1041–1057 (2018). https://doi.org/10.1007/s00419-018-1357-y

    Article  Google Scholar 

  8. Brito, W.K.F., Maia, C.D., Mendonca, A.V.: Bending analysis of elastically connected Euler-Bernoulli double-beam system using the direct boundary element method. Appl. Math. Model 74, 387–408 (2019). https://doi.org/10.1016/j.apm.2019.04.049

    Article  MathSciNet  MATH  Google Scholar 

  9. Liu, S., Yang, B.: A closed-form analytical solution method for vibration analysis of elastically connected double-beam systems. Compos. Struct. 212, 598–608 (2019). https://doi.org/10.1016/j.compstruct.2019.01.038

    Article  Google Scholar 

  10. Zhao, X., Chen, B., Li, Y.H., Zhu, W.D., Nkiegaing, F.J., Shao, Y.B.: Forced vibration analysis of Timoshenko double-beam system under compressive axial load by means of Green’s functions. J. Sound Vib. 464, 115001 (2020). https://doi.org/10.1016/j.jsv.2019.115001

    Article  Google Scholar 

  11. Yulin, F., Lizhong, J., Wangbao, Z.: Dynamic response of a three-beam system with intermediate elastic connections under a moving load/mass-spring. Earthq Eng. & Eng. Vib. 19(2), 377–395 (2020). https://doi.org/10.1007/s11803-020-0568-8

    Article  Google Scholar 

  12. Li, Y., Xiong, F., Xie, L., Sun, L.: State-space approach for transverse vibration of double-beam systems. Int. J. Mech. Sci. 189, 105974 (2021). https://doi.org/10.1016/j.ijmecsci.2020.105974

    Article  Google Scholar 

  13. Civalek, Ö., Dastjerdi, S., Akgöz, B.: Buckling and free vibrations of CNT-reinforced cross-ply laminated composite plates. Mech. Based Des. Struct. Mach. 50(6), 1914–1931 (2020). https://doi.org/10.1080/15397734.2020.1766494

    Article  Google Scholar 

  14. Chonan, S.: Moving load on initially stressed thick plates attached together by a flexible core. Ingenieur-Archiv 48(3), 143–154 (1979). https://doi.org/10.1007/BF00537268

    Article  MATH  Google Scholar 

  15. Kukla, S.: Application of Green’s function in free vibration analysis of a system of line connected rectangular plates. J. Sound Vib. 217(1), 1–15 (1998). https://doi.org/10.1006/jsvi.1998.1745

    Article  MATH  Google Scholar 

  16. Kukla, S.: Free vibration of a system of two elastically connected rectangular plates. J. Sound Vib. 225(1), 29–39 (1999). https://doi.org/10.1006/jsvi.1999.2196

    Article  Google Scholar 

  17. Oniszczuk, Z.: Free transverse vibrations of an elastically connected rectangular simply supported double-plate complex system. J. Sound Vib. 236(4), 595–608 (2000). https://doi.org/10.1006/jsvi.2000.2995

    Article  Google Scholar 

  18. Oniszczuk, Z.: Forced transverse vibrations of an elastically connected complex rectangular simply supported double-plate system. J. Sound Vib. 270(4), 997–1011 (2004). https://doi.org/10.1016/S0022-460X(03)00769-7

    Article  Google Scholar 

  19. Hedrih, K.: Transversal vibrations of double-plate systems. Acta Mech. Sin. 22(5), 487–501 (2006). https://doi.org/10.1007/s10409-006-0018-5

    Article  MATH  Google Scholar 

  20. Hedrih, K.: Double plate system with a discontinuity in the elastic bonding layer. Acta Mech. Sin. 23(2), 221–229 (2007). https://doi.org/10.1007/s10409-007-0061-x

    Article  MATH  Google Scholar 

  21. Stojanović, V., Kozić, P., Ristić, M.: Vibrations and stability analysis of multiple rectangular plates coupled with elastic layers based on different plate theories. Int. J. Mech. Sci. 92, 233–244 (2015). https://doi.org/10.1016/j.ijmecsci.2014.10.027

    Article  Google Scholar 

  22. Heidari, E., Ariaei, A.: A new approach for free vibration analysis of a system of elastically interconnected similar rectangular plates. Earthq Eng. & Eng. Vib. 21, 947–967 (2022). https://doi.org/10.1007/s11803-022-2129-9

    Article  Google Scholar 

  23. Kunukkasseril, V.X., Swamidas, A.S.J.: Normal modes of elastically connected circular plates. J. Sound Vib. 30(1), 99–108 (1973). https://doi.org/10.1016/S0022-460X(73)80053-7

    Article  MATH  Google Scholar 

  24. Swamidas, A.S.J., Kunukkasseril, V.X.: Free vibration of elastically connected circular plate systems. J. Sound Vib. 39(2), 229–235 (1975). https://doi.org/10.1016/S0022-460X(75)80221-5

    Article  MATH  Google Scholar 

  25. Kunukkasseril, V.X., Swamidas, A.S.J.: Stability of continuous double-plate systems. AIAA J 13(10), 1326–1332 (1975). https://doi.org/10.2514/3.6989

    Article  MATH  Google Scholar 

  26. Chonan, S.: The free vibrations of elastically connected circular plate systems with elastically restrained edges and radial tensions. J. Sound Vib. 49(1), 129–136 (1976). https://doi.org/10.1016/0022-460X(76)90762-8

    Article  MATH  Google Scholar 

  27. Chonan, S.: Resonance frequencies and mode shapes of elastically restrained, prestressed annular plates attached together by flexible cores. J. Sound Vib. 67(4), 487–500 (1979). https://doi.org/10.1016/0022-460X(79)90440-1

    Article  MATH  Google Scholar 

  28. Irie, T., Yamada, G., Muramoto, Y.: The axisymmetrical steady-state response of internally damped annular double-plate systems. J. Appl. Mech. 49(2), 417–424 (1982). https://doi.org/10.1115/1.3162103

    Article  MATH  Google Scholar 

  29. Tang, S., Liu, S., Zhao, D., Ren, X., Zhang, W., Liu, Y.: Vibration response analysis of plate with microfloating raft arrays under multi-point random excitation. Arch. Appl. Mech. 91(10), 4081–4096 (2021). https://doi.org/10.1007/s00419-021-02028-7

    Article  Google Scholar 

  30. Rao, S.S.: Vibration of continuous systems, 2nd edn. John Wiley & Sons (2019)

    Book  Google Scholar 

  31. Zhang, G., Ge, J.: Vibration and acoustic radiation characteristics of simply supported curved plate in thermal environment. Arch. Appl. Mech. 92(11), 3163–3177 (2022). https://doi.org/10.1007/s00419-022-02229-8

    Article  MathSciNet  Google Scholar 

  32. Dastjerdi, S., Alibakhshi, A., Akgöz, B., Civalek, Ö.: On a comprehensive analysis for mechanical problems of spherical structures. Int. J. Eng. Sci. 183, 103796 (2023). https://doi.org/10.1016/j.ijengsci.2022.103796

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mechanical Engineering, Faculty of Engineering, University of Isfahan, Isfahan, Iran

    A. Mirian & A. Ariaei

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  1. A. Mirian
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Appendix 1

Appendix 1

The boundary conditions for a clamped annular plate are as follows [30]:

$$ \begin{aligned} \left. {u_{p} } \right|_{a = 1} = & \,\,0 \to A_{{{\text{mgp}}}}^{\left( 1 \right)} J_{m} \left( {\eta_{{{\text{mgp}}}} } \right) + A_{{{\text{mgp}}}}^{\left( 2 \right)} Y_{m} \left( {\eta_{{{\text{mgp}}}} } \right) + A_{{{\text{mgp}}}}^{\left( 3 \right)} I_{m} \left( {\eta_{{{\text{mgp}}}} } \right) \\ + & \,\,A_{{{\text{mgp}}}}^{\left( 4 \right)} K_{m} \left( {\eta_{{{\text{mgp}}}} } \right) = 0 \\ \end{aligned} $$
(54)
$$ \begin{aligned} \left. {u_{p} } \right|_{a = e} = & \,\,0 \to A_{{{\text{mgp}}}}^{\left( 1 \right)} J_{m} \left( {e\eta_{{{\text{mgp}}}} } \right) + A_{mgp}^{\left( 2 \right)} Y_{m} \left( {e\eta_{{{\text{mgp}}}} } \right) + A_{{{\text{mgp}}}}^{\left( 3 \right)} I_{m} \left( {e\eta_{{{\text{mgp}}}} } \right) \\ + & \,\,A_{{{\text{mgp}}}}^{\left( 4 \right)} K_{m} \left( {e\eta_{{{\text{mgp}}}} } \right) = 0 \\ \end{aligned} $$
(55)
$$ \begin{aligned} \left. {\frac{{\partial u_{p} }}{\partial a}} \right|_{a = 1} = & \,\,0 \to A_{{{\text{mgp}}}}^{\left( 1 \right)} \left[ {\left( {\eta_{{{\text{mgp}}}} } \right)J_{m - 1} \left( {\eta_{{{\text{mgp}}}} } \right) - mJ_{m} \left( {\eta_{{{\text{mgp}}}} } \right)} \right] \\ + & \,\,A_{{{\text{mgp}}}}^{\left( 2 \right)} \left[ {\left( {\eta_{{{\text{mgp}}}} } \right)Y_{m - 1} \left( {\eta_{{{\text{mgp}}}} } \right) - mY_{m} \left( {\eta_{{{\text{mgp}}}} } \right)} \right] \\ + & \,\,A_{{{\text{mgp}}}}^{\left( 3 \right)} \left[ {\left( {\eta_{{{\text{mgp}}}} } \right)I_{m - 1} \left( {\eta_{{{\text{mgp}}}} } \right) - mI_{m} \left( {\eta_{{{\text{mgp}}}} } \right)} \right] \\ + & \,\,A_{{{\text{mgp}}}}^{\left( 4 \right)} \left[ { - \left( {\eta_{{{\text{mgp}}}} } \right)K_{m - 1} \left( {\eta_{{{\text{mgp}}}} } \right) - mK_{m} \left( {\eta_{{{\text{mgp}}}} } \right)} \right] = 0 \\ \end{aligned} $$
(56)
$$ \begin{aligned} \left. {\frac{{\partial u_{p} }}{\partial a}} \right|_{a = e} = & \,\,0 \to A_{{{\text{mgp}}}}^{\left( 1 \right)} \left[ {\left( {e\eta_{{{\text{mgp}}}} } \right)J_{m - 1} \left( {e\eta_{{{\text{mgp}}}} } \right) - mJ_{m} \left( {e\eta_{{{\text{mgp}}}} } \right)} \right] \\ + & \,\,A_{{{\text{mgp}}}}^{\left( 2 \right)} \left[ {\left( {e\eta_{{{\text{mgp}}}} } \right)Y_{m - 1} \left( {e\eta_{{{\text{mgp}}}} } \right) - mY_{m} \left( {e\eta_{{{\text{mgp}}}} } \right)} \right] \\ + & \,\,A_{{{\text{mgp}}}}^{\left( 3 \right)} \left[ {\left( {e\eta_{{{\text{mgp}}}} } \right)I_{m - 1} \left( {e\eta_{{{\text{mgp}}}} } \right) - mI_{m} \left( {e\eta_{{{\text{mgp}}}} } \right)} \right] \\ + & \,\, A_{{{\text{mgp}}}}^{\left( 4 \right)} \left[ { - \left( {e\eta_{{{\text{mgp}}}} } \right)K_{m - 1} \left( {e\eta_{{{\text{mgp}}}} } \right) - mK_{m} \left( {e\eta_{{{\text{mgp}}}} } \right)} \right] = 0 \\ \end{aligned} $$
(57)

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Mirian, A., Ariaei, A. Free and forced vibrations of an elastically interconnected annular plates system. Arch Appl Mech 93, 3025–3043 (2023). https://doi.org/10.1007/s00419-023-02413-4

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