DOI QR코드

DOI QR Code

Nonlinear vibration of laminated piezoelectric layered plates with nonlinear viscoelastic support using different DQM techniques

  • Ola Ragb (Department of Engineering Mathematics and Physics, Faculty of Engineering, Zagazig University) ;
  • Mohamed Abd Elkhalek (Department of Engineering Mathematics and Physics, Faculty of Engineering, Zagazig University) ;
  • M.S. Matbuly (Department of Engineering Mathematics and Physics, Faculty of Engineering, Zagazig University) ;
  • Mohamed Salah (Department of Engineering Mathematics and Physics, Faculty of Engineering, Zagazig University) ;
  • Mohamed Eltaher (Mechanical Engineering Department, Faculty of Engineering, King Abdulaziz University) ;
  • Tharwat Osman (Department of Engineering Mathematics and Physics, Faculty of Engineering, Zagazig University)
  • 투고 : 2022.01.07
  • 심사 : 2024.09.23
  • 발행 : 2024.10.10

초록

This work presents the effectiveness of differential quadrature shape functions (i.e., Lagrange interpolation polynomial, Cardinal sine function, Delta Lagrange kernel and Regularized Shannon kernel) in the solution of nonlinear vibration of multilayers piezoelectric plates with nonlinear elastic support. A piezoelectric composite laminated plate is rested on nonlinear Winkler and Visco-Pasternak elastic foundations problems. Based on 3D elasticity theory and piezoelectricity, the governing equations of motion are derived. Differential quadrature methods based on four shape functions are presented as numerical techniques for solving this problem. The perturbation method is implemented to solve the obtained nonlinear eigenvalue problem. A MATLAB code is written for each technique for solving this problem and extract the numerical results. To validate these methods, the computed results are we compare with the previous exact results. In addition, parametric analyses are offered to investigate the influence of length to thickness ratio, elastic foundation parameters, various boundary conditions, and piezoelectric layers thickness on the natural frequencies and mode shapes. Consequently, it is discovered that the obtained results via the proposed schemes can be applied in structural health monitoring.

키워드

참고문헌

  1. Abdelrahman, A.A., Abdel-Mottaleb, H.E., Aljabri, A., Mahmoud, E.R. and Eltaher, M.A. (2024), "Modeling of size dependent buckling behavior of piezoelectric sandwich perforated nanobeams rested on elastic foundation with flexoelectricity", Mech. Based Des. Struct. Machines, 1-27. https://doi.org/10.1080/15397734.2024.2365918.
  2. Abdelrahman, A.A., Saleem, H.A., Abdelhaffez, G.S. and Eltaher, M.A. (2023), "On bending of piezoelectrically layered perforated nanobeams embedded in an elastic foundation with flexoelectricity", Mathematics, 11(5), 1162. https://doi.org/10.3390/math11051162.
  3. Ahmed, R.A., Khalaf, B.S., Raheef, K.M., Fenjan, R.M. and Faleh, N.M. (2021), "Investigating dynamic response of nonlocal functionally graded porous piezoelectric plates in thermal environment", Steel Compos. Struct., 40(2), 243-254. https://doi.org/10.12989/scs.2021.40.2.243.
  4. Akhavan Alavi, S.M., Mohammadimehr, M. and Ejtahed, S.H. (2021), "Vibration analysis and control of micro porous beam integrated with FG-CNT distributed piezoelectric sensor and actuator", Steel Compos. Struct., 41(4), 595-608. https://doi.org/10.12989/scs.2021.41.4.595.
  5. Alibeigloo, A. and Madoliat, R. (2009), "Static analysis of cross-ply laminated plates with integrated surface piezoelectric layers using differential quadrature", Compos. Struct., 88(3), 342-353. https://doi.org/10.1016/j.compstruct.2008.04.018.
  6. Alnujaie, A., Daikh, A.A., Ghazwani, M.H., Assie, A.E. and Eltaher, M.A. (2024), "Size-dependent free vibration of coated functionally graded graphene reinforced nanoplates rested on viscoelastic medium", Adv. Nano Res., 17(2), 181. https://doi.org/10.12989/anr.2024.17.2.181.
  7. Arefi, M. and Zenkour, A.M. (2017), "Size-dependent free vibration and dynamic analyses of piezo-electro-magnetic sandwich nanoplates resting on viscoelastic foundation", Physica B: Condensed Matter, 521, 188-197. https://doi.org/10.1016/j.physb.2017.06.066.
  8. Arefi, M., Pourjamshidian, M. and Arani, A.G. (2019), "Dynamic instability region analysis of sandwich piezoelectric nano-beam with FG-CNTRCs face-sheets based on various high-order shear deformation and nonlocal strain gradient theory", Steel Compos. Struct., 32(2), 157-171. https://doi.org/10.12989/scs.2019.32.2.157.
  9. Aslan, T.A., Noori, A.R. and Temel, B. (2018), "Dynamic response of viscoelastic tapered cycloidal rods", Mech. Res. Commun., 92, 8-14. https://doi.org/10.1016/j.mechrescom.2018.06.006.
  10. Aslan, T.A., Noori, A.R. and Temel, B. (2023), "An efficient approach for free vibration analysis of functionally graded sandwich beams of variable cross-section", Structures, 58, 105397. https://doi.org/10.1016/j.istruc.2023.105397.
  11. Assie, A., Akbas, S.D., Kabeel, A.M., Abdelrahman, A.A. and Eltaher, M.A. (2022), "Dynamic analysis of porous functionally graded layered deep beams with viscoelastic core", Steel Compos. Struct., 43(1), 79-90. https://doi.org/10.12989/scs.2022.43.1.079.
  12. Assie, A., Mohamed, S., Abdelrahman, A.A. and Eltaher, M.A. (2023), "Mathematical formulations for static behavior of bidirectional FG porous plates rested on elastic foundation including middle/neutral-surfaces", Steel Compos. Struct., 48(2), 113-130.
  13. Baghaee, M., Farrokhabadi, A. and Jafari-Talookolaei, R.A. (2019), "A solution method based on Lagrange multipliers and Legendre polynomial series for free vibration analysis of laminated plates sandwiched by two MFC layers", J. Sound Vib., 447, 42-60. https://doi.org/10.1016/j.jsv.2019.01.037.
  14. Behera, S. and Kumari, P. (2020), "Free vibration analysis of Levy-Type smart hybrid plates using three-dimensional extended Kantorovich Method", Struct. Integrity Assessment: Proceedings of ICONS 2018, 467-477. https://doi.org/10.1007/978-981-13-8767-8_39.
  15. Chanda, A.G., Kontoni, D.P.N. and Sahoo, R. (2023), "Development of analytical and FEM solutions for static and dynamic analysis of smart piezoelectric laminated composite plates on elastic foundation", J. Eng. Mathem., 138(1), 12. https://doi.org/10.1007/s10665-022-10251-6.
  16. Chen, W.Q. and Lu, C.F. (2005), "3D free vibration analysis of cross-ply laminated plates with one pair of opposite edges simply supported", Compos. Struct., 69(1), 77-87. https://doi.org/10.1016/j.compstruct.2004.05.015.
  17. Civalek, O. and Akgoz, B. (2010), "Free vibration analysis of microtubules as cytoskeleton components: Nonlocal euler-bernoulli beam modeling", Scientia Iranica, 17(5).
  18. Dogan, A. and Sahan, M.F. (2023), "Viscoelastic damped response of laminated composite shells subjected to various dynamic loads", Mech. Based Des. Struct. Machines, 51(8), 4685-4708. https://doi.org/10.1080/15397734.2021.1975296.
  19. Eltaher, M.A., Omar, F.A., Abdalla, W.S., Kabeel, A.M. and Alshorbagy, A.E. (2020), "Mechanical analysis of cutout piezoelectric nonlocal nanobeam including surface energy effects", Struct. Eng. Mech., 76(1), 141-151.
  20. Feri, M., Alibeigloo, A. and Pasha Zanoosi, A.A. (2016), "Three dimensional static and free vibration analysis of cross-ply laminated plate bonded with piezoelectric layers using differential quadrature method", Meccanica, 51, 921-937. https://doi.org/10.1007/s11012-015-0246-5.
  21. Feri, M., Krommer, M. and Alibeigloo, A. (2022), "Three-dimensional thermoelasticity analysis of viscoelastic FGM plate embedded in piezoelectric layers under thermal load", Appl. Sci., 13(1), 353.
  22. Feri, M., Krommer, M. and Alibeigloo, A. (2023), "Three-dimensional static analysis of a viscoelastic rectangular functionally graded material plate embedded between piezoelectric sensor and actuator layers", Mech. Based Des. Struct. Machines, 51(7), 3843-3867. https://doi.org/10.1080/15397734.2021.1943673.
  23. Goodarzi, M., Mohammadi, M., Khooran, M. and Saadi, F. (2016), "Thermo-mechanical vibration analysis of FG circular and annular nanoplate based on the visco-pasternak foundation", J. Solid Mech., 8(4), 788-805.
  24. Hachemi, M. (2022), "Layer-wise solutions for variable stiffness composite laminated sandwich plate using curvilinear fibers", Mech. Adv. Mater. Struct., 29(26), 5460-5477. https://doi.org/10.1080/15376494.2021.1956028.
  25. Heyliger, P. and Brooks, S. (1995), "Free vibration of piezoelectric laminates in cylindrical bending", Int. J. Solids Struct., 32(20), 2945-2960. https://doi.org/10.1016/0020-7683(94)00270-7.
  26. Kapuria, S. (2004), "A coupled zig-zag third-order theory for piezoelectric hybrid cross-ply plates", J. Appl. Mech., 71(5), 604-614.
  27. Kapuria, S. and Achary, G.G.S. (2005), "Exact 3D piezoelasticity solution of hybrid cross-ply plates with damping under harmonic electro-mechanical loads", J. Sound Vib., 282(3-5), 617-634. https://doi.org/10.1016/j.jsv.2004.03.030.
  28. Kapuria, S. and Achary, G.G.S. (2005), "Exact 3D piezoelasticity solution of hybrid cross-ply plates with damping under harmonic electro-mechanical loads", J. Sound Vib., 282(3-5), 617-634. https://doi.org/10.1016/j.jsv.2004.03.030.
  29. Kapuria, S. and Kulkarni, S.D. (2008), "An efficient quadrilateral element based on improved zigzag theory for dynamic analysis of hybrid plates with electroded piezoelectric actuators and sensors", J. Sound Vib., 315(1-2), 118-145. https://doi.org/10.1016/j.jsv.2008.01.053.
  30. Khalid, H.M., Ojo, S.O. and Weaver, P.M. (2023), "Inverse differential quadrature solutions for free vibration of arbitrary shaped laminated plate structures", Appl. Mathem. Modelling, 115, 778-802. https://doi.org/10.1016/j.apm.2022.11.013.
  31. Khdeir, A.A. (1988), "Free vibration and buckling of symmetric cross-ply laminated plates by an exact method", J. Sound Vib., 126(3), 447-461. https://doi.org/10.1016/0022-460X(88)90223-4.
  32. Kumar, R. and Kumar, A. (2023), "Free vibration analysis of laminated composite porous plate", Asian J. Civil Eng., 24(5), 1181-1198. https://doi.org/10.1007/s42107-022-00561-6.
  33. Kumari, P., Behera, S. and Kapuria, S. (2016), "Coupled three-dimensional piezoelasticity solution for edge effects in Levy-type rectangular piezolaminated plates using mixed field extended Kantorovich method", Compos. Struct., 140, 491-505. https://doi.org/10.1016/j.compstruct.2015.12.029.
  34. Matbuly, M.S., Ragb, O. and Nassar, M. (2009), "Natural frequencies of a functionally graded cracked beam using the differential quadrature method", Appl. Mathem. Comput., 215(6), 2307-2316. https://doi.org/10.1016/j.amc.2009.08.026.
  35. Matsunaga, H. (2000), "Vibration and stability of cross-ply laminated composite plates according to a global higher-order plate theory", Compos. Struct., 48(4), 231-244. https://doi.org/10.1016/S0263-8223(99)00110-5.
  36. Mei, B., Alamri, S., Jalil, A.T., Hadrawi, S.K., Khan, I. and Baghaei, S. (2022), "Wave propagation and vibration analysis of sandwich structure with a bio-based flexible core and composite face sheets subjected to visco-Pasternak foundation and magnetic field", Compos. Struct., 300, 116132. https://doi.org/10.1016/j.compstruct.2022.116132.
  37. Mohamed, S.A., Eltaher, M.A., Mohamed, N. and Abo-bakr, R.M. (2024), "Nonlinear dynamics and forced vibrations of simply-supported fractional viscoelastic microbeams using a fractional differential quadrature method", Mech. Based Des. Struct. Machines, 1-20. https://doi.org/10.1080/15397734.2024.2353321.
  38. Moleiro, F., Soares, C.M., Carrera, E. and Reddy, J.N. (2020), "Evaluation of exact electro-elastic static and free vibration solutions of multilayered plates for benchmarking: piezoelectric composite laminates and soft core sandwich plates", Compos. Part C: Open Access, 2, 100038. https://doi.org/10.1016/j.jcomc.2020.100038.
  39. Nassar, M., Matbuly, M.S. and Ragb, O. (2013), "Vibration analysis of structural elements using differential quadrature method", J. Adv. Res., 4(1), 93-102. https://doi.org/10.1016/j.jare.2012.01.009.
  40. Noori, A.R., Aslan, T.A. and Temel, B. (2018), "Damped transient response of in-plane and out-of-plane loaded stepped curved rods", J. Brazil. Society Mech. Sci. Eng., 40, 1-25. https://doi.org/10.1007/s40430-017-0949-8.
  41. Olunloyo, V., Osheku, C. and Olayiwola, P. (2016), "Concerning the effect of a viscoelastic foundation on the dynamic stability of a pipeline system conveying an incompressible fluid", J. Appl. Comput. Mech., 2(2), 96-117. https://doi.org/10.22055/jacm.2016.12393.
  42. Osman, T., Matbuly, M. S., Mohamed, S.A. and Nassar, M. (2013), "Analysis of cracked plates using localized multi-domain differential quadrature method", Appl Comput. Math., 2, 109-114. https://doi.org/10.11648/j.acm.20130204.12.
  43. Osman, T., Mohamed, S.A., Eltaher, M.A., Alazwari, M.A. and Mohamed, N. (2024), "Vibration of bio-inspired laminated composite beams under varying axial loads", Steel Compos. Struct., 50(1), 25.
  44. Ragb, O. and Matbuly, M.S. (2022), "Nonlinear vibration analysis of elastically supported multi-layer composite plates using efficient quadrature techniques", Int. J. Comput. Meth. Eng. Sci. Mech., 23(2), 129-146. https://doi.org/10.1080/15502287.2021.1921882.
  45. Ragb, O., Mohamed, M. and Matbuly, M.S. (2019), "Free vibration of a piezoelectric nanobeam resting on nonlinear Winkler-Pasternak foundation by quadrature methods", Heliyon, 5(6).
  46. Ragb, O., Mohamed, M., Matbuly, M.S. and Civalek, O. (2021), "An accurate numerical approach for studying perovskite solar cells", Int. J. Energy Res., 45(11), 16456-16477. https://doi.org/10.1002/er.6892.
  47. Ragb, O., Salah, M., Matbuly, M.S. and Amer, R.B.M. (2020), "Vibration analysis of piezoelectric composite plate resting on nonlinear elastic foundations using sinc and discrete singular convolution differential quadrature techniques", Mathem. Prob. Eng., 2020(1), 7592302. https://doi.org/10.1155/2020/7592302.
  48. Ragb, O., Salah, M., Matbuly, M.S. and Amer, R.M. (2019), "Vibration analysis of piezoelectric composite using sinc and discrete singular convolution differential quadrature techniques", J. Eng. Appl. Sci., 14(17), 6540-6553.
  49. Ragb, O., Seddek, L.F. and Matbuly, M.S. (2017), "Iterative differential quadrature solutions for Bratu problem", Comput. Mathem. Appl., 74(2), 249-257. https://doi.org/10.1016/j.camwa.2017.03.033.
  50. Rahmani, A., Faroughi, S. and Friswell, M.I. (2021), "Vibration analysis for anti-symmetric laminated composite plates resting on visco-elastic foundation with temperature effects", Appl. Mathem. Modelling, 94, 421-445. https://doi.org/10.1016/j.apm.2021.01.026.
  51. Rouzegar, J., Salmanpour, N., Abad, F. and Li, L. (2022), "An analytical state-space solution for free vibration of sandwich piezoelectric plate with functionally graded core", Scientia Iranica, 29(2), 502-533. https://doi.org/10.24200/sci.2021.56480.4741.
  52. Sahan, M.F. (2017), "Viscoelastic damped response of cross-ply laminated shallow spherical shells subjected to various impulsive loads", Mech. Time-Dependent Mater., 21, 499-518. https://doi.org/10.1007/s11043-017-9339-y.
  53. Shu, C. (2000), Differential Quadrature and Its Application in Engineering: Springer Science & Business Media. http://dx.doi.org/10.1007/978-1-4471-0407-0.
  54. Shu, C. and Du, H. (1997), "Implementation of clamped and simply supported boundary conditions in the GDQ free vibration analysis of beams and plates", Int. J. Solids Struct., 34(7), 819-835. https://doi.org/10.1016/S0020-7683(96)00057-1.
  55. Sui, S., Zhu, C., Li, C. and Lei, Z. (2023), "Free vibration of axially traveling moderately thick FG plates resting on elastic foundations", J. Vib. Eng. Technol., 11(1), 329-341. https://doi.org/10.1007/s42417-022-00582-0.
  56. Vel, S.S., Mewer, R.C. and Batra, R.C. (2004), "Analytical solution for the cylindrical bending vibration of piezoelectric composite plates", Int. J. Solids Struct., 41(5-6), 1625-1643. https://doi.org/10.1016/j.ijsolstr.2003.10.012.
  57. Wang, M., Xu, Y.G., Qiao, P. and Li, Z.M. (2022), "Buckling and free vibration analysis of shear deformable graphene-reinforced composite laminated plates", Compos. Struct., 280, 114854. https://doi.org/10.1016/j.compstruct.2021.114854.
  58. Wu, C.P. and Chen, W.Y. (1994), "Vibration and stability of laminated plates based on a local high order plate theory", J. Sound Vib., 177(4), 503-520. https://doi.org/10.1006/jsvi.1994.1448.
  59. Yang, C., Huang, B., Guo, Y. and Wang, J. (2021), "Characterization of delamination effects on free vibration and impact response of composite plates resting on visco-Pasternak foundations", Int. J. Mech. Sci., 212, 106833. https://doi.org/10.1016/j.ijmecsci.2021.106833.
  60. Yas, M.H., Jodaei, A., Irandoust, S. and Nasiri Aghdam, M. (2012), "Three-dimensional free vibration analysis of functionally graded piezoelectric annular plates on elastic foundations", Meccanica, 47, 1401-1423. https://doi.org/10.1007/s11012-011-9525-y.
  61. Ye, W., Zang, Q., Liu, J., Yang, F. and Lin, G. (2023), "Three-dimensional bending and free vibration analyses of laminated cylindrical panel with/without elastic foundation using two-dimensional discrete method", Soil Dyn. Earthq. Eng., 168, 107831. https://doi.org/10.1016/j.soildyn.2023.107831.
  62. Zamani, H.A., Aghdam, M.M. and Sadighi, M. (2017), "Free vibration analysis of thick viscoelastic composite plates on visco-Pasternak foundation using higher-order theory", Compos. Struct., 182, 25-35. https://doi.org/10.1016/j.compstruct.2017.08.101.
  63. Zenkour, A.M. and Alghanmi, R.A. (2019), "Bending of exponentially graded plates integrated with piezoelectric fiber-reinforced composite actuators resting on elastic foundations", Europ. J. Mech.A/Solids, 75, 461-471. https://doi.org/10.1016/j.euromechsol.2019.03.003.
  64. Zhang, Z., Feng, C. and Liew, K.M. (2006), "Three-dimensional vibration analysis of multilayered piezoelectric composite plates", Int. J. Eng. Sci., 44(7), 397-408. https://doi.org/10.1016/j.ijengsci.2006.02.002.
  65. Zong, Z., Lam, K.Y. and Zhang, Y.Y. (2005), "A multidomain differential quadrature approach to plane elastic problems with material discontinuity", Mathem. Comput. Modelling, 41(4-5), 539-553. https://doi.org/10.1016/j.mcm.2003.11.009.