DOI QR코드

DOI QR Code

Green's function coupled with perturbation approach to dynamic analysis of inhomogeneous beams with eigenfrequency and rotational effect's investigations

  • Hamza Hameed (Abdus Salam School of Mathematical Sciences, Government College University) ;
  • Sadia Munir (Abdus Salam School of Mathematical Sciences, Government College University) ;
  • F.D. Zaman (Abdus Salam School of Mathematical Sciences, Government College University)
  • 투고 : 2024.01.01
  • 심사 : 2024.03.08
  • 발행 : 2024.03.25

초록

The elastic theory of beams is fundamental in engineering of design and structure. In this study, we construct Green's function for inhomogeneous fourth-order differential operators subjected to associated constraints that arises in dealing with dynamic problems in the Rayleigh beam. We obtain solutions for homogeneous and completely inhomogeneous beam problems using Green's function. This enables us to consider rotational influences in determining the eigenfrequency of beam vibrations. Additionally, we investigate the dynamic vibration model of inhomogeneous beams incorporating rotational effects. The eigenvalues of Rayleigh beams, including first-order correction terms, are also computed and displayed in tabular forms.

키워드

과제정보

The authors are grateful to (AS-SMS) Government College University, Lahore, Pakistan for supporting the research.

참고문헌

  1. Alshorbagy, A.E., Eltaher, M.A. and Mahmoud, F. (2011), "Free vibration characteristics of a functionally graded beam by finite element method", Appl. Math. Model., 35(1), 412-425. https://doi.org/10.1016/j.apm.2010.07.006.
  2. Bakalah, E.S., Zaman, F.D. and Saleh, K. (2018), "Linear and nonlinear vibrations of inhomogeneous Euler Bernoulli beam", Coupled Syst. Mech., 7(5), 635-647. https://doi.org/10.12989/csm.2018.7.5.635.
  3. Cheng, Z.Q. and Batra, R. C. (2000), "Exact correspondence between eigenvalues of membranes and functionally graded simply supported polygonal plates", J. Sound Vib., 229(4), 879-895. https://doi.org/10.1006/jsvi.1999.2525.
  4. Ebrahimi-Mamaghani, A., Sarparast, H. and Rezaei, M. (2020), "On the vibrations of axially graded Rayleigh beams under a moving load", Appl. Math. Model., 84(1), 554-570. https://doi.org/10.1016/j.apm.2020.04.002.
  5. Hadzalic, E., Ibrahimbegovic, A. and Dolarevic, S. (2020), "3D thermo-hydro-mechanical coupled discrete beam lattice model of saturated poro-plastic medium", Coupled Syst. Mech., 9(2), 125-145. https://doi.org/10.12989/csm.2020.9.2.125.
  6. Han, S.M., Benaroya, H. and Wei, T. (1999), "Dynamics of transversely vibrating beams using four engineering theories", J. Sound Vib., 225(5), 935-988. https://doi.org/10.1006/jsvi.1999.2257.
  7. Hong, K.S., Chen, L.Q., Pham, P.T. and Yang, X.D. (2022), Beam Model, Control of Axially Moving Systems, Springer, Singapore.
  8. Hoskoti, L., Misra, A. and Sucheendran, M.M. (2021), "Modal analysis of a rotating twisted and tapered Rayleigh beam", Arch. Appl. Mech., 91(1), 2535-2567. https://doi.org/10.1007/s00419-021-01902-8.
  9. Ibrahimbegovic, A. and Nava, R.A.M. (2021), "Heterogeneities and material-scales providing physically based damping to replace Rayleigh damping for any structure size", Coupled Syst. Mech., 10(3), 201-216. https://doi.org/10.12989/csm.2021.10.3.201.
  10. Ibrahimbegovic, A., Mejia-Nava, R.A., Hajdo, E. and Limnios, N. (2022), "Instability of (heterogeneous) Euler beam: Deterministic vs. stochastic reduced model approach", Coupled Syst. Mech., 11(2), 167-198. https://doi.org/10.12989/csm.2022.11.2.167.
  11. Jo𝑐̌kovi𝑐́, M., Radenkovi𝑐́, G., Nefovska-Danilovi𝑐́, M. and Baitsch, M. (2019), "Free vibration analysis of spatial Bernoulli-Euler and Rayleigh curved beams using isogeometric approach", Appl. Math. Model., 71(1), 152-172. https://doi.org/10.1016/j.apm.2019.02.002.
  12. Kato, T. (2013), Perturbation Theory for Linear Operators, Science & Business Media, Springer, New-York.
  13. Lata, P. and Himanshi (2022), "Effect of rotation on Stoneley waves in orthotropic magneto-thermoela- stic media", Wind Struct., 35(6), 395-403. https://doi.org/10.12989/was.2022.35.6.395.
  14. Logan, J.D. (2013), Applied Mathematics, John Wiley & Sons, Hoboken, New Jersey.
  15. Love, A.E.H. (1892), A Treatise on the Mathematical Theory of Elasticity, Cambridge University Press, United Kingdom.
  16. Molina-Villegas, J.C., Ortega, J.E.B. and Mart'inez, G.M. (2023), "Closed-form solution for non-uniform Euler-Bernoulli beams and frames", Eng. Struct., 292(1). https://doi.org/10.1016/j.engstruct.2023.116381.
  17. Nieves, M.J. and Movchan, A.B. (2023), "Asymptotic theory of generalised Rayleigh beams and the dynamic coupling", Mechanics of High-Contrast Elastic Solids, 626(1) 173-200. https://doi.org/10.1007/978-3-031-24141-311.
  18. Olotu, O.T., Gbadeyan, J.A. and Agboola, O.O. (2023), "Free vibration analysis of tapered Rayleigh beams resting on variable two-parameter elastic foundation", Forces in Mech., 12(1). https://doi.org/10.1016/j.finmec.2023.100215.
  19. Orucoglu, K. (2005), "A new Green function concept for fourth-order differential equations", Electronic J. Differential Equations, 28(1), 1-12. http://ejde.math.txstate.edu.
  20. Panchore, V. (2022), "Meshless local Petrov-Galerkin method for rotating Rayleigh beam", Struct. Eng. Mech., 81(5), 607-616. https://doi.org/10.1007/s42417-022-00719-1.
  21. Rellich, F. (1969), Perturbation Theory of Eigenvalue Problems, Gordon and Breach Science Publishers, New-York.
  22. Rizov, V.I. (2018), "Non-linear longitudinal fracture in a functionally graded beam", Coupled Syst. Mech., 7(4), 441-453. https://doi.org/10.12989/csm.2018.7.4.441.
  23. Rizov, V.I. (2020), "Investigation of two parallel lengthwise cracks in an inhomogeneous beam of varying thickness", Coupled Syst. Mech., 9(4), 381-396. https://doi.org/10.12989/csm.2020.9.4.381.
  24. Rizov, V.I. (2021), "Delamination analysis of multilayered beams exhibiting creep under torsion", Coupled Syst. Mech., 10(4), 317-331. https://doi.org/10.12989/csm.2021.10.4.317.
  25. Russillo, A.F. and Failla, G. (2022), "Wave propagation in stress-driven nonlocal Rayleigh beam lattices", Int. J. Mech. Sci., 215(1). https://doi.org/10.1016/j.ijmecsci.2021.106901.
  26. Sankar, B.V. (2001), "An elasticity solution for functionally graded beams", Compos. Sci. Technol., 61(5), 689-696. https://doi.org/10.1016/S0266-3538(01)00007-0.
  27. Shariati, A., Jung, D.W., Mohammad-Sedighi, H., Zur, K.K., Habibi, M. and Safa, M. (2020), "Stability and dynamics of viscoelastic moving Rayleigh beams with an asymmetrical distribution of material parameters", Symmetry, 12(4), 586-598. https://doi.org/10.3390/sym12040586.
  28. Stakgold, I. and Holst, M.J. (2011), Green's Functions and Boundary Value Problems. John Wiley & Sons, Hoboken, New Jersey.
  29. Teterina, O. A. (2013), "The Green's function method for solutions of fourth order nonlinear boundary value problem", MS Dissertation University of Tennessee, Knoxville, USA.
  30. Truesdell, C. (1960), "Outline of the history of flexible or elastic bodies to 1788", J. Acoust. Soc. Am., 32(12), 1647-1656. https://doi.org/10.1121/1.1907980.
  31. Xu, M. and Ma, R. (2017), "On a fourth-order boundary value problem at resonance", J. Function Spaces, 2017(1). https://doi.org/10.1155/2017/2641856.
  32. Yang, S., Hu, H., Mo, G., Zhang, X., Qin, J., Yin, S. and Zhang, J. (2021), "Dynamic modeling and analysis of an axially moving and spinning Rayleigh beam based on a time-varying element", Appl. Math. Model., 95(1), 409-434. https://doi.org/10.1016/j.apm.2021.01.049.
  33. Yigit, G., Sahin, A. and Bayram, M. (2016), "Modeling of vibration for functionally graded beams", Open Mathematics, 14(1), 661-672. https://doi.org/10.1515/math-2016-0057