References
- A. Berkani, N. Tatar, and A. Khemmoudj, “Control of a viscoelastic translational Euler-Bernoulli beam,” Math. Meth. Appl. Sci., vol. 40, no. 1, pp. 237–254, 2017. DOI: https://doi.org/10.1002/mma.3985.
- A. Berkani, “Stabilization of a viscoelastic rotating Euler-Bernoulli beam,” Math. Meth. Appl. Sci., vol. 41, no. 8, pp. 2939–2960, 2018. DOI: https://doi.org/10.1002/mma.4793.
- M. Bayat, I. Pakar, and M. Bayat, “Analytical study on the vibration frequencies of tapered beams,” Lat. Am. J. Solids Struct., vol. 8, no. 2, pp. 149–162, 2011. DOI: https://doi.org/10.1590/S1679-78252011000200003.
- H. Salmani, G. H. Rahimi, and S. A. Hosseini Kordkheili, “An exact analytical solution to exponentially tapered piezoelectric energy harvester,” Shock Vib. vol. 2015, pp. 1–13, 2015. DOI: https://doi.org/10.1155/2015/426876.
- Y. K. Cheung and D. Zhou, “The free vibrations of tapered rectangular plates using a new set of beam functions with the Rayleigh-Ritz method,” J. Sound Vib., vol. 223, no. 5, pp. 703-722, 1999. DOI: https://doi.org/10.1006/jsvi.1998.2160.
- M. Boiangiu, V. Ceausu, and C. D. Untaroiu, “A transfer matrix method for free vibration analysis of Euler-Bernoulli beams with variable cross section,” J. Vib. Control, vol. 22, no. 11, pp. 2591–2602, 2016. DOI: https://doi.org/10.1177/1077546314550699.
- K. Mazanoglu, “Natural frequency analyses of segmented Timoshenko–Euler beams using the Rayleigh–Ritz method,” J. Vib. Control, vol. 23, no. 13, pp. 2135–2154, 2017. DOI: https://doi.org/10.1177/1077546315611525.
- M. H. Ghayesh, “Resonant dynamics of axially functionally graded imperfect tapered Timoshenko beams,” J. Vib. Control, vol. 25, no. 2, pp. 336–350, 2019. DOI: https://doi.org/10.1177/1077546318777591.
- D. Shin, S. Choi, G. W. Jang, and Y. Y. Kim, “Finite element beam analysis of tapered thin-walled box beams,” Thin-Walled Struct., vol. 102, pp. 205-214, 2016. DOI: https://doi.org/10.1016/j.tws.2016.01.028.
- S. M. R. Khalili, A. A. Jafari, and S. A. A. Eftekhari, “A mixed Ritz-DQ method for forced vibration of functionally graded beams carrying moving loads,” Compos. Struct., vol. 92, no. 10, pp. 2497-2511, 2010. DOI: https://doi.org/10.1016/j.compstruct.2010.02.012.
- X. Tong, B. Tabarrok, and K. Y. Yeh, “Vibration analysis of Timoshenko beams with non-homogeneity and varying cross-section,” J. Sound Vib., vol. 186, no. 5, pp. 821–835, 1995. DOI: https://doi.org/10.1006/jsvi.1995.0490.
- S. K. Jang and C. W. Bert, “Free vibration of stepped beams: higher mode frequencies and effects of steps on frequency,” J. Sound Vib., vol. 132, no. 1, pp. 164–168, 1989. DOI: https://doi.org/10.1016/0022-460X(89)90882-1.
- C. D. Bailey, “Direct analytical solutions to non-uniform beam problems,” J. Sound Vib., vol. 56, no. 4, pp. 501–507, 1978. DOI: https://doi.org/10.1016/0022-460X(78)90292-4.
- H. H. Mabie and C. B. Rogers, “Transverse vibrations of double‐tapered cantilever beams,” J. Acoust. Soc. Am., vol. 51, no. 5B, pp. 1771–1774, 1972. DOI: https://doi.org/10.1121/1.1913028.
- H. P. Lee, “Dynamic stability of a tapered cantilever beam on an elastic foundation subjected to a follower force,” Int. J. Solids Struct., vol. 33, no. 10, pp. 1409–1424, 1996. DOI: https://doi.org/10.1016/0020-7683(95)00108-5.
- A. K. Gupta, N. Agarwal, and H. Kaur, “Transverse vibration of nonhomogeneous orthotropic viscoelastic circular plate of varying parabolic thickness,” Math. Meth. Appl. Sci., vol. 34, no. 16, pp. 2065–2076, 2011. DOI: https://doi.org/10.1002/mma.1521.
- S. Abrate, “Vibration of non-uniform rods and beams,” J. Sound Vib., vol. 185, no. 4, pp. 703–716, 1995. DOI: https://doi.org/10.1006/jsvi.1995.0410.
- P. A. A. Laura, R. H. Gutierrez, and R. E. Rossi, “Free vibrations of beams of bilinearly varying thickness,” Ocean Eng., vol. 23, no. 1, pp. 1–6, 1996. DOI: https://doi.org/10.1016/0029-8018(95)00029-K.
- D. Zhou and Y. K. Cheung, “Free vibration of a type of tapered beams,” Comput. Methods Appl. Mech. Eng., vol. 188, no. 1–3, pp. 203–216, 2000. DOI: https://doi.org/10.1016/S0045-7825(99)00148-6.
- N. M. Auciello, “On the transverse vibrations of non-uniform beams with axial loads and elastically restrained ends,” Int. J. Mech. Sci., vol. 43, no. 1, pp. 193–208, 2001. DOI: https://doi.org/10.1016/S0020-7403(99)00110-1.
- K. K. Pradhan and S. Chakraverty, “Free vibration of Euler and Timoshenko functionally graded beams by Rayleigh-Ritz method,” Compos. Part B Eng., vol. 51, pp. 175–184, 2013. DOI: https://doi.org/10.1016/j.compositesb.2013.02.027.
- D. Zhou and Y. K. Cheung, “Vibrations of tapered Timoshenko beams in terms of static Timoshenko beam functions,” J. Appl. Mech., vol. 68, no. 4, pp. 596–602, 2001. DOI: https://doi.org/10.1115/1.1357164.
- R. D. Firouz-Abadi, H. Haddadpour, and A. B. Novinzadeh, “An asymptotic solution to transverse free vibrations of variable-section beams,” J. Sound Vib., vol. 304, no. 3–5, pp. 530–540, 2007. DOI: https://doi.org/10.1016/j.jsv.2007.02.030.
- İ. Çelik, “Free vibration of non-uniform Euler–Bernoulli beam under various supporting conditions using Chebyshev wavelet collocation method,” Appl. Math Model., vol. 54, pp. 268–280, 2018. DOI: https://doi.org/10.1016/j.apm.2017.09.041.
- J. W. Lee and J. Y. Lee, “Free vibration analysis using the transfer-matrix method on a tapered beam,” Comput. Struct., vol. 164, pp. 75–82, 2016. DOI: https://doi.org/10.1016/j.compstruc.2015.11.007.
- Q. Mao and S. Pietrzko, “Free vibration analysis of a type of tapered beams by using Adomian decomposition method,” Appl. Math. Comput., vol. 219, no. 6, pp. 3264–3271, 2012. DOI: https://doi.org/10.1016/j.amc.2012.09.069.
- H. Y. Lai, C. K. Chen, and J. C. Hsu, “Free vibration of non-uniform Euler-Bernoulli beams by the Adomian modified decomposition method,” Comput. Model. Eng. Sci., vol. 34, pp. 87–113, 2008. DOI: https://doi.org/10.1016/j.jsv.2008.05.010.
- J. R. Banerjee, “Free vibration of centrifugally stiffened uniform and tapered beams using the dynamic stiffness method,” J. Sound Vib., vol. 233, no. 5, pp. 857–875, 2000. DOI: https://doi.org/10.1006/jsvi.1999.2855.
- J. R. Banerjee, “Dynamic stiffness formulation and free vibration analysis of centrifugally stiffened Timoshenko beams,” J. Sound Vib., vol. 247, no. 1, pp. 97–115, 2001. DOI: https://doi.org/10.1006/jsvi.2001.3716.
- J. R. Banerjee, H. Su, and D. R. Jackson, “Free vibration of rotating tapered beams using the dynamic stiffness method,” J. Sound Vib., vol. 298, no. 4-5, pp. 1034–1054, 2006. DOI: https://doi.org/10.1016/j.jsv.2006.06.040.
- J. R. Banerjee and D. R. Jackson, “Free vibration of a rotating tapered Rayleigh beam: a dynamic stiffness method of solution,” Comput. Struct., vol. 124, pp. 11–20, 2013. DOI: https://doi.org/10.1016/j.compstruc.2012.11.010.
- R. D. Firouz-Abadi, M. Rahmanian, and M. Amabili, “Exact solutions for free vibrations and buckling of double tapered columns with elastic foundation and tip mass,” J. Vib. Acoust., vol. 135, no. 5, pp. 51017, 2013. DOI: https://doi.org/10.1115/1.4023991.
- O. Ozdemir Ozgumus and M. O. Kaya, “Flapwise bending vibration analysis of a rotating double-tapered Timoshenko beam,” Arch. Appl. Mech., vol. 78, no. 5, pp. 379–392, 2008. DOI: https://doi.org/10.1007/s00419-007-0158-5.
- S. Rajasekaran, “Differential transformation and differential quadrature methods for centrifugally stiffened axially functionally graded tapered beams,” Int. J. Mech. Sci., vol. 74, pp. 15–31, 2013. DOI: https://doi.org/10.1016/j.ijmecsci.2013.04.004.
- M. Ghafarian and A. Ariaei, “Free vibration analysis of a system of elastically interconnected rotating tapered Timoshenko beams using differential transform method,” Int. J. Mech. Sci., vol. 107, pp. 93–109, 2016. DOI: https://doi.org/10.1016/j.ijmecsci.2015.12.027.
- M. H. Ghayesh, “Mechanics of viscoelastic functionally graded microcantilevers,” Eur. J. Mech. A/Solids, vol. 73, pp. 492–499, 2019. DOI: https://doi.org/10.1016/j.euromechsol.2018.09.001.
- M. H. Ghayesh, “Viscoelastic mechanics of Timoshenko functionally graded imperfect microbeams,” Compos. Struct., vol. 225, pp. 110974, 2019. DOI: https://doi.org/10.1016/j.compstruct.2019.110974.
- M. H. Ghayesh, “Nonlinear vibrations of axially functionally graded Timoshenko Tapered beams,” J. Comput. Nonlinear Dyn., vol. 13, pp. 041002, 2018. DOI: https://doi.org/10.1115/1.4039191.
- A. Keshmiri, N. Wu, and Q. Wang, “Free vibration analysis of a nonlinearly tapered cone beam by Adomian decomposition method,” Int. J. Str. Stab. Dyn., vol. 18, no. 7, pp. 1850101, 2018. DOI: https://doi.org/10.1142/S0219455418501018.
- Y. Huang and X.-F. Li, “A new approach for free vibration of axially functionally graded beams with non-uniform cross-section,” J. Sound Vib., vol. 329, no. 11, pp. 2291–2303, 2010. DOI: https://doi.org/10.1016/j.jsv.2009.12.029.
- D. Ghazaryan, V. N. Burlayenko, A. Avetisyan, and A. Bhaskar, “Free vibration analysis of functionally graded beams with non-uniform cross-section using the differential transform method,” J. Eng. Math., vol. 110, no. 1, pp. 97–121, 2018. DOI: https://doi.org/10.1007/s10665-017-9937-3.
- M. A. Mahmoud, “Natural frequency of axially functionally graded, tapered cantilever beams with tip masses,” Eng. Struct., vol. 187, pp. 34–42, 2019. DOI: https://doi.org/10.1016/j.engstruct.2019.02.043.
- P. Liu, K. Lin, H. Liu, and R. Qin, “Free transverse vibration analysis of axially functionally graded tapered Euler-Bernoulli beams through spline finite point method,” Shock Vib., vol. 2016, pp. 1–23, 2016. DOI: https://doi.org/10.1155/2016/5891030.