Initial value method for free vibration of axially loaded functionally graded Timoshenko beams with nonuniform cross section

References

• Abramovich, H. 1992. Natural frequencies of timoshenko beams under compressive axial loads. Journal of Sound and Vibration 157 (1):183–189. doi: 10.1016/0022-460x(92)90574-h.
   Web of Science ®Google Scholar
• Alshorbagy, A. E., M. Eltaher, and F. Mahmoud. 2011. Free vibration characteristics of a functionally graded beam by finite element method. Applied Mathematical Modelling 35 (1):412–425. doi: 10.1016/j.apm.2010.07.006.
   Web of Science ®Google Scholar
• Auciello, N. M., and A. Ercolano. 2004. A general solution for dynamic response of axially loaded non-uniform timoshenko beams. International Journal of Solids and Structures 41 (18–19):4861–4874. doi: 10.1016/j.ijsolstr.2004.04.036.
   Web of Science ®Google Scholar
• Bokaian, A. 1988. Natural frequencies of beams under compressive axial loads. Journal of Sound and Vibration 126 (1):49–65. doi: 10.1016/0022-460x(88)90397-5.
   Web of Science ®Google Scholar
• Bokaian, A. 1990. Natural frequencies of beams under tensile axial loads. Journal of Sound and Vibration 142 (3):481–498. doi: 10.1016/0022-460x(90)90663-k.
   Web of Science ®Google Scholar
• Ding, J., L. Chu, L. Xin, and G. Dui. 2018. Nonlinear vibration analysis of functionally graded beams considering the influences of the rotary inertia of the cross section and neutral surface position. Mechanics Based Design of Structures and Machines 46 (2):225–237. doi: 10.1080/15397734.2017.1329020.
   Web of Science ®Google Scholar
• Elishakoff, I., and Z. Guede. 2004. Analytical polynomial solutions for vibrating axially graded beams. Mechanics of Advanced Materials and Structures 11 (6):517–533. doi: 10.1080/15376490490452669.
   Web of Science ®Google Scholar
• Esmailzadeh, E., and A. R. Ohadi. 2000. Vibration and stability analysis of non-uniform timoshenko beams under axial and distributed tangential loads. Journal of Sound and Vibration 236 (3):443–456. doi: 10.1006/jsvi.2000.2999.
   Web of Science ®Google Scholar
• Ghannadiasl, A., and M. Mofid. 2014. Dynamic green function for response of timoshenko beam with arbitrary boundary conditions. Mechanics Based Design of Structures and Machines 42 (1):97–110. doi: 10.1080/15397734.2013.836063.
   Web of Science ®Google Scholar
• Ghayesh, M. H. 2018. Nonlinear vibrations of axially functionally graded timoshenko tapered beams. Journal of Computational and Nonlinear Dynamics 13 (4):041002. doi: 10.1115/1.4039191.
   Web of Science ®Google Scholar
• Ghayesh, M. H. 2018. Nonlinear vibration analysis of axially functionally graded shear-deformable tapered beams. Applied Mathematical Modelling 59:583–596. doi: 10.1016/j.apm.2018.02.017.
   Web of Science ®Google Scholar
• Griffiths, D. F., and D. J. Higham. 2010. Numerical methods for ordinary differential equations: Initial value problems. London: Springer.
   Google Scholar
• Hein, H., and L. Feklistova. 2011. Free vibrations of non-uniform and axially functionally graded beams using haar wavelets. Engineering Structures 33 (12):3696–3701. doi: 10.1016/j.engstruct.2011.08.006.
   Web of Science ®Google Scholar
• Hijmissen, J. W., and W. T. van Horssen. 2008. On transverse vibrations of a vertical timoshenko beam. Journal of Sound and Vibration 314 (1–2):161–179. doi: 10.1016/j.jsv.2007.12.039.
   Web of Science ®Google Scholar
• Huang, Y., and X. F. Li. 2011. Buckling analysis of nonuniform and axially graded columns with varying flexural rigidity. Journal of Engineering Mechanics 137 (1):73–81. doi: 10.1061/(asce)em.1943-7889.0000206.
   Web of Science ®Google Scholar
• Huang, Y., and X. F. Li. 2010. A new approach for free vibration of axially functionally graded beams with non-uniform cross-section. Journal of Sound and Vibration 329 (11):2291–2303. doi: 10.1016/j.jsv.2009.12.029.
   Web of Science ®Google Scholar
• Huang, Y., L. E. Yang, and Q. Z. Luo. 2013. Free vibration of axially functionally graded timoshenko beams with non-uniform cross-section. Composites Part B: Engineering 45 (1):1493–1498. doi: 10.1016/j.compositesb.2012.09.015.
   Web of Science ®Google Scholar
• Larbi, L. O., A. Kaci, M. S. A. Houari, and A. Tounsi. 2013. An efficient shear deformation beam theory based on neutral surface position for bending and free vibration of functionally graded beams. Mechanics Based Design of Structures and Machines 41 (4):421–433. doi: 10.1080/15397734.2013.763713.
   Web of Science ®Google Scholar
• Li, S., J. Hu, C. Zhai, and L. Xie. 2013. A unified method for modeling of axially and/or transversally functionally graded beams with variable cross-section profile. Mechanics Based Design of Structures and Machines 41 (2):168–188. doi: 10.1080/15397734.2012.709466.
   Web of Science ®Google Scholar
• Li, X. F., A. Y. Tang, and L. Y. Xi. 2013. Vibration of a rayleigh cantilever beam with axial force and tip mass. Journal of Constructional Steel Research 80:15–22. doi: 10.1016/j.jcsr.2012.09.015.
   Web of Science ®Google Scholar
• Li, X. F. 2008. A unified approach for analyzing static and dynamic behaviors of functionally graded Timoshenko and euler – bernoulli beams. Journal of Sound and Vibration 318(4–5):1210–1229. doi: 10.1016/j.jsv.2008.04.056.
   Web of Science ®Google Scholar
• Li, X. F., Y. A. Kang, and J. X. Wu. 2013. Exact frequency equations of free vibration of exponentially functionally graded beams. Applied Acoustics 74 (3):413–420. doi: 10.1016/j.apacoust.2012.08.003.
   Web of Science ®Google Scholar
• Li, X. F., and K. Y. Lee. 2018. Effects of Engesser’s and Haringx’s hypotheses on buckling of timoshenko and higher-order shear-deformable columns. Journal of Engineering Mechanics 144 (1):04017150. doi: 10.1061/(asce)em.1943-7889.0001363.
   Web of Science ®Google Scholar
• Naguleswaran, S. 2004. Transverse vibration of an uniform Euler–Bernoulli beam under linearly varying axial force. Journal of Sound and Vibration 275 (1–2):47–57. doi: 10.1016/s0022-460x(03)00741-7.
   Web of Science ®Google Scholar
• Nguyen, T.-K., T. P. Vo, and H.-T. Thai. 2013. Static and free vibration of axially loaded functionally graded beams based on the first-order shear deformation theory. Composites Part B: Engineering 55:147–157. doi: 10.1016/j.compositesb.2013.06.011.
   Web of Science ®Google Scholar
• Pradhan, K. K., and S. Chakraverty. 2013. Free vibration of Euler and Timoshenko functionally graded beams by rayleigh – ritz method. Composites Part B: Engineering 51:175–184. doi: 10.1016/j.compositesb.2013.02.027.
   Web of Science ®Google Scholar
• Rajasekaran, S. 2013. Free vibration of centrifugally stiffened axially functionally graded tapered timoshenko beams using differential transformation and quadrature methods. Applied Mathematical Modelling 37 (6):4440–4463. doi: 10.1016/j.apm.2012.09.024.
   Web of Science ®Google Scholar
• Schafer, B. 1985. Free vibrations of a gravity-loaded clamped-free beam. Ingenieur-archiv 55 (1):66–80. doi: 10.1007/bf00539551.
   Google Scholar
• Shahba, A., R. Attarnejad, M. T. Marvi, and S. Hajilar. 2011. Free vibration and stability analysis of axially functionally graded tapered timoshenko beams with classical and non-classical boundary conditions. Composites Part B: Engineering 42 (4):801–808. doi: 10.1016/j.compositesb.2011.01.017.
   Web of Science ®Google Scholar
• Shahba, A., and S. Rajasekaran. 2012. Free vibration and stability of tapered euler – bernoulli beams made of axially functionally graded materials. Applied Mathematical Modelling 36 (7):3094–3111. doi: 10.1016/j.apm.2011.09.073.
   Web of Science ®Google Scholar
• Sun, D. L., X. F. Li, and C. Y. Wang. 2016. Buckling of standing tapered timoshenko columns with varying flexural rigidity under combined loadings. International Journal of Structural Stability and Dynamics 16 (06):1550017. doi: 10.1142/s0219455415500170.
   Google Scholar
• Tang, A. Y., J. X. Wu, X. F. Li, and K. Y. Lee. 2014. Exact frequency equations of free vibration of exponentially non-uniform functionally graded Timoshenko beams. International Journal of Mechanical Sciences 89:1–11. doi: 10.1016/j.ijmecsci.2014.08.017.
   Web of Science ®Google Scholar
• Virgin, L. N. 2007. Vibration of axially-loaded structures. New York: Cambridge University Press. doi: 10.1017/cbo9780511619236.
   Google Scholar
• Virgin, L. N., S. T. Santillan, and D. B. Holland. 2007. Effect of gravity on the vibration of vertical cantilevers. Mechanics Research Communications 34 (3):312–317. doi: 10.1016/j.mechrescom.2006.12.006.
   Web of Science ®Google Scholar
• Wang, C. Y. 2012. Influence of gravity and taper on the vibration of a standing column. Advances in Applied Mathematics and Mechanics 4 (04):483–495. doi: 10.4208/aamm.11-m11175.
   Google Scholar
• Wattanasakulpong, N., and A. Chaikittiratana. 2016. Adomian-modified decomposition method for large-amplitude vibration analysis of stepped beams with elastic boundary conditions. Mechanics Based Design of Structures and Machines 44 (3):270–282. doi: 10.1080/15397734.2015.1055762.
   Web of Science ®Google Scholar
• Weaver, W., Jr, S. P. Timoshenko, and D. H. Young. 1990. Vibration problems in engineering. New York: Wiley.
   Google Scholar
• Yokoyama, T. 1990. Vibrations of a hanging Timoshenko beam under gravity. Journal of Sound and Vibration 141 (2):245–258. doi: 10.1016/0022-460x(90)90838-q.
   Web of Science ®Google Scholar
• Yuan, J., Y.-H. Pao, and W. Chen. 2016. Exact solutions for free vibrations of axially inhomogeneous timoshenko beams with variable cross section. Acta Mechanica 227 (9):2625–2643. doi: 10.1007/s00707-016-1658-6.
   Web of Science ®Google Scholar
• Zhao, Y., Y. Huang, and M. Guo. 2017. A novel approach for free vibration of axially functionally graded beams with non-uniform cross-section based on Chebyshev polynomials theory. Composite Structures 168:277–284. doi: 10.1016/j.compstruct.2017.02.012.
   Web of Science ®Google Scholar