Abstract
We use the semi-inverse method to find the solutions to the dynamic equation of an inhomogeneous, functionally graded two-span beam with overhang. Here, the cross-sectional area of this beam is assumed to be constant, and the corresponding natural frequency is given. With both the fundamental mode and density given as polynomial functions in terms of a non-dimensional axial coordinate, we then determine Young’s modulus for the beam expanded as a polynomial function of this axial coordinate. In addition, existence criteria for solutions are obtained. The conditions for positivity of the given density functions and the obtained Young’s modulus are investigated.
Similar content being viewed by others
References
Loy C.T., Lam K.Y., Reddy J.N.: Vibration of functionally graded cylindrical shells. Int. J. Mech. Sci. 41, 309–324 (1999)
Yang J., Shen M.S.: Free vibration and parametric resonance of shear deformable functionally graded cylindrical panels. J. Sound Vib. 261, 871–893 (2003)
Yang J., Shen M.S.: Dynamic response of initially stressed functionally graded rectangular thin plates. Compos. Struct. 54, 497–508 (2001)
Chen W.Q., Ding H.J.: On free vibration of a functionally graded piezoelectric rectangular plate. Acta Mech. 153, 207–216 (2002)
Kim K.S., Noda N.: Green’s function approach to unsteady thermal stresses in an infinite hollow cylinder of functionally graded material. Acta Mech. 156, 145–161 (2002)
Akgöz, B., Civalek, Ö.: Buckling analysis of functionally graded microbeams based on the strain gradient theory. Acta Mech. doi:10.1007/s00707-013-0883-5 (2013)
Elishakoff I., Candan S.: Apparently first closed-form solution for frequencies of deterministically and/or stochastically inhomogeneous simply supported beams. J. Appl. Mech. 68, 176–185 (2001)
Elishakoff I., Candan S.: Apparently first closed-form solution for vibrating inhomogeneous beams. Int. J. Solids Struct. 38, 3411–3441 (2001)
Guede Z., Elishakoff I.: Apparently first closed-form solutions for inhomogeneous vibrating beams under axial loading. Proc. R Soc. Lond. A 457, 623–649 (2001)
Wu L., Wang Q., Elishakoff I.: Semi-inverse method for axially functionally graded beams with an anti-symmetric vibration mode. J. Sound Vib. 284, 1190–1202 (2005)
Wu L., Zhang L., Wang Q., Elishakoff I.: Reconstructing cantilever beams via vibration mode with a given node location. Acta Mech. 217, 135–148 (2011)
Cetin D., Simsek M.: Free vibration of an axially functionally graded pile with pinned ends embedded in Winkler-Pasternak elastic medium. Struct. Eng. Mech. 40, 583–594 (2011)
Shahba A., Attarnejad R., Hajilar S.: Free vibration and stability of axially functionally graded tapered Euler–Bernoulli beams. Shock Vib. 18, 683–696 (2011)
Nie G.J., Zhong Z.: Vibration analysis of functionally graded annular sectorial plates with simply supported radial edges. Compos. Struct. 84, 167–176 (2008)
Khalili S.M.R., Jafari A.A., Eftekhari S.A.: A mixed Ritz–DQ method for forced vibration of functionally graded beams carrying moving loads. Compos. Struct. 92, 2497–2511 (2010)
Wang Q., Wang D., He M. et al.: Some qualitative properties of the vibration modes of the continuous system of a beam with one or two overhangs. J. Eng. Mech. 138, 945–952 (2012)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
He, M., Zhang, L. & Wang, Q. Reconstructing cross-sectional physical parameters for two-span beams with overhang using fundamental mode. Acta Mech 225, 349–359 (2014). https://doi.org/10.1007/s00707-013-0963-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00707-013-0963-6