Free vibration analysis of functionally graded beams with non-uniform cross-section using the differential transform method
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Free vibrations of non-uniform cross-section and axially functionally graded Euler–Bernoulli beams with various boundary conditions were studied using the differential transform method. The method was applied to a variety of beam configurations that are either axially non-homogeneous or geometrically non-uniform along the beam length or both. The governing equation of an Euler–Bernoulli beam with variable coefficients was reduced to a set of simpler algebraic recurrent equations by means of the differential transformations. Then, transverse natural frequencies were determined by requiring the non-trivial solution of the eigenvalue problem stated for a transformed function of the transverse displacement with appropriately transformed its high derivatives and boundary conditions. To show the generality and effectiveness of this approach, natural frequencies of various beams with variable profiles of cross-section and functionally graded non-homogeneity were calculated and compared with analytical and numerical results available in the literature. The benefit of the differential transform method to solve eigenvalue problems for beams with arbitrary axial geometrical non-uniformities and axial material gradient profiles is clearly demonstrated.
KeywordsDifferential transform method Free vibrations Functionally graded material Non-uniform cross-sectional beam
This research has been carried out under the financial support of the Erasmus Mundus post-doctoral exchange program ACTIVE, Grant Agreement No. 2013-2523/001-001 at the University of Southampton.
- 2.Sadowski T (2009) Non-symmetric thermal shock in ceramic matrix composite (CMC) materials. In: de Borst R, Sadowski T (eds) Solid mechanics and its applications. Lecture notes on composite materials—current topics and achievements, vol 154. Springer, Netherlands, pp 99–148Google Scholar
- 4.Burlayenko VN (2016) Modelling thermal shock in functionally graded plates with finite element method. Adv Mater Sci Eng 2016:7514638Google Scholar
- 26.Pukhov GE (1982) Differential analysis of circuits. Naukova Dumka, Kiev (in Russian). http://catalog.lib.tpu.ru/catalogue/document/RU%5CTPU%5Cbook%5C53501
- 27.Pukhov GE (1980) Differential transformations of functions and equations. Naukova Dumka, Kiev (in Russian). http://catalog.lib.tpu.ru/catalogue/document/RU%5CTPU%5Cbook%5C211741
- 28.Pukhov GE (1986) Differential transformations and mathematical modeling of physical processes. Naukova Dumka, Kiev (in Russian). http://catalog.lib.tpu.ru/catalogue/document/RU%5CTPU%5Cbook%5C90103
- 29.Pukhov GE (1988) Approximate methods of mathematical modelling based on the use of differential T-transformations. Naukova Dumka, Kiev (in Russian). http://catalog.lib.tpu.ru/catalogue/document/RU%5CTPU%5Cbook%5C90073
- 30.Pukhov GE (1990) Differential spectrums and models. Naukova Dumka, Kiev (in Russian). http://urss.ru/cgi-bin/db.pl?lang=Ru&blang=ru&page=Book&id=101456
- 33.Avetisyan AG, Simonyan SH, Ghazaryan DA (2009) Solution of linear time optimal control problems in domain of differential transformations. Bull Tomsk Politech Univ 315(5):5–13Google Scholar
- 34.Avetisyan AG, Avinyan VR, Ghazaryan DA (2013) A method for solving silvester type parametric matrix equation. Proc NAS RA SEUA 66(4):376–383Google Scholar
- 43.Weaver W Jr, Timoshenko SP, Young DH (1990) Vibration problems in engineering, 5th edn. Wiley, New YorkGoogle Scholar