Vibration characteristics of stepped beams made of FGM using differential transformation method

Abstract

The present paper is given to investigate free vibration analysis of stepped beams produced from functionally graded materials (FGMs). The differential transformation method is employed to solve the governing differential equations of the beams to obtain their natural frequencies and mode shapes. The power law distribution is used and modified for describing material compositions across the thickness of the stepped beams made of FGM. Two main types of the stepped FGM beams in which their material compositions can be described by the modified power law distribution are selected to investigate the free vibration behaviour. The significant parametric studies such as step ratio, step location, boundary conditions and material volume fraction are also covered in this paper.

Appendices

Appendices

1.1 Appendix 1

See Table 8.

Table 8 Matrix elements used for vibration analysis of stepped FGM beams with other types of boundary conditions

1.2 Appendix 2

Mode shape functions for other types of boundary conditions

For S–S:

$$ W(x) = \left\{\!\! \begin{gathered} W_{1} (x) = \sum\limits_{r = 0}^{\infty } {\frac{{I_{01}^{r} \omega^{2r} }}{{\lambda_{1}^{r} (4r + 1)!}}x^{(4r + 1)} + C_{2} \sum\limits_{r = 0}^{\infty } {\frac{{I_{01}^{r} \omega^{2r} }}{{\lambda_{1}^{r} (4r + 3)!}}x^{(4r + 3)} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x \in [0,L_{1} ]} } \hfill \\ W_{2} (L - x) = C_{3} \sum\limits_{r = 0}^{\infty } {\frac{{\mu^{r} I_{01}^{r} \omega^{2r} }}{{\lambda_{1}^{r} (4r + 1)!}}(L - x)^{(4r + 1)} + C_{4} \sum\limits_{r = 0}^{\infty } {\frac{{\mu^{r} I_{01}^{r} \omega^{2r} }}{{\lambda_{1}^{r} (4r + 3)!}}(L - x)^{(4r + 3)} \,\,\,\,\,\,x \in [L_{1} ,L]} } \hfill \\ \end{gathered} \right. $$

For C–C:

$$ W(x) = \left\{\!\! \begin{gathered} W_{1} (x) = \sum\limits_{r = 0}^{\infty } {\frac{{I_{01}^{r} \omega^{2r} }}{{\lambda_{1}^{r} (4r + 2)!}}x^{(4r + 2)} + C_{2} \sum\limits_{r = 0}^{\infty } {\frac{{I_{01}^{r} \omega^{2r} }}{{\lambda_{1}^{r} (4r + 3)!}}x^{(4r + 3)} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x \in [0,L_{1} ]} } \hfill \\ W_{2} (L - x) = C_{3} \sum\limits_{r = 0}^{\infty } {\frac{{\mu^{r} I_{01}^{r} \omega^{2r} }}{{\lambda_{1}^{r} (4r + 2)!}}(L - x)^{(4r + 2)} + C_{4} \sum\limits_{r = 0}^{\infty } {\frac{{\mu^{r} I_{01}^{r} \omega^{2r} }}{{\lambda_{1}^{r} (4r + 3)!}}(L - x)^{(4r + 3)} \,\,\,\,\,\,x \in [L_{1} ,L]} } \hfill \\ \end{gathered} \right. $$

For C–S:

$$ W(x) = \left\{\!\! \begin{gathered} W_{1} (x) = \sum\limits_{r = 0}^{\infty } {\frac{{I_{01}^{r} \omega^{2r} }}{{\lambda_{1}^{r} (4r + 2)!}}x^{(4r + 2)} + C_{2} \sum\limits_{r = 0}^{\infty } {\frac{{I_{01}^{r} \omega^{2r} }}{{\lambda_{1}^{r} (4r + 3)!}}x^{(4r + 3)} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x \in [0,L_{1} ]} } \hfill \\ W_{2} (L - x) = C_{3} \sum\limits_{r = 0}^{\infty } {\frac{{\mu^{r} I_{01}^{r} \omega^{2r} }}{{\lambda_{1}^{r} (4r + 1)!}}(L - x)^{(4r + 1)} + C_{4} \sum\limits_{r = 0}^{\infty } {\frac{{\mu^{r} I_{01}^{r} \omega^{2r} }}{{\lambda_{1}^{r} (4r + 3)!}}(L - x)^{(4r + 3)} \,\,\,\,\,\,x \in [L_{1} ,L]} } \hfill \\ \end{gathered} \right. $$