Free vibration analysis of functionally graded beams with non-uniform cross-section using the differential transform method

Abstract

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.

Appendices

Appendix A

The coefficients $B_K(\omega )$, $C_K(\omega )$, $G_K(\omega )$ and $H_K(\omega )$ denoted in (8) are presented by the following recurrent expressions:

$$\begin{aligned} B_K(\omega )= & {} \frac{1}{(K+1)(K+2)(K+3)(K+4)}\nonumber \\&\times \left[ \omega ^2\left( \sum _{p=0}^{K-4}B_{p}M(K-4-p) +M(K)\right) -\sum _{p=0}^{K-1}(p+2)(p+3)(p+4)B_pD_1(K-1-p)\right. \nonumber \\&\left. -\sum _{p=0}^{K-2}(p+3)(p+4)B_pD_2(K-2-p) \right] , \end{aligned}$$

(A.1)

$$\begin{aligned} C_K(\omega )= & {} \frac{1}{(K+1)(K+2)(K+3)(K+4)}\cdot \left[ \omega ^2\left( \sum _{p=0}^{K-4}C_{p}M(K-4-p) +M(K-1)\right) \right. \nonumber \\&\left. -\sum _{p=0}^{K-1}(p+2)(p+3)(p+4)C_pD_1(K-1-p) -\sum _{p=0}^{K-2}(p+3)(p+4)C_pD_2(K-2-p) \right] , \end{aligned}$$

(A.2)

$$\begin{aligned} G_K(\omega )= & {} \frac{1}{(K+1)(K+2)(K+3)(K+4)}\cdot \left[ \omega ^2\left( \sum _{p=0}^{K-4}G_{p}M(K-4-p) +M(K-2)\right) \right. \nonumber \\&\left. -\sum _{p=0}^{K-1}(p+2)(p+3)(p+4)G_pD_1(K-1-p) \right. \nonumber \\&\left. -\left( \sum _{p=0}^{K-2}(p+3)(p+4)G_pD_2(K-2-p)+2D_2(K)\right) \right] , \end{aligned}$$

(A.3)

$$\begin{aligned} H_K(\omega )= & {} \frac{1}{(K+1)(K+2)(K+3)(K+4)}\cdot \left[ \omega ^2\left( \sum _{p=0}^{K-4}H_{p}M(K-4-p) +M(K-3)\right) \right. \nonumber \\&-\left( \sum _{p=0}^{K-1}(p+2)(p+3)(p+4)H_pD_1(K-1-p) +6D_1(K)\right) \nonumber \\&\left. -\left( \sum _{p=0}^{K-2}(p+3)(p+4)H_pD_2(K-2-p)+6D_2(K-1)\right) \right] . \end{aligned}$$

(A.4)

Appendix B

The constants taken from [19] for calculations of the natural frequencies presented in Tables 4, 5, and 6 have been used in the following forms:

$$\begin{aligned} b_{00}= & {} 26a_0, {~} b_{01}=16a_0,\quad b_{02}=6a_0,\quad b_{03}=-4a_0, b_{04}=a_0,\quad {\mathrm{where}}\; a_0=1, \end{aligned}$$

(B.1)

$$\begin{aligned} b_{10}= & {} \frac{2(71a_1+91a_0)}{5},\quad b_{11}=\frac{2(51a_1+56a_0)}{5},\quad b_{12}=\frac{2(31a_1+21a_0)}{5}, {~} \nonumber \\ b_{13}= & {} \frac{2(11a_1-14a_0)}{5},\quad b_{14}=\frac{-18a_1+7a_0}{5},\quad b_{15}= a_1, \nonumber \\&\text { where } a_0=1,\quad a_1=1 \end{aligned}$$

(B.2)

$$\begin{aligned} b_{20}= & {} \frac{465a_2+568a_1+728a_0}{15},\quad b_{21}=\frac{2(181a_2+204a_1+224a_0)}{15},\quad \nonumber \\ b_{22}= & {} \frac{259a_2+248a_1+168a_0}{15},\quad b_{23}=\frac{4(39a_2+22a_1-28a_0)}{15},\quad \nonumber \\ b_{24}= & {} \frac{53a_2-72a_1+28a_0}{5},\quad b_{25}=\frac{2(-5a_2+2a_1)}{3},\quad b_{26}= a_2, \nonumber \\&\text { where } a_0=1.5954, \quad a_1=0.04, {~} a_2=1 \end{aligned}$$

(B.3)