Analysis of Natural Frequencies in Non-uniform Cross-section Functionally Graded Porous Beams

Abstract

Purpose

The study aims to analyze the free vibration behavior of functionally graded porous beams with non-uniform rectangular cross-sections, investigating four distinct porosity distribution across the beam's thickness.

Methods

Utilizing the Euler–Bernoulli beam model and Hamilton’s principle, the equation of motion is derived. Four different boundary conditions (clamped–clamped, clamped-free, clamped–pinned, and pinned–pinned) are considered, and the resulting equation is solved using the differential transform method. Verification of accuracy is ensured through comparison with solutions for natural frequencies found in existing literature.

Results and Conclusion

The study provides validated natural frequency solutions for functionally graded porous beams with non-uniform rectangular cross-sections, confirming the accuracy of the proposed method through literature comparison. A comprehensive parametric study reveals significant insights into the influence of various factors on natural frequencies, including porosity volume fractions, types of porosity distribution, taper ratios, and boundary conditions. These findings deepen our understanding of free vibration analysis for functionally graded porous beams, offering valuable guidance for engineering design and structural optimization in relevant applications.

Appendix A

The averaged Young’s modulus $E_z(x)$ and mass density $\rho _z(x)$ in (16) determined by analytically integrating Eq. (11) over the rectangular domain A(x) for every porosity profile type (3)–(6) are presented as follows:

$$\begin{aligned} \begin{aligned} E_z(x) = E_0 \left( 1 - e_0 \alpha (x) \right) ,\\ \rho _z(x) = \rho _0 \left( 1 - e_m \alpha _{*}(x) \right) \\ \end{aligned} \end{aligned}$$

• Type I:

  $$\begin{aligned} \begin{aligned} \alpha (x)&= \frac{1}{e_0} - \frac{1}{e_0}\left( \frac{2}{\pi }\sqrt{1 - e_0} - \frac{2}{\pi } + 1 \right) ^2, \\&\alpha _{*}(x) = \frac{2}{\pi } \\ \end{aligned} \end{aligned}$$

  (A.1)

• Type II:

  $$\begin{aligned} \begin{aligned} \alpha (x)&= \frac{3}{\delta (x)}\Bigg [ \left( 1 - \frac{2}{\delta ^2(x)} \right) \sin \delta (x) + \frac{2}{\delta (x)}\cos \delta (x) \Bigg ],\\&\alpha _{*}(x) = \frac{\sin \delta (x) }{\delta (x)}, \\ \end{aligned} \end{aligned}$$

  (A.2)

  where $\delta (x)= \frac{\pi h(x)}{2 h_0}$

• Type III:

  $$\begin{aligned} \begin{aligned} \alpha (x)&= \frac{3}{\delta (x)}\Bigg [ -\cos \delta (x) + \frac{2}{\delta (x)}\sin \delta (x) - \frac{2}{\delta ^2(x)}\left( 1 - \cos \delta (x) \right) \Bigg ],\\&\alpha _{*}(x) = \frac{1 - \cos \delta (x) }{\delta (x)} \\ \end{aligned} \end{aligned}$$

  (A.3)

• Type IV:

  $$\begin{aligned} \begin{aligned} \alpha (x)&= \frac{3}{\delta _1(x)}\Bigg [ \left( 1 - \frac{2}{\delta _1^2(x)} \right) \sin \delta _1(x) + \frac{2}{\delta _1(x)}\cos \delta _1(x) \Bigg ]\cos \alpha _1,\\&\alpha _{*}(x) = \frac{\sin \delta _1(x) }{\delta _1(x)}\cos \alpha _1, \\ \end{aligned} \end{aligned}$$

  (A.4)

  where $\delta _1(x) = \frac{\delta (x)}{2}\cos \alpha _1$