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Exponential functionally graded plates resting on Winkler–Pasternak foundation: free vibration analysis by dynamic stiffness method

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Abstract

In this present work, the dynamic stiffness method (DSM) formulation is used to investigate the natural vibration of an exponential functionally graded plate (E-FGM) within the framework of Kirchhoff’s plate theory. The material property continuously varies along the transverse direction of the E-FGM plate by applying the exponential law. The governing partial differential equation of motion is derived by implementing Hamilton's principle based on the physical neutral surface of the FGM plate. The Wittrick–Williams algorithm is applied as a solution method to solve the complex nature of the dynamic stiffness matrix and compute the natural vibration frequencies of the E-FGM rectangular plates. The effect of different numerical parameter values (exponential volume fraction index, aspect ratio, boundary conditions density ratio, modulus ratio, elastic foundation parameters) on natural frequency is also reported. The obtained natural frequencies are compared with those available in the literature. Finally, this study presents a new set of DSM frequency results for different Levy-type boundary conditions for E-FGM plates resting on a Winkler–Pasternak elastic foundation.

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Authors and Affiliations

  1. Bennett University, Greater Noida, 201310, India

    Manish Chauhan

  2. Rowan University, Glassboro, NJ, 08028, USA

    Sarvagya Dwivedi, Pawan Mishra & Vinayak Ranjan

  3. Kaunas University of Technology, 51368, Kaunas, Lithuania

    Minvydas Ragulskis

  4. Silesian University of Technology, Gliwice, Poland

    Rafal Burdzik

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Appendices

Appendix A

The mathematical representation of \({D}_{FGM}\) and \({I}_{0}\)

$$\begin{array}{c}{\left({D}_{FGM}\right)}_{E}={\int }_{-h/2-{z}_{0}}^{h/2-{z}_{0}} \left({z}_{ns}^{2}\right){Q}_{11}\left({z}_{ns}\right)d{z}_{ns}={\int }_{-h/2}^{h/2} {Q}_{11}(z){\left(z-{z}_{0}\right)}^{2}dz\\ =\frac{{h}^{3}}{\mathrm{log}\left({E}_{c}/{E}_{m}\right)\left(1-{v}^{2}\right)}\left[\frac{{E}_{c}-{E}_{m}}{4}-\frac{{E}_{m}+{E}_{c}}{\mathrm{log}\left({E}_{c}/{E}_{m}\right)}+\frac{2\left({E}_{c}-{E}_{m}\right)}{\mathrm{log}{\left({E}_{c}/{E}_{m}\right)}^{2}}\right.\\ \left.+2\left(\frac{{z}_{0}}{h}\right)\left\{\frac{{E}_{m}+{E}_{c}}{2}-\frac{{E}_{c}-{E}_{m}}{\mathrm{log}\left({E}_{c}/{E}_{m}\right)}\right\}+{\left(\frac{{z}_{0}}{h}\right)}^{2}\left\{{E}_{c}-{E}_{m}\right\}\right]\\ =\frac{12{D}_{c}}{\mathrm{log}\left({E}_{rat}\right){E}_{rat}}\left[\frac{{E}_{rat}-1}{4}-\frac{1-{E}_{rat}}{\mathrm{log}\left({E}_{rat}\right)}+\frac{2\left({E}_{rat}-1\right)}{\mathrm{log}{\left({E}_{rat}\right)}^{2}}\right.\\ \left.+2\left(\frac{{z}_{0}}{h}\right)\left\{\frac{1+{E}_{rat}}{2}-\frac{1-{E}_{rat}}{\mathrm{log}\left({E}_{rat}\right)}\right\}+{\left(\frac{{z}_{0}}{h}\right)}^{2}\left\{{E}_{rat}-1\right\}\right]\end{array}$$
(A1)
$${\left({I}_{0}\right)}_{E}={\int }_{-h/2-{z}_{0}}^{h/2-{z}_{0}} \rho \left({z}_{ns}\right)d{z}_{ns}={\int }_{-h/2}^{h/2} \rho (z)dz=h{\rho }_{c}\left(\frac{{\rho }_{rat}-1}{\mathrm{log}\left({\rho }_{rat}\right)}\right)$$
(A2)

Appendix B

The mathematical expressions of DS matrix in Eq. (36)

This is given in the main body \({(S}_{vv}, {S}_{vm}, {F}_{vv}, {F}_{vm}, {S}_{mm}, {S}_{vn}).\)

Case 1:

$$\begin{aligned}{S}_{vv}& =\left({r}_{2m}{R}_{1}-{r}_{1m}{R}_{2}\right)\left({r}_{1m}{C}_{h2}{S}_{h1}+{r}_{2m}{C}_{h1}{S}_{h2}\right)/\Delta \\ {S}_{Dm} & ={r}_{1m}\left\{-{R}_{2}\left({C}_{h1}^{2}-{C}_{h1}{C}_{h2}+{S}_{h1}^{2}\right)-{R}_{2}{S}_{h1}{S}_{h2}\right\}\\ &\qquad-{r}_{2m}\left\{{R}_{2}{S}_{h1}{S}_{h2}+{R}_{1}\left(\left({C}_{h1}-{C}_{h2}\right){C}_{h2}+{S}_{h2}^{2}\right)\right\}/\Delta \\ {S}_{mm} & =\left({L}_{2}-{L}_{1}\right)\left({r}_{2m}{C}_{h2}{S}_{h1}-{r}_{1m}{C}_{h1}{S}_{h2}\right)/\Delta \\ {f}_{\nu v} & =\left({r}_{2m}{R}_{1}-{r}_{1m}{R}_{2}\right)\left({r}_{2m}{S}_{h2}-{r}_{1m}{S}_{h1}\right)/\Delta \\ {f}_{Dm} & =\left({C}_{h2}-{C}_{h1}\right)\left({r}_{2m}{R}_{1}-{r}_{1m}{R}_{2}\right)/\Delta \\ {f}_{mm} &=\left({L}_{1}-{L}_{2}\right)\left({r}_{2m}{S}_{h1}-{r}_{1m}{S}_{h2}\right)/\Delta ,\end{aligned}$$
(B1)

where \({r}_{im}, {S}_{hi}, {C}_{hi}, {S}_{i}, {C}_{i}, {L}_{i}, {R}_{i}\) (with suffix represents \(i=\mathrm{1,2}\)) are expressed in Eqs. (18), (31) and (33), respectively, and the \(\Delta \) defined by

$$\Delta ={S}_{h1}{S}_{h2}\left({r}_{1m}^{2}-{r}_{2m}^{2}\right)+{r}_{1m}{r}_{2m}\left\{{\left({C}_{h1}-{C}_{h2}\right)}^{2}-{S}_{h1}^{2}-{S}_{h2}^{2}\right\}$$
(B2)

Case 2:

$$\begin{aligned} {S}_{wv}& = \left({r}_{2m}{R}_{1}+{r}_{1m}{R}_{2}\right)\left({r}_{2m}{C}_{h1}{S}_{2}+{r}_{1m}{C}_{2}{S}_{h1}\right)/\Delta ,\\ {S}_{vm}& = -{r}_{2m}\left\{{R}_{1}\left({C}_{2}^{2}-{C}_{2}{C}_{h1}+{S}_{2}^{2}\right)-{R}_{2}{S}_{2}{S}_{h1}\right\}\\ &\quad -{r}_{1m}\left\{{R}_{1}{S}_{2}{S}_{h1}-{R}_{2}\left(\left({C}_{2}-{C}_{h1}\right){C}_{h1}+{S}_{h1}^{2}\right)\right\}/\Delta ,\\ {S}_{mm}&= \left({L}_{1}-{L}_{2}\right)\left({r}_{1m}{C}_{h1}{S}_{2}-{r}_{2m}{C}_{2}{S}_{h1}\right)/\Delta ,\\ {f}_{wV}& = \left({r}_{2m}{R}_{1}-{r}_{1m}{R}_{2}\right)\left({r}_{2m}{S}_{2}+{r}_{1m}{S}_{h1}\right)/\Delta ,\\ {f}_{vm}& = \left({C}_{h2}-{C}_{h1}\right)\left({r}_{2m}{R}_{1}-{r}_{1m}{R}_{2}\right)/\Delta \\ {f}_{mm}& = \left({L}_{1}-{L}_{2}\right)\left({r}_{2m}{S}_{h1}-{r}_{1m}{S}_{h2}\right)/\Delta ,\end{aligned}$$
(B3)

where \({r}_{im}, {S}_{hi}, {C}_{hi}, {S}_{i}, {C}_{i}, {L}_{i}, {R}_{i}\) (with suffix represents \(i=\mathrm{1,2}\)) are expressed in Eqs. (22), and (31), respectively, and the other given parameters can be expressed as:

$${R}_{i}={D}_{FGM}\left[{r}_{im}^{3}-{\alpha }^{2}{r}_{im}\left(2-v\right)\right]+{r}_{im}{I}_{2}{\omega }^{2},$$
$${L}_{i}={D}_{FGM}\left({r}_{im}^{3}-{\alpha }^{2}v\right)\mathrm{with }i=\mathrm{1,2}$$
(B4)

and \(\Delta \) given by

$$\Delta ={S}_{2}{S}_{h1}\left({r}_{1m}^{2}-{r}_{2m}^{2}\right)+{r}_{1m}{r}_{2m}\left\{{\left({C}_{2}-{C}_{h1}\right)}^{2}+{S}_{2}^{2}-{S}_{h1}^{2}\right\}$$
(B5)

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Chauhan, M., Dwivedi, S., Mishra, P. et al. Exponential functionally graded plates resting on Winkler–Pasternak foundation: free vibration analysis by dynamic stiffness method. Arch Appl Mech 93, 2483–2509 (2023). https://doi.org/10.1007/s00419-023-02392-6

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