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Free and forced vibration analysis of moderately thick orthotropic plates in thermal environment and resting on elastic supports

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Abstract

This paper investigates the free and forced vibration of moderately thick orthotropic plates under thermal environment and resting on elastic supports. Three kinds of elastic supports, namely non-homogeneous elastic foundations, point elastic supports and line elastic supports, are considered in the present study. The first-order shear deformation theory is employed to formulate the strain and kinetic energy functions of the structures, and then the stiffness and mass matrices can be obtained by applying the Hamilton’s principle. The modified Fourier method is adopted to solve the dynamic problems of moderately thick orthotropic plates with different combinations of temperature variations, elastic supports and boundary conditions. The accuracy and reliability of the proposed formulation are validated by comparing the obtained results with the finite element method results. Finally, the effects of some key parameters including temperature variation and stiffness values of the elastic supports on the modal and dynamic characteristics of the plates are analyzed in detail. In views of the versatility of the developed method, it offers an efficient tool for the structural analysis of moderately thick orthotropic plates under thermal environment and resting on elastic supports.

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Acknowledgements

The authors gratefully acknowledge the financial supports from the National Key Basic Research Program of China (No. 2014CB046302).

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Authors and Affiliations

  1. Institute of Vibration, Shock and Noise, State Key Laboratory of Mechanical System and Vibration, Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration, Shanghai Jiao Tong University, Dongchuan Road 800, Shanghai, 200240, China

    Kai Zhou, Jinpeng Su & Hongxing Hua

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  1. Kai Zhou
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  2. Jinpeng Su
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  3. Hongxing Hua
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Correspondence to Kai Zhou.

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Appendices

Appendix A

The displacement components can be written as follows, where H is the vector of modified Fourier series terms and q\(_{i}\) (i=1, 2, 3) denotes the unknown Fourier coefficient eigenvector

$$\begin{aligned} \varphi _x (x,y,t)= & {} \mathbf{Hq}_1 e^{j\omega t} \nonumber \\ \varphi _y (x,y,t)= & {} \mathbf{Hq}_2 e^{j\omega t} \nonumber \\ w(x,y,t)= & {} \mathbf{Hq}_3 e^{j\omega t} \end{aligned}$$
(A1)

The vector of modified Fourier series terms is defined by

$$\begin{aligned} \mathbf{H}= & {} [\mathbf{H}_1 ,\mathbf{H}_2 ,\mathbf{H}_3 ]\nonumber \\ \mathbf{H}_1= & {} [\cos \lambda _0 x\cos \lambda _0 y,\ldots , \cos \lambda _m x\cos \lambda _n y,\ldots , \cos \lambda _M x\cos \lambda _N y]\nonumber \\ \mathbf{H}_2= & {} [\xi _{1b} (y)\cos \lambda _0 x,\ldots ,\xi _{1b} (y)\cos \lambda _M x,\xi _{2b} (y)\cos \lambda _0 x, \ldots ,\xi _{2b} (y)\cos \lambda _M x]\nonumber \\ \mathbf{H}_3= & {} [\xi _{1a} (x)\cos \lambda _0 y,\ldots , \xi _{1a} (x)\cos \lambda _N y,\xi _{2a} (x)\cos \lambda _0 y,\ldots , \xi _{2a} (x)\cos \lambda _N y] \end{aligned}$$
(A2)
$$\begin{aligned} \xi _{1a} (x)= & {} L_1 \left( \frac{x}{L_1 }\right) \left( \frac{x}{L_1 }-1\right) ^{2},\xi _{2a} (x)=L_1 \left( \frac{x}{L_1 }\right) ^{2}\left( \frac{x}{L_1 }-1\right) \nonumber \\ \xi _{1b} (y)= & {} L_2 \left( \frac{y}{L_2 }\right) \left( \frac{y}{L_2 }-1\right) ^{2},\xi _{2b} (y)=L_2 \left( \frac{y}{L_2 }\right) ^{2}\left( \frac{y}{L_2 }-1\right) \end{aligned}$$
(A3)

The unknown Fourier coefficient eigenvector can be written by

$$\begin{aligned} \mathbf{q}_1= & {} [\mathbf{q}_{11} ,\mathbf{q}_{12} ,\mathbf{q}_{13} ]^{T} \,\,\mathbf{q}_2 =[\mathbf{q}_{21} ,\mathbf{q}_{22} ,\mathbf{q}_{23} ]^{T}\quad \mathbf{q}_3 =[\mathbf{q}_{31} ,\mathbf{q}_{32} ,\mathbf{q}_{33} ]^{T}\nonumber \\ \mathbf{q}_{11}= & {} [A_00 ,\ldots , A_mn ,\ldots , A_{MN} ]^{T} \nonumber \\ \mathbf{q}_{12}= & {} [d_{10}^1 ,\ldots , d_{1m}^1 ,\ldots , d_{1M}^1 ,d_{20}^1 ,\ldots , d_{2m}^1 ,\ldots , d_{2M}^1 ]^{T}\nonumber \\ \mathbf{q}_{13}= & {} [f_{10}^1 ,\ldots , f_{1n}^1 ,\ldots , f_{1N}^1 ,f_{20}^1 ,\ldots , f_{2n}^1 ,\ldots , f_{2N}^1 ]^{T}\nonumber \\ \mathbf{q}_{21}= & {} [B_00 ,\ldots , B_mn ,\ldots , B_{MN} ]^{T}\nonumber \\ \mathbf{q}_{22}= & {} [d_{10}^2 ,\ldots , d_{1m}^2 ,\ldots , d_{1M}^2 ,d_{20}^2 ,\ldots , d_{2m}^2 ,\ldots , d_{2M}^2 ]^{T}\nonumber \\ \mathbf{q}_{23}= & {} [f_{10}^2 ,\ldots , f_{1n}^2 ,\ldots , f_{1N}^2 ,f_{20}^2 ,\ldots , f_{2n}^2 ,\ldots , f_{2N}^2 ]^{T}\nonumber \\ \mathbf{q}_{31}= & {} [C_00 ,\ldots , C_mn ,\ldots , C_{MN} ]^{T}\nonumber \\ \mathbf{q}_{32}= & {} [d_{10}^3 ,\ldots , d_{1m}^3 ,\ldots , d_{1M}^3 ,d_{20}^3 ,\ldots , d_{2m}^3 ,\ldots , d_{2M}^3 ]^{T}\nonumber \\ \mathbf{q}_{33}= & {} [f_{10}^3 ,\ldots , f_{1n}^3 ,\ldots , f_{1N}^3 ,f_{20}^3 ,\ldots , f_{2n}^3 ,\ldots , f_{2N}^3 ]^{T} \end{aligned}$$
(A4)

Appendix B

The detail expressions in the stiffness and mass matrices are listed as follows

$$\begin{aligned} \mathbf{K}_{\varphi _x \varphi _x }= & {} \int _V {\left[ {\begin{array}{l} \frac{E_1 (1-(\alpha _{xx} +\nu _{21} \alpha _{yy} )\Delta T)z^{2}}{1-\nu _{12} \nu _{21} }\left( \frac{\partial \mathbf{H}}{\partial x}\right) ^{T}\left( \frac{\partial \mathbf{H}}{\partial x}\right) +kG_{13} \mathbf{H}^{T}{} \mathbf{H} \\ +(G_{12} -\frac{E_2 (\alpha _{yy} +\nu _{12} \alpha _{xx} )\Delta T}{1-\nu _{12} \nu _{21} })z^{2}\left( \frac{\partial \mathbf{H}}{\partial y}\right) ^{T}\left( \frac{\partial \mathbf{H}}{\partial y}\right) \\ \end{array}} \right] } dV \nonumber \\&+\int _0^{L_2 } {\left( \left. {K_{xx0} \mathbf{H}^{T}{} \mathbf{H}} \right| _{x=0} +\left. {K_{xxL_1 } \mathbf{H}^{T}{} \mathbf{H}} \right| _{x=L_1 } \right) } dy+\int _0^{L_1 } {\left( \left. {K_{xy0} \mathbf{H}^{T}{} \mathbf{H}} \right| _{y=0} +\left. {K_{xyL_2 } \mathbf{H}^{T}{} \mathbf{H}} \right| _{y=L_2 } \right) } \hbox {d}x\nonumber \\ \end{aligned}$$
(B1)
$$\begin{aligned} \mathbf{K}_{\varphi _x \varphi _y }= & {} \int _V {[\frac{\nu _{21} E_1 z^{2}}{1-\nu _{12} \nu _{21}} \left( \frac{\partial \mathbf{H}}{\partial x}\right) ^{T}\left( \frac{\partial \mathbf{H}}{\partial y}\right) +G_{12} z^{2}\left( \frac{\partial \mathbf{H}}{\partial y}\right) ^{T}\left( \frac{\partial \mathbf{H}}{\partial x}\right) ]} dV \end{aligned}$$
(B2)
$$\begin{aligned} \mathbf{K}_{\varphi _x w}= & {} \int _V {[kG_{13} \mathbf{H}^{T}\frac{\partial \mathbf{H}}{\partial x}]} dV \end{aligned}$$
(B3)
$$\begin{aligned} \mathbf{K}_{\varphi _y \varphi _y }= & {} \int _V {\left[ {\begin{array}{l} \frac{E_2 (1-(\alpha _{yy} +\nu _{12} \alpha _{xx} )\Delta T)z^{2}}{1-\nu _{12} \nu _{21} }\left( \frac{\partial \mathbf{H}}{\partial y}\right) ^{T}\left( \frac{\partial \mathbf{H}}{\partial y}\right) +kG_{23} \mathbf{H}^{T}{} \mathbf{H} \\ +(G_{12} -\frac{E_1 (\alpha _{xx} +\nu _{21} \alpha _{yy} )\Delta T}{1-\nu _{12} \nu _{21} })z^{2}\left( \frac{\partial \mathbf{H}}{\partial x}\right) ^{T}\left( \frac{\partial \mathbf{H}}{\partial x}\right) \\ \end{array}} \right] } dV \nonumber \\&+\int _0^{L_2 } {\left( \left. {K_{yx0} \mathbf{H}^{T}{} \mathbf{H}} \right| _{x=0} +\left. {K_{yxL_1 } \mathbf{H}^{T}{} \mathbf{H}} \right| _{x=L_1 } \right) } dy+\int _0^{L_1 } {\left( \left. {K_{yy0} \mathbf{H}^{T}{} \mathbf{H}} \right| _{y=0} +\left. {K_{yyL_2 } \mathbf{H}^{T}{} \mathbf{H}} \right| _{y=L_2 } \right) } \hbox {d}x\nonumber \\ \end{aligned}$$
(B4)
$$\begin{aligned} \mathbf{K}_{\varphi _y w}= & {} \int _V {[kG_{23} \mathbf{H}^{T}\frac{\partial \mathbf{H}}{\partial y}]} dV \end{aligned}$$
(B5)

For elastic foundations

$$\begin{aligned} \mathbf{K}_{ww}= & {} \int _V {\left[ {\begin{array}{c} (kG_{13} -\frac{E_1 (\alpha _{xx} +\nu _{21} \alpha _{yy} )\Delta T}{1-\nu _{12} \nu _{21} })(\frac{\partial \mathbf{H}}{\partial x})^{T}(\frac{\partial \mathbf{H}}{\partial x}) \\ +(kG_{23} -\frac{E_2 (\alpha _{yy} +\nu _{12} \alpha _{xx} )\Delta T}{1-\nu _{12} \nu _{21} })(\frac{\partial \mathbf{H}}{\partial y})^{T}(\frac{\partial \mathbf{H}}{\partial y}) \\ \end{array}} \right] } dV \nonumber \\&+\int _0^{L_2 } {\left( \left. {K_{wx0} \mathbf{H}^{T}{} \mathbf{H}} \right| _{x=0} +\left. {K_{wxL_1 } \mathbf{H}^{T}{} \mathbf{H}} \right| _{x=L_1 } \right) } dy+\int _0^{L_1 } {\left( \left. {K_{wy0} \mathbf{H}^{T}{} \mathbf{H}} \right| _{y=0} +\left. {K_{wyL_2 } \mathbf{H}^{T}{} \mathbf{H}} \right| _{y=L_2 } \right) } \hbox {d}x \nonumber \\&+\sum _{i=1}^q {\int _{Ls_{i-1} }^{Ls_i } {\int _0^{L_2 } {(K_{efi} \mathbf{H}^{T}{} \mathbf{H})} } \hbox {d}x\hbox {d}y} \end{aligned}$$
(B6)

For point elastic supports

$$\begin{aligned} \mathbf{K}_{ww}= & {} \int _V {\left[ {\begin{array}{c} (kG_{13} -\frac{E_1 (\alpha _{xx} +\nu _{21} \alpha _{yy} )\Delta T}{1-\nu _{12} \nu _{21} })\left( \frac{\partial \mathbf{H}}{\partial x}\right) ^{T}\left( \frac{\partial \mathbf{H}}{\partial x}\right) \\ +(kG_{23} -\frac{E_2 (\alpha _{yy} +\nu _{12} \alpha _{xx} )\Delta T}{1-\nu _{12} \nu _{21} })\left( \frac{\partial \mathbf{H}}{\partial y}\right) ^{T}\left( \frac{\partial \mathbf{H}}{\partial y}\right) \\ \end{array}} \right] } dV \nonumber \\&+\int _0^{L_2 } {\left( \left. {K_{wx0} \mathbf{H}^{T}{} \mathbf{H}} \right| _{x=0} +\left. {K_{wxL_1 } \mathbf{H}^{T}{} \mathbf{H}} \right| _{x=L_1} \right) } dy+\int _0^{L_1 } {\left( \left. {K_{wy0} \mathbf{H}^{T}{} \mathbf{H}} \right| _{y=0} +\left. {K_{wyL_2 } \mathbf{H}^{T}{} \mathbf{H}} \right| _{y=L_2 } \right) } \hbox {d}x \nonumber \\&+\sum _{i=1}^q {\int _0^{L_2 } {\int _0^{L_1 } {K_{psi} \mathbf{H}^{T}{} \mathbf{H}} } \delta (x-x_i ,y-y_i )\hbox {d}x\hbox {d}y} \end{aligned}$$
(B7)

For line elastic supports

$$\begin{aligned} \mathbf{K}_{ww}= & {} \int _V {\left[ {\begin{array}{c} (kG_{13} -\frac{E_1 (\alpha _{xx} +\nu _{21} \alpha _{yy} )\Delta T}{1-\nu _{12} \nu _{21} })\left( \frac{\partial \mathbf{H}}{\partial x}\right) ^{T}\left( \frac{\partial \mathbf{H}}{\partial x}\right) \\ +(kG_{23} -\frac{E_2 (\alpha _{yy} +\nu _{12} \alpha _{xx} )\Delta T}{1-\nu _{12} \nu _{21} })\left( \frac{\partial \mathbf{H}}{\partial y}\right) ^{T}\left( \frac{\partial \mathbf{H}}{\partial y}\right) \\ \end{array}} \right] } dV \nonumber \\&+\int _0^{L_2 } {(\left. {K_{wx0} \mathbf{H}^{T}{} \mathbf{H}} \right| _{x=0} +\left. {K_{wxL_1 } \mathbf{H}^{T}{} \mathbf{H}} \right| _{x=L_1 } )} dy+\int _0^{L_1 } {(\left. {K_{wy0} \mathbf{H}^{T}{} \mathbf{H}} \right| _{y=0} +\left. {K_{wyL_2 } \mathbf{H}^{T}{} \mathbf{H}} \right| _{y=L_2 } )} \hbox {d}x \nonumber \\&+\sum _{i=1}^{q_1 } {\int _0^{L_2 } {\int _0^{L_1 } {K_{xlsi} } \mathbf{H}^{T}{} \mathbf{H}\delta (y-y_i )\hbox {d}x\hbox {d}y} } +\sum _{i=1}^{q_2 } {\int _0^{L_2 } {\int _0^{L_1 } {K_{ylsi} } } \mathbf{H}^{T}{} \mathbf{H}\delta (x-x_i )\hbox {d}x\hbox {d}y}\nonumber \\ \end{aligned}$$
(B8)
$$\begin{aligned} \mathbf{M}_{\varphi _x \varphi _x }= & {} \int _V {[\rho z^{2}{} \mathbf{H}^{T}{} \mathbf{H}]} dV \end{aligned}$$
(B9)
$$\begin{aligned} \mathbf{M}_{\varphi _y \varphi _y }= & {} \int _V {[\rho z^{2}{} \mathbf{H}^{T}{} \mathbf{H}]} dV \end{aligned}$$
(B10)
$$\begin{aligned} \mathbf{M}_{ww}= & {} \int _V {[\rho \mathbf{H}^{T}{} \mathbf{H}]} dV \end{aligned}$$
(B11)

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Zhou, K., Su, J. & Hua, H. Free and forced vibration analysis of moderately thick orthotropic plates in thermal environment and resting on elastic supports. Arch Appl Mech 88, 855–873 (2018). https://doi.org/10.1007/s00419-018-1346-1

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