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Nonlinear vibration analysis of isotropic plate with inclined part-through surface crack

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Abstract

The four modes of vibration of an isotropic rectangular plate with an inclined crack are investigated. It is assumed that the crack remains continuous and its center is located at the center of the plate. The governing nonlinear equation of the transverse vibration of the plate with the plate boundary conditions being simply-supported on all edges is developed. The multiple scale perturbation method is utilized as the solution procedure to find the steady-state frequency response equations for all the four modes of vibration. The equations for the free and forced vibrations are derived and their frequency responses are presented. A special case of large-scale excitation force has also been considered. The parameter sensitivity analysis for the angle of crack, length of crack and the position of the external applied excitation force is performed. It has been shown that according to the aspect ratio of the plate, the vibration modes can have either nonlinear hardening effect or nonlinear softening behavior.

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Acknowledgments

The financial support provided by the Natural Science and Engineering Research Council (NSERC) of Canada to complete this research is gratefully acknowledged.

Author information

Authors and Affiliations

  1. Faculty of Science and Applied Science, University of Ontario Institute of Technology, Oshawa, ON, L1H 7K4, Canada

    F. Diba & E. Esmailzadeh

  2. Center of Excellence in Railway Transportation, Iran University of Science and Technology, Farjam Street, Tehran, 1684613114, Iran

    D. Younesian

Authors
  1. F. Diba
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  2. E. Esmailzadeh
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  3. D. Younesian
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Corresponding author

Correspondence to E. Esmailzadeh.

Appendices

Appendices

1.1 Appendix 1

Common coefficient expressions:

$$\begin{aligned} \hat{{\mu }}_c =\frac{\mu }{2\rho h}\qquad \,\,\,\, \alpha _c =\frac{3}{2}\frac{D\pi ^{2}}{\rho h^{2}l_1^2 l_2^2 } \end{aligned}$$

Coefficients of the first mode of vibration:

$$\begin{aligned} \omega _{11}^2&= \frac{D\pi ^{4}}{\rho hl_1^2 l_2^2}\left[ 2+\frac{l_1^4 +l_2^4 }{l_1^2 l_2^2 }-2\gamma _{M_{xy} }\left( {1-\nu }\right) \right. \\&-\left. \gamma _{M{_y}} \left( {\frac{l_1^2 }{l_2^2}+\nu }\right) \right] \end{aligned}$$
$$\begin{aligned}&\alpha _{d1} =\frac{64}{3}\frac{D}{h^{2}l_1^3 l_2^3 }\frac{A_{22}}{A_{11} }\gamma _{F_{xy}}\\&\alpha _{f1} =\frac{4}{\pi ^{2}A_{11}l_1 l_2 }\sin \frac{\pi x_0 }{l_1 }\sin \frac{\pi y_0 }{l_2 }\\&\alpha _{11} =\left( {\frac{\nu l_1^2 +l_2^2 }{l_1^2 }-\gamma _{F{_y}} \frac{l_1^2 +\nu l_2^2 }{l_2^2 }}\right) A_{11}^2 \\&\alpha _{12}=\left( {\frac{4\nu l_1^2 +l_2^2 }{l_1^2 }-\gamma _{F{_y }}\frac{4l_1^2 +\nu l_2^2 }{l_2^2 }}\right) A_{12}^2\\&\alpha _{13} =\left( {\frac{\upsilon l_{1}^{2} +4l_2^2 }{l_1^2 }-\gamma _{F{_y}} \frac{l_1^2 +4\upsilon l_{2}^2 }{l_2^2 }}\right) A_{21}^2\\&\alpha _{14} =\left( {\frac{4\upsilon l_{1}^{2} +4l_2^2 }{l_1^2}-\gamma _{F{_y} } \frac{4l_1^2 +4\upsilon l_{2}^{2} }{l_2^2}}\right) A_{22}^2\\&\alpha _{15} =\left( {l_1^2 +\nu l_2^2 }\right) A_{11}^2 \qquad \,\,\alpha _{16}=\left( {4l_1^2 +\nu l_2^2 }\right) A_{12}^2\\&\alpha _{17}=\left( {l_1^2 +4\nu l_2^2 }\right) A_{21}^2 \qquad \,\,\alpha _{18}=\left( {4l_1^2 +4\nu l_2^2 }\right) A_{22}^2 \end{aligned}$$

Second mode coefficients:

$$\begin{aligned}&\omega _{12}^2 =\frac{D\pi ^{4}}{\rho hl_1^2 l_2^2}\left[ 8+\frac{16l_1^4 +l_2^4 }{l_1^2 l_2^2 }-8\gamma _{M_{xy} }\left( {1-\nu }\right) \right. \\&\qquad \quad -\left. \gamma _{M{_y}} \left( {\frac{16l_1^2 }{l_2^2}+4\nu }\right) \right] \\&\alpha _{d2} =\frac{64}{3}\frac{D}{h^{2}l_1^3 l_2^3 }\frac{A_{21}}{A_{12} }\gamma _{F_{xy} }\\&\alpha _{f2} =\frac{4}{\pi ^{2}A_{12}l_1 l_2 }\sin \frac{\pi x_0 }{l_1 }\sin \frac{2\pi y_0 }{l_2 }\\&\alpha _{21} =\left( {\frac{\nu l_1^2 +l_2^2 }{l_1^2 }-4\gamma _{F{_y}} \frac{l_1^2 +\nu l_2^2 }{l_2^2 }}\right) A_{11}^2 \\&\alpha _{22} =\left( {\frac{4\nu l_1^2 +l_2^2 }{l_1^2 }-4\gamma _{F{_y}}\frac{4l_1^2 +\nu l_2^2 }{l_2^2 }}\right) A_{12}^2\\&\alpha _{23} =\left( {\frac{\upsilon l_1^2 +4l_2^2 }{l_1^2 }-4\gamma _{F_{y} } \frac{l_1^2 +4\upsilon l_2^2 }{l_2^2 }}\right) A_{21}^2\\&\alpha _{24} =\left( {\frac{4\upsilon l_1^2 +4l_2^2 }{l_1^2 }-4\gamma _{F{_y}} \frac{4l_1^2 +4\upsilon l_2^2 }{l_2^2 }}\right) A_{22}^2\\&\alpha _{25} =\left( {l_1^2 +\nu l_2^2 }\right) A_{11}^2 \qquad \,\,\alpha _{26} =\left( {4l_1^2 +\nu l_2^2 }\right) A_{12}^2\\&\alpha _{27}=\left( {l_1^2 +4\nu l_2^2 }\right) A_{21}^2 \qquad \,\,\alpha _{28}=\left( {4l_1^2 +4\nu l_2^2 }\right) A_{22}^2 \end{aligned}$$

Coefficients of the third mode of vibration:

$$\begin{aligned}&\omega _{21}^2 =\frac{D\pi ^{4}}{\rho hl_1^2 l_2^2}\left[ 8+\frac{l_1^4 +16l_2^4 }{l_1^2 l_2^2 }-8\gamma _{M_{xy} }\left( {1-\nu }\right) \right. \\&\qquad \quad -\left. \gamma _{M{_y}} \left( {\frac{l_1^2 }{l_2^2}+4\nu }\right) \right] \\&\alpha _{d3} =\frac{64}{3}\frac{D}{h^{2}l_1^3 l_2^3 }\frac{A_{12}}{A_{21} }\gamma _{F_{xy} }\\&\alpha _{f3} =\frac{4}{\pi ^{2}A_{21}l_1 l_2 }\sin \frac{2\pi x_0 }{l_1 }\sin \frac{\pi y_0 }{l_2 }\\&\alpha _{31} =\left( {4\frac{\nu l_1^2 +l_2^2 }{l_1^2 }-\gamma _{F{_y }}\frac{l_1^2 +\nu l_2^2 }{l_2^2 }}\right) A_{11}^2\\&\alpha _{32}=\left( {4\frac{4\nu l_1^2 +l_2^2 }{l_1^2 }-\gamma _{F{_y}}\frac{4l_1^2 +\nu l_2^2 }{l_2^2 }}\right) A_{12}^2\\&\alpha _{33} =\left( {4\frac{\upsilon l_1^2 +4l_2^2 }{l_1^2}-\gamma _{F{_y}} \frac{l_1^2 +4\upsilon l_2^2 }{l_2^2}}\right) A_{21}^2 \\&\alpha _{34} =\left( {4\frac{4\upsilon l_1^2+4l_2^2 }{l_1^2 }-\gamma _{F{_y}} \frac{4l_1^2 +4\upsilon l_2^2}{l_2^2 }}\right) A_{22}^2\\&\alpha _{35} =\left( {l_1^2 +\nu l_2^2 }\right) A_{11}^2 \qquad \,\,\alpha _{36} =\left( {4l_1^2 +\nu l_2^2 }\right) A_{12}^2 \\&\alpha _{37}=\left( {l_1^2 +4\nu l_2^2 }\right) A_{21}^2 \qquad \,\,\alpha _{38}=\left( {4l_1^2 +4\nu l_2^2 }\right) A_{22}^2 \end{aligned}$$

Fourth mode coefficients:

$$\begin{aligned}&\omega _{22}^2 =\frac{D\pi ^{4}}{\rho hl_1^2 l_2^2}\left[ 32+16\frac{l_1^4 +l_2^4 }{l_1^2 l_2^2 }-32\gamma _{M_{xy} }\left( {1-\nu }\right) \right. \\&\qquad \quad -\left. 16\gamma _{M{_y}} \left( {\frac{l_1^2 }{l_2^2}+\nu }\right) \right] \\&\alpha _{d4}=\frac{64}{3}\frac{D}{h^{2}l_1^3 l_2^3 }\frac{A_{11} }{A_{22}}\gamma _{F_{xy} }\\&\alpha _{f4} =\frac{4}{\pi ^{2}A_{22} l_1 l_2}\sin \frac{2\pi x_0 }{l_1 }\sin \frac{2\pi y_0 }{l_2 }\\&\alpha _{41} =4\left( {\frac{\nu l_1^2 +l_2^2 }{l_1^2 }-\gamma _{F{_y} } \frac{l_1^2 +\nu l_2^2 }{l_2^2 }}\right) A_{11}^2\\&\alpha _{42} =4\left( {\frac{4\nu l_1^2 +l_2^2 }{l_1^2}-\gamma _{F{_y}} \frac{4l_1^2 +\nu l_2^2 }{l_2^2 }}\right) A_{12}^2\\&\alpha _{43} =4\left( {\frac{\upsilon l_1^2 +4l_2^2 }{l_{1}^{2}}-\gamma _{F{_y}} \frac{l_1^2 +4\upsilon l_{2}^{2} }{l_{2}^{2}}}\right) A_{21}^2 \\&\alpha _{44} =4\left( {\frac{4\upsilon l_1^2 +4l_2^2 }{l_1^2}-\gamma _{F{_y}} \frac{4l_1^2 +4\upsilon l_2^2 }{l_2^2}}\right) A_{22}^2\\&\alpha _{45} =\left( {l_1^2 +\nu l_2^2 }\right) A_{11}^2 \qquad \,\,\alpha _{46} =\left( {4l_1^2 +\nu l_2^2 }\right) A_{12}^2\\&\alpha _{47}=\left( {l_1^2 +4\nu l_2^2 }\right) A_{21}^2 \qquad \,\,\alpha _{48}=\left( {4l_1^2 +4\nu l_2^2 }\right) A_{22}^2 \end{aligned}$$

1.2 Appendix 2

The procedure of deriving Eq. (15)–(18) from Eq. (1) has been presented here. In the first step, by substituting Eqs. (3), (4), (8) and (9) into Eq. (1) one will find:

$$\begin{aligned}&D\left( {\frac{\partial ^{4}w}{\partial x^{4}}+2\frac{\partial ^{4}w}{\partial x^{2}\partial y^{2}}+\frac{\partial ^{4}w}{\partial y^{4}}}\right) \nonumber \\&\quad =-\rho h\frac{\partial ^{2}w}{\partial t^{2}}-\mu \frac{\partial w}{\partial t}+F_x\frac{\partial ^{2}w}{\partial x^{2}} \nonumber \\&\qquad -\,\gamma _{F{_y}} F_{y} \frac{\partial ^{2}w}{\partial y^{2}}-\gamma _{M{_y}}\frac{\partial ^{2}M_y }{\partial y^{2}}-2\gamma _{M{_{xy}} } \frac{\partial ^{2}M_{xy} }{\partial x\partial y}\nonumber \\&\qquad -\,2\gamma _{F_{xy}} F_y \frac{\partial ^{2}w}{\partial x\partial y}+q_z \end{aligned}$$
(94)

Then by substituting Eqs. (5), (6), (7) and (10) into Eq. (94) leads into:

$$\begin{aligned}&D\left( {\frac{\partial ^{4}w}{\partial x^{4}}+2\frac{\partial ^{4}w}{\partial x^{2}\partial y^{2}}+\frac{\partial ^{4}w}{\partial y^{4}}}\right) \nonumber \\&\ =-\rho h\frac{\partial ^{2}w}{\partial t^{2}}-\mu \frac{\partial w}{\partial t}+q_z \nonumber \\&\ \ +\frac{6D}{h^{2}l_1 l_2 }\frac{\partial ^{2}w}{\partial x^{2}}\int \limits _0^{l_1 } \mathop \int \limits _0^{l_2 }\left( {\left( {\frac{\partial w}{\partial x}}\right) ^{2}+\nu \left( {\frac{\partial w}{\partial y}}\right) ^{2}}\right) \mathrm{d}x\mathrm{d}y\nonumber \\&\ \ -\gamma _{F{_y}} \frac{6D}{h^{2}l_1 l_2 }\frac{\partial ^{2}w}{\partial y^{2}}\nonumber \\&\quad \times \int \limits _0^{l_1 } \mathop \int \limits _0^{l_2 }\left( {\left( {\frac{\partial w}{\partial y}}\right) ^{2}+\nu \left( {\frac{\partial w}{\partial x}}\right) ^{2}}\right) \mathrm{d}x\mathrm{d}y \nonumber \\&\ \ +\,\gamma _{M{_y}} D\left( {\frac{\partial ^{2}w}{\partial y^{4}}+\nu \frac{\partial ^{2}w}{\partial x^{2}\partial y^{2}}}\right) \nonumber \\&\ \ +\,2\gamma _{M{_{xy}} } D\left( {1-\nu }\right) \frac{\partial ^{2}w}{\partial x^{2}\partial y^{2}} \nonumber \\&\ \ -\,\gamma _{F{_{xy}} } \frac{12D}{h^{2}l_1 l_2 }D\frac{\partial ^{2}w}{\partial x\partial y}\nonumber \\&\quad \times \mathop \int \limits _0^{l_1 } \int \limits _0^{l_2 } \left( {\left( {\frac{\partial w}{\partial y}}\right) ^{2}+\nu \left( {\frac{\partial w}{\partial x}}\right) ^{2}}\right) \mathrm{d}x\mathrm{d}y \end{aligned}$$
(95)

Based on the Galerkin’s method, the general form of the transverse deflection of the plate from Eq. (11) has been substituted into Eq. (95):

$$\begin{aligned}&D\left( \frac{\partial ^{4}X_i }{\partial x^{4}}A_{ij} Y_j \psi _{ij} (t)+2\frac{\partial ^{4}X_i Y_j }{\partial x^{2}\partial y^{2}}A_{ij} \psi _{ij} (t)\right. \nonumber \\&\quad +\left. \frac{\partial ^{4}Y_j }{\partial y^{4}}A_{ij} X_i \psi _{ij} (t)\right) \nonumber \\&\ \ =-\rho hA_{ij} X_i Y_j \ddot{\psi }_{ij} (t)-\mu A_{ij} X_i Y_j \dot{\psi }_{ij}(t) \nonumber \\&\quad +\frac{6D}{h^{2}l_1 l_2 }\frac{\partial ^{2}w}{\partial x^{2}}\int \limits _0^{l_1 } \int \limits _0^{l_2 }\left( \left( {\frac{\partial X_i }{\partial x}A_{ij} Y_j \psi _{ij} (t)}\right) ^{2} \right. \nonumber \\&\quad +\left. \nu \left( {\frac{\partial Y_j }{\partial y}A_{ij} X_i \psi _{ij} (t)}\right) ^{2}\right) \mathrm{d}x\mathrm{d}y \nonumber \\&\ \ -\gamma _{F{_y}} \frac{6D}{h^{2}l_1 l_2 }\frac{\partial ^{2}Y_j }{\partial y^{2}}A_{ij} X_i \psi _{ij} (t)\nonumber \\&\quad \times \int \limits _0^{l_1 } \int \limits _0^{l_2 } \left( \left( {\frac{\partial Y_j }{\partial y}A_{ij} X_i \psi _{ij} (t)}\right) ^{2}\right. \nonumber \\&\qquad +\left. \nu \left( {\frac{\partial X_i }{\partial x}A_{ij} Y_j \psi _{ij} (t)}\right) ^{2}\right) \mathrm{d}x\mathrm{d}y \nonumber \\&\ \ +\gamma _{M{_y}} D\left( {\frac{\partial ^{2}Y_j }{\partial y^{4}}A_{ij} X_i \psi _{ij} (t)+\nu \frac{\partial ^{2}X_i Y_j }{\partial x^{2}\partial y^{2}}A_{ij} \psi _{ij} (t)}\right) \nonumber \\&\ \ +2\gamma _{M {_{xy}} } D\left( {1-\nu }\right) \frac{\partial ^{2}X_i Y_j }{\partial x^{2}\partial y^{2}}A_{ij} \psi _{ij}(t) \nonumber \\&\ \ -\gamma _{F{_{xy}} } \frac{12D}{h^{2}l_1 l_2 }D\frac{\partial ^{2}X_i Y_j }{\partial x\partial y}A_{ij} \psi _{ij} (t)\nonumber \\&\quad \times \int \limits _0^{l_1 } \int \limits _0^{l_2 } \left( \left( {\frac{\partial Y_j }{\partial y}A_{ij} X_i \psi _{ij} (t)}\right) ^{2}\right. \nonumber \\&\qquad +\left. \nu \left( {\frac{\partial X_i}{\partial x}A_{ij} Y_j \psi _{ij} (t)}\right) ^{2}\right) \mathrm{d}x\mathrm{d}y \nonumber \\&\ \qquad +q(t)\delta \left( {x-x_0 }\right) \left( {y-y_0 }\right) \end{aligned}$$
(96)

In this Appendix the derivation of the equation of the vibration for first mode has been described in detail. The same procedure has been employed to derive the equation of the vibration for other three modes. To derived Eq. (15), the modal functions of the first mode have to be substituted in to Eq. (96). These modal functions based on the Eqs. (12) and (13) would be:

$$\begin{aligned} X_1&= \mathrm{sin}\left( {\frac{\pi x}{l_1 }}\right) \end{aligned}$$
(97)
$$\begin{aligned} Y_1&= \mathrm{sin}\left( {\frac{\pi y}{l_2 }}\right) \end{aligned}$$
(98)

By substituting Eqs. (97) and (98) into Eq. (96) and multiply by \(X_{1}Y_{1}\) and taking integral over the plate surface, one can obtain:

$$\begin{aligned}&D\frac{l_1 l_2 }{4}\left( {\frac{\pi ^{4}}{l_1^{4}}A_{11} \psi _{11}+2\frac{\pi ^{4}}{l_1^{2}l_2^{2}}A_{11} \psi _{11} +\frac{\pi ^{4}}{l_2^{4}}A_{11} \psi _{11} }\right) \nonumber \\&\quad =-\rho hA_{11} \frac{l_1 l_2 }{4}\ddot{\psi }_{11} -\mu A_{11} \frac{l_1 l_2 }{4}\dot{\psi }_{11} \nonumber \\&\qquad -\,\frac{3}{8}\frac{\pi ^{4}D}{h^{2}l_1^{3}l_2 }A_{11} \psi _{11}\nonumber \\&\qquad \times \left[ {\begin{array}{l} \left( {\nu l_1^{2}+l_2^{2}}\right) A_{11}^2 \psi _{11}^2+\left( {4\nu l_1^{2}+l_2^{2}}\right) A_{12}^2 \psi _{12}^2 \\ +\left( {\nu l_1^{2}+4l_2^{2}}\right) A_{21}^2 \psi _{21}^2\\ +\left( {4\nu l_1^{2}+4l_2^{2}}\right) A_{22}^2 \psi _{22}^2 \end{array}}\right] \nonumber \\&\qquad +\,\gamma _{F_{y}} \frac{3}{8}\frac{\pi ^{4}D}{h^{2}l_1 l_2^{3}}A_{11} \psi _{11}\nonumber \\&\qquad \times \left[ {\begin{array}{l} \left( {l_1^{2}+\nu l_2^{2}}\right) A_{11}^2 \psi _{11}^2+\left( {4l_1^{2}+\nu l_2^{2}}\right) A_{12}^2 \psi _{12}^2 \\ +\left( {l_1^{2}+4\nu l_2^{2}}\right) A_{21}^2 \psi _{21}^2\\ +\left( {4l_1^{2}+4\nu l_2^{2}}\right) A_{22}^2 \psi _{22}^2 \end{array}}\right] \nonumber \\&\qquad +\,\gamma _{M_{y}} D\frac{l_1 l_2 }{4}A_{11} \psi _{11}\left[ {\frac{\pi ^{4}}{l_1^{4}}+\nu \frac{\pi ^{4}}{l_1^{2}l_2^{2}}}\right] \nonumber \\&\qquad +\,2\gamma _{M_{xy} } D\left( {1-\nu }\right) \frac{l_1 l_2 }{4}\left[ {\frac{\pi ^{4}}{l_1^{2}l_2^{2}}}\right] \nonumber \\&\qquad -\,\gamma _{F_{xy}} \frac{16}{3}\frac{\pi ^{2}D}{h^{2}l_1^{2}l_2^{2}}A_{22} \psi _{22}\nonumber \\&\qquad \times \left[ {\begin{array}{l} \left( {l_1^{2}+\nu l_2^{2}}\right) A_{11}^2 \psi _{11}^2+\left( {4l_1^{2}+\nu l_2^{2}}\right) A_{12}^2 \psi _{12}^2\nonumber \\ +\left( {l_1^{2}+4\nu l_2^{2}}\right) A_{21}^2 \psi _{21}^2\\ +\left( {4l_1^{2}+4\nu l_2^{2}}\right) A_{22}^2 \psi _{22}^2 \end{array}}\right] \nonumber \\&\qquad +\,q(t)\mathrm{sin}\left( {\frac{\pi x_0 }{l_1 }}\right) \mathrm{sin}\left( {\frac{\pi y_0 }{l_2 }}\right) \end{aligned}$$
(99)

In the final step, by factorizing terms and cancelling \(\frac{l_1 l_2 }{4}A_{11} \pi ^{2}\) from Eq. (99) one finds:

$$\begin{aligned}&\frac{\rho h}{\pi ^{2}}\ddot{\psi }_{11}+\left[ D\left( {\frac{\pi ^{4}}{l_1^{4}}+2\frac{\pi ^{4}}{l_1^{2}l_2^{2}}+\frac{\pi ^{4}}{l_2^{4}}}\right) \right. \nonumber \\&-\gamma _{M_{y}} D\left( {\frac{\pi ^{4}}{l_1^{4}}+\nu \frac{\pi ^{4}}{l_1^{2}l_2^{2}}}\right) \nonumber \\&-\left. 2\gamma _{M_{xy} }D\left( {1-\nu }\right) \left( {\frac{\pi ^{4}}{l_1^{2}l_2^{2}}}\right) \right] \psi _{11}=-\frac{\mu }{\pi ^{2}}\dot{\psi }_{11} \nonumber \\&-\frac{3}{2}\frac{\pi ^{2}D}{h^{2}l_1^{2}l_2^{2}} \left[ \!\!{\begin{array}{l} \left( {\frac{\nu l_1^{2}+l_2^{2}}{l_1^{2}}-\gamma _{F_{y}}\frac{l_1^{2}+\nu l_2^{2}}{l_2^{2}}}\right) A_{11}^2 \psi _{11}^3\\ +\left( {\frac{4\nu l_1^{2}+l_2^{2}}{l_1^{2}}-\gamma _{F_{y}}\frac{4l_1^{2}+\nu l_2^{2}}{l_2^{2}}}\right) A_{12}^2 \psi _{11}\psi _{12}^2 \\ +\left( {\frac{\nu l_1^{2}+4l_2^{2}}{l_1^{2}}-\gamma _{F_{y}}\frac{l_1^{2}+\nu 4l_2^{2}}{l_2^{2}}}\right) A_{21}^2 \psi _{11}\psi _{21}^2\\ +\left( {\frac{4\nu l_1^{2}+4l_2^{2}}{l_1^{2}}-\gamma _{F_{y}} \frac{4l_1^{2}+4\nu l_2^{2}}{l_2^{2}}}\right) A_{22}^2 \psi _{11} \psi _{22}^2 \\ \end{array}}\!\right] \nonumber \\&-\frac{64}{3}\frac{D}{h^{2}l_1^{3}l_2^{3}}\gamma _{F_{xy}} \left[ {\begin{array}{l} \left( {l_1^{2}+\nu l_2^{2}}\right) A_{11}^2 \psi _{11}^2 \psi _{22}\\ +\left( {4l_1^{2}+\nu l_2^{2}}\right) A_{12}^2 \psi _{12}^2 \psi _{22}\\ +\left( {l_1^{2}+4\nu l_2^{2}}\right) A_{21}^2 \psi _{21}^2 \psi _{22}\\ +\left( {4l_1^{2}+4\nu l_2^{2}}\right) A_{22}^2 \psi _{22}^3 \\ \end{array}}\right] \nonumber \\&+\frac{4}{\pi ^{2}A_{11} l_1 l_2 }q(t)\mathrm{sin}\left( {\frac{\pi x_0 }{l_1 }}\right) \mathrm{sin}\left( {\frac{\pi y_0 }{l_2 }}\right) \end{aligned}$$
(100)

The coefficients of Eq. (100) have been listed in Appendix I.

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Diba, F., Esmailzadeh, E. & Younesian, D. Nonlinear vibration analysis of isotropic plate with inclined part-through surface crack. Nonlinear Dyn 78, 2377–2397 (2014). https://doi.org/10.1007/s11071-014-1595-7

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