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Investigation on the 1:2 internal resonance of an FGM blade

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Abstract

This paper focuses on the internal resonance analysis of a functionally graded material (FGM) blade under centrifugal and aerodynamic forces. Based on von Karman large deflection geometry and Reddy third-order shear deformation theory, the ordinary differential equations for transverse vibration of the first two modes are formulated. The parameter conditions of internal resonances under different blade length, width, thickness, speed ratio and material composition are obtained. Accordingly, in the case of 1:2 internal resonance, the amplitude responses of the first two modes of the system affected by the material component parameter, rotation perturbation amplitude, air velocity are analyzed, respectively. A variety of nonlinear phenomena such as hardening spring characteristic, the frequency band variation of the response curve and nested regions of multi-stability are obtained, which are discussed in detail through the amplitude–frequency response diagrams of single oscillator solution and dual oscillator solution. Most prominently, for the dual oscillator solution, the whole amplitude–frequency response curve can be divided into two branches according to the second-order mode amplitude: One reflects the internal resonance energy transfer mechanism, and the other has the similar law to the single oscillator solution. Moreover, it can be found that the sensitivities of the two branches to different parameters are different. This finding indicates that the energy transfer in the internal resonance can be designed by adjusting the parameters. The results obtained will help to further understand the internal resonance behavior in the process of blade rotation and provide the basis for revealing the complex nonlinear behavior.

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Acknowledgements

It is very grateful for the financial supports from the National Natural Science Foundation of China (Nos. 11972129, 11732005) and the National Major Science and Technology Projects of China (No. 2017-IV-0008-0045).

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Authors and Affiliations

  1. School of Astronautics, Harbin Institute of Technology, Harbin, 150001, People’s Republic of China

    Zeyuan Chang, Lei Hou & Yushu Chen

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  1. Zeyuan Chang
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  2. Lei Hou
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  3. Yushu Chen
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Correspondence to Lei Hou.

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Appendix

Appendix

The coefficients in Eq. (18a) are given as follows

$$\begin{aligned} {a_1}&= 1,\,{a_2} = \frac{{{a^2}}}{{{b^2}}}\frac{{{\bar{A}_{66}}}}{{{\bar{A}_{11}}}},\,\\{a_3}&= \frac{{{\bar{A}_{12}} + {\bar{A}_{66}}}}{{{\bar{A}_{11}}}},\,{a_4} = - \frac{{{h^2}}}{{{a^2}}}\frac{{{\bar{E}_{11}}{\bar{c}_1}}}{{{\bar{A}_{11}}}},\,\\ {a_5}&= \frac{{{h^2}}}{{{b^2}}}\left( \frac{{ - {\bar{E}_{12}}{\bar{c}_1} - 2{\bar{E}_{66}}{\bar{c}_1}}}{{{\bar{A}_{11}}}}\right) ,\,{a_6} = \frac{{{h^2}}}{{{a^2}}},\,\\ {a_7}&= \frac{{{h^2}}}{{{b^2}}}\frac{{{\bar{A}_{66}}}}{{{\bar{A}_{11}}}},\\ {a_8}&= \frac{{{h^2}}}{{{b^2}}}\frac{{{\bar{A}_{12}} + {\bar{A}_{66}}}}{{{\bar{A}_{11}}}},\,\\ {a_9}&= \frac{h}{a}(\frac{{{\bar{B}_{11}} - {\bar{E}_{11}}{\bar{c}_1}}}{{{\bar{A}_{11}}}}),\,\\ {a_{10}}&= \frac{{ah}}{{{b^2}}}(\frac{{{\bar{B}_{66}} - {\bar{E}_{66}}{\bar{c}_1}}}{{{\bar{A}_{11}}}}),\,\\ {a_{11}}&= \frac{h}{b}\left( \frac{{{\bar{B}_{12}} + {\bar{B}_{66}} - {\bar{E}_{66}}{\bar{c}_1} - {\bar{E}_{12}}{\bar{c}_1}}}{{{\bar{A}_{11}}}}\right) ,\\ {a_{12}}&= \frac{{{a^2}}}{{{h^2}}}\frac{{{\bar{I}_0}}}{{{\bar{A}_{11}}}},\,\\{a_{13}}&= - \frac{{ab\sqrt{ab} }}{{{h^3}}}\frac{{{\bar{I}_3}{\bar{c}_1}}}{{{\bar{A}_{11}}}},\\{a_{14}}&= \frac{{a\sqrt{ab} }}{{{h^2}}}\left( \frac{{{\bar{I}_1}}}{{{\bar{A}_{11}}}} - \frac{{ab{\bar{I}_3}{\bar{c}_1}}}{{{h^2}{\bar{A}_{11}}}}\right) ,\\{a_{15}}&= \frac{a}{{{h^2}}}\frac{{{\bar{I}_0}}}{{{\bar{A}_{11}}}}({R_0} + a\bar{x})\\ {b_1}&= 1,\,{b_2} = \frac{{{a^2}}}{{{b^2}}}\frac{{{\bar{A}_{11}}}}{{{\bar{A}_{66}}}},\\{b_3}&= \frac{{{a^2}}}{{{b^2}}}\frac{{{\bar{A}_{12}} + {\bar{A}_{66}}}}{{{\bar{A}_{66}}}},\\{b_4}&= - \frac{{{a^2}{h^2}}}{{{b^4}}}\frac{{{\bar{E}_{11}}{\bar{c}_1}}}{{{\bar{A}_{66}}}},\\{b_5}&= \frac{{{h^2}}}{{{b^2}}}\left( \frac{{ - {\bar{E}_{12}}{\bar{c}_1} - 2{\bar{E}_{66}}{\bar{c}_1}}}{{{\bar{A}_{66}}}}\right) ,\\ {b_6}&= \frac{{{h^2}}}{{{b^2}}}\frac{{{\bar{A}_{12}} + {\bar{A}_{66}}}}{{{\bar{A}_{66}}}},\\{b_7}&= \frac{{{a^2}{h^2}}}{{{b^4}}}\frac{{{\bar{A}_{11}}}}{{{\bar{A}_{66}}}},\\{b_8}&= \frac{{{h^2}}}{{{b^2}}},\,{b_9} = \frac{{ah}}{{{b^2}}}\left( \frac{{{\bar{B}_{12}} + {\bar{B}_{66}} - {\bar{E}_{66}}{\bar{c}_1} - {\bar{E}_{12}}{\bar{c}_1}}}{{{\bar{A}_{66}}}}\right) ,\\{b_{10}}&= \frac{h}{b}\left( \frac{{{\bar{B}_{66}} - {\bar{E}_{66}}{\bar{c}_1}}}{{{\bar{A}_{66}}}}\right) ,\\ {b_{11}}&= \frac{{{a^2}h}}{{{b^3}}}\left( \frac{{{\bar{B}_{11}} - {\bar{E}_{11}}{\bar{c}_1}}}{{{\bar{A}_{66}}}}\right) ,\\{b_{12}}&= \frac{{{a^2}}}{{{h^2}}}\frac{{{\bar{I}_0}}}{{{\bar{A}_{66}}}},\\{b_{13}}&= - \frac{a}{{{h^2}}}\frac{{{\bar{I}_0}}}{{{\bar{A}_{66}}}}({R_0} + a\bar{x}),\\{b_{14}}&= \frac{a}{{{h^2}}}\frac{{{\bar{I}_0}}}{{{\bar{A}_{66}}}}({R_0} + \frac{a}{2} - {R_0}\bar{x} - \frac{a}{2}{\bar{x}^2}),\\ {b_{15}}&= - \frac{{{a^3}\sqrt{ab} }}{{b{h^3}}}\frac{{{\bar{I}_3}{\bar{c}_1}}}{{{\bar{A}_{66}}}},\\{b_{16}}&= \frac{{{a^2}\sqrt{ab} }}{{b{h^2}}}\left( \frac{{{\bar{I}_1}}}{{{\bar{A}_{66}}}} - \frac{{ab{\bar{I}_3}{\bar{c}_1}}}{{{h^2}{\bar{A}_{11}}}}\right) ,\,{b_{17}} = \frac{{{a^2}}}{{{h^2}}}\frac{{{\bar{I}_0}}}{{{\bar{A}_{66}}}}\\ {\lambda _1}&= {\bar{H}_{11}}{\bar{c}_1}^2 - 2{\bar{F}_{11}}{\bar{c}_1} + {\bar{D}_{11}}\\ {d_1}&= - 1,\,{d_2} = \frac{{{a^2}}}{{{\lambda _1}{b^2}}}( - {\bar{H}_{66}}{\bar{c}_1}^2 + 2{\bar{F}_{66}}{\bar{c}_1} - {\bar{D}_{66}}),\\{d_3}&= \frac{{{a^2}}}{{{\lambda _1}{h^2}}}(9{\bar{F}_{66}}{\bar{c}_1}^2 - 6{\bar{D}_{66}}{\bar{c}_1} + {\bar{A}_{66}}),\\ {d_4}&= \frac{a}{{{\lambda _1}b}}( - {\bar{D}_{12}} - {\bar{D}_{66}}\\&\quad - {\bar{H}_{12}}{\bar{c}_1}^2 - {\bar{H}_{66}}{\bar{c}_1}^2 + 2{\bar{F}_{12}}{\bar{c}_1} + 2{\bar{F}_{66}}{\bar{c}_1}),\\{d_5}&= \frac{a}{{{\lambda _1}h}}({\bar{E}_{11}}{\bar{c}_1} - {\bar{B}_{11}}),\\ {d_6}&= \frac{{{a^3}}}{{{\lambda _1}{b^2}h}}({\bar{E}_{66}}{\bar{c}_1} - {\bar{B}_{66}}),\,\\ {d_7}&= \frac{a}{{{\lambda _1}h}}({\bar{E}_{12}}{\bar{c}_1} + {\bar{E}_{66}}{\bar{c}_1} - {\bar{B}_{12}} - {\bar{B}_{66}}),\\{d_8}&= \frac{h}{{{\lambda _1}a}}( - {\bar{H}_{11}}{\bar{c}_1}^2 + {\bar{F}_{11}}{\bar{c}_1}),\\ {d_9}&= \frac{{ah}}{{{\lambda _1}{b^2}}}\left( - {\bar{H}_{12}}{\bar{c}_1}^2 - 2{\bar{H}_{66}}{\bar{c}_1}^2 + {\bar{F}_{12}}{\bar{c}_1} + 2{\bar{F}_{66}}{\bar{c}_1}\right) ,\\{d_{10}}&= \frac{h}{{{\lambda _1}a}}({\bar{E}_{11}}{\bar{c}_1} - {\bar{B}_{11}}),\,{d_{11}} = \frac{{ah}}{{{\lambda _1}{b^2}}}({\bar{E}_{66}}{\bar{c}_1} - {\bar{B}_{66}}),\\ {d_{12}}&= \frac{{ah}}{{{\lambda _1}{b^2}}}({\bar{E}_{12}}{\bar{c}_1} + {\bar{E}_{66}}{\bar{c}_1} - {\bar{B}_{12}} - {\bar{B}_{66}}),\,{d_{13}}\\&= \frac{a}{{{\lambda _1}h}}(9{\bar{F}_{66}}{\bar{c}_1}^2 - 6{\bar{D}_{66}}{\bar{c}_1} + {\bar{A}_{66}}),\\ {d_{14}}&= \frac{{{a^3}b}}{{{\lambda _1}{h^4}}}\left( - {\bar{I}_2} + \frac{{2ab{\bar{I}_4}{\bar{c}_1}}}{{{h^2}}} - \frac{{{a^2}{b^2}{\bar{I}_6}{\bar{c}_1}^2}}{{{h^4}}}\right) ,\\{d_{15}}&= \frac{{{a^3}\sqrt{ab} }}{{{\lambda _1}{h^4}}}(\frac{{ab{\bar{I}_3}{\bar{c}_1}}}{{{h^2}}} - {\bar{I}_1}),\\{d_{16}}&= \frac{{{a^3}{b^2}}}{{{\lambda _1}{h^5}}}({\bar{I}_4}{\bar{c}_1} - \frac{{ab{\bar{I}_6}{\bar{c}_1}^2}}{{{h^2}}}),\\ {d_{17}}&= \frac{{{a^2}\sqrt{ab} }}{{{\lambda _1}{h^4}}}\Bigg ( - {R_0}{\bar{I}_1} + \frac{{ab{R_0}{\bar{I}_3}{\bar{c}_1}}}{{{h^2}}} - a{\bar{I}_1}\bar{x} \\&\quad + \frac{{{a^2}b{\bar{I}_3}{\bar{c}_1}}}{{{h^2}}}\bar{x}\Bigg )\\ {e_1}&= - 1,\,{e_2} = \frac{{{b^2}}}{{{\lambda _1}{a^2}}}( - {\bar{H}_{66}}{\bar{c}_1}^2 + 2{\bar{F}_{66}}{\bar{c}_1} - {\bar{D}_{66}}),\,{e_3}\\&= \frac{{{b^2}}}{{{\lambda _1}{h^2}}}(9{\bar{F}_{66}}{\bar{c}_1}^2 - 6{\bar{D}_{66}}{\bar{c}_1} + {\bar{A}_{66}}),\,\\ {e_4}&= \frac{b}{{{\lambda _1}a}}( - {\bar{D}_{12}} - {\bar{D}_{66}} - {\bar{H}_{12}}{\bar{c}_1}^2 - {\bar{H}_{66}}{\bar{c}_1}^2 + 2{\bar{F}_{12}}{\bar{c}_1} \nonumber \\ {}&\quad + 2{\bar{F}_{66}}{\bar{c}_1}),\,{e_5}\\&= \frac{b}{{{\lambda _1}h}}({\bar{E}_{12}}{\bar{c}_1} + {\bar{E}_{66}}{\bar{c}_1} - {\bar{B}_{12}} - {\bar{B}_{66}}),\,\\ {e_6}&= \frac{{{b^3}}}{{{\lambda _1}{a^2}h}}({\bar{E}_{66}}{\bar{c}_1} - {\bar{B}_{66}}),\,{e_7} = \frac{b}{{{\lambda _1}h}}({\bar{E}_{11}}{\bar{c}_1} - {\bar{B}_{11}}),\,\\{e_8}&= \frac{{bh}}{{{\lambda _1}{a^2}}}( - {\bar{H}_{12}}{\bar{c}_1}^2 - 2{\bar{H}_{66}}{\bar{c}_1}^2 + {\bar{F}_{12}}{\bar{c}_1} + 2{\bar{F}_{66}}{\bar{c}_1}),\,\\ {e_9}&= \frac{h}{{{\lambda _1}b}}( - {\bar{H}_{11}}{\bar{c}_1}^2 + {\bar{F}_{11}}{\bar{c}_1}),\,{e_{10}}\\&= \frac{{bh}}{{{\lambda _1}{a^2}}}({\bar{E}_{12}}{\bar{c}_1} + {\bar{E}_{66}}{\bar{c}_1} - {\bar{B}_{12}} - {\bar{B}_{66}}),\,{e_{11}}\\&= \frac{h}{{{\lambda _1}b}}({\bar{E}_{11}}{\bar{c}_1} - {\bar{B}_{11}}),\\ {e_{12}}&= \frac{{bh}}{{{\lambda _1}{a^2}}}({\bar{E}_{66}}{\bar{c}_1} - {\bar{B}_{66}}),\nonumber \\ {}&\quad \,{e_{13}} = \frac{b}{{{\lambda _1}h}}(9{\bar{F}_{66}}{\bar{c}_1}^2 - 6{\bar{D}_{66}}{\bar{c}_1} + {\bar{A}_{66}}),\,\\{e_{14}}&= \frac{{a{b^3}}}{{{\lambda _1}{h^4}}}\left( - {\bar{I}_2} + \frac{{2ab{\bar{I}_4}{\bar{c}_1}}}{{{h^2}}} - \frac{{{a^2}{b^2}{\bar{I}_6}{\bar{c}_1}^2}}{{{h^4}}}\right) ,\,\\ {e_{15}}&= \frac{{{b^3}\sqrt{ab} }}{{{\lambda _1}{h^4}}}\left( \frac{{ab{\bar{I}_3}{\bar{c}_1}}}{{{h^2}}} - {\bar{I}_1}\right) ,\,\nonumber \\ {}&\quad {e_{16}} = \frac{{{a^2}{b^3}}}{{{\lambda _1}{h^5}}}\left( {\bar{I}_4}{\bar{c}_1} - \frac{{ab{\bar{I}_6}{\bar{c}_1}^2}}{{{h^2}}}\right) ,\\{e_{17}}&= \frac{{{b^3}\sqrt{ab} }}{{{\lambda _1}{h^4}}}\left( - {\bar{I}_1}\bar{y} + \frac{{ab{\bar{I}_3}{\bar{c}_1}}}{{{h^2}}}\bar{y}\right) \end{aligned}$$

The transverse dimensionless equation is expressed as

$$\begin{aligned}&{c_1}{\bar{\varphi ''}_{y,y}} + {c_2}{\bar{\varphi }_{y,y}} + {c_3}{\bar{\varphi }_{x,x}} + {c_4}{\bar{\varphi ''}_{x,x}} + {c_5}{\bar{\varphi }_{x,xxx}} \nonumber \\&\quad + ({c_6}{\bar{\varphi }_{x,y}}\nonumber \\\&\quad + {c_7}{\bar{\varphi }_{y,x}} - {c_8}{\bar{u}_{0,y}} - {c_9}{\bar{v}_{0,x}}\nonumber \\&\quad + {c_{10}}{\bar{w}_{0,x}}{\bar{w}_{0,y}}){\bar{w}_{0,xy}}\nonumber \\&\quad + {c_{11}}{\bar{u}_{0,xyy}} + ({c_{12}}{\bar{\varphi }_{y,y}}\nonumber \\&\quad + {c_{13}}{\bar{w}_{0,x}} + {c_{14}}{\bar{w}_{0,xx}} + {c_{15}}{\bar{w}_{0,yy}} + {c_{16}}{\bar{\varphi }_{x,x}} - {c_{17}}\nonumber \\&\quad - {c_{18}}{\bar{v}_{0,y}} - {c_{19}}{\bar{u}_{0,x}}){\bar{\Omega }^2}\nonumber \\&\quad + ({c_{20}}{\bar{w}_{0,yy}} + {c_{21}}{\bar{\varphi }_{x,x}}\nonumber \\&\quad + {c_{22}}{\bar{\varphi }_{y,y}} - {c_{23}}{\bar{u}_{0,x}} - {c_{24}}{\bar{v}_{0,y}}\nonumber \\&\quad - {c_{25}}\bar{w}_{0,x}^2 + {c_{26}}\bar{w}_{0,y}^2 + {c_{27}}){\bar{w}_{0,xx}}\nonumber \\&\quad + ({c_{28}}{\bar{\varphi }_{x,x}} + {c_{29}}{\bar{\varphi }_{y,y}}\nonumber \\&\quad - {c_{30}}{\bar{u}_{0,x}} - {c_{31}}{\bar{v}_{0,y}} + {c_{32}}\bar{w}_{0,x}^2\nonumber \\&\quad + {c_{33}} - {c_{34}}\bar{w}_{0,y}^2){\bar{w}_{0,yy}}\nonumber \\&\quad + ({c_{35}}{\bar{v'}_{0,x}} + {c_{36}}{\bar{u'}_{0,y}}\nonumber \\&\quad + {c_{37}}{\bar{\varphi '}_{x,y}} + {c_{38}}{\bar{\varphi '}_{y,x}})\bar{\Omega }+ {c_{39}}\bar{w}_{0,xy}^2 - {c_{40}}{\bar{w}_{0,y}}{\bar{v}_{0,xx}}\nonumber \\&\quad - {c_{41}}{\bar{w''}_{0,yy}} - {c_{42}}{\bar{w''}_{0,xx}}\nonumber \\&\quad + ({c_{43}}{\bar{\varphi }_{x,y}} + {c_{44}}{\bar{\varphi }_{y,x}}\nonumber \\&\quad + {c_{45}}{\bar{u}_{0,y}} - {c_{46}}{\bar{v}_{0,x}})\bar{\Omega }' + {c_{47}}{\bar{v}_{0,xxy}} + {c_{48}}{\bar{w}_{0,xxyy}}\nonumber \\&\quad + {c_{49}}{\bar{\varphi }_{x,xyy}} + {c_{50}}{\bar{\varphi }_{y,xxy}}\nonumber \\&\quad + {c_{51}}{\bar{\varphi }_{y,yyy}} + {c_{52}}{\bar{w''}_0}\nonumber \\&\quad + {c_{53}}{\bar{w}_{0,y}}{\bar{u}_{0,xy}} + {c_{54}}{\bar{w}_{0,x}}{\bar{v}_{0,xy}}\nonumber \\&\quad + {c_{55}}{\bar{w}_{0,y}}{\bar{\varphi }_{y,yy}}\nonumber \\&\quad + {c_{56}}{\bar{w}_{0,x}}{\bar{\varphi }_{y,xy}}\nonumber \\&\quad + {c_{57}}{\bar{w}_{0,x}}{\bar{\varphi }_{x,yy}}\nonumber \\&\quad + {c_{58}}{\bar{w}_{0,y}}{\bar{\varphi }_{y,xx}}\nonumber \\&\quad + {c_{59}}{\bar{w}_{0,y}}{\bar{\varphi }_{x,xy}}\nonumber \\&\quad + {c_{60}}{\bar{w}_{0,x}}{\bar{\varphi }_{x,xx}} + {c_{61}}{\bar{w}_{0,xxxx}}\nonumber \\&\quad + {c_{62}}{\bar{w}_{0,yyyy}} - {c_{63}}{\bar{v}_{0,yyy}}\nonumber \\&\quad - {c_{64}}{\bar{u}_{0,xxx}}\nonumber \\&\quad - {c_{65}}{\bar{w}_{0,x}}{\bar{u}_{0,yy}} - {c_{66}}{\bar{w}_{0,y}}{\bar{v}_{0,yy}}\nonumber \\&\quad - {c_{67}}{\bar{w}_{0,x}}{\bar{u}_{0,xx}} + {c_{68}}{\bar{u''}_{0,x}}\nonumber \\&\quad + {c_{69}}{\bar{v''}_{0,y}} - {c_{70}}({c_{701}}{q_1}{\bar{w}_{0,y}} + {c_{702}}{q_2}{\bar{w'}_0}) = 0 \end{aligned}$$
(A-1)

where coefficients are given as follows

$$\begin{aligned} {\lambda _2}&= 9{\bar{F}_{66}}{\bar{c}_1}^2\\&- 6{\bar{D}_{66}}{\bar{c}_1} + {\bar{A}_{66}}\\ {c_1}&= \frac{{{a^4}b}}{{{\lambda _2}{h^5}}}\left( {\bar{I}_4}{\bar{c}_1} - \frac{{ab{\bar{I}_6}{\bar{c}_1}^2}}{{{h^2}}}\right) ,\\ {c_2}&= - \frac{{{a^2}}}{{bh}},{c_3} = - \frac{a}{h},\\ {c_4}&= \frac{{{a^3}{b^2}}}{{{\lambda _2}{h^5}}}\left( {\bar{I}_4}{\bar{c}_1} - \frac{{ab{\bar{I}_6}{\bar{c}_1}^2}}{{{h^2}}}\right) ,\\ {c_5}&= \frac{h}{{{\lambda _2}a}}\left( {\bar{H}_{11}}{\bar{c}_1}^2 - {\bar{F}_{11}}{\bar{c}_1}\right) ,\\ {c_6}&= \frac{{2ah}}{{{\lambda _2}{b^2}}}\left( {\bar{E}_{66}}{\bar{c}_1} - {\bar{B}_{66}}\right) ,\\ {c_7}&= \frac{{2h}}{{{\lambda _2}b}}({\bar{E}_{66}}{\bar{c}_1} - {\bar{B}_{66}}),\\ {c_8}&= \frac{{2{a^2}}}{{{\lambda _2}{b^2}}}{\bar{A}_{66}},\\ {c_9}&= \frac{2}{{{\lambda _2}}}{\bar{A}_{66}},{c_{10}} = \frac{{2{h^2}}}{{{\lambda _2}{b^2}}}( - {\bar{A}_{12}} - 2{\bar{A}_{66}})\\ {c_{11}}&= \frac{{{a^2}}}{{{\lambda _2}{b^2}}}( - 2{\bar{E}_{66}}{\bar{c}_1} - {\bar{E}_{12}}{\bar{c}_1}),\\ {c_{12}}&= \frac{{{a^4}b}}{{{\lambda _2}{h^5}}}( - {\bar{I}_4}{\bar{c}_1}\\&\quad + \frac{{ab{\bar{I}_6}{\bar{c}_1}^2}}{{{h^2}}}),{c_{13}} = \frac{a}{{{\lambda _2}{h^2}}}({\bar{I}_0}{R_0} + a{\bar{I}_0}\bar{x}),\\ {c_{14}}&= \frac{a}{{{\lambda _2}{h^2}}}\Bigg (\frac{{a{\bar{I}_0}{\bar{x}^2} - a{\bar{I}_0} - 2{\bar{I}_0}{R_0} + 2{\bar{I}_0}{R_0}\bar{x}}}{2}\\&\quad + \frac{{{a^2}{b^3}{\bar{I}_6}{\bar{c}_1}^2}}{{{h^4}}}\Bigg ),\\ {c_{15}}&= \frac{{{a^5}b}}{{{\lambda _2}{h^6}}}{\bar{I}_6}{\bar{c}_1}^2,{c_{16}} = \frac{{{a^3}{b^2}}}{{{\lambda _2}{h^5}}}\left( - {\bar{I}_4}{\bar{c}_1} + \frac{{ab{\bar{I}_6}{\bar{c}_1}^2}}{{{h^2}}}\right) ,\nonumber \\ {}&\quad {c_{17}} = \frac{{2{a^3}b\sqrt{ab} }}{{{\lambda _2}{h^5}}}{\bar{I}_3}{\bar{c}_1},\\ {c_{18}}&= \frac{{{a^3}b\sqrt{ab} }}{{{\lambda _2}{h^5}}}{\bar{I}_3}{\bar{c}_1},\\ {c_{19}}&= \frac{{{a^3}b\sqrt{ab} }}{{{\lambda _2}{h^5}}}{\bar{I}_3}{\bar{c}_1},\\ {c_{20}}&= \frac{{2{h^2}}}{{{\lambda _2}{b^2}}}( - {\bar{E}_{66}}{\bar{c}_1} + {\bar{E}_{12}}{\bar{c}_1}),\nonumber \\&\quad {c_{21}} = \frac{h}{{{\lambda _2}a}}({\bar{E}_{11}}{\bar{c}_1} - {\bar{B}_{11}}),\\ {c_{22}}&= \frac{h}{{{\lambda _2}b}}({\bar{E}_{12}}{\bar{c}_1} - {\bar{B}_{12}}),\\ {c_{23}}&= \frac{1}{{{\lambda _2}}}{\bar{A}_{11}},{c_{24}} = \frac{1}{{{\lambda _2}}}{\bar{A}_{12}}, \end{aligned}$$
$$\begin{aligned} {c_{25}}&= \frac{{3{h^2}}}{{2{\lambda _2}{a^2}}}{\bar{A}_{11}},\\ {c_{26}}&= \frac{1}{{{\lambda _2}}}\left( - {\bar{A}_{66}} - \frac{{{\bar{A}_{12}}}}{2}\right) ,\\ {c_{27}}&= - 1,{c_{28}} = \frac{{ah}}{{{\lambda _2}{b^2}}}({\bar{E}_{12}}{\bar{c}_1} - {\bar{B}_{12}}),\\ {c_{29}}&= \frac{{{a^2}h}}{{{\lambda _2}{b^3}}}({\bar{E}_{11}}{\bar{c}_1} - {\bar{B}_{11}}),\\ {c_{30}}&= \frac{{{a^2}}}{{{\lambda _2}{b^2}}}{\bar{A}_{12}},\\ {c_{31}}&= \frac{{{a^2}}}{{{\lambda _2}{b^2}}}{\bar{A}_{11}},\\ {c_{32}}&= \frac{{{h^2}}}{{{\lambda _2}{b^2}}}( - {\bar{A}_{66}} - \frac{{{\bar{A}_{12}}}}{2}),\\ {c_{33}}&= - \frac{{{a^2}}}{{{b^2}}},{c_{34}} = \frac{{3{a^2}{h^2}}}{{2{\lambda _2}{b^4}}}{\bar{A}_{11}},\\ {c_{35}}&= - \frac{{2{a^2}{b^2}\sqrt{ab} }}{{{\lambda _2}{h^5}}}{\bar{I}_3}{\bar{c}_1},\\ {c_{36}}&= \frac{{2{a^4}\sqrt{ab} }}{{{\lambda _2}{h^5}}}{\bar{I}_3}{\bar{c}_1},\\ {c_{37}}&= \frac{{2{a^4}b}}{{{\lambda _2}{h^5}}}\left( {\bar{I}_4}{\bar{c}_1} - \frac{{ab{\bar{I}_6}{\bar{c}_1}^2}}{{{h^2}}}\right) ,\\ {c_{38}}&= \frac{{2{a^3}{b^2}}}{{{\lambda _2}{h^5}}}\left( {\bar{I}_4}{\bar{c}_1} - \frac{{ab{\bar{I}_6}{\bar{c}_1}^2}}{{{h^2}}}\right) ,\\ {c_{39}}&= \frac{{2{h^2}}}{{{\lambda _2}{b^2}}}({\bar{E}_{66}}{\bar{c}_1} - {\bar{E}_{12}}{\bar{c}_1}),\\ {c_{40}}&= \frac{1}{{{\lambda _2}}}{\bar{A}_{66}},{c_{41}} = \frac{{{a^5}b}}{{{\lambda _2}{h^6}}}{\bar{I}_6}{\bar{c}_1}^2,\\ {c_{42}}&= \frac{{{a^3}{b^3}}}{{{\lambda _2}{h^6}}}{\bar{I}_6}{\bar{c}_1}^2,\\ {c_{43}}&= \frac{{{a^4}b}}{{{\lambda _2}{h^5}}}\left( {\bar{I}_4}{\bar{c}_1} - \frac{{ab{\bar{I}_6}{\bar{c}_1}^2}}{{{h^2}}}\right) ,\\ {c_{44}}&= \frac{{{a^3}{b^2}}}{{{\lambda _2}{h^5}}}\left( - {\bar{I}_4}{\bar{c}_1} + \frac{{ab{\bar{I}_6}{\bar{c}_1}^2}}{{{h^2}}}\right) ,\\ {c_{45}}&= \frac{{{a^4}\sqrt{ab} }}{{{\lambda _2}{h^5}}}{\bar{I}_3}{\bar{c}_1},\\ {c_{46}}&= \frac{{{a^2}{b^2}\sqrt{ab} }}{{{\lambda _2}{h^5}}}{\bar{I}_3}{\bar{c}_1},\\ {c_{47}}&= \frac{1}{{{\lambda _2}}}( - 2{\bar{E}_{66}}{\bar{c}_1} - {\bar{E}_{12}}{\bar{c}_1}),\\ {c_{48}}&= \frac{{2{h^2}}}{{{\lambda _2}{b^2}}}({\bar{H}_{12}}{\bar{c}_1}^2 + 2{\bar{H}_{66}}{\bar{c}_1}^2),\\ {c_{49}}&= \frac{{ah}}{{{\lambda _2}{b^2}}}({\bar{H}_{12}}{\bar{c}_1}^2 - 2{\bar{F}_{66}}{\bar{c}_1} - {\bar{F}_{12}}{\bar{c}_1} + 2{\bar{H}_{66}}{\bar{c}_1}^2),\\ {c_{50}}&= \frac{h}{{{\lambda _2}b}}({\bar{H}_{12}}{\bar{c}_1}^2 - 2{\bar{F}_{66}}{\bar{c}_1} - {\bar{F}_{12}}{\bar{c}_1} + 2{\bar{H}_{66}}{\bar{c}_1}^2),\\ {c_{51}}&= \frac{{{a^2}h}}{{{\lambda _2}{b^3}}}({\bar{H}_{11}}{\bar{c}_1}^2 - {\bar{F}_{11}}{\bar{c}_1}),\\ {c_{52}}&= \frac{{{a^2}}}{{{\lambda _2}{h^2}}}{\bar{I}_0},\\ {c_{53}}&= \frac{{{a^2}}}{{{\lambda _2}{b^2}}}( - {\bar{A}_{12}} - {\bar{A}_{66}}),\\ {c_{54}}&= \frac{1}{{{\lambda _2}}}( - {\bar{A}_{12}} - {\bar{A}_{66}}),\\ {c_{55}}&= \frac{{{a^2}h}}{{{\lambda _2}{b^3}}}({\bar{E}_{11}}{\bar{c}_1} - {\bar{B}_{11}}),\\ {c_{56}}&= \frac{h}{{{\lambda _2}b}}({\bar{E}_{12}}{\bar{c}_1} + {\bar{E}_{66}}{\bar{c}_1} - {\bar{B}_{12}} - {\bar{B}_{66}}),\\ {c_{57}}&= \frac{{ah}}{{{\lambda _2}{b^2}}}({\bar{E}_{66}}{\bar{c}_1} - {\bar{B}_{66}}),\\ {c_{58}}&= \frac{h}{{{\lambda _2}b}}({\bar{E}_{66}}{\bar{c}_1} - {\bar{B}_{66}}),\\ {c_{59}}&= \frac{{ah}}{{{\lambda _2}{b^2}}}({\bar{E}_{12}}{\bar{c}_1} + {\bar{E}_{66}}{\bar{c}_1} - {\bar{B}_{12}} - {\bar{B}_{66}}),\\ {c_{60}}&= \frac{h}{{{\lambda _2}a}}({\bar{E}_{11}}{\bar{c}_1} - {\bar{B}_{11}}),\\ {c_{61}}&= \frac{{{h^2}}}{{{\lambda _2}{a^2}}}{\bar{H}_{11}}{\bar{c}_1}^2,\\ {c_{62}}&= \frac{{{a^2}{h^2}}}{{{\lambda _2}{b^4}}}{\bar{H}_{11}}{\bar{c}_1}^2,\\ {c_{63}}&= \frac{{{a^2}}}{{{\lambda _2}{b^2}}}{\bar{E}_{11}}{\bar{c}_1},\\ {c_{64}}&= \frac{1}{{{\lambda _2}}}{\bar{E}_{11}}{\bar{c}_1},{c_{65}} = \frac{{{a^2}}}{{{\lambda _2}{b^2}}}{\bar{A}_{66}},\\ {c_{66}}&= \frac{{{a^2}}}{{{\lambda _2}{b^2}}}{\bar{A}_{11}},{c_{67}} = \frac{1}{{{\lambda _2}}}{\bar{A}_{11}},\\ {c_{68}}&= \frac{{{a^3}b\sqrt{ab} }}{{{\lambda _2}{h^5}}}{\bar{I}_3}{\bar{c}_1},{c_{69}} = \frac{{{a^3}b\sqrt{ab} }}{{{\lambda _2}{h^5}}}{\bar{I}_3}{\bar{c}_1},\\ {c_{70}}&= \frac{{{a^2}\sqrt{ab} }}{{{E_m}{\lambda _2}{h^3}}},{c_{701}} = \frac{h}{b}{E_m},{c_{702}} = \frac{h}{{\sqrt{ab} }}{E_m} \end{aligned}$$

The relationship between \(\bar{u_1}\), \(\bar{u_2}\), \(\bar{v_1}\), \(\bar{v_2}\), \(\bar{\varphi _{11}}\), \(\bar{\varphi _{12}}\), \(\bar{\varphi _{21}}\), \(\bar{\varphi _{22}}\) and \(\bar{w_1}\), \(\bar{w_2}\) takes the following form

$$\begin{aligned} \pmb {P}_1 = \pmb {K}\pmb {P}_2 \end{aligned}$$
(A-2)

where \(\pmb {P}_1 = \begin{bmatrix}{\bar{u}_1}&{\bar{u}_2}&{\bar{v}_1}&{\bar{v}_2}&{\bar{\varphi }_{11}}&{\bar{\varphi }_{12}}&{\bar{\varphi }_{21}}&{\bar{\varphi }_{22}}\end{bmatrix}^T\), \(\pmb {P}_2 = \begin{bmatrix}{\bar{w}_1}&{\bar{w}_2}&{\bar{w}_1}{\bar{w}_2}&\bar{w_1^2}&\bar{w_2^2}&1\end{bmatrix}^T\) and \(\pmb {K} = \begin{bmatrix} {{k_{11}}}&{}{{k_{12}}}&{} \cdots &{}{{k_{16}}}\\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ {{k_{41}}}&{}{{k_{42}}}&{} \cdots &{}{{k_{46}}}\\ {{k_{71}}}&{}{{k_{72}}}&{} \cdots &{}{{k_{76}}}\\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ {{k_{101}}}&{}{{k_{102}}}&{} \cdots &{}{{k_{106}}} \end{bmatrix}_{8 \times 6}\). Substituting Eq. (A-2) into Eq. (A-1), Eq. (A-1) can be written as

$$\begin{aligned} \int _0^1 {\int _0^1 {f({\bar{w}_1}^{\prime \prime },\,{\bar{w}_2}^{\prime \prime },\,{\bar{w}_1}^\prime ,\,{\bar{w}_2}^{\prime },\,{\bar{w}_1},\,{\bar{w}_2}{\mathrm{, }}\bar{x},\,\bar{y})} C{C_i}{\mathrm{d}}} \bar{y}{\mathrm{d}}\bar{x} = 0 \end{aligned}$$
(A-3)

where \(C{C_i} = {\varphi _i}(\bar{x}){\psi _i}(\bar{y})\), \((i = 1,2)\). The coefficient of each term in Eq. (A-3) can be expressed as

$$\begin{aligned} \int _0^1 {\int _0^1 {{\bar{u}_{0,xxx}}} C{C_i}{\mathrm{d}}} \bar{y}{\mathrm{d}}\bar{x} = n_{11}^i\bar{u_1}+n_{12}^i\bar{u_2} \end{aligned}$$
(A-4)

Replace the integrand on the left side of Eq. (A-4), we can obtain

$$\begin{aligned}&{\bar{u}_{0,xxx}} \sim n_{11}^i{\bar{u}_1}+n_{12}^i{\bar{u}_2},\,{\bar{u}_{0,xyy}}\\&\quad \sim n_{21}^i{\bar{u}_1}+n_{22}^i{\bar{u}_2},\,{\bar{u}_{0,x}}\\&\quad \sim n_{31}^i{\bar{u}_1}+n_{32}^i{\bar{u}_2},\,{\bar{u}_{0,y}}\\&\quad \sim n_{41}^i{\bar{u}_1}+n_{42}^i{\bar{u}_2},\,\\&\quad {\bar{v}_{0,yyy}} \sim n_{51}^i{\bar{v}_1}+n_{52}^i{\bar{v}_2},\,{\bar{v}_{0,xxy}}\\&\quad \sim n_{61}^i{\bar{v}_1}+n_{62}^i{\bar{v}_2},\,{\bar{v}_{0,x}}\\&\quad \sim n_{71}^i{\bar{v}_1}+n_{72}^i{\bar{v}_2},\,{\bar{v}_{0,y}}\\&\quad \sim n_{81}^i{\bar{v}_1}+n_{82}^i{\bar{v}_2},\,\\&\quad {\bar{\varphi }_{x,xxx}} \sim n_{91}^i{\bar{\varphi }_{11}}+n_{92}^i{\bar{\varphi }_{12}},\,{\bar{\varphi }_{x,xyy}}\\&\quad \sim n_{101}^i{\bar{\varphi }_{11}}+n_{102}^i{\bar{\varphi }_{12}},\,{\bar{\varphi }_{x,x}}\\&\quad \sim n_{111}^i{\bar{\varphi }_{11}}+n_{112}^i{\bar{\varphi }_{12}},\,{\bar{\varphi }_{x,y}}\\&\quad \sim n_{121}^i{\bar{\varphi }_{11}}+n_{122}^i{\bar{\varphi }_{12}},\\&\quad {\bar{\varphi }_{y,yyy}} \sim n_{131}^i{\bar{\varphi }_{21}}+n_{132}^i{\bar{\varphi }_{22}},\,{\bar{\varphi }_{y,xxy}} \sim n_{141}^i{\bar{\varphi }_{21}}\\ {}&\quad +n_{142}^i{\bar{\varphi }_{22}},\,{\bar{\varphi }_{y,x}}\\&\quad \sim n_{151}^i{\bar{\varphi }_{21}}+n_{152}^i{\bar{\varphi }_{22}},\,{\bar{\varphi }_{y,y}} \sim n_{161}^i{\bar{\varphi }_{21}}+n_{162}^i{\bar{\varphi }_{22}},\\&\quad 1 \sim n_{171}^i,\,{\bar{w}_0} \sim n_{181}^i{\bar{w}_1}+n_{182}^i{\bar{w}_2},\,{\bar{w}_{0,xxxx}}\\&\quad \sim n_{191}^i{\bar{w}_1}+n_{192}^i{\bar{w}_2},\,{\bar{x}^2}{\bar{w}_{0,xx}}\\&\quad \sim n_{201}^i{\bar{w}_1}+n_{202}^i{\bar{w}_2},\,\bar{x}{\bar{w}_{0,xx}} \sim n_{211}^i{\bar{w}_1}+n_{212}^i{\bar{w}_2},\,\\&\quad {\bar{w}_{0,xx}} \sim n_{221}^i{\bar{w}_1}+n_{222}^i{\bar{w}_2},\,\bar{x}{\bar{w}_{0,x}}\\&\quad \sim n_{231}^i{\bar{w}_1}+n_{232}^i{\bar{w}_2},\,{\bar{w}_{0,x}} \sim n_{241}^i{\bar{w}_1}+n_{242}^i{\bar{w}_2},\,\\&\quad {\bar{u}_{0,y}}{\bar{w}_{0,xy}} \sim n_{251}^i{\bar{u}_1}{\bar{w}_1}+n_{252}^i{\bar{u}_1}{\bar{w}_2}\\ {}&\quad + n_{253}^i{\bar{u}_2}{\bar{w}_1}+n_{254}^i{\bar{u}_2}{\bar{w}_2},\,{\bar{v}_{0,x}}{\bar{w}_{0,xy}}\\&\quad \sim n_{261}^i{\bar{v}_1}{\bar{w}_1}+n_{262}^i{\bar{v}_1}{\bar{w}_2} + n_{263}^i{\bar{v}_2}{\bar{w}_1}+n_{264}^i{\bar{v}_2}{\bar{w}_2},\,\\&\quad {\bar{\varphi }_{x,y}}{\bar{w}_{0,xy}} \sim n_{271}^i{\bar{\varphi }_{11}}{\bar{w}_1}+n_{272}^i{\bar{\varphi }_{11}}{\bar{w}_2}\\ {}&\quad + n_{273}^i{\bar{\varphi }_{12}}{\bar{w}_1}+n_{274}^i{\bar{\varphi }_{12}}{\bar{w}_2},\\&\quad \,{\bar{\varphi }_{y,x}}{\bar{w}_{0,xy}} \sim n_{281}^i{\bar{\varphi }_{21}}{\bar{w}_1}+n_{282}^i{\bar{\varphi }_{21}}{\bar{w}_2} \\ {}&\quad + n_{283}^i{\bar{\varphi }_{22}}{\bar{w}_1}+n_{284}^i{\bar{\varphi }_{22}}{\bar{w}_2},\,\\&\quad {\bar{u}_{0,x}}{\bar{w}_{0,xx}} \sim n_{291}^i{\bar{u}_1}{\bar{w}_1}+n_{292}^i{\bar{u}_1}{\bar{w}_2} + n_{293}^i{\bar{u}_2}{\bar{w}_1}\\ {}&\quad +n_{294}^i{\bar{u}_2}{\bar{w}_2},\,{\bar{v}_{0,y}}{\bar{w}_{0,xx}}\\&\quad \sim n_{301}^i{\bar{v}_1}{\bar{w}_1}+n_{302}^i{\bar{v}_1}{\bar{w}_2} + n_{303}^i{\bar{v}_2}{\bar{w}_1}+n_{304}^i{\bar{v}_2}{\bar{w}_2},\,\\&\quad {\bar{\varphi }_{x,x}}{\bar{w}_{0,xx}} \sim n_{311}^i{\bar{\varphi }_{11}}{\bar{w}_1}+n_{312}^i{\bar{\varphi }_{11}}{\bar{w}_2}\\ {}&\quad + n_{313}^i{\bar{\varphi }_{12}}{\bar{w}_1}+n_{314}^i{\bar{\varphi }_{12}}{\bar{w}_2},\\&\quad \,{\bar{\varphi }_{y,y}}{\bar{w}_{0,xx}} \sim n_{321}^i{\bar{\varphi }_{21}}{\bar{w}_1}+n_{322}^i{\bar{\varphi }_{21}}{\bar{w}_2} \\ {}&\quad + n_{323}^i{\bar{\varphi }_{22}}{\bar{w}_1}+n_{324}^i{\bar{\varphi }_{22}}{\bar{w}_2},\,\\&\quad {\bar{u}_{0,xx}}{\bar{w}_{0,x}} \sim n_{331}^i{\bar{u}_1}{\bar{w}_1}+n_{332}^i{\bar{u}_1}{\bar{w}_2} \\ {}&\quad + n_{333}^i{\bar{u}_2}{\bar{w}_1}+n_{334}^i{\bar{u}_2}{\bar{w}_2},\,{\bar{u}_{0,yy}}{\bar{w}_{0,x}}\\&\quad \sim n_{341}^i{\bar{u}_1}{\bar{w}_1}+n_{342}^i{\bar{u}_1}{\bar{w}_2} + n_{343}^i{\bar{u}_2}{\bar{w}_1}+n_{344}^i{\bar{u}_2}{\bar{w}_2},\,\\&\quad {\bar{v}_{0,xy}}{\bar{w}_{0,x}} \sim n_{351}^i{\bar{v}_1}{\bar{w}_1}+n_{352}^i{\bar{v}_1}{\bar{w}_2}\\ {}&\quad + n_{353}^i{\bar{v}_2}{\bar{w}_1}+n_{354}^i{\bar{v}_2}{\bar{w}_2},\,{\bar{\varphi }_{x,xx}}{\bar{w}_{0,x}}\\&\quad \sim n_{361}^i{\bar{\varphi }_{11}}{\bar{w}_1}+n_{362}^i{\bar{\varphi }_{11}}{\bar{w}_2} + n_{363}^i{\bar{\varphi }_{12}}{\bar{w}_1}\\ {}&\quad +n_{364}^i{\bar{\varphi }_{12}}{\bar{w}_2},\,\\&\quad {\bar{\varphi }_{x,yy}}{\bar{w}_{0,x}} \sim n_{371}^i{\bar{\varphi }_{11}}{\bar{w}_1}+n_{372}^i{\bar{\varphi }_{11}}{\bar{w}_2} \\ {}&\quad + n_{373}^i{\bar{\varphi }_{12}}{\bar{w}_1}+n_{374}^i{\bar{\varphi }_{12}}{\bar{w}_2},\\&\quad \,{\bar{\varphi }_{y,xy}}{\bar{w}_{0,x}} \sim n_{381}^i{\bar{\varphi }_{21}}{\bar{w}_1}+n_{382}^i{\bar{\varphi }_{21}}{\bar{w}_2} \\ {}&\quad + n_{383}^i{\bar{\varphi }_{22}}{\bar{w}_1}+n_{384}^i{\bar{\varphi }_{22}}{\bar{w}_2},\,\\&\quad {\bar{u}_{0,xy}}{\bar{w}_{0,y}} \sim n_{391}^i{\bar{u}_1}{\bar{w}_1}+n_{392}^i{\bar{u}_1}{\bar{w}_2}\\ {}&\quad + n_{393}^i{\bar{u}_2}{\bar{w}_1}+n_{394}^i{\bar{u}_2}{\bar{w}_2},\,{\bar{v}_{0,yy}}{\bar{w}_{0,y}}\\&\quad \sim n_{401}^i{\bar{v}_1}{\bar{w}_1}+n_{402}^i{\bar{v}_1}{\bar{w}_2} + n_{403}^i{\bar{v}_2}{\bar{w}_1}+n_{404}^i{\bar{v}_2}{\bar{w}_2},\,\\&\quad {\bar{v}_{0,xx}}{\bar{w}_{0,y}} \sim n_{411}^i{\bar{v}_1}{\bar{w}_1}+n_{412}^i{\bar{v}_1}{\bar{w}_2} + n_{413}^i{\bar{v}_2}{\bar{w}_1}\\ {}&\quad +n_{424}^i{\bar{v}_2}{\bar{w}_2},\,{\bar{\varphi }_{x,xy}}{\bar{w}_{0,y}}\\&\quad \sim n_{421}^i{\bar{\varphi }_{11}}{\bar{w}_1}+n_{422}^i{\bar{\varphi }_{11}}{\bar{w}_2} + n_{423}^i{\bar{\varphi }_{12}}{\bar{w}_1}\\ {}&\quad +n_{424}^i{\bar{\varphi }_{12}}{\bar{w}_2},\,\\&\quad {\bar{\varphi }_{y,xx}}{\bar{w}_{0,y}} \sim n_{431}^i{\bar{\varphi }_{21}}{\bar{w}_1}+n_{432}^i{\bar{\varphi }_{21}}{\bar{w}_2} + n_{433}^i{\bar{\varphi }_{22}}{\bar{w}_1}\\ {}&\quad +n_{434}^i{\bar{\varphi }_{22}}{\bar{w}_2},\\&\quad \,{\bar{\varphi }_{y,yy}}{\bar{w}_{0,y}} \sim n_{441}^i{\bar{\varphi }_{21}}{\bar{w}_1}+n_{442}^i{\bar{\varphi }_{21}}{\bar{w}_2} \\ {}&\quad + n_{443}^i{\bar{\varphi }_{22}}{\bar{w}_1}+n_{444}^i{\bar{\varphi }_{22}}{\bar{w}_2},\,\\&\quad \bar{w}_{0,xx}^2 \sim n_{451}^i\bar{w}_1^2+n_{452}^i{\bar{w}_1}{\bar{w}_2} + n_{453}^i\bar{w}_2^2,\,\bar{w}_{0,xy}^2\\&\quad \sim n_{461}^i\bar{w}_1^2+n_{462}^i{\bar{w}_1}{\bar{w}_2} + n_{463}^i\bar{w}_2^2,\\&\quad \,{\bar{w}_{0,x}}{\bar{w}_{0,xxx}} \sim n_{471}^i\bar{w}_1^2+n_{472}^i{\bar{w}_1}{\bar{w}_2}\\ {}&\quad + n_{473}^i\bar{w}_2^2,\,{\bar{w}_{0,y}}{\bar{w}_{0,xxy}}\\&\quad \sim n_{481}^i\bar{w}_1^2+n_{482}^i{\bar{w}_1}{\bar{w}_2} + n_{483}^i\bar{w}_2^2,\\&\quad \,\bar{w}_{0,x}^2{\bar{w}_{0,xx}} \sim n_{491}^i\bar{w}_1^3+n_{492}^i\bar{w}_1^2{\bar{w}_2} \\ {}&\quad + n_{493}^i{\bar{w}_1}\bar{w}_2^2 + n_{494}^i\bar{w}_2^3,\,\bar{w}_{0,y}^2{\bar{w}_{0,xx}}\\&\quad \sim n_{501}^i\bar{w}_1^3+n_{502}^i\bar{w}_1^2{\bar{w}_2} + n_{503}^i{\bar{w}_1}\bar{w}_2^2 + n_{504}^i\bar{w}_2^3,\\&\quad \,{\bar{w}_{0,x}}{\bar{w}_{0,y}}{\bar{w}_{0,xx}} \sim n_{511}^i\bar{w}_1^3+n_{512}^i\bar{w}_1^2{\bar{w}_2} \\ {}&\quad + n_{513}^i{\bar{w}_1}\bar{w}_2^2 + n_{514}^i\bar{w}_2^3,\,{\bar{w}_{0,y}}\\&\quad \sim n_{521}^i{\bar{w}_1}+n_{522}^i{\bar{w}_2} \end{aligned}$$

The coefficients in Eq. (19a) are given as follows

$$\begin{aligned} {\alpha _0}&= {c_1}{k_{101}}n_{162}^1 + {c_1}{k_{91}}n_{161}^1 + {c_4}{k_{71}}n_{111}^1 + {c_4}{k_{81}}n_{112}^1\\&\quad + {c_{68}}{k_{11}}n_{31}^1 + {c_{68}}{k_{21}}n_{32}^1 + {c_{69}}{k_{31}}n_{81}^1\\&\quad + {c_{69}}{k_{41}}n_{82}^1 - {c_{42}}n_{221}^1 + {c_{52}}n_{181}^1\\ {\alpha _1}&= ({c_2}n_{162}^1 + {c_7}n_{152}^1 + {c_{50}}n_{142}^1 + {c_{51}}n_{132}^1){k_{101}} + ({c_{22}}n_{323}^1\\&\quad + {c_{55}}n_{443}^1 + {c_{56}}n_{383}^1 + {c_{58}}n_{433}^1){k_{106}}\\&\quad + ( - {c_{23}}n_{291}^1 + {c_{53}}n_{391}^1 - {c_{65}}n_{341}^1 - {c_{67}}n_{331}^1){k_{16}}\\&\quad + ( - {c_{23}}n_{293}^1 + {c_{53}}n_{393}^1 - {c_{65}}n_{343}^1 - {c_{67}}n_{333}^1){k_{26}}\\&\quad + ( - {c_{24}}n_{301}^1 - {c_{40}}n_{411}^1 + {c_{54}}n_{351}^1 - {c_{66}}n_{401}^1){k_{36}} \\&\quad + ( - {c_{24}}n_{303}^1 - {c_{40}}n_{413}^1 + {c_{54}}n_{353}^1 - {c_{66}}n_{403}^1){k_{46}}\\&\quad + ({c_3}n_{111}^1 + {c_5}n_{91}^1 + {c_6}n_{121}^1 + {c_{49}}n_{10}^1){k_{71}}\\&\quad + ({c_{21}}n_{311}^1 + {c_{57}}n_{371}^1 + {c_{59}}n_{421}^1 + {c_{60}}n_{361}^1){k_{76}}\\&\quad + ({c_3}n_{112}^1 + {c_5}n_{92}^1 + {c_6}n_{122}^1 + {c_{49}}n_{102}^1){k_{81}}\\&\quad + ({c_{21}}n_{313}^1 + {c_{57}}n_{373}^1 + {c_{59}}n_{423}^1 + {c_{60}}n_{363}^1){k_{86}}\\&\quad + ({c_2}n_{161}^1 + {c_7}n_{151}^1 + {c_{50}}n_{141}^1 + {c_{51}}n_{131}^1){k_{91}}\\&\quad + ({c_{22}}n_{321}^1 + {c_{55}}n_{441}^1 + {c_{56}}n_{381}^1 + {c_{58}}n_{431}^1){k_{96}}\\&\quad + ( - {c_8}n_{41}^1 + {c_{11}}n_{21}^1 - {c_{64}}n_{11}^1){k_{11}} + ( - {c_8}n_{42}^1\\&\quad + {c_{11}}n_{22}^1 - {c_{64}}n_{12}^1){k_{21}} + ( - {c_9}n_{71}^1 + {c_{47}}n_{61}^1 - {c_{63}}n_{51}^1){k_{31}}\\&\quad + ( - {c_9}n_{72}^1 + {c_{47}}n_{62}^1 - {c_{63}}n_{52}^1){k_{41}} + {c_{27}}n_{221}^1 + {c_{61}}n_{191}^1\\ {\alpha _2}&= {c_{43}}{k_{71}}n_{121}^1 + {c_{43}}{k_{81}}n_{122}^1 + {c_{44}}{k_{101}}n_{152}^1 + {c_{44}}{k_{91}}n_{151}^1\\&\quad + {c_{45}}{k_{11}}n_{41}^1 + {c_{45}}{k_{21}}n_{42}^1 - {c_{46}}{k_{31}}n_{71}^1 - {c_{46}}{k_{41}}n_{72}^1\\ {\alpha _3}&= {c_{12}}{k_{101}}n_{162}^1 + {c_{12}}{k_{91}}n_{161}^1 + {c_{16}}{k_{71}}n_{111}^1\\&\quad + {c_{16}}{k_{81}}n_{112}^1 - {c_{18}}{k_{31}}n_{81}^1 - {c_{18}}{k_{41}}n_{82}^1 \\ {}&\quad - {c_{19}}{k_{11}}n_{31}^1 - {c_{19}}{k_{21}}n_{32}^1 \\&\quad + {c_{13}}n_{241}^1+ {c_{130}}n_{231}^1 + {c_{14}}n_{221}^1 + {c_{140}}n_{231}^1 + {c_{141}}n_{201}^1\\ {\alpha _4}&= {c_{35}}{k_{31}}n_{71}^1 + {c_{35}}{k_{41}}n_{72}^1\\&\quad + {c_{36}}{k_{11}}n_{41}^1 + {c_{36}}{k_{21}}n_{42}^1\\&\quad + {c_{37}}{k_{71}}n_{121}^1 + {c_{37}}{k_{81}}n_{122}^1\\&\quad + {c_{38}}{k_{101}}n_{152}^1 + {c_{38}}{k_{91}}n_{151}^1 \end{aligned}$$
$$\begin{aligned} {\alpha _5}&= ({c_{22}}n_{323}^1 + {c_{55}}n_{443}^1 + {c_{56}}n_{383}^1 + {c_{58}}n_{433}^1){k_{101}}\\&\quad + ({c_2}n_{162}^1 + {c_7}n_{152}^1 + {c_{50}}n_{142}^1 + {c_{51}}n_{132}^1){k_{104}}\\&\quad + ( - {c_{23}}n_{291}^1 + {c_{53}}n_{391}^1 - {c_{65}}n_{341}^1 - {c_{67}}n_{331}^1){k_{11}}\\&\quad + ( - {c_{23}}n_{293}^1 + {c_{53}}n_{393}^1 - {c_{65}}n_{343}^1 - {c_{67}}n_{333}^1){k_{21}}\\&\quad + ( - {c_{24}}n_{301}^1 - {c_{40}}n_{411}^1 + {c_{54}}n_{351}^1 - {c_{66}}n_{401}^1){k_{31}}\\&\quad + ( - {c_{24}}n_{303}^1 - {c_{40}}n_{413}^1 + {c_{54}}n_{353}^1 - {c_{66}}n_{403}^1){k_{41}}\\&\quad + ({c_{21}}n_{311}^1 + {c_{57}}n_{371}^1 + {c_{59}}n_{421}^1 + {c_{60}}n_{361}^1){k_{71}}\\&\quad + ({c_3}n_{111}^1 + {c_5}n_{91}^1 + {c_6}n_{121}^1 + {c_{49}}n_{10}^1){k_{74}}\\&\quad + ({c_{21}}n_{313}^1 + {c_{57}}n_{373}^1 + {c_{59}}n_{423}^1 + {c_{60}}n_{363}^1){k_{81}}\\&\quad + ({c_3}n_{112}^1 + {c_5}n_{92}^1 + {c_6}n_{122}^1 + {c_{49}}n_{102}^1){k_{84}}\\&\quad + ({c_{22}}n_{321}^1 + {c_{55}}n_{441}^1 + {c_{56}}n_{381}^1 + {c_{58}}n_{431}^1){k_{91}}\\&\quad + ({c_2}n_{161}^1 + {c_7}n_{151}^1 + {c_{50}}n_{141}^1 + {c_{51}}n_{131}^1){k_{94}}\\&\quad + ( - {c_8}n_{41}^1 + {c_{11}}n_{21}^1 - {c_{64}}n_{11}^1){k_{14}}\\&\quad + ( - {c_8}n_{42}^1 + {c_{11}}n_{22}^1 - {c_{64}}n_{12}^1){k_{24}}\\&\quad + ( - {c_9}n_{71}^1 + {c_{47}}n_{61}^1 - {c_{63}}n_{51}^1){k_{34}}\\&\quad + ( - {c_9}n_{72}^1 + {c_{47}}n_{62}^1 - {c_{63}}n_{52}^1){k_{44}} + {c_{39}}n_{461}^1\\ {\alpha _6}&= {c_{43}}{k_{74}}n_{121}^1 + {c_{43}}{k_{84}}n_{122}^1 + {c_{44}}{k_{104}}n_{152}^1\\&\quad + {c_{44}}{k_{94}}n_{151}^1 + {c_{45}}{k_{14}}n_{41}^1 + {c_{45}}{k_{24}}n_{42}^1 \\ {}&\quad - {c_{46}}{k_{34}}n_{71}^1 - {c_{46}}{k_{44}}n_{72}^1\\ {\alpha _7}&= {c_{12}}{k_{104}}n_{162}^1 + {c_{12}}{k_{94}}n_{161}^1 + {c_{16}}{k_{74}}n_{111}^1 + {c_{16}}{k_{84}}n_{112}^1\\&\quad - {c_{18}}{k_{34}}n_{81}^1 - {c_{18}}{k_{44}}n_{82}^1 - {c_{19}}{k_{14}}n_{31}^1 - {c_{19}}{k_{24}}n_{32}^1\\ {\alpha _8}&= ( - {c_9}n_{71}^1 + {c_{47}}n_{61}^1 - {c_{63}}n_{51}^1){k_{33}}\\&\quad + ( - {c_9}n_{72}^1 + {c_{47}}n_{62}^1 - {c_{63}}n_{52}^1){k_{43}}\\&\quad + ( - {k_{11}}n_{342}^1 - {k_{12}}n_{341}^1 - {k_{21}}n_{344}^1 - {k_{22}}n_{343}^1){c_{65}}\\&\quad + ({c_2}n_{161}^1 + {c_7}n_{151}^1 + {c_{50}}n_{141}^1 + {c_{51}}n_{131}^1){k_{93}}\\&\quad + ( - {c_8}n_{41}^1 + {c_{11}}n_{21}^1 - {c_{64}}n_{11}^1){k_{13}}\\&\quad + ( - {c_8}n_{42}^1 + {c_{11}}n_{22}^1 - {c_{64}}n_{12}^1){k_{23}}\\&\quad + ({k_{71}}n_{362}^1 + {k_{72}}n_{361}^1 + {k_{81}}n_{364}^1 + {k_{82}}n_{363}^1){c_{60}}\\&\quad + ( - {k_{31}}n_{402}^1 - {k_{32}}n_{401}^1 - {k_{41}}n_{404}^1 - {k_{42}}n_{403}^1){c_{66}}\\&\quad + ({c_3}n_{111}^1 + {c_5}n_{91}^1 + {c_6}n_{121}^1 + {c_{49}}n_{10}^1){k_{73}}\\&\quad + ({c_3}n_{112}^1 + {c_5}n_{92}^1 + {c_6}n_{122}^1 + {c_{49}}n_{102}^1){k_{83}}\\&\quad + ({c_{22}}n_{324}^1 + {c_{55}}n_{444}^1 + {c_{56}}n_{384}^1 + {c_{58}}n_{434}^1){k_{101}}\\&\quad + ({c_{22}}n_{323}^1 + {c_{55}}n_{443}^1 + {c_{56}}n_{383}^1 + {c_{58}}n_{433}^1){k_{102}}\\&\quad + ( - {k_{11}}n_{332}^1 - {k_{12}}n_{331}^1 - {k_{21}}n_{334}^1 - {k_{22}}n_{333}^1){c_{67}}\\&\quad + ({c_2}n_{162}^1 + {c_7}n_{152}^1 + {c_{50}}n_{142}^1 + {c_{51}}n_{132}^1){k_{103}}\\&\quad + ({k_{71}}n_{312}^1 + {k_{72}}n_{311}^1 + {k_{81}}n_{314}^1 + {k_{82}}n_{313}^1){c_{21}}\\&\quad + ({k_{91}}n_{322}^1 + {k_{92}}n_{321}^1){c_{22}}\\&\quad + ( - {k_{11}}n_{292}^1 - {k_{12}}n_{291}^1 - {k_{21}}n_{294}^1 - {k_{22}}n_{293}^1){c_{23}}\\&\quad + ( - {k_{31}}n_{302}^1 - {k_{32}}n_{301}^1 - {k_{41}}n_{304}^1 - {k_{42}}n_{303}^1){c_{24}}\\&\quad + ( - {k_{31}}n_{412}^1 - {k_{32}}n_{411}^1 - {k_{41}}n_{414}^1\\&\quad - {k_{42}}n_{413}^1){c_{40}} + ({k_{11}}n_{392}^1 + {k_{12}}n_{391}^1 + {k_{21}}n_{394}^1 \\ {}&\quad + {k_{22}}n_{393}^1){c_{53}}\\&\quad + ({k_{31}}n_{352}^1 + {k_{32}}n_{351}^1 + {k_{41}}n_{354}^1 + {k_{42}}n_{353}^1){c_{54}}\\&\quad + ({k_{91}}n_{442}^1 + {k_{92}}n_{441}^1){c_{55}} + ({k_{91}}n_{382}^1 + {k_{92}}n_{381}^1){c_{56}}\\&\quad + ({k_{71}}n_{372}^1 + {k_{72}}n_{371}^1 + {k_{81}}n_{374}^1 + {k_{82}}n_{373}^1){c_{57}}\\&\quad + ({k_{91}}n_{432}^1 + {k_{92}}n_{431}^1){c_{58}}\\&\quad + ({k_{71}}n_{422}^1 + {k_{72}}n_{421}^1 + {k_{81}}n_{424}^1 + {k_{82}}n_{423}^1){c_{59}}\\&\quad + {c_{39}}n_{462}^1\\ {\alpha _9}&= {c_{43}}{k_{73}}n_{121}^1 + {c_{43}}{k_{83}}n_{122}^1 + {c_{44}}{k_{103}}n_{152}^1 + {c_{44}}{k_{93}}n_{151}^1 \\&\quad + {c_{45}}{k_{13}}n_{41}^1+ {c_{45}}{k_{23}}n_{42}^1 - {c_{46}}{k_{33}}n_{71}^1 - {c_{46}}{k_{43}}n_{72}^1\\ {\alpha _{10}}&= {c_{12}}{k_{103}}n_{162}^1 + {c_{12}}{k_{93}}n_{161}^1 + {c_{16}}{k_{73}}n_{111}^1 + {c_{16}}{k_{83}}n_{112}^1 \\&\quad - {c_{18}}{k_{33}}n_{81}^1- {c_{18}}{k_{43}}n_{82}^1 - {c_{19}}{k_{13}}n_{31}^1 - {c_{19}}{k_{23}}n_{32}^1\\ {\alpha _{11}}&= ({c_{22}}n_{324}^1 + {c_{55}}n_{444}^1 + {c_{56}}n_{384}^1 + {c_{58}}n_{434}^1){k_{103}} + ({c_{22}}n_{323}^1\\&\quad + {c_{55}}n_{443}^1 + {c_{56}}n_{383}^1 + {c_{58}}n_{433}^1){k_{105}}\\&\quad + ({k_{73}}n_{312}^1 + {k_{75}}n_{311}^1 + {k_{83}}n_{314}^1 + {k_{85}}n_{313}^1){c_{21}}\\&\quad + ({k_{93}}n_{322}^1 + {k_{95}}n_{321}^1){c_{22}}\\&\quad + ( - {k_{13}}n_{292}^1 - {k_{15}}n_{291}^1 - {k_{23}}n_{294}^1 - {k_{25}}n_{293}^1){c_{23}} \\&\quad + ( - {k_{33}}n_{302}^1- {k_{35}}n_{301}^1 - {k_{43}}n_{304}^1 - {k_{45}}n_{303}^1){c_{24}}\\&\quad + ( - {k_{33}}n_{412}^1 - {k_{35}}n_{411}^1 - {k_{43}}n_{414}^1\\&\quad - {k_{45}}n_{413}^1){c_{40}} + ({k_{13}}n_{392}^1 + {k_{15}}n_{391}^1 + {k_{23}}n_{394}^1 \\&\quad + {k_{25}}n_{393}^1){c_{53}}+ ({k_{33}}n_{352}^1 + {k_{35}}n_{351}^1 + {k_{43}}n_{354}^1\\&\quad + {k_{45}}n_{353}^1){c_{54}} + ({k_{93}}n_{442}^1 + {k_{95}}n_{441}^1){c_{55}}\\&\quad + ({k_{93}}n_{382}^1 + {k_{95}}n_{381}^1){c_{56}}\\&\quad + ({k_{73}}n_{372}^1 + {k_{75}}n_{371}^1 + {k_{83}}n_{374}^1 + {k_{85}}n_{373}^1){c_{57}}\\&\quad + ({k_{93}}n_{432}^1 + {k_{95}}n_{431}^1){c_{58}}\\&\quad + ({k_{73}}n_{422}^1 + {k_{75}}n_{421}^1 + {k_{83}}n_{424}^1 + {k_{85}}n_{423}^1){c_{59}}\\&\quad + ({k_{73}}n_{362}^1 + {k_{75}}n_{361}^1 + {k_{83}}n_{364}^1 + {k_{85}}n_{363}^1){c_{60}}\\&\quad + ( - {k_{13}}n_{342}^1 - {k_{15}}n_{341}^1 - {k_{23}}n_{344}^1 - {k_{25}}n_{343}^1){c_{65}}\\&\quad + ( - {k_{33}}n_{402}^1 - {k_{35}}n_{401}^1 - {k_{43}}n_{404}^1 - {k_{45}}n_{403}^1){c_{66}}\\&\quad + ( - {k_{13}}n_{332}^1 - {k_{15}}n_{331}^1 - {k_{23}}n_{334}^1 - {k_{25}}n_{333}^1){c_{67}}\\&\quad + {c_{10}}n_{513}^1 - {c_{25}}n_{493}^1 + {c_{26}}n_{503}^1\\ {\alpha _{12}}&= ({c_{22}}n_{323}^1 + {c_{55}}n_{443}^1 + {c_{56}}n_{383}^1 + {c_{58}}n_{433}^1){k_{103}} + ({c_{22}}n_{324}^1\\&\quad + {c_{55}}n_{444}^1 + {c_{56}}n_{384}^1 + {c_{58}}n_{434}^1){k_{104}}\\&\quad + ({k_{73}}n_{311}^1 + {k_{74}}n_{312}^1 + {k_{83}}n_{313}^1 + {k_{84}}n_{314}^1){c_{21}}\\&\quad + ({k_{93}}n_{321}^1 + {k_{94}}n_{322}^1){c_{22}}\\&\quad + {( - {k_{13}}n_{291}^1 - {k_{14}}n_{292}^1 - {k_{23}}n_{293}^1 - {k_{24}}n_{294}^1)_{c23}}\\&\quad + ( - {k_{33}}n_{301}^1 - {k_{34}}n_{302}^1 - {k_{43}}n_{303}^1 - {k_{44}}n_{304}^1){c_{24}}\\&\quad + ( - {k_{33}}n_{411}^1 - {k_{34}}n_{412}^1 - {k_{43}}n_{413}^1 - {k_{44}}n_{414}^1){c_{40}}\\&\quad + ({k_{13}}n_{391}^1 + {k_{14}}n_{392}^1 + {k_{23}}n_{393}^1 + {k_{_{24}}}n_{394}^1){c_{53}}\\&\quad + ({k_{33}}n_{351}^1 + {k_{34}}n_{352}^1 + {k_{43}}n_{353}^1 + {k_{44}}n_{354}^1){c_{54}}\\&\quad + ({k_{93}}n_{441}^1 + {k_{94}}n_{442}^1){c_{55}}\\&\quad + ({k_{93}}n_{381}^1 + {k_{94}}n_{382}^1){c_{56}} + ({k_{73}}n_{371}^1 + {k_{74}}n_{372}^1\\&\quad + {k_{83}}n_{373}^1 + {k_{84}}n_{374}^1){c_{57}}\\&\quad + ({k_{93}}n_{431}^1 + {k_{94}}n_{432}^1){c_{58}} + ({k_{73}}n_{421}^1 + {k_{74}}n_{422}^1\\&\quad + {k_{83}}n_{423}^1 + {k_{84}}n_{424}^1){c_{59}}\\&\quad + ({k_{73}}n_{361}^1 + {k_{74}}n_{362}^1 + {k_{83}}n_{363}^1 + {k_{84}}n_{364}^1){c_{60}}\\&\quad + ( - {k_{13}}n_{341}^1 - {k_{14}}n_{342}^1 - {k_{23}}n_{343}^1 - {k_{24}}n_{344}^1){c_{65}}\\&\quad + ( - {k_{33}}n_{401}^1 - {k_{34}}n_{402}^1 - {k_{43}}n_{403}^1 - {k_{44}}n_{404}^1){c_{66}}\\&\quad + ( - {k_{13}}n_{331}^1 - {k_{14}}n_{332}^1 - {k_{23}}n_{333}^1 - {k_{24}}n_{334}^1){c_{67}} \\&\quad + {c_{10}}n_{512}^1 - {c_{25}}n_{492}^1 + {c_{26}}n_{502}^1\\ \end{aligned}$$
$$\begin{aligned} {\alpha _{13}}&= ({c_{22}}n_{323}^1 + {c_{55}}n_{443}^1 + {c_{56}}n_{383}^1 + {c_{58}}n_{433}^1){k_{104}}\\&\quad + ( - {c_{23}}n_{291}^1 + {c_{53}}n_{391}^1 - {c_{65}}n_{341}^1 - {c_{67}}n_{331}^1){k_{14}}\\&\quad + ( - {c_{23}}n_{293}^1 + {c_{53}}n_{393}^1 - {c_{65}}n_{343}^1 - {c_{67}}n_{333}^1){k_{24}}\\&\quad + ( - {c_{24}}n_{301}^1 - {c_{40}}n_{411}^1 + {c_{54}}n_{351}^1 - {c_{66}}n_{401}^1){k_{34}}\\&\quad + ( - {c_{24}}n_{303}^1 - {c_{40}}n_{413}^1 + {c_{54}}n_{353}^1 - {c_{66}}n_{403}^1){k_{44}}\\&\quad + ({c_{21}}n_{311}^1 + {c_{57}}n_{371}^1 + {c_{59}}n_{421}^1 + {c_{60}}n_{361}^1){k_{74}}\\&\quad + ({c_{21}}n_{313}^1 + {c_{57}}n_{373}^1 + {c_{59}}n_{423}^1 + {c_{60}}n_{363}^1){k_{84}}\\&\quad + ({c_{22}}n_{321}^1 + {c_{55}}n_{441}^1 + {c_{56}}n_{381}^1 + {c_{58}}n_{431}^1){k_{94}} \\&\quad + {c_{10}}n_{511}^1 - {c_{25}}n_{491}^1 + {c_{26}}n_{501}^1\\ {\alpha _{14}}&= ({c_{22}}n_{324}^1 + {c_{55}}n_{444}^1 + {c_{56}}n_{384}^1 + {c_{58}}n_{434)}^1{k_{102}}\\&\quad + ({c_2}n_{162}^1 + {c_7}n_{152}^1 + {c_{50}}n_{142}^1 + {c_{51}}n_{132}^1){k_{105}}\\&\quad + ( - {c_{23}}n_{292}^1 + {c_{53}}n_{392}^1 - {c_{65}}n_{342}^1 - {c_{67}}n_{332}^1){k_{12}}\\&\quad + ( - {c_{23}}n_{294}^1 + {c_{53}}n_{394}^1 - {c_{65}}n_{344}^1 - {c_{67}}n_{334}^1){k_{22}}\\&\quad + ( - {c_{24}}n_{302}^1 - {c_{40}}n_{412}^1 + {c_{54}}n_{352}^1 - {c_{66}}n_{402}^1){k_{32}}\\&\quad + ( - {c_{24}}n_{304}^1 - {c_{40}}n_{414}^1 + {c_{54}}n_{354}^1 - {c_{66}}n_{404}^1){k_{42}}\\&\quad + ({c_{21}}n_{312}^1 + {c_{57}}n_{372}^1 + {c_{59}}n_{422}^1 + {c_{60}}n_{362}^1){k_{72}}\\&\quad + ({c_3}n_{111}^1 + {c_5}n_{91}^1 + {c_6}n_{121}^1 + {c_{49}}n_{10}^1){k_{75}}\\&\quad + ({c_{21}}n_{314}^1 + {c_{57}}n_{374}^1 + {c_{59}}n_{424}^1 + {c_{60}}n_{364}^1){k_{82}}\\&\quad + ({c_3}n_{112}^1 + {c_5}n_{92}^1 + {c_6}n_{122}^1 + {c_{49}}n_{102}^1){k_{85}}\\&\quad + ({c_{22}}n_{322}^1 + {c_{55}}n_{442}^1 + {c_{56}}n_{382}^1 + {c_{58}}n_{432}^1){k_{92}}\\&\quad + ({c_2}n_{161}^1 + {c_7}n_{151}^1 + {c_{50}}n_{141}^1 + {c_{51}}n_{131}^1){k_{95}}\\&\quad + ( - {c_8}n_{41}^1 + {c_{11}}n_{21}^1 - {c_{64}}n_{11}^1){k_{15}}\\&\quad + ( - {c_8}n_{42}^1 + {c_{11}}n_{22}^1 - {c_{64}}n_{12}^1){k_{25}}\\&\quad + ( - {c_9}n_{71}^1 + {c_{47}}n_{61}^1 - {c_{63}}n_{51}^1){k_{35}}\\&\quad + ( - {c_9}n_{72}^1 + {c_{47}}n_{62}^1 - {c_{63}}n_{52}^1){k_{45}} + {c_{39}}n_{463}^1\\ {\alpha _{15}}&= {c_{43}}{k_{75}}n_{121}^1 + {c_{43}}{k_{85}}n_{122}^1 + {c_{44}}{k_{105}}n_{152}^1\\&\quad + {c_{44}}{k_{95}}n_{151}^1 + {c_{45}}{k_{15}}n_{41}^1 + {c_{45}}{k_{25}}n_{42}^1 - {c_{46}}{k_{35}}n_{71}^1\\ {}&\quad - {c_{46}}{k_{45}}n_{72}^1\\ {\alpha _{16}}&= {c_{12}}{k_{105}}n_{162}^1 + {c_{12}}{k_{95}}n_{161}^1 + {c_{16}}{k_{75}}n_{111}^1 + {c_{16}}{k_{85}}n_{112}^1\\&\quad - {c_{18}}{k_{35}}n_{81}^1 - {c_{18}}{k_{45}}n_{82}^1 - {c_{19}}{k_{15}}n_{31}^1 - {c_{19}}{k_{25}}n_{32}^1\\ {\alpha _{17}}&= ({c_{22}}n_{324}^1 + {c_{55}}n_{444}^1 + {c_{56}}n_{384}^1 + {c_{58}}n_{434}^1){k_{105}}\\&\quad + ( - {c_{23}}n_{292}^1 + {c_{53}}n_{392}^1 - {c_{65}}n_{342}^1 - {c_{67}}n_{332}^1){k_{15}}\\&\quad + ( - {c_{23}}n_{294}^1 + {c_{53}}n_{394}^1 - {c_{65}}n_{344}^1 - {c_{67}}n_{334}^1){k_{25}}\\&\quad + ( - {c_{24}}n_{302}^1 - {c_{40}}n_{412}^1 + {c_{54}}n_{352}^1 - {c_{66}}n_{402}^1){k_{35}}\\ {}&\quad + {c_{10}}n_{514}^1\\&\quad + ( - {c_{24}}n_{304}^1 - {c_{40}}n_{414}^1 + {c_{54}}n_{354}^1 - {c_{66}}n_{404}^1){k_{45}}\\&\quad + ({c_{21}}n_{312}^1 + {c_{57}}n_{372}^1 + {c_{59}}n_{422}^1 + {c_{60}}n_{362}^1){k_{75}}- {c_{25}}n_{494}^1\\&\quad + ({c_{21}}n_{314}^1 + {c_{57}}n_{374}^1 + {c_{59}}n_{424}^1 + {c_{60}}n_{364}^1){k_{85}}\\&\quad + ({c_{22}}n_{322}^1 + {c_{55}}n_{442}^1 + {c_{56}}n_{382}^1 + {c_{58}}n_{432}^1){k_{95}}+ {c_{26}}n_{504}^1 \\ {\alpha _{18}}&= ({c_2}n_{162}^1 + {c_7}n_{152}^1 + {c_{50}}n_{142}^1 + {c_{51}}n_{132}^1){k_{102}}\\&\quad + ({c_{22}}n_{324}^1 + {c_{55}}n_{444}^1 + {c_{56}}n_{384}^1 + {c_{58}}n_{434}^1){k_{106}}\\&\quad + ( - {c_{23}}n_{292}^1 + {c_{53}}n_{392}^1 - {c_{65}}n_{342}^1 - {c_{67}}n_{332}^1){k_{16}}\\&\quad + ( - {c_{23}}n_{294}^1 + {c_{53}}n_{394}^1 - {c_{65}}n_{344}^1 - {c_{67}}n_{334}^1){k_{26}}\\&\quad + ( - {c_{24}}n_{302}^1 - {c_{40}}n_{412}^1 + {c_{54}}n_{352}^1 - {c_{66}}n_{402}^1){k_{36}}\\&\quad + ( - {c_{24}}n_{304}^1 - {c_{40}}n_{414}^1 + {c_{54}}n_{354}^1 - {c_{66}}n_{404}^1){k_{46}}\\&\quad + ({c_3}n_{111}^1 + {c_5}n_{91}^1 + {c_6}n_{121}^1 + {c_{49}}n_{10}^1){k_{72}}\\&\quad + ({c_{21}}n_{312}^1 + {c_{57}}n_{372}^1 + {c_{59}}n_{422}^1 + {c_{60}}n_{362}^1){k_{76}}\\&\quad + ({c_3}n_{112}^1 + {c_5}n_{92}^1 + {c_6}n_{122}^1 + {c_{49}}n_{102}^1){k_{82}}\\&\quad + ({c_{21}}n_{314}^1 + {c_{57}}n_{374}^1 + {c_{59}}n_{424}^1 + {c_{60}}n_{364}^1){k_{86}}\\&\quad + ({c_2}n_{161}^1 + {c_7}n_{151}^1 + {c_{50}}n_{141}^1 + {c_{51}}n_{131}^1){k_{92}}\\&\quad + ({c_{22}}n_{322}^1 + {c_{55}}n_{442}^1 + {c_{56}}n_{382}^1 + {c_{58}}n_{432}^1){k_{96}} \\&\quad + ( - {c_8}n_{41}^1 + {c_{11}}n_{21}^1 - {c_{64}}n_{11}^1){k_{12}}\\&\quad + ( - {c_8}n_{42}^1 + {c_{11}}n_{22}^1 - {c_{64}}n_{12}^1){k_{22}}\\&\quad + ( - {c_9}n_{71}^1 + {c_{47}}n_{61}^1 - {c_{63}}n_{51}^1){k_{32}} \\&\quad + ( - {c_9}n_{72}^1 + {c_{47}}n_{62}^1 - {c_{63}}n_{52}^1){k_{42}}\\&\quad + {c_{27}}n_{222}^1 + {c_{61}}n_{192}^1\\ {\alpha _{19}}&= {c_{43}}{k_{72}}n_{121}^1 + {c_{43}}{k_{82}}n_{122}^1\\&\quad + {c_{44}}{k_{102}}n_{152}^1 + {c_{44}}{k_{92}}n_{151}^1 + {c_{45}}{k_{12}}n_{41}^1 + {c_{45}}{k_{_{22}}}n_{42}^1 \\ {}&\quad -- {c_{46}}{k_{32}}n_{71}^1 {c_{46}}{k_{42}}n_{72}^1\\ {\alpha _{20}}&= {c_{12}}{k_{102}}n_{162}^1 + {c_{12}}{k_{92}}n_{161}^1\\&\quad + {c_{16}}{k_{72}}n_{111}^1 + {c_{16}}{k_{82}}n_{112}^1 - {c_{18}}{k_{32}}n_{81}^1 \\ {}&\quad - {c_{18}}{k_{42}}n_{82}^1 - {c_{19}}{k_{12}}n_{31}^1 - {c_{19}}{k_{22}}n_{32}^1 \\&\quad + {c_{13}}n_{242}^1 + {c_{130}}n_{232}^1+ {c_{14}}n_{222}^1 + {c_{140}}n_{232}^1\\&\quad + {c_{141}}n_{202}^1\\ {\alpha _{21}}&= {c_{35}}{k_{32}}n_{71}^1 + {c_{35}}{k_{42}}n_{72}^1 + {c_{36}}{k_{12}}n_{41}^1\\&\quad + {c_{36}}{k_{22}}n_{42}^1 + {c_{37}}{k_{72}}n_{121}^1 + {c_{37}}{k_{82}}n_{122}^1 \\ {}&\quad + {c_{38}}{k_{102}}n_{152}^1 + {c_{38}}{k_{92}}n_{151}^1\\ {\alpha _{22}}&= {c_1}{k_{102}}n_{162}^1 + {c_1}{k_{92}}n_{161}^1 + {c_4}{k_{72}}n_{111}^1 + {c_4}{k_{82}}n_{112}^1 \\ {}&\quad + {c_{68}}{k_{12}}n_{31}^1 + {c_{68}}{k_{22}}n_{32}^1 + {c_{69}}{k_{32}}n_{81}^1\\&\quad + {c_{69}}{k_{42}}n_{82}^1 - {c_{42}}n_{222}^1 + {c_{52}}n_{182}^1\\ {\alpha _{23}}&= ({c_2}n_{162}^1 + {c_7}n_{152}^1 + {c_{50}}n_{142}^1 + {c_{51}}n_{132}^1){k_{106}}\\ {}&\quad + ({c_3}n_{111}^1 + {c_5}n_{91}^1 + {c_6}n_{121}^1 + {c_{49}}n_{10}^1){k_{76}}\\&\quad + ({c_3}n_{112}^1 + {c_5}n_{92}^1 + {c_6}n_{122}^1 + {c_{49}}n_{102}^1){k_{86}}\\ {}&\quad + ({c_2}n_{161}^1 + {c_7}n_{151}^1 + {c_{50}}n_{141}^1 + {c_{51}}n_{131}^1){k_{96}}\\&\quad + ( - {c_8}n_{41}^1 + {c_{11}}n_{21}^1 - {c_{64}}n_{11}^1){k_{16}}\\&\quad + ( - {c_8}n_{42}^1 + {c_{11}}n_{22}^1 - {c_{64}}n_{12}^1){k_{26}} \\ {}&\quad + ( - {c_9}n_{71}^1 + {c_{47}}n_{61}^1 - {c_{63}}n_{51}^1){k_{36}}\\&\quad - ({c_9}n_{72}^1 - {c_{47}}n_{62}^1 + {c_{63}}n_{52}^1){k_{46}}\\ {\alpha _{24}}&= {c_{43}}{k_{76}}n_{121}^1 + {c_{43}}{k_{86}}n_{122}^1 + {c_{44}}{k_{106}}n_{152}^1 \\ {}&\quad + {c_{44}}{k_{96}}n_{151}^1 + {c_{45}}{k_{16}}n_{41}^1\\&\quad + {c_{45}}{k_{26}}n_{42}^1 - {c_{46}}{k_{36}}n_{71}^1 - {c_{46}}{k_{46}}n_{72}^1\\ {\alpha _{25}}&= {c_{12}}{k_{106}}n_{162}^1 + {c_{12}}{k_{96}}n_{161}^1 + {c_{16}}{k_{76}}n_{111}^1 \\ {}&\quad + {c_{16}}{k_{86}}n_{112}^1 - {c_{18}}{k_{36}}n_{81}^1 - {c_{18}}{k_{46}}n_{82}^1\\&\quad - {c_{19}}{k_{16}}n_{31}^1 - {c_{19}}{k_{26}}n_{32}^1 - {c_{17}}n_{17}^1\\ {\alpha _{26}}&= {c_{70}}{c_{701}}{q_1}n_{521}^1,{\alpha _{27}} = {c_{70}}{c_{701}}{q_1}n_{522}^1,{\alpha _{28}}\\&\quad = {c_{70}}{c_{702}}{q_2}n_{181}^1,{\alpha _{29}} = {c_{70}}{c_{702}}{q_2}n_{182}^1\\ \end{aligned}$$

where \({\beta _0} \sim {\beta _{29}}\) can be obtained by changing the prime i of the corresponding term above.

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Chang, Z., Hou, L. & Chen, Y. Investigation on the 1:2 internal resonance of an FGM blade. Nonlinear Dyn 107, 1937–1964 (2022). https://doi.org/10.1007/s11071-021-07070-2

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