Optimal Design of a Composite Lattice Rectangular Plate for Solar Panels of a High-Agility Satellite

Abstract

This paper proposes that a composite lattice rectangular plate is applied to the solar panel for a high-agility satellite. We define the modified effective membrane stiffness, bending stiffness considering the directionally dependent mechanical properties to an intersection and mode shape function of the composite lattice rectangular plate, which is assumed to be a Kirchhoff–Love plate. We subsequently present an approximate method of conducting vibration, buckling analyses of the lattice plate for the solar panel with a torsional spring using the Ritz method. This method considers the buckling as well as the vibration characteristics (natural frequencies and modes). The validity of the present method is verified by comparing the results of the finite-element analysis. Finally, we apply the method to optimize the lattice plate for a solar panel in a high-agility satellite minimizing mass with a limited area. Consequently, we conclude that the present method is very useful not only vibration and buckling analyses of the composite lattice structures but also optimization due to their relative simplicity and computational efficiency.

Appendices

Appendix A

$$\begin{aligned} A_{11}= & {} \left\{ {2V_{11} Q_{11, d} \cos ^{4}\phi +2V_{21} Q_{22, d} \sin ^{4}\phi } \right. \nonumber \\&\quad \left. {+2\left[ {\left( {V_{11} +V_{21} } \right) Q_{12, d} +4V_{31} Q_{66, d} } \right] \cos ^{2}\phi \sin ^{2}\phi } \right\} H, \end{aligned}$$

(26.1)

$$\begin{aligned} A_{12}= & {} \left[ {2(V_{11} Q_{11, d} +V_{21} Q_{22, d} -4V_{31} Q_{66, d} )\cos ^{2}\phi \sin ^{2}\phi } \right. \nonumber \\&\quad +2Q_{12, d} (V_{11} \cos ^{4}\phi +V_{21} \sin ^{4}\phi ) \left. {+Q_{12, t} V_{13} } \right] H, \end{aligned}$$

(26.2)

$$\begin{aligned} A_{21}= & {} A_{12} , \end{aligned}$$

(26.3)

$$\begin{aligned} A_{22}= & {} \left\{ {2V_{11} Q_{11, d} \sin ^{4}\phi +2V_{21} Q_{22, d} \cos ^{4}\phi +V_{13} Q_{11, t} } \right. \nonumber \\&\quad \left. {+2\left[ {\left( {V_{11} +V_{21} } \right) Q_{12, d} +4V_{31} Q_{66, d} } \right] \cos ^{2}\phi \sin ^{2}\phi } \right\} H, \end{aligned}$$

(26.4)

$$\begin{aligned} A_{66}= & {} \left\{ {2V_{31} Q_{66, d} \left( {\cos ^{4}\phi +\sin ^{4}\phi } \right) +V_{33} Q_{66, t} } \right. \nonumber \\&\quad 2\left[ {V_{11} Q_{11, d} +V_{21} Q_{22, d} -\left( {V_{11} +V_{21} } \right) Q_{12, d} } \right. \nonumber \\&\quad \left. {\left. {-2V_{31} Q_{66, d} } \right] \cos ^{2}\phi \sin ^{2}\phi } \right\} H, \end{aligned}$$

(26.5)

$$\begin{aligned} A_{16}= & {} A_{26} =A_{61} =A_{62} =0. \end{aligned}$$

(26.6)

Appendix B

$$\begin{aligned} P_{11}= & {} \frac{144D_{11} L_b }{5L_a ^{3}}, \end{aligned}$$

(46.1)

$$\begin{aligned} P_{13}= & {} \frac{104D_{11} L_b }{L_a ^{3}}\quad , \end{aligned}$$

(46.2)

$$\begin{aligned} P_{12}= & {} P_{14} =P_{15} =0, \end{aligned}$$

(46.3)

$$\begin{aligned} P_{22}= & {} \frac{48D_{11} L_b }{5L_a ^{3}}+\frac{1152D_{66} }{7L_a L_b }, \end{aligned}$$

(46.4)

$$\begin{aligned} P_{24}= & {} -\frac{392D_{11} L_b }{L_a ^{3}}-\frac{31592D_{66} }{7L_a L_b }, \end{aligned}$$

(46.5)

$$\begin{aligned} P_{21}= & {} P_{23} =P_{25} =0, \end{aligned}$$

(46.6)

$$\begin{aligned} P_{31}= & {} P_{13} , \end{aligned}$$

(46.7)

$$\begin{aligned} P_{33}= & {} \frac{2640D_{11} L_b }{7L_a ^{3}}, \end{aligned}$$

(46.8)

$$\begin{aligned} P_{32}= & {} P_{34} =P_{35} =0, \end{aligned}$$

(46.9)

$$\begin{aligned} P_{42}= & {} P_{24} , \end{aligned}$$

(46.10)

$$\begin{aligned} P_{44}= & {} \frac{880D_{11} L_b }{7L_a ^{3}}+\frac{141040D_{66} }{63L_a L_b }, \end{aligned}$$

(46.11)

$$\begin{aligned} P_{41}= & {} P_{43} =P_{45} =0, \end{aligned}$$

(46.12)

$$\begin{aligned} P_{51}= & {} P_{52} =P_{53} =P_{54} =0, \end{aligned}$$

(46.13)

$$\begin{aligned} P_{55}= & {} k_\mathrm{ts}, \end{aligned}$$

(46.14)

$$\begin{aligned} Q_{11}= & {} \frac{104L_a L_b H\rho _\mathrm{eq} }{45}, \end{aligned}$$

(47.1)

$$\begin{aligned} Q_{13}= & {} \frac{2644L_a L_b H\rho _\mathrm{eq} }{315}, \end{aligned}$$

(47.2)

$$\begin{aligned} Q_{15}= & {} \frac{13L_a ^{2}L_b H\rho _\mathrm{eq} }{15}, \end{aligned}$$

(47.3)

$$\begin{aligned} Q_{12}= & {} Q_{14} =0, \end{aligned}$$

(47.4)

$$\begin{aligned} Q_{22}= & {} \frac{104L_a ^{2}L_b H\rho _\mathrm{eq} }{135}, \end{aligned}$$

(47.5)

$$\begin{aligned} Q_{24}= & {} \frac{2644L_a L_b H\rho _\mathrm{eq} }{945}, \end{aligned}$$

(47.6)

$$\begin{aligned} Q_{21}= & {} Q_{23} =Q_{25} =0, \end{aligned}$$

(47.7)

$$\begin{aligned} Q_{31}= & {} Q_{13} , \end{aligned}$$

(47.8)

$$\begin{aligned} Q_{33}= & {} \frac{21128L_a L_b H\rho _\mathrm{eq} }{693}, \end{aligned}$$

(47.9)

$$\begin{aligned} Q_{35}= & {} \frac{22L_a ^{2}L_b H\rho _\mathrm{eq} }{7}, \end{aligned}$$

(47.10)

$$\begin{aligned} Q_{32}= & {} Q_{34} =0, \end{aligned}$$

(47.11)

$$\begin{aligned} Q_{42}= & {} Q_{24} , \end{aligned}$$

(47.12)

$$\begin{aligned} Q_{44}= & {} \frac{21128L_a L_b H\rho _\mathrm{eq} }{2079}, \end{aligned}$$

(47.13)

$$\begin{aligned} Q_{41}= & {} Q_{43} =Q_{45} =0, \end{aligned}$$

(47.14)

$$\begin{aligned} Q_{51}= & {} Q_{15} , \end{aligned}$$

(47.15)

$$\begin{aligned} Q_{53}= & {} Q_{35} , \end{aligned}$$

(47.16)

$$\begin{aligned} Q_{55}= & {} \frac{L_a ^{3}L_b H\rho _\mathrm{eq} }{3}, \end{aligned}$$

(47.17)

$$\begin{aligned} Q_{52}= & {} Q_{54}. \end{aligned}$$

(47.18)

Appendix C

$$\begin{aligned} R_{11}= & {} \frac{9D_{11} L_b \left( {24D_{11} ^{2}-6D_{11} L_a k_\mathrm{ts} +L_a ^{2}k_\mathrm{ts} ^{2}} \right) }{5L_a ^{3}\left( {3D_{11} -L_a k_\mathrm{ts} } \right) ^{2}}\quad , \end{aligned}$$

(60.1)

$$\begin{aligned} R_{12}= & {} \frac{D_{11} L_b \left( {108D_{11} ^{2}-30D_{11} L_a k_\mathrm{ts} +5L_a ^{2}k_\mathrm{ts} ^{2}} \right) }{L_a ^{3}\left( {3D_{11} -L_a k_\mathrm{ts} } \right) ^{2}}, \end{aligned}$$

(60.2)

$$\begin{aligned} R_{13}= & {} \frac{6D_{11} L_b \left( {225D_{11} ^{2}-66D_{11} L_a k_\mathrm{ts} +11L_a ^{2}k_\mathrm{ts} ^{2}} \right) }{7L_a ^{3}\left( {3D_{11} -L_a k_\mathrm{ts} } \right) ^{2}}, \end{aligned}$$

(60.3)

$$\begin{aligned} R_{14}= & {} \frac{3D_{11} L_b \left( {99D_{11} ^{2}-30D_{11} L_a k_\mathrm{ts} +5L_a ^{2}k_\mathrm{ts} ^{2}} \right) }{L_a ^{3}\left( {3D_{11} -L_a k_\mathrm{ts} } \right) ^{2}}, \end{aligned}$$

(60.4)

$$\begin{aligned} R_{21}= & {} R_{12} , \end{aligned}$$

(60.5)

$$\begin{aligned} R_{22}= & {} \frac{3D_{11} L_b \left( {640D_{11} ^{2}-198D_{11} L_a k_\mathrm{ts} +33L_a ^{2}k_\mathrm{ts} ^{2}} \right) }{7L_a ^{3}\left( {3D_{11} -L_a k_\mathrm{ts} } \right) ^{2}}, \end{aligned}$$

(60.6)

$$\begin{aligned} R_{23}= & {} \frac{9D_{11} L_b \left( {55D_{11} ^{2}-18D_{11} L_a k_\mathrm{ts} +3L_a ^{2}k_\mathrm{ts} ^{2}} \right) }{L_a ^{3}\left( {3D_{11} -L_a k_\mathrm{ts} } \right) ^{2}}, \end{aligned}$$

(60.7)

$$\begin{aligned} R_{24}= & {} \frac{2D_{11} L_b \left( {1152D_{11} ^{2}-390D_{11} L_a k_\mathrm{ts} +65L_a ^{2}k_\mathrm{ts} ^{2}} \right) }{3L_a ^{3}\left( {3D_{11} -L_a k_\mathrm{ts} } \right) ^{2}}, \end{aligned}$$

(60.8)

$$\begin{aligned} R_{31}= & {} R_{13} , \end{aligned}$$

(60.9)

$$\begin{aligned} R_{32}= & {} R_{23}, \end{aligned}$$

(60.10)

$$\begin{aligned} R_{33}= & {} \frac{4D_{11} L_b \left( {225D_{11} ^{2}-78D_{11} L_a k_\mathrm{ts} +13L_a ^{2}k_\mathrm{ts} ^{2}} \right) }{L_a ^{3}\left( {3D_{11} -L_a k_\mathrm{ts} } \right) ^{2}}, \end{aligned}$$

(60.11)

$$\begin{aligned} R_{34}= & {} \frac{12D_{11} L_b \left( {117D_{11} ^{2}-42D_{11} L_a k_\mathrm{ts} +7L_a ^{2}k_\mathrm{ts} ^{2}} \right) }{L_a ^{3}\left( {3D_{11} -L_a k_\mathrm{ts} } \right) ^{2}}, \end{aligned}$$

(60.12)

$$\begin{aligned} R_{41}= & {} R_{14}, \end{aligned}$$

(60.13)

$$\begin{aligned} R_{42}= & {} R_{24}, \end{aligned}$$

(60.14)

$$\begin{aligned} R_{43}= & {} R_{34}, \end{aligned}$$

(60.15)

$$\begin{aligned} R_{44}= & {} \frac{12D_{11} L_b \left( {2016D_{11} ^{2}-750D_{11} L_a k_\mathrm{ts} +125L_a ^{2}k_\mathrm{ts} ^{2}} \right) }{11L_a ^{3}\left( {3D_{11} -L_a k_\mathrm{ts} } \right) ^{2}}, \end{aligned}$$

(60.16)

$$\begin{aligned} S_{11}= & {} \frac{3L_b \left( {51D_{11} ^{2}-13D_{11} L_a k_\mathrm{ts} +L_a ^{2}k_\mathrm{ts} ^{2}} \right) }{35L_a \left( {3D_{11} -L_a k_\mathrm{ts} } \right) ^{2}}, \end{aligned}$$

(61.1)

$$\begin{aligned} S_{12}= & {} \frac{L_b \left( {765D_{11} ^{2}-208D_{11} L_a k_\mathrm{ts} +17L_a ^{2}k_\mathrm{ts} ^{2}} \right) }{70L_a \left( {3D_{11} -L_a k_\mathrm{ts} } \right) ^{2}}, \end{aligned}$$

(61.2)

$$\begin{aligned} S_{13}= & {} \frac{L_b \left( {16380D_{11} ^{2}-4611D_{11} L_a k_\mathrm{ts} +389L_a ^{2}k_\mathrm{ts} ^{2}} \right) }{840L_a \left( {3D_{11} -L_a k_\mathrm{ts} } \right) ^{2}}, \end{aligned}$$

(61.3)

$$\begin{aligned} S_{14}= & {} \frac{L_b \left( {3600D_{11} ^{2}-1035D_{11} L_a k_\mathrm{ts} +89L_a ^{2}k_\mathrm{ts} ^{2}} \right) }{120L_a \left( {3D_{11} -L_a k_\mathrm{ts} } \right) ^{2}}, \end{aligned}$$

(61.4)

$$\begin{aligned} S_{21}= & {} S_{12} , \end{aligned}$$

(61.5)

$$\begin{aligned} S_{22}= & {} \frac{L_b \left( {8640D_{11} ^{2}-2505D_{11} L_a k_\mathrm{ts} +218L_a ^{2}k_\mathrm{ts} ^{2}} \right) }{315L_a \left( {3D_{11} -L_a k_\mathrm{ts} } \right) ^{2}}, \end{aligned}$$

(61.6)

$$\begin{aligned} S_{23}= & {} \frac{L_b \left( {13740D_{11} ^{2}-4125D_{11} L_a k_\mathrm{ts} +371L_a ^{2}k_\mathrm{ts} ^{2}} \right) }{280L_a \left( {3D_{11} -L_a k_\mathrm{ts} } \right) ^{2}}, \end{aligned}$$

(61.7)

$$\begin{aligned} S_{24}= & {} \frac{L_b \left( {698880D_{11} ^{2}-214339D_{11} L_a k_\mathrm{ts} +19665L_a ^{2}k_\mathrm{ts} ^{2}} \right) }{9240L_a \left( {3D_{11} -L_a k_\mathrm{ts} } \right) ^{2}}, \end{aligned}$$

(61.8)

$$\begin{aligned} S_{31}= & {} S_{13} , \end{aligned}$$

(61.9)

$$\begin{aligned} S_{32}= & {} S_{23} , \end{aligned}$$

(61.10)

$$\begin{aligned} S_{33}= & {} \frac{3L_b \left( {22575D_{11} ^{2}-7020D_{11} L_a k_\mathrm{ts} +653L_a ^{2}k_\mathrm{ts} ^{2}} \right) }{770L_a \left( {3D_{11} -L_a k_\mathrm{ts} } \right) ^{2}}, \end{aligned}$$

(61.11)

$$\begin{aligned} S_{34}= & {} \frac{L_b \left( {57015D_{11} ^{2}-18117D_{11} L_a k_\mathrm{ts} +1720L_a ^{2}k_\mathrm{ts} ^{2}} \right) }{420L_a \left( {3D_{11} -L_a k_\mathrm{ts} } \right) ^{2}}, \end{aligned}$$

(61.12)

$$\begin{aligned} S_{41}= & {} S_{14} , \end{aligned}$$

(61.13)

$$\begin{aligned} S_{42}= & {} S_{24} , \end{aligned}$$

(61.14)

$$\begin{aligned} S_{43}= & {} S_{34} , \end{aligned}$$

(61.15)

$$\begin{aligned} S_{44}= & {} \frac{L_b \left( {81792D_{11} ^{2}-26565D_{11} L_a k_\mathrm{ts} +2575L_a ^{2}k_\mathrm{ts} ^{2}} \right) }{390L_a \left( {3D_{11} -L_a k_\mathrm{ts} } \right) ^{2}}. \end{aligned}$$

(61.16)