Closed form solutions for nonlinear static response of curled cantilever micro-/nanobeams including both the fringing field and van der Waals force effect

Abstract

In this paper, the nonlinear static responses of curled cantilever micro-/nanobeams are investigated. The beams are subjected to a one-sided electrostatic actuation, and the effects of both fringing field and van der Waals force are also included. Based on the combination of the Galerkin method and the choice of the shape function of the beam deformation, the analytical approximate solutions are established. The Pull-In voltages which determine the stability of the curled beam actuators are also obtained. When van der Waals force is neglected, the Pull-In parameters are explicitly presented. These approximate solutions show excellent agreements with numerical solutions obtained by the shooting method and the experimental data for a wide range of beam length. Expressions of these analytical approximate solutions are brief and could easily be used to derive the effects of various factors, such as fringing field, van der Waals force, the width of the beam and the gap, on micro-/nanobeams.

Appendix

$$ \begin{aligned} B_{1} = &\,\left\{ {\left( {15940696127569920000 - 5259888473162400000\pi } \right)} \right.c^{4} + c^{3} \left[ { - 20214118900039680000 \, } \right. \\ & \quad + \, 7205401210807800000\pi + \left( {32072765077566259200} \right. - 4623912780987220875\pi \\ & \quad - 2526764862504960000\pi^{2} + \left. {300225050450325000\pi^{3} } \right)\left. {\rho \varOmega^{2} } \right] + c^{2} \left[ {6738039633346560000} \right. \\ & \quad - 3260484855936000000\pi + \left( { - 28595162967559372800} \right. + 4827328967260800000\pi \\ & \quad + 1684509908336640000\pi^{2} - \left. {271707071328000000\pi^{3} } \right)\rho \varOmega^{2} + \left( {91183675520488308736} \right. \\ & \quad - 4235045288859000000\pi - 10723186112834764800\pi^{2} + 603416120907600000\pi^{3} \\ & \quad + 105281869271040000\pi^{4} - \left. {10189015174800000\pi^{5} } \right)\left. {\rho^{2} \varOmega^{4} } \right] + c\left[ {543414142656000000\pi } \right. \\ & \quad + \left( {5409989686886400000} \right. - 1206832241815200000\pi + \left. {67926767832000000\pi^{3} } \right)\rho \varOmega^{2} \\ & \quad - 36066597912576000000 + 2117522644429500000\pi + 4057492265164800000\pi^{2} \\ & \quad - 301708060453800000\pi^{3} + \left. {5094507587400000\pi^{5} } \right)\rho^{2} \varOmega^{4} + \left( {182336689446912000000} \right. \\ & \quad - 2743974771160218750\pi - 22541623695360000000\pi^{2} + 441150550922812500\pi^{3} \\ & \quad + 422655444288000000\pi^{4} - 18856753778362500\pi^{5} + \left. {151622249625000\pi^{7} } \right) \\ & \quad \times {{\left. {\left. {\rho^{3} \varOmega^{6} } \right]} \right\}} \mathord{\left/ {\vphantom {{\left. {\left. {\rho^{3} \varOmega^{6} } \right]} \right\}} {2103884878233600000}}} \right. \kern-0pt} {2103884878233600000}} \\ \end{aligned} $$

$$ \begin{aligned} B_{2} &= \left\{ {72c^{2} F \, (} \right. - 11997568 + 4143195\pi ) - 735c\left[ {( - 8960 + 3447\pi )} \right.\left( {1 + 2F} \right) + F(1304576 - 41559\pi \\ & \quad - 161280\pi^{2} + \left. {20682\pi^{3} } \right)\left. {\rho \varOmega^{2} } \right] + 686\left[ {8640(1 + F) \, ( - 40 + 21\pi )} \right. + 120(1 + 2F) \, (2912 - 360\pi^{2} \\ & \quad + 63\pi^{3} )\rho \varOmega^{2} + F( - 2099200 + 262080\pi^{2} - 5400\pi^{4} + 567\pi^{5} ){{\left. {\left. {\rho^{2} \varOmega^{4} } \right]} \right\}} \mathord{\left/ {\vphantom {{\left. {\left. {\rho^{2} \varOmega^{4} } \right]} \right\}} {248935680}}} \right. \kern-0pt} {248935680}} \\ \end{aligned} $$

$$ B_{3} = ( - 40 \, + \, 21\pi ){\beta \mathord{\left/ {\vphantom {\beta {42}}} \right. \kern-0pt} {42}} $$

$$ \begin{aligned} D_{1} &=\,420078960000\pi + 3(929359872000 - 207316746000\pi + 11668860000\pi^{3} )\rho \varOmega^{2} \, \\ & \quad + \, 3( - 3097866240000 + 181880251875\pi + 348509952000\pi^{2} \, \\ & \quad - 25914593250\pi^{3} + 437582250\pi^{5} )\rho^{2} \varOmega^{4} \\ \end{aligned} $$

$$ \begin{aligned} D_{2} = &\,3(1157500108800 - 560105280000\pi ) + 3 \, ( - 2456122703872 + 414633492000\pi \\ & \;\;\;\; + 144687513600\pi^{2} - 23337720000\pi^{3} )\rho \varOmega^{2} \\ \end{aligned} $$

$$ D_{3} =\,3( - 1736250163200 + 618893115750\pi ) $$

$$ H = 38416000\left[ {\left( {1 + F} \right)} \right.\left( {21168\pi - 40320} \right) + F(40768 - 5040\pi^{2} + 882\pi^{3} )\left. {\rho \varOmega^{2} } \right] $$

$$ H_{1} = 38416000(80640F - 31023F\pi ). $$