静电驱动微浅拱梁跳跃和吸合特性的尺寸效应研究
|
摘要
基于修正偶应力理论研究了静电驱动微浅拱梁在考虑Casimir力时的跳跃和吸合特性。利用最小势能原理得到了微浅拱梁弯曲的控制方程和边界条件,应用广义微分求积法(GDQM)和广义积分求积法(GIQM)求解得到静电驱动微浅拱梁跳跃电压和吸合电压以及无量纲跳跃位移和吸合位移的数值解。结果表明,静电驱动微浅拱梁的跳跃特性和吸合特性具有明显的尺寸效应;基于修正偶应力理论的静电驱动微浅拱梁的跳跃和吸合电压低于经典理论值;Casimir力降低了微浅拱梁的跳跃和吸合电压以及无量纲跳跃和吸合位移;微浅拱梁初始拱高对其跳跃和吸合特性有着重要的影响。
The snap-through and pull-in instability of the electrostatically actuated micro shallow arches incorporating Casimir force was investigated based on the modified couple stress theory. The nonlinear governing equation and boundary conditions were derived by using the principle of minimum total potential energy. The snap-through voltages, the pull-in voltages, the non-dimensional snap-through displacements and the non-dimensional pull-in displacements of the electrostatically actuated micro shallow arches were calculated by applying the generalized differential quadrature method(GDQM) and the generalized integral quadrature method(GIQM). The results show that the snap-through and the pull-in instability of the electrostatically actuated micro shallow arches are size-dependent. The snap-through and pull-in voltages of the electrostatically actuated micro shallow arches based on the modified couple stress theory are smaller than the classical results. The Casimir force can reduce the snap-through voltages, the pull-in voltages, the non-dimensional snap-through and the pull-in displacements of the micro shallow arches. The initial rise of the micro shallow arches affects the snap-through and pull-in instability.
The snap-through and pull-in instability of the electrostatically actuated micro shallow arches incorporating Casimir force was investigated based on the modified couple stress theory. The nonlinear governing equation and boundary conditions were derived by using the principle of minimum total potential energy. The snap-through voltages, the pull-in voltages, the non-dimensional snap-through displacements and the non-dimensional pull-in displacements of the electrostatically actuated micro shallow arches were calculated by applying the generalized differential quadrature method(GDQM) and the generalized integral quadrature method(GIQM). The results show that the snap-through and the pull-in instability of the electrostatically actuated micro shallow arches are size-dependent. The snap-through and pull-in voltages of the electrostatically actuated micro shallow arches based on the modified couple stress theory are smaller than the classical results. The Casimir force can reduce the snap-through voltages, the pull-in voltages, the non-dimensional snap-through and the pull-in displacements of the micro shallow arches. The initial rise of the micro shallow arches affects the snap-through and pull-in instability.
引文
[1] G M Rebeiz.RF MEMS:Theory,Design,and Technology[M].Hoboken:John Wiley & Sons Inc,2004:1-15.
[2] W Wang,S Soper A.Bio-MEMS:Technologies and Applications[M].Boca Raton:CRC Press,2006:375-398.
[3] Medina L,Gilat R,Ilic B,et al.Experimental investigation of the snap-through buckling of electrostatically actuated initially curved pre-stressed microbeams[J].Sensors Actuators A,2014,220:323-332.
[4] 赵亚溥.纳米与介观力学[M].北京:科学出版社,2014:425-442.ZHAO YaPu.Nano and Mesoscopic Mechanics [M].Beijing:Science Press,2014:425-442 (In Chinese).
[5] 赵剑,贾建援,王洪喜等.双稳态屈曲梁的非线性跳跃特性研究[J].西安电子科技大学学报,2007,34(3):458-462.ZHAO Jian,JIA JianYuan,WANG HongXi et.al.Bistable buckling of beam nonlinear jump characteristic research[J].Journal of Xi 'an University of Electronic Science and Technology,2007,34(3):458-462 (In Chinese).
[6] 慕昆,于海鹏,基于双稳态机制的低g值微加速度开关[J].仪表技术与传感器,2014(6):16-19.MU Kun,YU HaiPeng.Low Value g Micro Acceleration Switch Based on Bistable Mechanism[J].Instrument Technique and Sensor,2014(6):16-19 (In Chinese).
[7] Krylov S,Dick N.Dynamic stability of electrostatically actuated initially curved shallow micro beams[J].Continuum Mechanics & Thermodynamics.2010,22(6-8):445-468.
[8] S Krylov,B R Ilic,S Lulinsky.Bistability of curved microbeams actuated by fringing electrostatic fields[J].Nonlinear Dynamics,2011(66):403-426.
[9] Younis M I,Ouakad H M,Alsaleem F M,et al.Nonlinear dynamics of MEMS arches under harmonic electrostatic actuation[J] Journal of Microelectromechanical Systems.2010,19(3):647-656.
[10] H M Ouakad,M I Younis.The dynamic behavior of MEMS arch resonators actuated electrically[J].International Journal of Non-Linear Mechanics.2010,45(7):704-713.
[11] Das K,Batra R C.Nonlinear behavior for nanoscales electrostatic actuators with Casimir force.Chaos,Solitons & Fractals,2005(23):1777-1785.
[12] K Das and R C Batra.Pull-in and snapthrough instabilities in transient deformations of microelectromechanical systems[J].Journal of Micromechanics and Microengineering.2009,19:035008
[13] Lin WenHui,Zhao YaPu.Stability and bifurcation behavior of electrostatic torsional NEMS varactor influenced by dispersion forces[J].Journal of Physics D-Applied Physics,2007,40 (6):1649-1654.
[14] D C C Lam,F Yang,A C M Chong,et al.Experiments and theory in strain gradient elasticity[J].Journal of the Mechanics and Physics of Solids,2003,51(8):1477-1508.
[15] A W McFarland,J S Colton.Role of material microstructure in plate stiffness with relevance to microcantilever sensors[J].Journal of Micromechanics and Microengineering,2005,15(5):1060-1067.
[16] Zhang Yin,Zhao YaPu.Measuring the nonlocal effects of a micro/nanobeam by the shifts of resonant frequencies.International Journal of Solids and Structures,2016(102-103):259-266.
[17] Yang F,Chong A C M,Lam D C C,et al.Couple stress based strain gradient theory for elasticity[J].International Journal of Solids and structures,2002,39(10):2731-2743.
[18] 赵亚溥.近代连续介质力学[M].北京:科学出版社,2016:455-463.ZHAO YaPu.Modern Continuum Mechanics [M].Beijing:Science Press,2016:455-463 (In Chinese).
[19] 孔胜利,周慎杰,聂志峰,等.基于偶应力理论的压杆屈曲载荷的尺寸效应[J].机械强度,2009,31(1):136-139.KONG ShengLi,ZHOU ShenJie,NIE ZhiFeng,et al.Size effect on the buckling loadings of slender columns based on a modified couple stress theory[J].Journal of mechanical strength,2009,31(1):136-139 (In Chinese).
[20] 赵俊峰,周慎杰,王炳雷,等.应变梯度对静电力驱动微梁吸合电压的影响[J].机械强度,2012,34(4):510-515.ZHAO JunFeng,ZHOU ShenJie,WANG BingLei,et al.Effect of strain gradient on the pull-in voltage of micro-beam actuated by electrostatic force[J].Journal of mechanical strength,2012,34(4):510-515 (In Chinese).
[21] Ghayesh M,Farokhi H,Alici G.Size-dependent electro-elasto-mechanics of MEMS with initially curved deformable electrodes[J].International Journal of Mechanical Sciences,2015,103:247-264.
[22] Shojaeian M,Beni Y T,Ataei H.Size-dependent snap-through and pull-in instabilities of initially curved pre-stressed electrostatic nano-bridges[J].Journal of Physics D-Applied Physics.2016,49(29):1-14.
[23] C Shu.Differential Quadrature and its Application in Engineering [M].London:Springer,2000:186-196.
[24] C Shu,H Du.A generalized approach for implementing general boundary conditions in the GDQ free vibration analysis of plates[J].International Journal of Solids & Structures,1997,34(7):837-846.
[25] H A Tilmans,R Legtenberg.Electrostatically driven vacuum-encapsulated polysilicon resonators:part II.Theory and performance[J].Sensors Actuators A.,1994,45:67-84.
[26] A H Nayfeh,S A Emam.Exact solution and stability of postbuckling configurations of beams[J].Nonlinear Dynamics,2008(54):395-408.
[2] W Wang,S Soper A.Bio-MEMS:Technologies and Applications[M].Boca Raton:CRC Press,2006:375-398.
[3] Medina L,Gilat R,Ilic B,et al.Experimental investigation of the snap-through buckling of electrostatically actuated initially curved pre-stressed microbeams[J].Sensors Actuators A,2014,220:323-332.
[4] 赵亚溥.纳米与介观力学[M].北京:科学出版社,2014:425-442.ZHAO YaPu.Nano and Mesoscopic Mechanics [M].Beijing:Science Press,2014:425-442 (In Chinese).
[5] 赵剑,贾建援,王洪喜等.双稳态屈曲梁的非线性跳跃特性研究[J].西安电子科技大学学报,2007,34(3):458-462.ZHAO Jian,JIA JianYuan,WANG HongXi et.al.Bistable buckling of beam nonlinear jump characteristic research[J].Journal of Xi 'an University of Electronic Science and Technology,2007,34(3):458-462 (In Chinese).
[6] 慕昆,于海鹏,基于双稳态机制的低g值微加速度开关[J].仪表技术与传感器,2014(6):16-19.MU Kun,YU HaiPeng.Low Value g Micro Acceleration Switch Based on Bistable Mechanism[J].Instrument Technique and Sensor,2014(6):16-19 (In Chinese).
[7] Krylov S,Dick N.Dynamic stability of electrostatically actuated initially curved shallow micro beams[J].Continuum Mechanics & Thermodynamics.2010,22(6-8):445-468.
[8] S Krylov,B R Ilic,S Lulinsky.Bistability of curved microbeams actuated by fringing electrostatic fields[J].Nonlinear Dynamics,2011(66):403-426.
[9] Younis M I,Ouakad H M,Alsaleem F M,et al.Nonlinear dynamics of MEMS arches under harmonic electrostatic actuation[J] Journal of Microelectromechanical Systems.2010,19(3):647-656.
[10] H M Ouakad,M I Younis.The dynamic behavior of MEMS arch resonators actuated electrically[J].International Journal of Non-Linear Mechanics.2010,45(7):704-713.
[11] Das K,Batra R C.Nonlinear behavior for nanoscales electrostatic actuators with Casimir force.Chaos,Solitons & Fractals,2005(23):1777-1785.
[12] K Das and R C Batra.Pull-in and snapthrough instabilities in transient deformations of microelectromechanical systems[J].Journal of Micromechanics and Microengineering.2009,19:035008
[13] Lin WenHui,Zhao YaPu.Stability and bifurcation behavior of electrostatic torsional NEMS varactor influenced by dispersion forces[J].Journal of Physics D-Applied Physics,2007,40 (6):1649-1654.
[14] D C C Lam,F Yang,A C M Chong,et al.Experiments and theory in strain gradient elasticity[J].Journal of the Mechanics and Physics of Solids,2003,51(8):1477-1508.
[15] A W McFarland,J S Colton.Role of material microstructure in plate stiffness with relevance to microcantilever sensors[J].Journal of Micromechanics and Microengineering,2005,15(5):1060-1067.
[16] Zhang Yin,Zhao YaPu.Measuring the nonlocal effects of a micro/nanobeam by the shifts of resonant frequencies.International Journal of Solids and Structures,2016(102-103):259-266.
[17] Yang F,Chong A C M,Lam D C C,et al.Couple stress based strain gradient theory for elasticity[J].International Journal of Solids and structures,2002,39(10):2731-2743.
[18] 赵亚溥.近代连续介质力学[M].北京:科学出版社,2016:455-463.ZHAO YaPu.Modern Continuum Mechanics [M].Beijing:Science Press,2016:455-463 (In Chinese).
[19] 孔胜利,周慎杰,聂志峰,等.基于偶应力理论的压杆屈曲载荷的尺寸效应[J].机械强度,2009,31(1):136-139.KONG ShengLi,ZHOU ShenJie,NIE ZhiFeng,et al.Size effect on the buckling loadings of slender columns based on a modified couple stress theory[J].Journal of mechanical strength,2009,31(1):136-139 (In Chinese).
[20] 赵俊峰,周慎杰,王炳雷,等.应变梯度对静电力驱动微梁吸合电压的影响[J].机械强度,2012,34(4):510-515.ZHAO JunFeng,ZHOU ShenJie,WANG BingLei,et al.Effect of strain gradient on the pull-in voltage of micro-beam actuated by electrostatic force[J].Journal of mechanical strength,2012,34(4):510-515 (In Chinese).
[21] Ghayesh M,Farokhi H,Alici G.Size-dependent electro-elasto-mechanics of MEMS with initially curved deformable electrodes[J].International Journal of Mechanical Sciences,2015,103:247-264.
[22] Shojaeian M,Beni Y T,Ataei H.Size-dependent snap-through and pull-in instabilities of initially curved pre-stressed electrostatic nano-bridges[J].Journal of Physics D-Applied Physics.2016,49(29):1-14.
[23] C Shu.Differential Quadrature and its Application in Engineering [M].London:Springer,2000:186-196.
[24] C Shu,H Du.A generalized approach for implementing general boundary conditions in the GDQ free vibration analysis of plates[J].International Journal of Solids & Structures,1997,34(7):837-846.
[25] H A Tilmans,R Legtenberg.Electrostatically driven vacuum-encapsulated polysilicon resonators:part II.Theory and performance[J].Sensors Actuators A.,1994,45:67-84.
[26] A H Nayfeh,S A Emam.Exact solution and stability of postbuckling configurations of beams[J].Nonlinear Dynamics,2008(54):395-408.