Abstract

A nanowaveguide resonator driven by a tunable optical gradient force can easily enter into the nonlinear oscillation regime, where the resonance frequency will shift. In this work, a continuum elastic model of the optoresonator is presented and solved analytically using the method of multiple scales. The effects of the optical gradient force on the resonance frequency and dynamic behavior are investigated. The results theoretically figure out why and when the nonlinear behavior of spring softening and spring hardening can occur. It is shown that the nonlinear phenomenon of spring softening is generally more dominant than the hardening effect when the optical gradient force is strong. However, the nonlinear cubic mechanical stiffness of the waveguide makes the dynamic behavior of spring softening dominant when the optical force is not strong enough. Based on the logical derivation of the closed-form solution, it can be found that the decrease of resonance frequency is due to the bias term, which is inherent in the nature of the tunable optical gradient force. Additionally, the complex variations of the resonance frequency and maximum vibration amplitude with different waveguide widths, lengths, and initial gaps are investigated and discussed. The proposed solutions are also verified with the reported experimental results.

© 2013 Optical Society of America

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2013 (5)

Y. He and D. Mihalache, “Lattice solitons in optical media described by the complex Ginzburg-Landau model with PT-symmetric periodic potentials,” Phys. Rev. A 87, 013812 (2013).
[CrossRef]

Y. He, L. Sun, S. He, and X. Yang, “Deep subwavelength beam propagation in extremely loss-anisotropic metamaterials,” J. Opt. 15, 055105 (2013).
[CrossRef]

Z. Y. Zhong, W. M. Zhang, G. Meng, and J. Wu, “Inclination effects on the frequency tuning of comb-driven resonators,” J. Microelectromech. Syst. 22, 865–875 (2013).
[CrossRef]

E. Gil-Santos, D. Ramos, V. Pini, J. Llorens, M. Fernández-Regúlez, M. Calleja, J. Tamayo, and A. San Paulo, “Optical back-action in silicon nanowire resonators: bolometric versus radiation pressure effects,” New J. Phys. 15, 035001 (2013).
[CrossRef]

Y. Sun, T. P. White, and A. A. Sukhorukov, “Coupled-mode theory analysis of optical forces between longitudinally shifted periodic waveguides,” J. Opt. Soc. Am. B 30, 736–742 (2013).
[CrossRef]

2012 (7)

M. Muradoglu, W. S.-Y. Chiu, and T. W. Ng, “Optical force lateral push–pulling using focus positioning,” J. Opt. Soc. Am. B 29, 874–880 (2012).
[CrossRef]

Y. He, S. He, and X. Yang, “Optical field enhancement in nanoscale slot waveguides of hyperbolic metamaterials,” Opt. Lett. 37, 2907–2909 (2012).
[CrossRef]

Y. He, S. He, J. Gao, and X. Yang, “Nanoscale metamaterial optical waveguides with ultrahigh refractive indices,” J. Opt. Soc. Am. B 29, 2559–2566 (2012).
[CrossRef]

Y. R. He, S. L. He, J. Gao, and X. D. Yang, “Giant transverse optical forces in nanoscale slot waveguides of hyperbolic metamaterials,” Opt. Express 20, 22372–22382 (2012).
[CrossRef]

A. Haldar, S. B. Pal, B. Roy, S. D. Gupta, and A. Banerjee, “Self-assembly of microparticles in stable ring structures in an optical trap,” Phys. Rev. A 85, 033832 (2012).
[CrossRef]

S. Zaitsev, O. Shtempluck, E. Buks, and O. Gottlieb, “Nonlinear damping in a micromechanical oscillator,” Nonlinear Dyn. 67, 859–883 (2012).
[CrossRef]

X. Sun, X. Zhang, and H. X. Tang, “High-Q silicon optomechanical microdisk resonators at gigahertz frequencies,” Appl. Phys. Lett. 100, 173116 (2012).
[CrossRef]

2011 (4)

A. M. Elshurafa, K. Khirallah, H. H. Tawfik, A. Emira, A. K. S. A. Aziz, and S. M. Sedky, “Nonlinear dynamics of spring softening and hardening in folded-MEMS comb drive resonators,” J. Microelectromech. Syst. 20, 943–958 (2011).
[CrossRef]

J. M. A. Eichler, J. Chaste, M. Zdrojek, I. Wilson-Rae, and A. Bachtold, “Nonlinear damping in mechanical resonators made from carbon nanotubes and graphene,” Nat. Nanotechnol. 6, 339–342 (2011).
[CrossRef]

J. Roels, B. Maes, W. Bogaerts, R. Baets, and D. V. Thourhout, “Parametric instability of an integrated micromechanical oscillator by means of active optomechanical feedback,” Opt. Express 19, 13081–13088 (2011).
[CrossRef]

D. R. Burnham and D. McGloin, “Modeling of optical traps for aerosols,” J. Opt. Soc. Am. B 28, 2856–2864 (2011).
[CrossRef]

2010 (4)

D. V. Thourhout and J. Roels, “Optomechanical device actuation through the optical gradient force,” Nat. Photonics 4, 211–217 (2010).
[CrossRef]

Q. Lin, J. Rosenberg, D. Chang, R. Camacho, M. Eichenfield, K. J. Vahala, and O. Painter, “Coherent mixing of mechanical excitations in nano-optomechanical structures,” Nat. Photonics 4, 236–242 (2010).
[CrossRef]

M. Li, W. H. P. Pernice, and H. X. Tang, “Ultrahigh-frequency nano-optomechanical resonators in slot waveguide ring cavities,” Appl. Phys. Lett. 97, 183110 (2010).
[CrossRef]

D. V. Seletskiy, S. D. Melgaard, S. Bigotta, A. D. Lieto, M. Tonelli, and M. Sheik-Bahae, “Laser cooling of solids to cryogenic temperatures,” Nat. Photonics 4, 161–164 (2010).
[CrossRef]

2009 (9)

M. Li, W. H. P. Pernice, and H. X. Tang, “Reactive cavity optical force on microdisk-coupled nanomechanical beam waveguides,” Phys. Rev. Lett. 103, 223901 (2009).
[CrossRef]

M. Li, W. H. P. Pernice, and H. X. Tang, “Tunable bipolar optical interactions between guided lightwaves,” Nat. Photonics 3, 464–468 (2009).
[CrossRef]

J. Roels, I. de Vlaminck, L. Lagae, B. Maes, D. Van Thourhout, and R. Baets, “Tunable optical forces between nanophotonic waveguides,” Nat. Nanotechnol. 4, 510–513 (2009).
[CrossRef]

J. Rosenberg, Q. Lin, and O. Painter, “Static and dynamic wavelength routing via the gradient optical force,” Nat. Photonics 3, 478–483 (2009).
[CrossRef]

N. Kacem, S. Hentz, D. Pinto, B. Reig, and V. Nguyen, “Nonlinear dynamics of nanomechanical beam resonators: improving the performance of NEMS-based sensors,” Nanotechnology 20, 275501 (2009).
[CrossRef]

Q. Lin, J. Rosenberg, X. S. Jiang, K. J. Vahala, and O. Painter, “Mechanical oscillation and cooling actuated by the optical gradient force,” Phys. Rev. Lett. 103, 103601 (2009).
[CrossRef]

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459, 550–555 (2009).
[CrossRef]

X. Wei, C. Anthony, D. Lowe, and M. Ward, “Design and fabrication of a nonlinear micro impact oscillator,” Proc. Chem. 1, 855–858 (2009).
[CrossRef]

V. Liu, M. Povinelli, and S. Fan, “Resonance-enhanced optical forces between coupled photonic crystal slabs,” Opt. Express 17, 21897–21909 (2009).
[CrossRef]

2008 (1)

M. Li, W. H. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. X. Tang, “Harnessing optical forces in integrated photonic circuits,” Nature 456, 480–484 (2008).
[CrossRef]

2007 (2)

M. Sheik-Bahae and R. I. Epstein, “Optical refrigeration,” Nat. Photonics 1, 693–699 (2007).
[CrossRef]

M. Bao and H. Yang, “Squeeze film air damping in MEMS,” Sens. Actuators A 136, 3–27 (2007).
[CrossRef]

2005 (1)

Anthony, C.

X. Wei, C. Anthony, D. Lowe, and M. Ward, “Design and fabrication of a nonlinear micro impact oscillator,” Proc. Chem. 1, 855–858 (2009).
[CrossRef]

Aziz, A. K. S. A.

A. M. Elshurafa, K. Khirallah, H. H. Tawfik, A. Emira, A. K. S. A. Aziz, and S. M. Sedky, “Nonlinear dynamics of spring softening and hardening in folded-MEMS comb drive resonators,” J. Microelectromech. Syst. 20, 943–958 (2011).
[CrossRef]

Bachtold, A.

J. M. A. Eichler, J. Chaste, M. Zdrojek, I. Wilson-Rae, and A. Bachtold, “Nonlinear damping in mechanical resonators made from carbon nanotubes and graphene,” Nat. Nanotechnol. 6, 339–342 (2011).
[CrossRef]

Baehr-Jones, T.

M. Li, W. H. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. X. Tang, “Harnessing optical forces in integrated photonic circuits,” Nature 456, 480–484 (2008).
[CrossRef]

Baets, R.

J. Roels, B. Maes, W. Bogaerts, R. Baets, and D. V. Thourhout, “Parametric instability of an integrated micromechanical oscillator by means of active optomechanical feedback,” Opt. Express 19, 13081–13088 (2011).
[CrossRef]

J. Roels, I. de Vlaminck, L. Lagae, B. Maes, D. Van Thourhout, and R. Baets, “Tunable optical forces between nanophotonic waveguides,” Nat. Nanotechnol. 4, 510–513 (2009).
[CrossRef]

Banerjee, A.

A. Haldar, S. B. Pal, B. Roy, S. D. Gupta, and A. Banerjee, “Self-assembly of microparticles in stable ring structures in an optical trap,” Phys. Rev. A 85, 033832 (2012).
[CrossRef]

Bao, M.

M. Bao and H. Yang, “Squeeze film air damping in MEMS,” Sens. Actuators A 136, 3–27 (2007).
[CrossRef]

Bigotta, S.

D. V. Seletskiy, S. D. Melgaard, S. Bigotta, A. D. Lieto, M. Tonelli, and M. Sheik-Bahae, “Laser cooling of solids to cryogenic temperatures,” Nat. Photonics 4, 161–164 (2010).
[CrossRef]

Bogaerts, W.

Buks, E.

S. Zaitsev, O. Shtempluck, E. Buks, and O. Gottlieb, “Nonlinear damping in a micromechanical oscillator,” Nonlinear Dyn. 67, 859–883 (2012).
[CrossRef]

Burnham, D. R.

Calleja, M.

E. Gil-Santos, D. Ramos, V. Pini, J. Llorens, M. Fernández-Regúlez, M. Calleja, J. Tamayo, and A. San Paulo, “Optical back-action in silicon nanowire resonators: bolometric versus radiation pressure effects,” New J. Phys. 15, 035001 (2013).
[CrossRef]

Camacho, R.

Q. Lin, J. Rosenberg, D. Chang, R. Camacho, M. Eichenfield, K. J. Vahala, and O. Painter, “Coherent mixing of mechanical excitations in nano-optomechanical structures,” Nat. Photonics 4, 236–242 (2010).
[CrossRef]

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459, 550–555 (2009).
[CrossRef]

Capasso, F.

Chan, J.

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459, 550–555 (2009).
[CrossRef]

Chang, D.

Q. Lin, J. Rosenberg, D. Chang, R. Camacho, M. Eichenfield, K. J. Vahala, and O. Painter, “Coherent mixing of mechanical excitations in nano-optomechanical structures,” Nat. Photonics 4, 236–242 (2010).
[CrossRef]

Chaste, J.

J. M. A. Eichler, J. Chaste, M. Zdrojek, I. Wilson-Rae, and A. Bachtold, “Nonlinear damping in mechanical resonators made from carbon nanotubes and graphene,” Nat. Nanotechnol. 6, 339–342 (2011).
[CrossRef]

Chiu, W. S.-Y.

de Vlaminck, I.

J. Roels, I. de Vlaminck, L. Lagae, B. Maes, D. Van Thourhout, and R. Baets, “Tunable optical forces between nanophotonic waveguides,” Nat. Nanotechnol. 4, 510–513 (2009).
[CrossRef]

Eichenfield, M.

Q. Lin, J. Rosenberg, D. Chang, R. Camacho, M. Eichenfield, K. J. Vahala, and O. Painter, “Coherent mixing of mechanical excitations in nano-optomechanical structures,” Nat. Photonics 4, 236–242 (2010).
[CrossRef]

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459, 550–555 (2009).
[CrossRef]

Eichler, J. M. A.

J. M. A. Eichler, J. Chaste, M. Zdrojek, I. Wilson-Rae, and A. Bachtold, “Nonlinear damping in mechanical resonators made from carbon nanotubes and graphene,” Nat. Nanotechnol. 6, 339–342 (2011).
[CrossRef]

Elshurafa, A. M.

A. M. Elshurafa, K. Khirallah, H. H. Tawfik, A. Emira, A. K. S. A. Aziz, and S. M. Sedky, “Nonlinear dynamics of spring softening and hardening in folded-MEMS comb drive resonators,” J. Microelectromech. Syst. 20, 943–958 (2011).
[CrossRef]

Emira, A.

A. M. Elshurafa, K. Khirallah, H. H. Tawfik, A. Emira, A. K. S. A. Aziz, and S. M. Sedky, “Nonlinear dynamics of spring softening and hardening in folded-MEMS comb drive resonators,” J. Microelectromech. Syst. 20, 943–958 (2011).
[CrossRef]

Epstein, R. I.

M. Sheik-Bahae and R. I. Epstein, “Optical refrigeration,” Nat. Photonics 1, 693–699 (2007).
[CrossRef]

Fan, S.

Fernández-Regúlez, M.

E. Gil-Santos, D. Ramos, V. Pini, J. Llorens, M. Fernández-Regúlez, M. Calleja, J. Tamayo, and A. San Paulo, “Optical back-action in silicon nanowire resonators: bolometric versus radiation pressure effects,” New J. Phys. 15, 035001 (2013).
[CrossRef]

Gao, J.

Gil-Santos, E.

E. Gil-Santos, D. Ramos, V. Pini, J. Llorens, M. Fernández-Regúlez, M. Calleja, J. Tamayo, and A. San Paulo, “Optical back-action in silicon nanowire resonators: bolometric versus radiation pressure effects,” New J. Phys. 15, 035001 (2013).
[CrossRef]

Gottlieb, O.

S. Zaitsev, O. Shtempluck, E. Buks, and O. Gottlieb, “Nonlinear damping in a micromechanical oscillator,” Nonlinear Dyn. 67, 859–883 (2012).
[CrossRef]

Gupta, S. D.

A. Haldar, S. B. Pal, B. Roy, S. D. Gupta, and A. Banerjee, “Self-assembly of microparticles in stable ring structures in an optical trap,” Phys. Rev. A 85, 033832 (2012).
[CrossRef]

Haldar, A.

A. Haldar, S. B. Pal, B. Roy, S. D. Gupta, and A. Banerjee, “Self-assembly of microparticles in stable ring structures in an optical trap,” Phys. Rev. A 85, 033832 (2012).
[CrossRef]

He, S.

He, S. L.

He, Y.

Y. He, L. Sun, S. He, and X. Yang, “Deep subwavelength beam propagation in extremely loss-anisotropic metamaterials,” J. Opt. 15, 055105 (2013).
[CrossRef]

Y. He and D. Mihalache, “Lattice solitons in optical media described by the complex Ginzburg-Landau model with PT-symmetric periodic potentials,” Phys. Rev. A 87, 013812 (2013).
[CrossRef]

Y. He, S. He, and X. Yang, “Optical field enhancement in nanoscale slot waveguides of hyperbolic metamaterials,” Opt. Lett. 37, 2907–2909 (2012).
[CrossRef]

Y. He, S. He, J. Gao, and X. Yang, “Nanoscale metamaterial optical waveguides with ultrahigh refractive indices,” J. Opt. Soc. Am. B 29, 2559–2566 (2012).
[CrossRef]

He, Y. R.

Hentz, S.

N. Kacem, S. Hentz, D. Pinto, B. Reig, and V. Nguyen, “Nonlinear dynamics of nanomechanical beam resonators: improving the performance of NEMS-based sensors,” Nanotechnology 20, 275501 (2009).
[CrossRef]

Hochberg, M.

M. Li, W. H. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. X. Tang, “Harnessing optical forces in integrated photonic circuits,” Nature 456, 480–484 (2008).
[CrossRef]

Ibanescu, M.

Jiang, X. S.

Q. Lin, J. Rosenberg, X. S. Jiang, K. J. Vahala, and O. Painter, “Mechanical oscillation and cooling actuated by the optical gradient force,” Phys. Rev. Lett. 103, 103601 (2009).
[CrossRef]

Johnson, S. G.

Kacem, N.

N. Kacem, S. Hentz, D. Pinto, B. Reig, and V. Nguyen, “Nonlinear dynamics of nanomechanical beam resonators: improving the performance of NEMS-based sensors,” Nanotechnology 20, 275501 (2009).
[CrossRef]

Khirallah, K.

A. M. Elshurafa, K. Khirallah, H. H. Tawfik, A. Emira, A. K. S. A. Aziz, and S. M. Sedky, “Nonlinear dynamics of spring softening and hardening in folded-MEMS comb drive resonators,” J. Microelectromech. Syst. 20, 943–958 (2011).
[CrossRef]

Lagae, L.

J. Roels, I. de Vlaminck, L. Lagae, B. Maes, D. Van Thourhout, and R. Baets, “Tunable optical forces between nanophotonic waveguides,” Nat. Nanotechnol. 4, 510–513 (2009).
[CrossRef]

Li, M.

M. Li, W. H. P. Pernice, and H. X. Tang, “Ultrahigh-frequency nano-optomechanical resonators in slot waveguide ring cavities,” Appl. Phys. Lett. 97, 183110 (2010).
[CrossRef]

M. Li, W. H. P. Pernice, and H. X. Tang, “Tunable bipolar optical interactions between guided lightwaves,” Nat. Photonics 3, 464–468 (2009).
[CrossRef]

M. Li, W. H. P. Pernice, and H. X. Tang, “Reactive cavity optical force on microdisk-coupled nanomechanical beam waveguides,” Phys. Rev. Lett. 103, 223901 (2009).
[CrossRef]

M. Li, W. H. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. X. Tang, “Harnessing optical forces in integrated photonic circuits,” Nature 456, 480–484 (2008).
[CrossRef]

Lieto, A. D.

D. V. Seletskiy, S. D. Melgaard, S. Bigotta, A. D. Lieto, M. Tonelli, and M. Sheik-Bahae, “Laser cooling of solids to cryogenic temperatures,” Nat. Photonics 4, 161–164 (2010).
[CrossRef]

Lin, Q.

Q. Lin, J. Rosenberg, D. Chang, R. Camacho, M. Eichenfield, K. J. Vahala, and O. Painter, “Coherent mixing of mechanical excitations in nano-optomechanical structures,” Nat. Photonics 4, 236–242 (2010).
[CrossRef]

Q. Lin, J. Rosenberg, X. S. Jiang, K. J. Vahala, and O. Painter, “Mechanical oscillation and cooling actuated by the optical gradient force,” Phys. Rev. Lett. 103, 103601 (2009).
[CrossRef]

J. Rosenberg, Q. Lin, and O. Painter, “Static and dynamic wavelength routing via the gradient optical force,” Nat. Photonics 3, 478–483 (2009).
[CrossRef]

Liu, V.

Llorens, J.

E. Gil-Santos, D. Ramos, V. Pini, J. Llorens, M. Fernández-Regúlez, M. Calleja, J. Tamayo, and A. San Paulo, “Optical back-action in silicon nanowire resonators: bolometric versus radiation pressure effects,” New J. Phys. 15, 035001 (2013).
[CrossRef]

Loncar, M.

Lowe, D.

X. Wei, C. Anthony, D. Lowe, and M. Ward, “Design and fabrication of a nonlinear micro impact oscillator,” Proc. Chem. 1, 855–858 (2009).
[CrossRef]

Maes, B.

J. Roels, B. Maes, W. Bogaerts, R. Baets, and D. V. Thourhout, “Parametric instability of an integrated micromechanical oscillator by means of active optomechanical feedback,” Opt. Express 19, 13081–13088 (2011).
[CrossRef]

J. Roels, I. de Vlaminck, L. Lagae, B. Maes, D. Van Thourhout, and R. Baets, “Tunable optical forces between nanophotonic waveguides,” Nat. Nanotechnol. 4, 510–513 (2009).
[CrossRef]

McGloin, D.

Melgaard, S. D.

D. V. Seletskiy, S. D. Melgaard, S. Bigotta, A. D. Lieto, M. Tonelli, and M. Sheik-Bahae, “Laser cooling of solids to cryogenic temperatures,” Nat. Photonics 4, 161–164 (2010).
[CrossRef]

Meng, G.

Z. Y. Zhong, W. M. Zhang, G. Meng, and J. Wu, “Inclination effects on the frequency tuning of comb-driven resonators,” J. Microelectromech. Syst. 22, 865–875 (2013).
[CrossRef]

Mihalache, D.

Y. He and D. Mihalache, “Lattice solitons in optical media described by the complex Ginzburg-Landau model with PT-symmetric periodic potentials,” Phys. Rev. A 87, 013812 (2013).
[CrossRef]

Muradoglu, M.

Ng, T. W.

Nguyen, V.

N. Kacem, S. Hentz, D. Pinto, B. Reig, and V. Nguyen, “Nonlinear dynamics of nanomechanical beam resonators: improving the performance of NEMS-based sensors,” Nanotechnology 20, 275501 (2009).
[CrossRef]

Painter, O.

Q. Lin, J. Rosenberg, D. Chang, R. Camacho, M. Eichenfield, K. J. Vahala, and O. Painter, “Coherent mixing of mechanical excitations in nano-optomechanical structures,” Nat. Photonics 4, 236–242 (2010).
[CrossRef]

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459, 550–555 (2009).
[CrossRef]

J. Rosenberg, Q. Lin, and O. Painter, “Static and dynamic wavelength routing via the gradient optical force,” Nat. Photonics 3, 478–483 (2009).
[CrossRef]

Q. Lin, J. Rosenberg, X. S. Jiang, K. J. Vahala, and O. Painter, “Mechanical oscillation and cooling actuated by the optical gradient force,” Phys. Rev. Lett. 103, 103601 (2009).
[CrossRef]

Pal, S. B.

A. Haldar, S. B. Pal, B. Roy, S. D. Gupta, and A. Banerjee, “Self-assembly of microparticles in stable ring structures in an optical trap,” Phys. Rev. A 85, 033832 (2012).
[CrossRef]

Pernice, W. H.

M. Li, W. H. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. X. Tang, “Harnessing optical forces in integrated photonic circuits,” Nature 456, 480–484 (2008).
[CrossRef]

Pernice, W. H. P.

M. Li, W. H. P. Pernice, and H. X. Tang, “Ultrahigh-frequency nano-optomechanical resonators in slot waveguide ring cavities,” Appl. Phys. Lett. 97, 183110 (2010).
[CrossRef]

M. Li, W. H. P. Pernice, and H. X. Tang, “Tunable bipolar optical interactions between guided lightwaves,” Nat. Photonics 3, 464–468 (2009).
[CrossRef]

M. Li, W. H. P. Pernice, and H. X. Tang, “Reactive cavity optical force on microdisk-coupled nanomechanical beam waveguides,” Phys. Rev. Lett. 103, 223901 (2009).
[CrossRef]

Pini, V.

E. Gil-Santos, D. Ramos, V. Pini, J. Llorens, M. Fernández-Regúlez, M. Calleja, J. Tamayo, and A. San Paulo, “Optical back-action in silicon nanowire resonators: bolometric versus radiation pressure effects,” New J. Phys. 15, 035001 (2013).
[CrossRef]

Pinto, D.

N. Kacem, S. Hentz, D. Pinto, B. Reig, and V. Nguyen, “Nonlinear dynamics of nanomechanical beam resonators: improving the performance of NEMS-based sensors,” Nanotechnology 20, 275501 (2009).
[CrossRef]

Povinelli, M.

Povinelli, M. L.

Ramos, D.

E. Gil-Santos, D. Ramos, V. Pini, J. Llorens, M. Fernández-Regúlez, M. Calleja, J. Tamayo, and A. San Paulo, “Optical back-action in silicon nanowire resonators: bolometric versus radiation pressure effects,” New J. Phys. 15, 035001 (2013).
[CrossRef]

Reig, B.

N. Kacem, S. Hentz, D. Pinto, B. Reig, and V. Nguyen, “Nonlinear dynamics of nanomechanical beam resonators: improving the performance of NEMS-based sensors,” Nanotechnology 20, 275501 (2009).
[CrossRef]

Roels, J.

J. Roels, B. Maes, W. Bogaerts, R. Baets, and D. V. Thourhout, “Parametric instability of an integrated micromechanical oscillator by means of active optomechanical feedback,” Opt. Express 19, 13081–13088 (2011).
[CrossRef]

D. V. Thourhout and J. Roels, “Optomechanical device actuation through the optical gradient force,” Nat. Photonics 4, 211–217 (2010).
[CrossRef]

J. Roels, I. de Vlaminck, L. Lagae, B. Maes, D. Van Thourhout, and R. Baets, “Tunable optical forces between nanophotonic waveguides,” Nat. Nanotechnol. 4, 510–513 (2009).
[CrossRef]

Rosenberg, J.

Q. Lin, J. Rosenberg, D. Chang, R. Camacho, M. Eichenfield, K. J. Vahala, and O. Painter, “Coherent mixing of mechanical excitations in nano-optomechanical structures,” Nat. Photonics 4, 236–242 (2010).
[CrossRef]

J. Rosenberg, Q. Lin, and O. Painter, “Static and dynamic wavelength routing via the gradient optical force,” Nat. Photonics 3, 478–483 (2009).
[CrossRef]

Q. Lin, J. Rosenberg, X. S. Jiang, K. J. Vahala, and O. Painter, “Mechanical oscillation and cooling actuated by the optical gradient force,” Phys. Rev. Lett. 103, 103601 (2009).
[CrossRef]

Roy, B.

A. Haldar, S. B. Pal, B. Roy, S. D. Gupta, and A. Banerjee, “Self-assembly of microparticles in stable ring structures in an optical trap,” Phys. Rev. A 85, 033832 (2012).
[CrossRef]

San Paulo, A.

E. Gil-Santos, D. Ramos, V. Pini, J. Llorens, M. Fernández-Regúlez, M. Calleja, J. Tamayo, and A. San Paulo, “Optical back-action in silicon nanowire resonators: bolometric versus radiation pressure effects,” New J. Phys. 15, 035001 (2013).
[CrossRef]

Sedky, S. M.

A. M. Elshurafa, K. Khirallah, H. H. Tawfik, A. Emira, A. K. S. A. Aziz, and S. M. Sedky, “Nonlinear dynamics of spring softening and hardening in folded-MEMS comb drive resonators,” J. Microelectromech. Syst. 20, 943–958 (2011).
[CrossRef]

Seletskiy, D. V.

D. V. Seletskiy, S. D. Melgaard, S. Bigotta, A. D. Lieto, M. Tonelli, and M. Sheik-Bahae, “Laser cooling of solids to cryogenic temperatures,” Nat. Photonics 4, 161–164 (2010).
[CrossRef]

Sheik-Bahae, M.

D. V. Seletskiy, S. D. Melgaard, S. Bigotta, A. D. Lieto, M. Tonelli, and M. Sheik-Bahae, “Laser cooling of solids to cryogenic temperatures,” Nat. Photonics 4, 161–164 (2010).
[CrossRef]

M. Sheik-Bahae and R. I. Epstein, “Optical refrigeration,” Nat. Photonics 1, 693–699 (2007).
[CrossRef]

Shtempluck, O.

S. Zaitsev, O. Shtempluck, E. Buks, and O. Gottlieb, “Nonlinear damping in a micromechanical oscillator,” Nonlinear Dyn. 67, 859–883 (2012).
[CrossRef]

Smythe, E. J.

Sukhorukov, A. A.

Sun, L.

Y. He, L. Sun, S. He, and X. Yang, “Deep subwavelength beam propagation in extremely loss-anisotropic metamaterials,” J. Opt. 15, 055105 (2013).
[CrossRef]

Sun, X.

X. Sun, X. Zhang, and H. X. Tang, “High-Q silicon optomechanical microdisk resonators at gigahertz frequencies,” Appl. Phys. Lett. 100, 173116 (2012).
[CrossRef]

Sun, Y.

Tamayo, J.

E. Gil-Santos, D. Ramos, V. Pini, J. Llorens, M. Fernández-Regúlez, M. Calleja, J. Tamayo, and A. San Paulo, “Optical back-action in silicon nanowire resonators: bolometric versus radiation pressure effects,” New J. Phys. 15, 035001 (2013).
[CrossRef]

Tang, H. X.

X. Sun, X. Zhang, and H. X. Tang, “High-Q silicon optomechanical microdisk resonators at gigahertz frequencies,” Appl. Phys. Lett. 100, 173116 (2012).
[CrossRef]

M. Li, W. H. P. Pernice, and H. X. Tang, “Ultrahigh-frequency nano-optomechanical resonators in slot waveguide ring cavities,” Appl. Phys. Lett. 97, 183110 (2010).
[CrossRef]

M. Li, W. H. P. Pernice, and H. X. Tang, “Tunable bipolar optical interactions between guided lightwaves,” Nat. Photonics 3, 464–468 (2009).
[CrossRef]

M. Li, W. H. P. Pernice, and H. X. Tang, “Reactive cavity optical force on microdisk-coupled nanomechanical beam waveguides,” Phys. Rev. Lett. 103, 223901 (2009).
[CrossRef]

M. Li, W. H. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. X. Tang, “Harnessing optical forces in integrated photonic circuits,” Nature 456, 480–484 (2008).
[CrossRef]

Tawfik, H. H.

A. M. Elshurafa, K. Khirallah, H. H. Tawfik, A. Emira, A. K. S. A. Aziz, and S. M. Sedky, “Nonlinear dynamics of spring softening and hardening in folded-MEMS comb drive resonators,” J. Microelectromech. Syst. 20, 943–958 (2011).
[CrossRef]

Thourhout, D. V.

Timoshenko, S.

S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells (McGraw-Hill, 1959).

Tonelli, M.

D. V. Seletskiy, S. D. Melgaard, S. Bigotta, A. D. Lieto, M. Tonelli, and M. Sheik-Bahae, “Laser cooling of solids to cryogenic temperatures,” Nat. Photonics 4, 161–164 (2010).
[CrossRef]

Vahala, K. J.

Q. Lin, J. Rosenberg, D. Chang, R. Camacho, M. Eichenfield, K. J. Vahala, and O. Painter, “Coherent mixing of mechanical excitations in nano-optomechanical structures,” Nat. Photonics 4, 236–242 (2010).
[CrossRef]

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459, 550–555 (2009).
[CrossRef]

Q. Lin, J. Rosenberg, X. S. Jiang, K. J. Vahala, and O. Painter, “Mechanical oscillation and cooling actuated by the optical gradient force,” Phys. Rev. Lett. 103, 103601 (2009).
[CrossRef]

Van Thourhout, D.

J. Roels, I. de Vlaminck, L. Lagae, B. Maes, D. Van Thourhout, and R. Baets, “Tunable optical forces between nanophotonic waveguides,” Nat. Nanotechnol. 4, 510–513 (2009).
[CrossRef]

Ward, M.

X. Wei, C. Anthony, D. Lowe, and M. Ward, “Design and fabrication of a nonlinear micro impact oscillator,” Proc. Chem. 1, 855–858 (2009).
[CrossRef]

Wei, X.

X. Wei, C. Anthony, D. Lowe, and M. Ward, “Design and fabrication of a nonlinear micro impact oscillator,” Proc. Chem. 1, 855–858 (2009).
[CrossRef]

White, T. P.

Wilson-Rae, I.

J. M. A. Eichler, J. Chaste, M. Zdrojek, I. Wilson-Rae, and A. Bachtold, “Nonlinear damping in mechanical resonators made from carbon nanotubes and graphene,” Nat. Nanotechnol. 6, 339–342 (2011).
[CrossRef]

Woinowsky-Krieger, S.

S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells (McGraw-Hill, 1959).

Wu, J.

Z. Y. Zhong, W. M. Zhang, G. Meng, and J. Wu, “Inclination effects on the frequency tuning of comb-driven resonators,” J. Microelectromech. Syst. 22, 865–875 (2013).
[CrossRef]

Xiong, C.

M. Li, W. H. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. X. Tang, “Harnessing optical forces in integrated photonic circuits,” Nature 456, 480–484 (2008).
[CrossRef]

Yang, H.

M. Bao and H. Yang, “Squeeze film air damping in MEMS,” Sens. Actuators A 136, 3–27 (2007).
[CrossRef]

Yang, X.

Yang, X. D.

Zaitsev, S.

S. Zaitsev, O. Shtempluck, E. Buks, and O. Gottlieb, “Nonlinear damping in a micromechanical oscillator,” Nonlinear Dyn. 67, 859–883 (2012).
[CrossRef]

Zdrojek, M.

J. M. A. Eichler, J. Chaste, M. Zdrojek, I. Wilson-Rae, and A. Bachtold, “Nonlinear damping in mechanical resonators made from carbon nanotubes and graphene,” Nat. Nanotechnol. 6, 339–342 (2011).
[CrossRef]

Zhang, W. M.

Z. Y. Zhong, W. M. Zhang, G. Meng, and J. Wu, “Inclination effects on the frequency tuning of comb-driven resonators,” J. Microelectromech. Syst. 22, 865–875 (2013).
[CrossRef]

Zhang, X.

X. Sun, X. Zhang, and H. X. Tang, “High-Q silicon optomechanical microdisk resonators at gigahertz frequencies,” Appl. Phys. Lett. 100, 173116 (2012).
[CrossRef]

Zhong, Z. Y.

Z. Y. Zhong, W. M. Zhang, G. Meng, and J. Wu, “Inclination effects on the frequency tuning of comb-driven resonators,” J. Microelectromech. Syst. 22, 865–875 (2013).
[CrossRef]

Appl. Phys. Lett. (2)

M. Li, W. H. P. Pernice, and H. X. Tang, “Ultrahigh-frequency nano-optomechanical resonators in slot waveguide ring cavities,” Appl. Phys. Lett. 97, 183110 (2010).
[CrossRef]

X. Sun, X. Zhang, and H. X. Tang, “High-Q silicon optomechanical microdisk resonators at gigahertz frequencies,” Appl. Phys. Lett. 100, 173116 (2012).
[CrossRef]

J. Microelectromech. Syst. (2)

Z. Y. Zhong, W. M. Zhang, G. Meng, and J. Wu, “Inclination effects on the frequency tuning of comb-driven resonators,” J. Microelectromech. Syst. 22, 865–875 (2013).
[CrossRef]

A. M. Elshurafa, K. Khirallah, H. H. Tawfik, A. Emira, A. K. S. A. Aziz, and S. M. Sedky, “Nonlinear dynamics of spring softening and hardening in folded-MEMS comb drive resonators,” J. Microelectromech. Syst. 20, 943–958 (2011).
[CrossRef]

J. Opt. (1)

Y. He, L. Sun, S. He, and X. Yang, “Deep subwavelength beam propagation in extremely loss-anisotropic metamaterials,” J. Opt. 15, 055105 (2013).
[CrossRef]

J. Opt. Soc. Am. B (4)

Nanotechnology (1)

N. Kacem, S. Hentz, D. Pinto, B. Reig, and V. Nguyen, “Nonlinear dynamics of nanomechanical beam resonators: improving the performance of NEMS-based sensors,” Nanotechnology 20, 275501 (2009).
[CrossRef]

Nat. Nanotechnol. (2)

J. M. A. Eichler, J. Chaste, M. Zdrojek, I. Wilson-Rae, and A. Bachtold, “Nonlinear damping in mechanical resonators made from carbon nanotubes and graphene,” Nat. Nanotechnol. 6, 339–342 (2011).
[CrossRef]

J. Roels, I. de Vlaminck, L. Lagae, B. Maes, D. Van Thourhout, and R. Baets, “Tunable optical forces between nanophotonic waveguides,” Nat. Nanotechnol. 4, 510–513 (2009).
[CrossRef]

Nat. Photonics (6)

M. Li, W. H. P. Pernice, and H. X. Tang, “Tunable bipolar optical interactions between guided lightwaves,” Nat. Photonics 3, 464–468 (2009).
[CrossRef]

Q. Lin, J. Rosenberg, D. Chang, R. Camacho, M. Eichenfield, K. J. Vahala, and O. Painter, “Coherent mixing of mechanical excitations in nano-optomechanical structures,” Nat. Photonics 4, 236–242 (2010).
[CrossRef]

J. Rosenberg, Q. Lin, and O. Painter, “Static and dynamic wavelength routing via the gradient optical force,” Nat. Photonics 3, 478–483 (2009).
[CrossRef]

D. V. Seletskiy, S. D. Melgaard, S. Bigotta, A. D. Lieto, M. Tonelli, and M. Sheik-Bahae, “Laser cooling of solids to cryogenic temperatures,” Nat. Photonics 4, 161–164 (2010).
[CrossRef]

D. V. Thourhout and J. Roels, “Optomechanical device actuation through the optical gradient force,” Nat. Photonics 4, 211–217 (2010).
[CrossRef]

M. Sheik-Bahae and R. I. Epstein, “Optical refrigeration,” Nat. Photonics 1, 693–699 (2007).
[CrossRef]

Nature (2)

M. Li, W. H. Pernice, C. Xiong, T. Baehr-Jones, M. Hochberg, and H. X. Tang, “Harnessing optical forces in integrated photonic circuits,” Nature 456, 480–484 (2008).
[CrossRef]

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram- and nanometre-scale photonic-crystal optomechanical cavity,” Nature 459, 550–555 (2009).
[CrossRef]

New J. Phys. (1)

E. Gil-Santos, D. Ramos, V. Pini, J. Llorens, M. Fernández-Regúlez, M. Calleja, J. Tamayo, and A. San Paulo, “Optical back-action in silicon nanowire resonators: bolometric versus radiation pressure effects,” New J. Phys. 15, 035001 (2013).
[CrossRef]

Nonlinear Dyn. (1)

S. Zaitsev, O. Shtempluck, E. Buks, and O. Gottlieb, “Nonlinear damping in a micromechanical oscillator,” Nonlinear Dyn. 67, 859–883 (2012).
[CrossRef]

Opt. Express (3)

Opt. Lett. (2)

Phys. Rev. A (2)

A. Haldar, S. B. Pal, B. Roy, S. D. Gupta, and A. Banerjee, “Self-assembly of microparticles in stable ring structures in an optical trap,” Phys. Rev. A 85, 033832 (2012).
[CrossRef]

Y. He and D. Mihalache, “Lattice solitons in optical media described by the complex Ginzburg-Landau model with PT-symmetric periodic potentials,” Phys. Rev. A 87, 013812 (2013).
[CrossRef]

Phys. Rev. Lett. (2)

M. Li, W. H. P. Pernice, and H. X. Tang, “Reactive cavity optical force on microdisk-coupled nanomechanical beam waveguides,” Phys. Rev. Lett. 103, 223901 (2009).
[CrossRef]

Q. Lin, J. Rosenberg, X. S. Jiang, K. J. Vahala, and O. Painter, “Mechanical oscillation and cooling actuated by the optical gradient force,” Phys. Rev. Lett. 103, 103601 (2009).
[CrossRef]

Proc. Chem. (1)

X. Wei, C. Anthony, D. Lowe, and M. Ward, “Design and fabrication of a nonlinear micro impact oscillator,” Proc. Chem. 1, 855–858 (2009).
[CrossRef]

Sens. Actuators A (1)

M. Bao and H. Yang, “Squeeze film air damping in MEMS,” Sens. Actuators A 136, 3–27 (2007).
[CrossRef]

Other (1)

S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells (McGraw-Hill, 1959).

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Figures (7)

Fig. 1.
Fig. 1.

Schematic diagram of the optowaveguides.

Fig. 2.
Fig. 2.

(a) Calculated optical forces on the waveguide with different gaps g. Here L=15μm, b=h=220nm. (b) The variation of optical forces with the operating wavelength. Here L=15μm, b=h=g=220nm.

Fig. 3.
Fig. 3.

Frequency responses of the optoresonator with respect to varying input optical power. Here L=15μm, b=h=g=220nm.

Fig. 4.
Fig. 4.

Resonance frequency shift (black curve) and the vibration amplitude (red curve) of the optoresonator with respect to the varying input optical power. Inset: partial enlarged drawing when the optical power is smaller than the critical value P0. Here L=15μm, b=h=g=220nm.

Fig. 5.
Fig. 5.

Resonance frequency shift (red curve) and the vibration amplitude (blue curve) of the optoresonator with respect to the waveguide width, for different amounts of input optical power. Here L=15μm, h=g=220nm.

Fig. 6.
Fig. 6.

Resonance frequency shift (red curve) and the vibration amplitude (blue curve) of the optoresonator with (a) different waveguide lengths (b=h=g=220nm.) and (b) different initial gaps (L=15μm, b=h=220nm.) between the waveguides. Here, the input power value is set as10 mW.

Fig. 7.
Fig. 7.

(a) Frequency responses of the optoresonator for different amounts of input optical power. Here g=300nm. (b) Resonance response curves of a waveguide beam at varying modulation levels of the actuation light as presented in [17] and (c) frequency responses of the optoresonator for different amounts of input optical power. Here g=500nm.

Tables (1)

Tables Icon

Table 1. Physical Properties and Structural Dimensions Reported by Li et al. [17]

Equations (28)

Equations on this page are rendered with MathJax. Learn more.

foptP2bcneff,x1/εz+|εx|sinh2(γg/2),symmetric mode,fopt+P2bcneff,x|εx|cosh2(γg/2),anti-symmetric mode,
ftot=foptcos2(πΔLneff,x/λ)+fopt+sin2(πΔLneff,x/λ),
ftot=fopt+fopt+2+foptfopt+2cos(ωt+ϑ0),
fopt/Pf0f1wf2w2f3w3+0(w4),fopt+/Pf0++f1+w+f2+w2+f3+w3+0(w4),
ftot/P=f0+f0+2+f0f0+2cos(ωt+ϑ0)+[f1+f1+2+f1f1+2cos(ωt+ϑ0)]w+[f2+f2+2+f2f2+2cos(ωt+ϑ0)]w2+[f3+f3+2+f3f3+2cos(ωt+ϑ0)]w3.
EbI4w(x,t)x4+[EbA2L0L(w(x,t)x)2dx]2w(x,t)x2+ρbA2w(x,t)t2=Fa(w(x,t),w(x,t)t)+fopt(w(x,t),t),
me·d2u(t)/dt2+k1u(t)+k3u3(t)=[c0+c1u(t)+c2u2(t)].du(t)/dt+[F0++F1+u(t)+F2+u2(t)+F3+u3(t)]+[F0+F1u(t)+F2u2(t)+F3u3(t)]cos(ωt+ϑ0),
τ=ω0t,ω0=k1me,Ω=ωω0,X=ug0,β=g02k3k1=2g02b2,η0=c0meω0,η1=g0c1meω0,η2=g02c2meω0,0±=F0±g0k1,1±=F1±k1,2±=g0F2±k1,3±=g02F3±k1,X˙=dXdτ,X¨=d2Xdτ2.
X¨(τ)+X(τ)+βX3(τ)=[η0+η1X(τ)+η2X2(τ)]X˙(τ)+P[0++1+X(τ)+2+X2(τ)]+P[0+1X(τ)+2X2(τ)]cos(Ωτ+ϑ0).
X¨+X=ε{˜0cos(ΩT0+ϑ0)[η˜0+η˜1X+η˜2X2]X˙+P˜0++P[˜1++˜1cos(ΩT0+ϑ0)]X+P[˜2++˜2cos(ΩT0+ϑ0)]X2+P[˜3+β˜+˜3cos(ΩT0+ϑ0)]X3}.
X(τ;ε)=X0(T0,T1)+εX1(T0,T1)+.
D02X0+X0=0,D02X1+X1=2D0D1X0[η˜0+η˜1X0+η˜2X02]D0X0+P˜0++P˜0cos(ΩT0+ϑ0)+P[˜1++˜1cos(ΩT0+ϑ0)]X0+P[˜2++˜2cos(ΩT0+ϑ0)]X02+[P˜3+β˜+P˜3cos(ΩT0+ϑ0)]X03,
X0(T0,T1)=a(T1)cos[T0+δ(T1)]=A(T1)ejT0+cc,
D1a=12(η˜0+14η˜2a2)a+P(18˜2a2+˜02)sin(σT1+ϑ0δ)aD1δ=38(β˜P˜3+)a3P˜1+2aP(38˜2a2+˜02)cos(σT1+ϑ0δ).
(Ω138(βP3+)a¯2+P1+2)2+14(η0+14η2a¯2)2(32a¯2+40)2(2a¯2+40)2=P2a¯2(382a¯2+02)2.
Ω=1+38(βP3+)a¯2P1+2±P(382a¯2+02)1a¯214(η0+14η2a¯2)21P2(2a¯2/8+0/2)2.
a¯max=R+(P2)212η0η29η22R+P23η2,
R=1η2{23η0η2P2+2η22P0+127(2)3P3+89η2[3η2η03316(Pη02)2278η0η202P2+8116(Pη20)2+3160(2)3P4]12}13.
P˜=3βa¯max241++33+a¯max2.
Ω=1+38(β3+P)a¯max2121+P.
f0=12bcneff,xεx1+εx2sinh2(γg0/2),f0+=12bcneff,xεxcosh2(γg0/2),f1=14bcγneff,xεx3sinh(γg0)(1+εx2sinh2(γg0/2))2,f1+=14bcγneff,xsinh(γg0/2)εxcosh3(γg0/2),f2=18bcγ2neff,xεx3[cosh(γg0)1/4εx2(cosh(2γg0)+2cosh(γg0)3)](1+εx2sinh2(γg0/2))3,f2+=18bcγ2neff,x(cosh(γg0)2)εxcosh4(γg0/2),f3=1192bcγ3neff,xεx3{16sinh(γg0)16εx2[sinh(γg0)+sinh(2γg0)]+εx4[sinh(3γg0)+8sinh(2γg0)+19sinh(γg0)]}(1+εx2sinh2(γg0/2))4,f3+=112bcγ3neff,x(2sinh(γg0/2)sinh3(γg0/2))εxcosh5(γg0/2).
D1=22βμA1δ,
D2=(PaL2h4πg)2MmπRT,
Fa(g,g˙)=(22βμbδ+PLh4πg2MmπRT)g˙.
Fa(w,w˙,g0)i=02Caiwi·w˙=[22βμbδ+(PLh4πg0)2MmπRT(1+1g0w+1g02w2)]w˙,
u(t)EbI0Ld4Y(x)dx4Y(x)dx+u3(t)EbA2L0L0L(dY(ξ)dξ)2d2Y(x)dx2Y(x)dξdx+d2u(t)dt2ρbA0LY2(x)dx=du(t)dtCa00LY2(x)dxu(t)du(t)dtCa10LY3(x)dxu2(t)du(t)dtCa20LY4(x)dx+P[f0f0+2+f0+f0+2cos(ωt+ϑ0)]0LY(x)dx+P[f1f1+2+f1+f1+2cos(ωt+ϑ0)]u(t)0LY2(x)dx+P[f2f2+2+f2+f2+2cos(ωt+ϑ0)]u2(t)0LY3(x)dx+P[f3f3+2+f3+f3+2cos(ωt+ϑ0)]u3(t)0LY4(x)dx.
med2u(t)dt2+k1u(t)+k3u3(t)=(c0+c1u(t)+c2u2(t))du(t)dt+P(F0++F1+u(t)+F2+u2(t)+F3+u3(t))+P(F0+F1u(t)+F2u2(t)+F3u3(t))cos(ωt+ϑ0),
k1=0LEbI(d2Y(x)dx2)2dx=16EbIπ43L3,k3=EbA2L0L(dY(x)dx)2dx=8π4EbA9L3,me=ρbA0LY2(x)dx=ρbAL,c0=Ca00LY2(x)dx=Ca0L,c1=Ca10LY3(x)dx=5323Ca1L,c2=Ca20LY4(x)dx=3518Ca2L,F0±=23f0f0+2L,F1±=f1f1+2L,F2±=5323f2f2+2L,F3±=3518f3f3+2L.

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