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Asymptotic analysis of a vibrating cantilever with a nonlinear boundary

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Abstract

Nonlinear vibration of a cantilever in a contact atomic force microscope is analyzed via an asymptotic approach. The asymptotic solution is sought for a beam equation with a nonlinear boundary condition. The steady-state responses are determined in primary resonance and subharmonic resonance. The relations between the response amplitudes and the excitation frequencies and amplitudes are derived from the solvability condition. Multivaluedness occurs in the relations as a consequence of the nonlinearity. The stability of steady-state responses is analyzed by use of the Lyapunov linearized stability theory. The stability analysis predicts the jumping phenomenon for certain parameters. The curves of the response amplitudes changing with the excitation frequencies are numerically compared with those obtained via the method of multiple scales. The calculation results demonstrate that the two methods predict the same varying tendencies while there are small quantitative differences.

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References

  1. Binnig G, Quate C F, Gerber C. Atomic force microscope. Phys Rev Lett, 1986, 56: 930–933

    Article  ADS  Google Scholar 

  2. Garcia R, Perez R. Dynamics atomic force microscopy methods. Surf Sci Rep, 2002, 47: 197–301

    Article  MATH  ADS  Google Scholar 

  3. Jalili N, Laxminrayana K. A review of atomic force microscopy imaging systems: Application to molecular metrology and biological sciences. Mechatronics, 2004, 14: 907–945

    Article  Google Scholar 

  4. Cappella B, Dietler G. Force-distance curve by atomic force microscopy. Surf Sci Rep, 1999, 34: 1–104

    Article  Google Scholar 

  5. Wolf K, Gottlieb O. Nonlinear dynamics of a noncontacting atomic force microscope cantilever actuated by a piezoelectric layer. J Appl Phys, 2002, 91: 4701–4709

    Article  ADS  Google Scholar 

  6. Turner J A. Non-linear vibrations of a beam with cantilever-Hertzian contact boundary conditions. J Sound Vib, 2004, 275: 177–191

    Article  ADS  Google Scholar 

  7. Abdel-Rahman E M, Nayfeh A H. Contact force identification using the subharmonic resonance of a contact-mode atomic force microscopy. Nanotechnology, 2005, 16: 199–207

    Article  Google Scholar 

  8. Hu Q Q, Lim C W, Chen L Q. Nonlinear vibration of a cantilever with a Derjaguin-Muller-Toprov contact end. Int J Struct Sta Dyn, 2008, 8: 25–40

    Article  MathSciNet  Google Scholar 

  9. Nayfeh A H, Pai P H. Linear and Nonlinear Structural Mechanics. New Jersey: John Wiley and Sons, 2004

    Book  MATH  Google Scholar 

  10. Awrejcewicz J A, Andrianov J V, Manevitch L I. Asymptotic Approaches in Nonlinear Dynamics. Berlin: Springer-Verlag, 1998

    MATH  Google Scholar 

  11. Andrianov I V, Awrejcewicz J, Barantsev R G. Asymptotic approaches in mechanics: New parameters and procedures. Appl Mech Rev, 2003, 56: 87–110

    Article  Google Scholar 

  12. Boertjens G J, van Horssen W T. On mode interactions for a weakly nonlinear beam equation. Nonlinear Dyn, 1998, 12: 23–40

    Article  Google Scholar 

  13. Maccari A. The asymptotic perturbation method for nonlinear continuous systems. Nonlinear Dyn, 1999, 19: 1–18

    Article  MATH  MathSciNet  Google Scholar 

  14. Boertjens G J, van Horssen W T. An asymptotic theory for a weakly nonlinear beam equation with a quadratic perturbation. SIAM J Appl Math, 2000, 60: 602–632

    Article  MATH  MathSciNet  Google Scholar 

  15. Boertjens G J, van Horssen W T. On interactions of oscillation modes for a weakly non-linear undamped elastic beam with an external force. J Sound Vib, 2000, 235: 201–217

    Article  ADS  Google Scholar 

  16. Andrianov I V, Danishevs’kyy V V. Asymptotic approach for nonlinear periodical vibrations of continuous structures. J Sound Vib, 2002, 249: 465–481

    Article  ADS  MathSciNet  Google Scholar 

  17. Chen R M. Some nonlinear dispersive waves arising in compressible hyperelastic plates. Int J Eng Sci, 2006, 44: 1188–1204

    Article  Google Scholar 

  18. Chen L Q, Chen H, Lim C W. Asymptotic analysis of axially accelerating viscoelastic strings. Int J Eng Sci, 2008, 46: 976–985

    Article  MathSciNet  Google Scholar 

  19. Maccari A. Saddle-node bifurcations of cycles in a relief valve. Nonlinear Dyn, 2000, 22: 225–247

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mechanics, Shanghai University, Shanghai, 200444, China

    LiQun Chen

  2. Shanghai Institute of Applied Mathematics and Mechanics, Shanghai, 200072, China

    LiQun Chen, QingQuan Hu & Hu Ding

  3. Department of Building and Construction, City University of Hong Kong, Kowloon, Hong Kong, China

    C. W. Lim

  4. Shandong JiaoTong University, Jinan, 250023, China

    QingQuan Hu

Authors
  1. LiQun Chen
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  2. C. W. Lim
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  3. QingQuan Hu
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  4. Hu Ding
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Corresponding author

Correspondence to LiQun Chen.

Additional information

Supported by the National Outstanding Young Scientists Fund of China (Grant No. 10725209), the Shanghai Leading Academic Discipline Project (Grant No. S30106), and Shandong Jiaotong University Science Foundation (Grant No. Z200812)

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Chen, L., Lim, C.W., Hu, Q. et al. Asymptotic analysis of a vibrating cantilever with a nonlinear boundary. Sci. China Ser. G-Phys. Mech. Astron. 52, 1414–1422 (2009). https://doi.org/10.1007/s11433-009-0185-z

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  • DOI: https://doi.org/10.1007/s11433-009-0185-z

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