Advertisement

Nonlinear Dynamics

, Volume 92, Issue 4, pp 1955–1974 | Cite as

Optimizing the dynamical behavior of a dual-frequency parametric amplifier with quadratic and cubic nonlinearities

  • A. DolevEmail author
  • I. Bucher
Original Paper

Abstract

The paper describes a novel parametric excitation scheme that acts as a tunable amplifier by controlling two pumping signals and two nonlinear feedback terms. By modulating the stiffness of a mechanical oscillator with a digital signal processor, low-frequency inputs are projected onto a higher resonance frequency, thus exploiting the natural selective filtering of such structures. Described is an optimized dual-term nonlinear stiffness resonator that enhances the input signal level and the sensitivity to changes in both amplitude and phase, while limiting the obtained response to desired levels. This amplifier is geared to cases when the frequency of the input is known or measurable, like in rotating structures, while the amplitude and phase are too weak to be detected without amplification. It is shown that by tuning the cubic and quadratic feedback terms, the amplifier benefits from a nearly linear response behavior, while exploiting the benefits of nonlinear and pumping signal enhancements.

Keywords

Parametric amplifier Perturbation methods Tunable system Optimization Principal parametric resonance Combination resonance 

Notes

Acknowledgements

The first author would like to acknowledge the generous financial support of the Israeli Ministry of Science, Technology and Space for the Applied science scholarship for engineering Ph.D. students.

Compliance with ethical standards

Conflict of interest

Both authors declare that they have no conflict of interest.

References

  1. 1.
    Rugar, D., Grütter, P.: Mechanical parametric amplification and thermomechanical noise squeezing. Phys. Rev. Lett. 67, 699–702 (1991).  https://doi.org/10.1103/PhysRevLett.67.699 CrossRefGoogle Scholar
  2. 2.
    Yu, M.-F., Wagner, G.J., Ruoff, R.S., Dyer, M.J.: Realization of parametric resonances in a nanowire mechanical system with nanomanipulation inside a scanning electron microscope. Phys. Rev. B. 66, 73406 (2002).  https://doi.org/10.1103/PhysRevB.66.073406 CrossRefGoogle Scholar
  3. 3.
    Krylov, S., Gerson, Y., Nachmias, T., Keren, U.: Excitation of large-amplitude parametric resonance by the mechanical stiffness modulation of a microstructure. J. Micromech. Microeng. 20, 15041 (2009).  https://doi.org/10.1088/0960-1317/20/1/015041 CrossRefGoogle Scholar
  4. 4.
    Zalalutdinov, M., Olkhovets, A., Zehnder, A., Ilic, B., Czaplewski, D., Craighead, H.G., Parpia, J.M.: Optically pumped parametric amplification for micromechanical oscillators. Appl. Phys. Lett. 78, 3142–3144 (2001).  https://doi.org/10.1063/1.1371248 CrossRefGoogle Scholar
  5. 5.
    Oropeza-Ramos, L.A., Burgner, C.B., Turner, K.L.: Robust micro-rate sensor actuated by parametric resonance. Sensors Actuators A Phys. 152, 80–87 (2009).  https://doi.org/10.1016/j.sna.2009.03.010 CrossRefGoogle Scholar
  6. 6.
    Zaqarashvili, T.V., Oliver, R., Ballester, J.L.: Parametric amplification of magnetosonic waves by an external, transversal, periodic action. Astrophys. J. 569, 519–530 (2002).  https://doi.org/10.1086/339288 CrossRefGoogle Scholar
  7. 7.
    Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley, Weinheim (2008)zbMATHGoogle Scholar
  8. 8.
    Dolev, A., Bucher, I.: A parametric amplifier for weak, low-frequency forces. In: In ASME 2015 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, p. V006T10A046. ASME, Boston (2015)Google Scholar
  9. 9.
    Dolev, A., Bucher, I.: Experimental and numerical validation of digital, electromechanical, parametrically excited amplifiers. J. Vib. (2016).  https://doi.org/10.1115/1.4033897 Google Scholar
  10. 10.
    Dolev, A., Bucher, I.: Tuneable, non-degenerated, nonlinear, parametrically-excited amplifier. J. Sound Vib. 361, 176–189 (2016).  https://doi.org/10.1016/j.jsv.2015.09.048 CrossRefGoogle Scholar
  11. 11.
    Dolev, A., Bucher, I.: Dual frequency parametric excitation of a nonlinear, multi degree of freedom mechanical amplifier with electronically modified topology. J. Sound Vib. 419, 420–435 (2018).  https://doi.org/10.1016/j.jsv.2018.01.008
  12. 12.
    Tresser, S., Dolev, A., Bucher, I.: Dynamic balancing of super-critical rotating structures using slow-speed data via parametric excitation. J. Sound Vib. 415, 59–77 (2018).  https://doi.org/10.1016/j.jsv.2017.11.029
  13. 13.
    Tresser, S., Bucher, I.: A method for balancing high-speed rotors using low rotation speed measured data through parametric excitation. In: Proceedings of the 11th International Conference Vibration in Rotating Machinery (VIRM), Manchester (2016)Google Scholar
  14. 14.
    Ali, H.N., Walter, L.: On the discretization of spatially continuous systems with quadratic and cubic nonlinearities. JSME Int. J. Ser. C 41, 510–531 (1998).  https://doi.org/10.1299/jsmec.41.510 CrossRefGoogle Scholar
  15. 15.
    Nayfeh, A.H., Lacarbonara, W.: On the discretization of distributed-parameter systems with quadratic and cubic nonlinearities. Nonlinear Dyn. 13, 203–220 (1997).  https://doi.org/10.1023/A:1008253901255 MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Nayfeh, A., Bouguerra, H.: Non-linear response of a fluid valve. Int. J. Nonlinear Mech. 25, 433–449 (1990).  https://doi.org/10.1016/0020-7462(90)90031-4 CrossRefzbMATHGoogle Scholar
  17. 17.
    Nayfeh, A.H.: Perturbation Methods. Wiley, Weinheim (2008)Google Scholar
  18. 18.
    Rhoads, J.F., Shaw, S.W.: The impact of nonlinearity on degenerate parametric amplifiers Rhoads. Appl. Phys. Lett. 96, 234101 (2010).  https://doi.org/10.1063/1.3446851 CrossRefGoogle Scholar
  19. 19.
    DeMartini, B.E., Rhoads, J.F., Turner, K.L., Shaw, S.W., Moehlis, J.: Linear and nonlinear tuning of parametrically excited MEMS oscillators. J. Microelectromech. Syst. 16, 310–318 (2007).  https://doi.org/10.1109/JMEMS.2007.892910 CrossRefGoogle Scholar
  20. 20.
    Rhoads, J.F., Shaw, S.W., Turner, K.L., Moehlis, J., DeMartini, B.E., Zhang, W.: Generalized parametric resonance in electrostatically actuated microelectromechanical oscillators. J. Sound Vib. 296, 797–829 (2006).  https://doi.org/10.1016/j.jsv.2006.03.009 CrossRefGoogle Scholar
  21. 21.
    Maccari, A.: Multiple resonant or non-resonant parametric excitations for nonlinear oscillators. J. Sound Vib. 242, 855–866 (2001).  https://doi.org/10.1006/jsvi.2000.3386 MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Troger, H., Hsu, C.: Response of a nonlinear system under combined parametric and forcing excitation. ASME Trans. Ser. E J. 44, 179–181 (1977)CrossRefGoogle Scholar
  23. 23.
    Perkins, N.: Modal interactions in the non-linear response of elastic cables under parametric/external excitation. Int. J. Nonlinear Mech. 27, 233–250 (1992).  https://doi.org/10.1016/0020-7462(92)90083-J CrossRefzbMATHGoogle Scholar
  24. 24.
    Szabelski, K., Warminski, J.: Self-excited system vibrations with parametric and external excitations. J. Sound Vib. 187, 595–607 (1995).  https://doi.org/10.1006/jsvi.1995.0547 CrossRefzbMATHGoogle Scholar
  25. 25.
    Zhang, W., Tang, Y.: Global dynamics of the cable under combined parametrical and external excitations. Int. J. Nonlinear Mech. 37, 505–526 (2002).  https://doi.org/10.1016/S0020-7462(01)00026-9 CrossRefzbMATHGoogle Scholar
  26. 26.
    El-Bassiouny, A., Abdelhafez, H.: Prediction of bifurcations for external and parametric excited one-degree-of-freedom system with quadratic, cubic and quartic non-linearities. Math. Comput. Simul. 57, 61–80 (2001).  https://doi.org/10.1016/S0378-4754(01)00292-0 MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Kim, C.H., Lee, C.-W., Perkins, N.C.: Nonlinear vibration of sheet metal plates under interacting parametric and external excitation during manufacturing. In: ASME 2003 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, pp. 2481–2489. ASME, Chicago (2003)Google Scholar
  28. 28.
    Zavodney, L.D., Nayfeh, A.H., Sanchez, N.E.: The response of a single-degree-of-freedom system with quadratic and cubic non-linearities to a principal parametric resonance. J. Sound Vib. 129, 417–442 (1989).  https://doi.org/10.1016/0022-460X(89)90433-1 CrossRefzbMATHGoogle Scholar
  29. 29.
    Zavodney, L.D., Nayfeh, A.H.: The response of a single-degree-of-freedom system with quadratic and cubic non-linearities to a fundamental parametric resonance. J. Sound Vib. 120, 63–93 (1988).  https://doi.org/10.1016/0022-460X(88)90335-5 CrossRefzbMATHGoogle Scholar
  30. 30.
    Nayfeh, A.H.: The response of single degree of freedom systems with quadratic and cubic non-linearities to a subharmonic excitation. J. Sound Vib. 89, 457–470 (1983).  https://doi.org/10.1016/0022-460X(83)90347-4 MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Chu, H.: Influence of large amplitudes on free flexural vibrations of rectangular elastic plates. J. Appl. Mech. 23, 532–540 (1956)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Marín, J., Perkins, N., Vorus, W.: Non-linear response of predeformed plates subject to harmonic in-plane edge loading. J. Sound Vib. 176, 515–529 (1994).  https://doi.org/10.1006/jsvi.1994.1393 CrossRefzbMATHGoogle Scholar
  33. 33.
    Bannon, F., Clark, J.: High-Q HF microelectromechanical filters. IEEE J. Solid State 35, 512–526 (2000).  https://doi.org/10.1109/4.839911 CrossRefGoogle Scholar
  34. 34.
    Billah, K.Y., Scanlan, R.H.: Resonance, Tacoma Narrows bridge failure, and undergraduate physics textbooks. Am. J. Phys. 59, 118–124 (1991).  https://doi.org/10.1119/1.16590 CrossRefGoogle Scholar
  35. 35.
    Kovacic, I., Brennan, M.J.: The Duffing Equation: Nonlinear Oscillators and Their Behaviour. Wiley (2011)Google Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Dynamics Laboratory, Faculty of Mechanical EngineeringTechnionHaifaIsrael

Personalised recommendations