Phase dependent nonlinear parametrically excited systems

Abstract

Nonlinear parametrically excited (NPE) systems govern the dynamics of many engineering applications, from cable-stayed bridges where vibrations need to be suppressed, to energy harvesters, transducers and acoustic amplifiers where vibrations need to be amplified. This work investigates the effect of different system parameters on the dynamics of a prototype NPE system. The NPE system in this work is a cantilever beam with an electromagnetic subsystem excited at its base. This system allows cubic stiffness, parametric stiffness, cubic parametric stiffness, and the phase difference between different sources of excitation to be varied independently to achieve different dynamic behaviors. A mathematical model is also derived, which provides theoretical understanding of the effects of these parameters and allows the analysis to be extended to other applications.

Keywords

Nonlinear parametrically excited base excitation relative phase

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References

Chen

Sammut

(2009) Vibration suppression of a principal parametric resonance. Journal of Vibration and Control 15(3): 439–463.

DeMartini B, Moehlis J, Turner K, et al. (2005) Modeling of parametrically excited microelectromechanical oscillator dynamics with application to filtering. In: Proceedings of the Fourth IEEE Conference on Sensors, 30 October–3 November 2005, Irvine, CA, USA. Piscataway, NJ: Institute of Electrical and Electronics Engineers, pp.345–348.

Hammond

(1997) Electromagnetism for Engineers: An Introductory Course, 4th edition. Oxford, United States: Oxford University Press, Oxford Science Publications.

Gallacher

Burdess

et al. (2011) A parametrically amplified mems rate gyroscope. Sensors and Actuators A: Physical 167(2): 249–260.

Kumar

Miller

Rhoads

(2011) Nonlinear parametric amplification and attenuation in a base-excited cantilever beam. Journal of Sound and Vibration 330(22): 5401–5409.

Liu

(2006) A tunable electromagnetic vibration absorber: Characterization and application. Journal of Sound and Vibration 295(3): 708–724.

Murdock

(1999) Perturbations: Theory and Methods, Volume 27, Philadelphia, Pennsylvania, USA: Society for Industrial and Applied Mathematics (SIAM).

Nayfeh

Mook

(2008) Nonlinear Oscillations, Chichester, UK: John Wiley & Sons.

Neumeyer

Sorokin

Thomsen

(2016) Effects of quadratic and cubic nonlinearities on a perfectly tuned parametric amplifier. Journal of Sound and Vibration 386: 327–335.

10.

Rhoads

Miller

Shaw

et al. (2008) Mechanical domain parametric amplification. Journal of Vibration and Acoustics 130(6): 061006-061006-7.

11.

Rhoads

Shaw

(2010) The impact of nonlinearity on degenerate parametric amplifiers. Applied Physics Letters 96(23): 234101-234101-3.

12.

Sneller

Mann

(2010) On the nonlinear electromagnetic coupling between a coil and an oscillating magnet. Journal of Physics D: Applied Physics 43(29): 295005.

13.

Tusset

Janzen

Piccirillo

et al. (2017) On nonlinear dynamics of a parametrically excited pendulum using both active control and passive rotational (MR) damper. Journal of Vibration and Control 24(9): 1587–1599.

14.

Zaghari B (2016) Dynamic analysis of a nonlinear parametrically excited system using electromagnets. PhD Thesis, Institute of Sound and Vibration Research (ISVR), University of Southampton, UK. Available at: https://eprints.soton.ac.uk/411397/1/BaharehZaghariPhDThesis.pdf (accessed 13 June 2018).

15.

Zaghari B, Ghandchi Tehrani M and Rustighi E (2014) Mechanical modelling of a vibration energy harvester with time-varying stiffness. In: EURODYN 2014: 9th International Conference on Structural Dynamics, Porto, Portugal, 30 June–2 July 2014, pp.2079–2086. Available at: https://eprints.soton.ac.uk/366807/1/288_MS11_ABS_1108EuroDyn2014.pdf (accessed 13 June 2018).

16.

Zaghari B, Rustighi E and Ghandchi Tehrani M (2016) Dynamic response of a nonlinear parametrically excited system subject to harmonic base excitation. Journal of Physics: Conference Series 744(1): 012125-012125-10.