Abstract
Parametric resonances can occur in internally stressed systems due to the periodic variation of the stiffness in time. Parametric resonance can lead to unstable dynamic behaviour; however, their response is limited by existing nonlinearities in the system, thus resulting in limit cycle oscillations (LCOs). This phenomenon can be exploited in the design of energy harvesters. The amplitude and frequency of the parametric excitation can be adjusted so that the vibration response of internally stressed systems is close to instability. In this paper, a cantilever beam is considered in vertical position and an axial excitation is applied to the base of the beam. The imposed kinematics of the base leads to internal stress along the beam, which produces a variation of the bending stiffness. If the frequency of the base excitation is twice the first natural frequency of the beam, the principal parametric resonances can occur. A quasi-linear FEM approach is adopted, together with a simplified single-degree-of-freedom model of the beam, in order to numerically simulate its dynamic behaviour, to identify unstable conditions and to obtain the Floquet diagram. An analytical approach is developed as well, using a multi-degree-of-freedom model of the beam, considering the system as autoparametric. Harmonic balance method is used to determine the Floquet diagram and to validate the numerical model.
Principal parametric resonance is observed experimentally. Harvesting energy from parametric resonance is therefore potentially very efficient, especially if the external source is not directly exploitable. Parametric resonance in this case acts as a power amplification.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Faraday M (1831) On a peculiar class of acoustical figures and on certain forms assumed by a group of particles upon vibrating elastic surfaces. Philos Trans R Soc Lond 121:299–318
Mathieu E (1868) Mémorie sur le movement vibratoire d’une membrane de forme elliptique. Journal de Mathématiques Pures et Appliquées 13:137–203
Butikov E (2005) Parametric resonance in a linear oscillator at square-wave modulation. Eur J Phys 26:157–174
Nayfeh AH, Mook DT (1979) Nonlinear oscillations. John Wiley & Sons Inc, New York
Zavodney LD, Nayfeh AH, Sanchez NE (1989) 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
van Laarhoven BJH (2009) Stability analysis of parametric roll resonance, DCT 2009.062, 2009
Oueini SS, Nayfeh AH (1999) Single-mode control of a cantilever beam under principal parametric excitation. J Sound Vib 224(1):33–47
Ghandchi Tehrani M, Kalkowski MK, Elliott SJ (2014) Active control of parametrically exited systems. J Intell Mater Syst Struct, submit on July 2014
Jia Y, Seshia AA (2013) Directly and parametrically excited bi-stable vibration energy harvester for broadband operation. In: Solid-state sensors, actuators and microsystems, IEEE, Barcelona, pp 454–457
Jia Y, Yan J, Soga K, Seshia AA (2012) Parametrically excited MEMS vibration energy harvesters. Dig Tech Pap Power MEMS 12:215–218
Jia Y, Yan J, Soga K, Seshia AA (2013) Multi-frequency operation of a MEMS vibration energy harvester by accessing five orders of parametric resonance. J Phys Conf Ser, 476. doi:10.1088/1742-6596/476/1/012126
Jia Y, Yan J, Soga K, Seshia AA (2014) Parametric resonance for vibration energy harvesting with design techniques to passively reduce the initiation threshold amplitude. Smart Mater Struct 23:1–10
Nayfeh AH (1987) Parametric excitation of two internally resonant oscillators. J Sound Vib 119(1):95–109
Bonisoli E, Scapolan M, Ghandchi Tehrani M (2014) Internal stresses and unstable dynamics for harvesting design. Mech Syst Signal Process, submit on Oct 2014
Zaghari B, Ghandchi Tehrani M, Rustighi E (2014) Mechanical modeling of a vibration energy harvester with time-varying stiffness. In: Proceedings of the 9th international conference on structural dynamics, EURODYN 2014. Porto, 30 June–2 July, pp 2079–2085
Zaghari B, Rustighi E, Ghandchi Tehrani M (2014) Experimental study on harvesting energy from a parametrically excited system In: The 12th international conference on motion and vibration, MOVIC 2014, Sapporo, pp 1–13
Zaghari B, Ghandchi Tehrani M, Rustighi E (2014) Energy harvesting from a parametrically excited cantilever beam. J Sound Vib, submit on July 2014
Peeters B, Van der Auweraer H, Guillaume P, Leuridan J (2004) The PolyMAX frequency-domain method: a new standard for modal parameter estimation? J Shock Vib 11(3–4):395–409
Acknowledgements
Dr Ghandchi Tehrani would like to acknowledge the support provided by EPSRC for her first grant (Ref: EP/K005456/1). The authors would like to acknowledge the support provided for Matteo Scapolan as a visiting student at ISVR, University of Southampton.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 The Society for Experimental Mechanics, Inc.
About this paper
Cite this paper
Tehrani, M.G., Bonisoli, E., Scapolan, M. (2015). Energy Harvesting Perspectives from Parametric Resonant Systems. In: Wicks, A. (eds) Shock & Vibration, Aircraft/Aerospace, and Energy Harvesting, Volume 9. Conference Proceedings of the Society for Experimental Mechanics Series. Springer, Cham. https://doi.org/10.1007/978-3-319-15233-2_23
Download citation
DOI: https://doi.org/10.1007/978-3-319-15233-2_23
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-15232-5
Online ISBN: 978-3-319-15233-2
eBook Packages: EngineeringEngineering (R0)