0
Research Papers

Floquet-Based Analysis of General Responses of the Mathieu Equation

[+] Author and Article Information
Gizem Acar

Department of Mechanical Engineering,
Michigan State University,
East Lansing, MI 48824
e-mail: dilbergi@egr.msu.edu

Brian F. Feeny

Department of Mechanical Engineering,
Michigan State University,
East Lansing, MI 48824
e-mail: feeny@egr.msu.edu

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received August 20, 2015; final manuscript received March 29, 2016; published online May 25, 2016. Assoc. Editor: Marco Amabili.

J. Vib. Acoust 138(4), 041017 (May 25, 2016) (9 pages) Paper No: VIB-15-1332; doi: 10.1115/1.4033341 History: Received August 20, 2015; Revised March 29, 2016

Solutions to the linear unforced Mathieu equation, and their stabilities, are investigated. Floquet theory shows that the solution can be written as a product between an exponential part and a periodic part at the same frequency or half the frequency of excitation. In the current work, an approach combining Floquet theory with the harmonic balance method is investigated. A Floquet solution having an exponential part with an unknown exponential argument and a periodic part consisting of a series of harmonics is assumed. Then, performing harmonic balance, frequencies of the response are found and stability of the solution is examined over a parameter set. The truncated solution is consistent with an existing infinite series solution for the undamped case. The truncated solution is then applied to the damped Mathieu equation and parametric excitation with two harmonics.

FIGURES IN THIS ARTICLE
<>
Copyright © 2016 by ASME
Your Session has timed out. Please sign back in to continue.

References

Ruby, L. , 1996, “ Applications of the Mathieu Equation,” Am. J. Phys., 64(1), pp. 39–44. [CrossRef]
Li, Y. , Fan, S. , Guo, Z. , Li, J. , Cao, L. , and Zhuang, H. , 2013, “ Mathieu Equation With Application to Analysis of Dynamic Characteristics of Resonant Inertial Sensors,” Commun. Nonlinear Sci. Numer. Simul., 18(2), pp. 401–410. [CrossRef]
Sofroniou, A. , and Bishop, S. , 2014, “ Dynamics of a Parametrically Excited System With Two Forcing Terms,” Mathematics, 2(3), pp. 172–195. [CrossRef]
Ramakrishnan, V. , and Feeny, B. F. , 2012, “ Resonances of a Forced Mathieu Equation With Reference to Wind Turbine Blades,” ASME J. Vib. Acoust., 134(6), p. 064501. [CrossRef]
Inoue, T. , Ishida, Y. , and Kiyohara, T. , 2012, “ Nonlinear Vibration Analysis of the Wind Turbine Blade (Occurrence of the Superharmonic Resonance in the Out of Plane Vibration of the Elastic Blade),” ASME J. Vib. Acoust., 134(3), p. 031009. [CrossRef]
Rhoads, J. F. , Miller, N. J. , Shaw, S. W. , and Feeny, B. F. , 2008, “ Mechanical Domain Parametric Amplification,” ASME J. Vib. Acoust., 130(6), p. 061006. [CrossRef]
Nayfeh, A. H. , and Mook, D. T. , 2008, Nonlinear Oscillations, Wiley, New York.
Taylor, J. H. , and Narendra, K. S. , 1969, “ Stability Regions for the Damped Mathieu Equation,” SIAM J. Appl. Math., 17(2), pp. 343–352. [CrossRef]
Younesian, D. , Esmailzadeh, E. , and Sedaghati, R. , 2005, “ Existence of Periodic Solutions for the Generalized Form of Mathieu Equation,” Nonlinear Dyn., 39(4), pp. 335–348. [CrossRef]
Thomson, W. , 1996, Theory of Vibration With Applications, CRC Press, Englewood Cliffs, NJ.
Nayfeh, A. H. , 2008, Perturbation Methods, Wiley, New York.
Benaroya, H. , and Nagurka, M. L. , 2011, Mechanical Vibration: Analysis, Uncertainties, and Control, CRC Press, Englewood Cliffs, NJ.
Turrittin, H. , 1952, “ Asymptotic Expansions of Solutions of Systems of Ordinary Linear Differential Equations Containing a Parameter,” Contributions to the Theory of Nonlinear Oscillations, Vol. II, Princeton University Press, Princeton, NJ, pp. 81–116.
Rand, R. H. , 1969, “ On the Stability of Hill's Equation With Four Independent Parameters,” ASME J. Appl. Mech., 36(4), pp. 885–886. [CrossRef]
Ishida, Y. , Inoue, T. , and Nakamura, K. , 2009, “ Vibration of a Wind Turbine Blade (Theoretical Analysis and Experiment Using a Single Rigid Blade Model),” J. Environ. Eng., 4(2), pp. 443–454. [CrossRef]
Ecker, H. , 2009, “ Parametric Excitation in Engineering Systems,” 20th International Congress of Mechanical Engineering (COBEM 2009), Gramado, Brazil, Nov. 15–20, pp. 15–20.
Ecker, H. , 2011, “ Beneficial Effects of Parametric Excitation in Rotor Systems,” IUTAM Symposium on Emerging Trends in Rotor Dynamics, New Delhi, India, Mar. 23–26, pp. 361–371.
Klotter, K. , and Kotowski, G. , 1943, “ Über die Stabilität der Lösungen Hillscher Differentialgleichungen mit drei unabhängigen Parametern,” ZAMM, 23(3), pp. 149–155. [CrossRef]
Stoker, J. J. , 1950, Nonlinear Vibrations in Mechanical and Electrical Systems, Vol. 2, Interscience Publishers, New York.
Ward, M. , 2010, “ Lecture Notes on Basic Floquet Theory,” http://www.emba.uvm.edu/~jxyang/teaching/
McLachlan, N. W. , 1961, Theory and Application of Mathieu Functions, Dover, New York.
Peterson, A. , and Bibby, M. , 2013, Accurate Computation of Mathieu Functions, Morgan & Claypool Publishers, San Rafael, CA.
Hodge, D. , 1972, The Calculation of the Eigenvalues and Eigenfunctions of Mathieu's Equation, Vol. 1937, National Aeronautics and Space Administration, Washington, DC.
Hagedorn, P. , 1988, Non-Linear Oscillations, Clarendon Press, Oxford, UK.
Hale, J. K. , 1963, Oscillations in Nonlinear Systems, McGraw-Hill, New York.
Hayashi, C. , 2014, Nonlinear Oscillations in Physical Systems, Princeton University Press, Princeton, NJ.
Schmidt, G. , and Tondl, A. , 1986, Non-Linear Vibrations, Vol. 66, Cambridge University Press, Cambridge, UK.
Hartog, J. P. D. , 1985, Mechanical Vibrations, Dover, New York.
Magnus, W. , and Winkler, S. , 1979, Hill's Equation, Dover, New York.
Rand, R. , 2012, “ Lecture Notes on Nonlinear Vibrations,” http://www.tam.cornell.edu/randdocs
Whittaker, E. T. , 1913, “ On the General Solution of Mathieu's Equation,” Proc. Edinburgh Math. Soc., 32, pp. 75–80. [CrossRef]
Malasoma, J.-M. , Lamarque, C.-H. , and Jezequel, L. , 1994, “ Chaotic Behavior of a Parametrically Excited Nonlinear Mechanical System,” Nonlinear Dyn., 5(2), pp. 153–160.
Coisson, R. , Vernizzi, G. , and Yang, X. , 2009, “ Mathieu Functions and Numerical Solutions of the Mathieu Equation,” IEEE International Workshop on Open-Source Software for Scientific Computation (OSSC), Guiyang, China, Sept. 18–20, pp. 3–10.
Insperger, T. , and Stépán, G. , 2002, “ Stability Chart for the Delayed Mathieu Equation,” Proc. R. Soc. London, Ser. A, 458(2024), pp. 1989–1998. [CrossRef]
Insperger, T. , and Stépán, G. , 2003, “ Stability of the Damped Mathieu Equation With Time Delay,” ASME J. Dyn. Syst., Meas., Control, 125(2), pp. 166–171. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

Transition curves for the n = 2 Floquet-based approximation, infinite series solution, and the n = 2 Hill's determinant solution

Grahic Jump Location
Fig. 2

Analytically predicted response frequencies for δ = 0.8, n = 2

Grahic Jump Location
Fig. 3

Numerical and theoretical (n = 2) solutions of the undamped Mathieu equation for n = 2 (a) ω = 0.6, δ = 0.4, (b) ω = 0.8, δ = 0.6, (c) ω = 1.3, δ = 0.8, and (d) ω = 2.5, δ = 0.7

Grahic Jump Location
Fig. 4

FFTs of numerical and theoretical (n = 2) solutions of the undamped Mathieu equation for n = 2 (a) ω = 0.6, δ = 0.4, (b) ω = 0.8, δ = 0.6, (c) ω = 1.3, δ = 0.8, and (d) ω = 2.5, δ = 0.7

Grahic Jump Location
Fig. 5

Free response and FFT plots for ω = 0.6, δ = 0.4, with n = 3 harmonics (a) time response for ω = 0.6, δ = 0.4 and (b) FFT for ω = 0.6, δ = 0.4

Grahic Jump Location
Fig. 6

Transition curves for the damped Mathieu equation for ζ = 0.005, ζ = 0.025, and ζ = 0.05, approximated with n = 2

Grahic Jump Location
Fig. 7

Decay and growth factors for ζ = 0.05 and δ = 0.8

Grahic Jump Location
Fig. 8

Numerical and theoretical (n = 2) solutions of the damped Mathieu equation (a) ω = 0.8, δ = 0.6, ζ = 0.025, (b) ω = 1, δ = 0.3, ζ = 0.025, (c) ω = 1.3, δ = 0.8, ζ = 0.025, and (d) ω = 2, δ = 0.15, ζ = 0.05

Grahic Jump Location
Fig. 9

FFT of numerical and theoretical (n = 2) stable solutions of the damped Mathieu equation (a) ω = 0.8, δ = 0.6, ζ = 0.025, (b) ω = 1, δ = 0.3, ζ = 0.025, (c) ω = 1.3, δ = 0.8, ζ = 0.025, and (d) ω = 2, δ = 0.15, ζ = 0.05

Grahic Jump Location
Fig. 10

Numerical and theoretical solutions of the two-frequency Mathieu equation (a) response for ω = 0.8, δ = 0.6, γ = 0.5, n = 3, (b) response for ω = 1.5, δ = 0.5, γ = 1, n = 2, (c) FFT for ω = 0.8, δ = 0.6, γ = 0.5, n = 3, and (d) FFT for ω = 1.5, δ = 0.5, γ = 1, n = 2

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In