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A symbolic algorithm for the automatic computation of multitone-input harmonic balance equations for nonlinear systems

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Abstract

A symbolic algorithm is developed for the automatic generation of harmonic balance equations for multitone input for a class of nonlinear differential systems with polynomial nonlinearities. Generalized expressions are derived for the construction of balance equations for a defined multitone signal form. Procedures are described for determining combinations for a given output frequency from the desired set obtained from box truncated spectra and their permutations to automate symbolic algorithm. An application of method is demonstrated using the well-known Duffing–Van der Pol equation. Then the obtained analytical results are compared with numerical simulations to show the accuracy of the approach. The computation times for both the numerical solutions of equations versus the number of frequency components and the symbolic generation of the equations versus the power of nonlinearity are also investigated.

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References

  1. Volterra, V.: Theory of Functionals and of Integral and Integro Differential Equations. Dover, New York (1959)

    MATH  Google Scholar 

  2. Schetzen, M.: The Volterra and Wiener Theories of Nonlinear Systems. Wiley, New York (1980)

    MATH  Google Scholar 

  3. Lozowicki, A.: On application of the describing function method for optimization of feedback control systems. Int. J. Robust Nonlinear Control. 7, 911–933 (1997). doi:10.1002/(SICI)1099-1239(199710)7:10<911::AID-RNC250>3.0.CO;2-3

    Article  MATH  MathSciNet  Google Scholar 

  4. Taylor, J.A.: Robust nonlinear control based on describing functions methods. In: ASME IMECE. Dynamic Systems and Control Division, vol. 64. Anaheim (1998)

  5. Kahlil, H.K.: Nonlinear Systems, 3rd edn. Prentice Hall International, New York (2001)

    Google Scholar 

  6. Cook, P.A.: Nonlinear dynamical systems. In: Grimble, M.J. (ed.) Prentice Hall International Series in Systems and Control Engineering. Prentice Hall International, New York (1994)

    Google Scholar 

  7. Peyton Jones, J.C., Zhuang, M., Çankaya, İ.: Symbolic computation of harmonic balance equations. Int. J. Control 68(3), 449–460 (1997). doi:10.1080/002071797223460

    Article  MATH  Google Scholar 

  8. Peyton Jones, J.C.: Automatic computation of harmonic balance equations for non-linear systems. Int. J. Control 76(4), 355–365 (2003). doi:10.1080/0020717031000079436

    Article  MATH  MathSciNet  Google Scholar 

  9. Peyton Jones, J.C., Çankaya, İ.: Polyharmonic balance analysis of nonlinear ship roll response. Nonlinear Dyn. 35, 123–146 (2004). doi:10.1023/B:NODY.0000021033.27607.fa

    Article  MATH  Google Scholar 

  10. Peyton Jones, J.C.: Practical frequency response analysis of non-linear time-delayed differential or difference equation models. Int. J. Control 78(1), 65–79 (2005). doi:10.1080/00207170412331330904

    Article  MATH  MathSciNet  Google Scholar 

  11. Dunne, J.F., Hayward, P.: A split-frequency harmonic balance method for non-linear oscillators with multi-harmonic forcing. J. Sound Vib. 295, 939–963 (2006). doi:10.1016/j.jsv.2006.01.050

    Article  MathSciNet  Google Scholar 

  12. Yang, S., Nayfeh, A.H., Mook, D.T.: Combination resonances in the response of the Duffing oscillator to a three-frequency excitation. Acta Mech. 131, 235–245 (1998). doi:10.1007/BF01177227

    Article  MATH  MathSciNet  Google Scholar 

  13. Pusenjak, R.R., Oblak, M.M.: Incremental harmonic balance method with multiple time variables for dynamical systems with cubic non-linearities. Int. J. Numer. Methods Eng. 59, 255–292 (2004). doi:10.1002/nme.875

    Article  MATH  Google Scholar 

  14. Lou, J.-J., He, Q.-W., Zhu, S.-J.: Chaos in the softening duffing system under multi-frequency periodic forces. Appl. Math. Mech. 25(12) (2004)

  15. Kim, C.H., Lee, C.-W., Perkins, N.C.: Nonlinear vibration of sheet metal plates under interacting parametric and external excitation during manufacturing. J. Vib. Acoust. 127, 36–43 (2005). doi:10.1115/1.1857924

    Article  Google Scholar 

  16. Kundert, K.S., White, J.K., Sangiovanni-Vincentelli, A.: Steady-State Methods for Simulating Analog and Microwave Circuits. Kluwer Academic, Boston (1990)

    MATH  Google Scholar 

  17. Dennis, J., Schnebel, R.: Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Prentice Hall, Englewood Cliffs (1983)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Electronics and Computer, Technical Education Faculty, Sakarya University, 54187, Esentepe, Adapazarı, Turkey

    D. Akgün & İ. Çankaya

  2. Department of Electrical and Computer Engineering, Villanova University, 800 Lancaster Ave, Villanova, PA, 19085, USA

    J. C. Peyton Jones

Authors
  1. D. Akgün
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  2. İ. Çankaya
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  3. J. C. Peyton Jones
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Correspondence to İ. Çankaya.

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Akgün, D., Çankaya, İ. & Peyton Jones, J.C. A symbolic algorithm for the automatic computation of multitone-input harmonic balance equations for nonlinear systems. Nonlinear Dyn 56, 179–191 (2009). https://doi.org/10.1007/s11071-008-9390-y

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  • DOI: https://doi.org/10.1007/s11071-008-9390-y

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