Energy-Rate Method

  • Reza N. Jazar


Determination of the stability chart and transition curves of parametric differential equations are extremely important in design, optimization, and study of parametric systems. Besides several approximate methods that are capable of determining the stability chart of parametric systems, the energy-rate method is the most exact one. In this chapter we review this analytics-numerical method to determine stable, unstable, and periodic response of differential equations that their stability depends on relation between parameters. We will use the Mathieu equation as the principal example to develop the method.


Energy-rate method Mathieu stability chart Mathieu equation Parametric vibrations Stability of Mathieu equation Stability chart 


  1. Argyris, J., Faust, G., & Haase, M. (1994). An exploration of chaos. Amsterdam: North-Holland.zbMATHGoogle Scholar
  2. Bolotin, V. V. (1964). The dynamic stability of elastic systems. San Francisco: Holden-Day.zbMATHGoogle Scholar
  3. Brigham, E. O. (1974). The fast Fourier transform. Englewood Cliffs: Prentice-Hall.zbMATHGoogle Scholar
  4. Christopherson, J., & Jazar, R. N. (2005). Optimization of classical hydraulic engine mounts based on RMS method. Journal of Shock and Vibration, 12(2), 119–147.CrossRefGoogle Scholar
  5. Christopherson, J., & Jazar, R. N. (2006). Dynamic behavior comparison of passive hydraulic engine mounts, part 1: Mathematical analysis. Journal of Sound and Vibration, 290(3–4), 1040–1070.CrossRefGoogle Scholar
  6. Cveticanin, L. (2014). Oscillators with time variable parameters. New York: Springer.CrossRefGoogle Scholar
  7. Cveticanin, L., & Kovacic, I. (2007). Parametrically excited vibrations of an oscillator with strong cubic negative nonlinearity. Journal of Sound and Vibration, 304(1–2), 201–212.CrossRefGoogle Scholar
  8. Esmailzadeh, E., & Jazar, R. N. (1997). Periodic solution of a Mathieu-Duffing type equation. International Journal of Nonlinear Mechanics, 32(5), 905–912.MathSciNetCrossRefGoogle Scholar
  9. Esmailzadeh, E., Mehri, B., & Jazar, R. N. (1996). Periodic solution of a second order, autonomous, nonlinear system. Journal of Nonlinear Dynamics, 10(4), 307–316.MathSciNetCrossRefGoogle Scholar
  10. Jazar, R. N. (2004). Stability chart of parametric vibrating systems using energy-rate method. International Journal of Non-Linear Mechanics, 39(8), 1319–1331.CrossRefGoogle Scholar
  11. Jazar, R. N. (2013). Advanced vibrations: A modern approach. New York: Springer.CrossRefGoogle Scholar
  12. Jazar, R. N., Mahinfalah, M., Mahmoudian, N., & Aagaah, M. R. (2009). Effects of nonlinearities on the steady state dynamic behavior of electric actuated microcantilever-based resonators. Journal of Vibration and Control, 15(9), 1283–1306.MathSciNetCrossRefGoogle Scholar
  13. Jazar, R. N., Mahinfalah, M., Mahmoudian, N., Aagaah, M. R., & Shiari, B. (2006). Behavior of Mathieu equation in stable regions. International Journal for Mechanics and Solids, 1(1), 1–18.Google Scholar
  14. Jazar, R. N., Mahinfalah, M., Mahmoudian, N., & Rastgaar, M. A. (2008). Energy-rate method and stability chart of parametric vibrating systems. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 30(3), 182–188.CrossRefGoogle Scholar
  15. Mahmoudian, N., Aagaah, M. R., Jazar, R. N., & Mahinfalah, M. (2004). Dynamics of a micro electro mechanical system (MEMS). In 2004 International Conference on MEMS, NANO, and Smart Systems (ICMENS 2004), Banff (pp. 688–693)Google Scholar
  16. McLachlan, N. W. (1947). Theory and application of Mathieu functions. Oxford, UK: Clarendon Press.zbMATHGoogle Scholar
  17. Platonov, A. V. (2018). On the asymptotic stability of nonlinear time-varying switched systems. Journal of Computer and Systems Sciences International, 57(6), 854–863.CrossRefGoogle Scholar
  18. Sheikhlou, M., Rezazadeh, G., & Shabani, R. (2013). Stability and torsional vibration analysis of a micro-shaft subjected to an electrostatic parametric excitation using variational iteration method. Meccanica, 48(2), 259–274.MathSciNetCrossRefGoogle Scholar
  19. Sochacki, W. (2008). The dynamic stability of a simply supported beam with additional discrete elements. Journal of Sound and Vibration, 314(1–2), 180–193.CrossRefGoogle Scholar
  20. Wilcox, R. M. (1967). Exponential operators and parameter differentiation in quantum physics. Journal of Mathematical Physics, 8(4), 962–982.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2020

Authors and Affiliations

  • Reza N. Jazar
    • 1
  1. 1.School of EngineeringRMIT UniversityMelbourneAustralia

Personalised recommendations