Decoupling of second-order linear systems by isospectral transformation

Abstract

We consider the class of real second-order linear dynamical systems that admit real diagonal forms with the same eigenvalues and partial multiplicities. The nonzero leading coefficient is allowed to be singular, and the associated quadratic matrix polynomial is assumed to be regular. We present a method and algorithm for converting any such n-dimensional system into a set of n mutually independent second-, first-, and zeroth-order equations. The solutions of these two systems are related by a real, time-dependent, and nonlinear n-dimensional transformation. Explicit formulas for computing the \(2n \times 2n\) real and time-invariant equivalence transformation that enables this conversion are provided. This paper constitutes a complete solution to the problem of diagonalizing a second-order linear system while preserving its associated Jordan canonical form.

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References

  1. 1.

    Meirovitch, L.: Principles and Techniques of Vibrations. Prentice Hall, Upper Saddle River (1997)

    Google Scholar 

  2. 2.

    Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1985)

    Google Scholar 

  3. 3.

    Caughey, T.K., O’Kelly, M.E.J.: Classical normal modes in damped linear dynamic systems. ASME J. Appl. Mech. 32(3), 583–588 (1965)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Ma, F., Caughey, T.K.: Analysis of linear nonconservative vibrations. ASME J. Appl. Mech. 62(3), 685–691 (1995)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Lancaster, P., Zaballa, I.: Parameterizing structure preserving transformations of matrix polynomials. Oper. Theory Adv. Appl. 218, 403–424 (2012)

    MATH  Google Scholar 

  6. 6.

    Garvey, S.D., Lancaster, P., Popov, A.A., Prells, U., Zaballa, I.: Filters connecting isospectral quadratic systems. Linear Algebra Appl. 438(4), 1497–1516 (2013)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Lancaster, P., Zaballa, I.: Diagonalizable quadratic eigenvalue problems. Mech. Syst. Signal Process. 23(4), 1134–1144 (2009)

    Article  Google Scholar 

  8. 8.

    Garvey, S.D., Friswell, M.I., Prells, U.: Co-ordinate transformations for second order systems. Part I: general transformations. J. Sound Vib. 258(5), 885–909 (2002)

    Article  Google Scholar 

  9. 9.

    Morzfeld, M., Ma, F., Parlett, B.N.: The transformation of second-order linear systems into independent equations. SIAM J. Appl. Math. 71(4), 1026–1043 (2011)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Zúñiga Anaya, J.C.: Diagonalization of quadratic matrix polynomials. Syst. Control Lett. 59(2), 105–113 (2010)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Gohberg, I., Lancaster, P., Rodman, L.: Matrix Polynomials. Academic Press, New York (1982)

    Google Scholar 

  12. 12.

    Lancaster, P., Tismenetsky, M.: The Theory of Matrices, 2nd edn. Academic Press, San Diego (1985)

    Google Scholar 

  13. 13.

    Tisseur, F., Meerbergen, K.: The quadratic eigenvalue problem. SIAM Rev. 43(2), 235–286 (2001)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Gohberg, I., Kaashoek, M.A., Lancaster, P.: General theory of regular matrix polynomials and band Toeplitz operators. Integral Equ. Oper. Theory 11(6), 776–882 (1988)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Mackey, D.S., Mackey, N., Mehl, C., Mehrmann, M.: Vector spaces of linearizations for matrix polynomials. SIAM J. Matrix Anal. Appl. 28(4), 971–1004 (2006)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Lancaster, P., Psarrakos, P.: A note on weak and strong linearizations of regular matrix polynomials. Numerical analysis report 470, Manchester Centre for Computational Mathematics, Manchester (2005)

  17. 17.

    Ma, F., Morzfeld, M., Imam, A.: The decoupling of damped linear systems in free or forced vibration. J. Sound Vib. 329(15), 3182–3202 (2010)

    Article  Google Scholar 

Download references

Acknowledgements

RGS was supported by a Science Without Borders fellowship from the CAPES Foundation (grant no. 99999.011952/2013-00).

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Correspondence to Daniel T. Kawano.

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Kawano, D.T., Salsa, R.G. & Ma, F. Decoupling of second-order linear systems by isospectral transformation. Z. Angew. Math. Phys. 69, 137 (2018). https://doi.org/10.1007/s00033-018-1030-x

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Keywords

  • Second-order linear differential equations
  • Quadratic matrix polynomials
  • Diagonalization
  • Isospectral systems

Mathematics Subject Classification

  • 15A22
  • 34A30
  • 70J10