Journal of Applied Mathematics

Solutions of a Quadratic Inverse Eigenvalue Problem for Damped Gyroscopic Second-Order Systems

Hong-Xiu Zhong, Guo-Liang Chen, and Xiang-Yun Zhang

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Given k pairs of complex numbers and vectors (closed under conjugation), we consider the inverse quadratic eigenvalue problem of constructing n × n real matrices M , D , G , and K , where M > 0 , K and D are symmetric, and G is skew-symmetric, so that the quadratic pencil Q ( λ ) = λ 2 M + λ ( D + G ) + K has the given k pairs as eigenpairs. First, we construct a general solution to this problem with k n . Then, with the special properties D = 0 and K < 0 , we construct a particular solution. Numerical results illustrate these solutions.

Article information

Source
J. Appl. Math., Volume 2014 (2014), Article ID 703178, 9 pages.

Dates
First available in Project Euclid: 26 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.jam/1395855289

Digital Object Identifier
doi:10.1155/2014/703178

Mathematical Reviews number (MathSciNet)
MR3166781

Zentralblatt MATH identifier
07010719

Citation

Zhong, Hong-Xiu; Chen, Guo-Liang; Zhang, Xiang-Yun. Solutions of a Quadratic Inverse Eigenvalue Problem for Damped Gyroscopic Second-Order Systems. J. Appl. Math. 2014 (2014), Article ID 703178, 9 pages. doi:10.1155/2014/703178. https://projecteuclid.org/euclid.jam/1395855289


Export citation

References

  • M. J. Balas, “Trends in large space structure control theory: fondest hopes, wildest dreams,” IEEE Transactions on Automatic Control, vol. 27, no. 3, pp. 522–535, 1982.
  • Z. Bai and Y. Su, “Soar: a second-order arnoldi method for the solution of the quadratic eigenvalue problem,” SIAM Journal on Matrix Analysis and Applications, vol. 26, no. 3, pp. 640–659, 2005.
  • C. Guo, “Numerical solution of a quadratic eigenvalue problem,” Linear Algebra and Its Applications, vol. 385, no. 1–3, pp. 391–406, 2004.
  • Z. Jia and Y. Sun, “A refined Jacobi-Davidson method for the quadratic eigenvalue problem,” in Proceedings of the 10th WSEAS International Confenrence on APPLIED MATHEMATICS, pp. 1150–3155, Dallas, Tex, USA, November 2006.
  • K. Meerbergen, “The quadratic arnoldi method for the solution of the quadratic eigenvalue problem,” SIAM Journal on Matrix Analysis and Applications, vol. 30, no. 4, pp. 1463–1482, 2008.
  • J. Qian and W. Lin, “A numerical method for quadratic eigenvalue problems of gyroscopic systems,” Journal of Sound and Vibration, vol. 306, no. 1-2, pp. 284–296, 2007.
  • F. Tisseur and K. Meerbergen, “The quadratic eigenvalue problem,” SIAM Review, vol. 43, no. 2, pp. 235–286, 2001.
  • L. Zhou, L. Bao, Y. Lin, Y. Wei, and Q. Wu, “Restarted generalized second-order krylov subspace methods for solving quadratic eigenvalue problems,” World Academy of Science, Engineering and Technology, vol. 67, pp. 429–436, 2010.
  • I. Gohberg, P. Lancaster, and L. Rodman, “On selfadjoint matrix polynomials,” Integral Equations and Operator Theory, vol. 2, no. 3, pp. 434–439, 1979.
  • M. Chu and S. Xu, “Spectral decomposition of real symmetric quadratic $\lambda $-matrices and its applications,” Mathematics of Computation, vol. 78, no. 265, pp. 293–313, 2009.
  • Z. Jia and M. Wei, “A real-valued spectral decomposition of the undamped gyroscopic system with applications,” SIAM Journal on Matrix Analysis and Applications, vol. 32, no. 2, pp. 584–604, 2011.
  • Y. Kuo, W. Lin, and S. Xu, “Solutions of the partially described inverse quadratic eigenvalue problem,” SIAM Journal on Matrix Analysis and Applications, vol. 29, no. 1, pp. 33–53, 2006.
  • Y. Cai, Y. Kuo, W. Lin, and S. Xu, “Solutions to a quadratic inverse eigenvalue problem,” Linear Algebra and Its Applications, vol. 430, no. 5-6, pp. 1590–1606, 2009.
  • Y. Yuan, “An inverse eigenvalue problem for damped gyroscopic second-order systems,” Mathematical Problems in Engineering, vol. 2009, Article ID 725616, 10 pages, 2009.
  • H. W. Braden, “The equations ${A}^{T}X\pm {X}^{T}A=B$,” Journal on Matrix Analysis and Applications, vol. 20, no. 2, pp. 295–302, 1999. \endinput