Eigensolution Variability of Asymmetric Damped Systems

Affiliation(s)

School of Mechanical and Electronic Engineering, Wuhan University of Technology, Wuhan, China.

School of Mechanical and Electronic Engineering, Wuhan University of Technology, Wuhan, China.

ABSTRACT

The characterization of energy dissipation or damping in rotor dynamic model is of fundamental importance. Noise and vibration are not only uncomfortable to the users, but also may lead to fatigue, fracture and even failure. During the design process of asymmetric damped systems, it is often required to make changes in the design variables such that the design is optimal. This paper is aimed at developing computationally efficient numerical methods for parametric sensitivity analysis. The algebraic method considered here computes the eigenvector sensitivity by assembling the derivatives of eigenproblems and the additional constraints into an algebraic equation. The coefficient matrix may be ill-conditioned since the elements of it are not all of the same order of magnitude. In this study, a new algebraic method is presented to compute the eigensolution variability of asymmetric damped systems. Some weight constants are introduced such that the proposed method is well-conditioned. The method is very compact and highly efficient, and the numerical stability is also demonstrated. Moreover, several special cases can be presented based on the similar idea of the proposed method. Finally, two numerical examples show the validity of the proposed method.

The characterization of energy dissipation or damping in rotor dynamic model is of fundamental importance. Noise and vibration are not only uncomfortable to the users, but also may lead to fatigue, fracture and even failure. During the design process of asymmetric damped systems, it is often required to make changes in the design variables such that the design is optimal. This paper is aimed at developing computationally efficient numerical methods for parametric sensitivity analysis. The algebraic method considered here computes the eigenvector sensitivity by assembling the derivatives of eigenproblems and the additional constraints into an algebraic equation. The coefficient matrix may be ill-conditioned since the elements of it are not all of the same order of magnitude. In this study, a new algebraic method is presented to compute the eigensolution variability of asymmetric damped systems. Some weight constants are introduced such that the proposed method is well-conditioned. The method is very compact and highly efficient, and the numerical stability is also demonstrated. Moreover, several special cases can be presented based on the similar idea of the proposed method. Finally, two numerical examples show the validity of the proposed method.

KEYWORDS

Design Sensitivity Analysis; Asymmetric Damped Systems; Rotor Dynamic Model; Condition Number

Design Sensitivity Analysis; Asymmetric Damped Systems; Rotor Dynamic Model; Condition Number

Cite this paper

J. Chen and Y. Tan, "Eigensolution Variability of Asymmetric Damped Systems,"*Journal of Analytical Sciences, Methods and Instrumentation*, Vol. 2 No. 3, 2012, pp. 140-148. doi: 10.4236/jasmi.2012.23023.

J. Chen and Y. Tan, "Eigensolution Variability of Asymmetric Damped Systems,"

References

[1] R. L. Fox and M. P. Kapoor, “Rates of Change of Eigenvalues and Eigenvectors,” AIAA Journal, Vol. 6, No. 12, 1968, pp. 2426-2429. doi:10.2514/3.5008

[2] L. C. Rogers, “Derivatives of Eigenvalues and Eigenvectors,” AIAA Journal, Vol. 11, No. 8, 1970, pp. 943-944. doi:10.2514/3.5795

[3] S. Garg, “Derivatives of Eigensolutions for a General Matrix,” AIAA Journal, Vol. 11, No. 8, 1973, pp. 1191-1194. doi:10.2514/3.6892

[4] R. H. Plaut and K. Huseyin, “Derivative of Eigenvalues and Eigenvectors in Non-Self-Adjoint Systems,” AIAA Journal, Vol. 11, No. 2, 1973, pp. 250-251. doi:10.2514/3.6740

[5] C. S. Rudisill, “Derivatives of Eigenvalues and Eigenvectors for a General Matrix,” AIAA Journal, Vol. 12, No. 5, 1974, pp. 721-722. doi:10.2514/3.49330

[6] S. Adhikari and M. I. Friswell, “Eigenderivative Analysis of Asymmetric Non-Conservative Systems,” International Journal for Numerical Methods in Engineering, Vol. 51, No. 6, 2001, pp. 709-733. doi:10.1002/nme.186

[7] Q. H. Zeng, “Highly Accurate Modal Method for Calculating Eigenvector Derivative in Viscous Damping Systems,” AIAA Journal, Vol. 33, No. 4, 1995, pp. 746-751. doi:10.2514/3.12453

[8] Y. J. Moon, B. W. Kim, M. G. Ko and I. W. Lee, “Modified Modal Methods for Calculating Eigenpair Sensitivity of Asymmetric Damped System,” International Journal for Numerical Methods in Engineering, Vol. 60, No. 11, 2004, pp. 1847-1860. doi:10.1002/nme.1025

[9] B. P. Wang, “Improved Approximate Methods for Computing Eigenvector Derivatives in Structural Dynamics,” AIAA Journal, Vol. 29, No. 6, 1992, pp. 1018-1020.

[10] R. B. Nelson, “Simplified Calculation of Eigenvector Derivatives,” AIAA Journal, Vol. 14, No. 9, 1976, pp. 1201-1205. doi:10.2514/3.7211

[11] T. R. Sutter, C. J. Camarda, J. L. Walsh and H. M. Adelman, “Comparison of Several Methods for Calculating Vibration Mode Shape Derivatives,” AIAA Journal, Vol. 26, No. 12, 1989, pp. 1506-1511. doi:10.2514/3.10070

[12] M. I. Friswell, S. Adhikari, “Derivatives of Complex Eigenvectors Using Nelson’s Method,” AIAA Journal, Vol. 38, No. 12, 2000, pp. 2355-2357. doi:10.2514/2.907

[13] N. Guedria, M. Chouchane and H. Smaoui, “Second-Order Eigensensitivity Analysis of Asymmetric Damped Systems Using Nelson’s Method,” Journal of Sound and Vibration, Vol. 300, No. 3-5, 2007, pp. 974-992. doi:10.1016/j.jsv.2006.09.003

[14] C. S. Rudisill and Y. Chu, “Numerical Methods for Evaluating the Derivatives of Eigenvalues and Eigenvectors,” AIAA Journal, Vol. 13, No. 6, 1975, pp. 834-837. doi:10.2514/3.60449

[15] I. W. Lee and G. H. Jung, “An Efficient Algebraic Method for the Computation of Natural Frequency and Mode Shape Sensitivities: Part I, Distinct Natural Frequencies,” Computers and Structures, Vol. 62, No. 3, 1997, pp. 429-435. doi:10.1016/S0045-7949(96)00206-4

[16] I. W. Lee, D. O. Kim and G. H. Jung, “Natural Frequency and Mode Shape Sensitivities of Damped Systems: Part I, Distinct Natural Frequencies,” Journal of Sound and Vibration, Vol. 223, No. 3, 1999, pp. 399-412. doi: doi:10.1006/jsvi.1998.2129

[17] N. Guedria, H. Smaoui and M. Chouchane, “A Direct Algebraic Method for Eigensolution Sensitivity Computation of Damped Asymmetric Systems,” International Journal for Numerical Methods in Engineering, Vol. 68, No. 6, 2006, pp. 674-689. doi:10.1002/nme.1732

[18] M. Chouchane, N. Guedria and H. Smaoui, “Eigensensitivity Computation of Asymmetric Damped Systems Using an Algebraic Approach,” Mechanical System and Signal Processing, Vol. 21, No. 7, 2007, pp. 2761-2776. doi:10.1002/nme.1732

[19] Z. H. Xu, H. X. Zhong, X. W. Zhu and B. S. Wu, “An Efficient Algebraic Method for Computing Eigensolution Sensitivity of Asymmetric Damped Systems,” Journal of Sound and Vibration, Vol. 327, No. 3-5, 2009, pp. 584-592. doi:10.1016/j.jsv.2009.07.013

[20] L. Li, Y. Hu and X. Wang, “A Parallel Way for Computing Eigenvector Sensitivity of Asymmetric Damped Systems with Distinct and Repeated Eigenvalues,” Mechanical System and Signal Processing, Vol. 30, No. 7, 2012, pp. 61-67. doi:10.1016/j.ymssp.2012.01.008

[21] D. V. Murthy and R. T. Haftka, “Derivatives of Eigenvalues and Eigenvectors of a General Complex Matrix,” International Journal for Numerical Methods in Engineering, Vol. 26, No. 2, 1988, pp. 293-311. doi:10.1002/nme.1620260202

[22] N. J. Higham, “Accuracy and Stability of Numerical Algorithms,” 2nd Edition, Siam, Philadelphia, 2002. doi:10.1137/1.9780898718027

[1] R. L. Fox and M. P. Kapoor, “Rates of Change of Eigenvalues and Eigenvectors,” AIAA Journal, Vol. 6, No. 12, 1968, pp. 2426-2429. doi:10.2514/3.5008

[2] L. C. Rogers, “Derivatives of Eigenvalues and Eigenvectors,” AIAA Journal, Vol. 11, No. 8, 1970, pp. 943-944. doi:10.2514/3.5795

[3] S. Garg, “Derivatives of Eigensolutions for a General Matrix,” AIAA Journal, Vol. 11, No. 8, 1973, pp. 1191-1194. doi:10.2514/3.6892

[4] R. H. Plaut and K. Huseyin, “Derivative of Eigenvalues and Eigenvectors in Non-Self-Adjoint Systems,” AIAA Journal, Vol. 11, No. 2, 1973, pp. 250-251. doi:10.2514/3.6740

[5] C. S. Rudisill, “Derivatives of Eigenvalues and Eigenvectors for a General Matrix,” AIAA Journal, Vol. 12, No. 5, 1974, pp. 721-722. doi:10.2514/3.49330

[6] S. Adhikari and M. I. Friswell, “Eigenderivative Analysis of Asymmetric Non-Conservative Systems,” International Journal for Numerical Methods in Engineering, Vol. 51, No. 6, 2001, pp. 709-733. doi:10.1002/nme.186

[7] Q. H. Zeng, “Highly Accurate Modal Method for Calculating Eigenvector Derivative in Viscous Damping Systems,” AIAA Journal, Vol. 33, No. 4, 1995, pp. 746-751. doi:10.2514/3.12453

[8] Y. J. Moon, B. W. Kim, M. G. Ko and I. W. Lee, “Modified Modal Methods for Calculating Eigenpair Sensitivity of Asymmetric Damped System,” International Journal for Numerical Methods in Engineering, Vol. 60, No. 11, 2004, pp. 1847-1860. doi:10.1002/nme.1025

[9] B. P. Wang, “Improved Approximate Methods for Computing Eigenvector Derivatives in Structural Dynamics,” AIAA Journal, Vol. 29, No. 6, 1992, pp. 1018-1020.

[10] R. B. Nelson, “Simplified Calculation of Eigenvector Derivatives,” AIAA Journal, Vol. 14, No. 9, 1976, pp. 1201-1205. doi:10.2514/3.7211

[11] T. R. Sutter, C. J. Camarda, J. L. Walsh and H. M. Adelman, “Comparison of Several Methods for Calculating Vibration Mode Shape Derivatives,” AIAA Journal, Vol. 26, No. 12, 1989, pp. 1506-1511. doi:10.2514/3.10070

[12] M. I. Friswell, S. Adhikari, “Derivatives of Complex Eigenvectors Using Nelson’s Method,” AIAA Journal, Vol. 38, No. 12, 2000, pp. 2355-2357. doi:10.2514/2.907

[13] N. Guedria, M. Chouchane and H. Smaoui, “Second-Order Eigensensitivity Analysis of Asymmetric Damped Systems Using Nelson’s Method,” Journal of Sound and Vibration, Vol. 300, No. 3-5, 2007, pp. 974-992. doi:10.1016/j.jsv.2006.09.003

[14] C. S. Rudisill and Y. Chu, “Numerical Methods for Evaluating the Derivatives of Eigenvalues and Eigenvectors,” AIAA Journal, Vol. 13, No. 6, 1975, pp. 834-837. doi:10.2514/3.60449

[15] I. W. Lee and G. H. Jung, “An Efficient Algebraic Method for the Computation of Natural Frequency and Mode Shape Sensitivities: Part I, Distinct Natural Frequencies,” Computers and Structures, Vol. 62, No. 3, 1997, pp. 429-435. doi:10.1016/S0045-7949(96)00206-4

[16] I. W. Lee, D. O. Kim and G. H. Jung, “Natural Frequency and Mode Shape Sensitivities of Damped Systems: Part I, Distinct Natural Frequencies,” Journal of Sound and Vibration, Vol. 223, No. 3, 1999, pp. 399-412. doi: doi:10.1006/jsvi.1998.2129

[17] N. Guedria, H. Smaoui and M. Chouchane, “A Direct Algebraic Method for Eigensolution Sensitivity Computation of Damped Asymmetric Systems,” International Journal for Numerical Methods in Engineering, Vol. 68, No. 6, 2006, pp. 674-689. doi:10.1002/nme.1732

[18] M. Chouchane, N. Guedria and H. Smaoui, “Eigensensitivity Computation of Asymmetric Damped Systems Using an Algebraic Approach,” Mechanical System and Signal Processing, Vol. 21, No. 7, 2007, pp. 2761-2776. doi:10.1002/nme.1732

[19] Z. H. Xu, H. X. Zhong, X. W. Zhu and B. S. Wu, “An Efficient Algebraic Method for Computing Eigensolution Sensitivity of Asymmetric Damped Systems,” Journal of Sound and Vibration, Vol. 327, No. 3-5, 2009, pp. 584-592. doi:10.1016/j.jsv.2009.07.013

[20] L. Li, Y. Hu and X. Wang, “A Parallel Way for Computing Eigenvector Sensitivity of Asymmetric Damped Systems with Distinct and Repeated Eigenvalues,” Mechanical System and Signal Processing, Vol. 30, No. 7, 2012, pp. 61-67. doi:10.1016/j.ymssp.2012.01.008

[21] D. V. Murthy and R. T. Haftka, “Derivatives of Eigenvalues and Eigenvectors of a General Complex Matrix,” International Journal for Numerical Methods in Engineering, Vol. 26, No. 2, 1988, pp. 293-311. doi:10.1002/nme.1620260202

[22] N. J. Higham, “Accuracy and Stability of Numerical Algorithms,” 2nd Edition, Siam, Philadelphia, 2002. doi:10.1137/1.9780898718027