Identification of Rotor Unbalance as Inverse Problem of Measurement

Affiliation(s)

Faculty of Mechanics & Mathematics, Dnepropetrovsk University, Dnepropetrovsk, Ukraine.

Faculty of Mechanics & Mathematics, Dnepropetrovsk University, Dnepropetrovsk, Ukraine.

ABSTRACT

In this paper, the problem of identification of the characteristics of the rotor unbalance on two supports is investigated as the inverse problem of measurement. The vibration of rotor supports in two mutually perpendicular directions used as the initial information. The inverse problem is considered, taking into account the error of the mathematical description of rotor-bearings system. To obtain estimates of real unbalance characteristics, the hypothesis as to the exact solutions is applied. The method of Tikhonov regularization is used to obtain stable results. Test calculations are given to illustrate the proposed approach.

KEYWORDS

Mathematical Model; Unbalance Identification; Solution Estimation; Regularization; Numerical Test

Mathematical Model; Unbalance Identification; Solution Estimation; Regularization; Numerical Test

Cite this paper

Y. Menshikov, "Identification of Rotor Unbalance as Inverse Problem of Measurement,"*Advances in Pure Mathematics*, Vol. 3 No. 9, 2013, pp. 20-25. doi: 10.4236/apm.2013.39A1004.

Y. Menshikov, "Identification of Rotor Unbalance as Inverse Problem of Measurement,"

References

[1] A. W. Lees and M. I. Friswell, “The Evaluation of Rotor Imbalance in Flexibly Mounted Machines,” Journal of Sound and Vibration, Vol. 208, No. 5, 1997, pp. 671-683.

http://dx.doi.org/10.1006/jsvi.1997.1260

[2] M. G. Smart, M. I. Friswell and A. W. Lees, “Estimating Turbogenerator Foundation Parameters: Model Selection and Regularization,” Proceedings of the Royal Society of London A, Vol. 456, 2000, pp. 1583-1607.

http://dx.doi.org/10.1098/rspa.2000.0577

[3] M. S. Darlow, “Balancing of High-Speed Machinery: Theory, Methods and Experimental Results,” Mechanical Systems and Signal Processing, Vol. 1, No. 1, 1982, pp. 105-134.

http://dx.doi.org/10.1016/0888-3270(87)90087-2

[4] Yu. L. Menshikov and N. V. Polyakov, “Operative Evaluation of Unbalance Characteristics of a Deformable Rotor,” Proceedings of 8th International Symposium on Technical Diagnostics (IMEKO), Dresden, 23-25 September 1992, pp. 399-408.

[5] Yu. L. Menshikov and N. V. Polyakov, “The New Statement of Problem of Unbalance Identification,” Proceedings of ICTAM 2004, Warsaw, 15-21 August 2004.

[6] A. N. Tikhonov and V. Y. Arsenin, “The Methods of Solution of the Incorrectly Formulated Problems,” Science, Moscow, 1979.

[7] A. V. Goncharskij, A. C. Leonov and A. G. Yagola, “About One Regularized Algorithm for Ill-Posed Problems with Approximate Given Operator,” Journal of Computational Mathematics and Mathematical Physics, Vol. 12, No. 6, 1972, pp. 1592-1594.

[8] Yu. L. Menshikov, “Useful Hypothesis in Inverse Problems of Interpretation,” Proceedings of International Conference Inverse Problems and Applications (IPA2013), Linkoping University, Linkoping, 2-6 April 2013.

[9] Yu. L. Menshikov, “Solution Estimations of Measurement’s Inverse Problem,” Proceedings of 4th Inverse Problems, Design and Optimization Symposium (IPDO-2013), Book of Abstracts, Albi, 26-28 June 2013.

[10] Yu. L. Menshikov, “Synthesis of Adequate Mathematical Description as Solution of Special Inverse Problems,” European Journal of Mathematical Sciences, Vol. 2, No 3, 2013, pp. 256-271.

[11] Yu. L. Menshikov, “The Reduction of Initial Date Inaccuracy in Ill-Posed Problems,” Proceedings of 15th IMACS World Congress on Scientific Computation, Modelling and Applied Mathematics, Vol. VI, Berlin, 24-29 August 1997, pp. 577-582.

[12] Yu. L. Menshikov, “Identification of Optimal Mathematical Model of External Impacts,” Proceedings of MATHMOD 03, Vienna, 5-7 February 2003.

[13] Yu. L. Menshikov, “Some New Statements of Inverse Problems of Recognition,” 10th International Conference on Mathematical Methods in Electromagnetic Theory, Kharkov, 14-17 September 2004, pp. 464-466.

[14] Yu. L. Menshikov, “Recognition of Rotor Machines Unbalance,” Differential Equations and Their Appendices in Physics. The Collection of the Proceedings, Dnepropetrovsk, 1988, pp. 44-56.

[15] Yu. L. Menshikov, “Identification of Rotor Unbalance in Minimax Statement,” 13th International Congress on Sound and Vibration, Vienna, 2-6 July 2006, pp. 345-352.

[1] A. W. Lees and M. I. Friswell, “The Evaluation of Rotor Imbalance in Flexibly Mounted Machines,” Journal of Sound and Vibration, Vol. 208, No. 5, 1997, pp. 671-683.

http://dx.doi.org/10.1006/jsvi.1997.1260

[2] M. G. Smart, M. I. Friswell and A. W. Lees, “Estimating Turbogenerator Foundation Parameters: Model Selection and Regularization,” Proceedings of the Royal Society of London A, Vol. 456, 2000, pp. 1583-1607.

http://dx.doi.org/10.1098/rspa.2000.0577

[3] M. S. Darlow, “Balancing of High-Speed Machinery: Theory, Methods and Experimental Results,” Mechanical Systems and Signal Processing, Vol. 1, No. 1, 1982, pp. 105-134.

http://dx.doi.org/10.1016/0888-3270(87)90087-2

[4] Yu. L. Menshikov and N. V. Polyakov, “Operative Evaluation of Unbalance Characteristics of a Deformable Rotor,” Proceedings of 8th International Symposium on Technical Diagnostics (IMEKO), Dresden, 23-25 September 1992, pp. 399-408.

[5] Yu. L. Menshikov and N. V. Polyakov, “The New Statement of Problem of Unbalance Identification,” Proceedings of ICTAM 2004, Warsaw, 15-21 August 2004.

[6] A. N. Tikhonov and V. Y. Arsenin, “The Methods of Solution of the Incorrectly Formulated Problems,” Science, Moscow, 1979.

[7] A. V. Goncharskij, A. C. Leonov and A. G. Yagola, “About One Regularized Algorithm for Ill-Posed Problems with Approximate Given Operator,” Journal of Computational Mathematics and Mathematical Physics, Vol. 12, No. 6, 1972, pp. 1592-1594.

[8] Yu. L. Menshikov, “Useful Hypothesis in Inverse Problems of Interpretation,” Proceedings of International Conference Inverse Problems and Applications (IPA2013), Linkoping University, Linkoping, 2-6 April 2013.

[9] Yu. L. Menshikov, “Solution Estimations of Measurement’s Inverse Problem,” Proceedings of 4th Inverse Problems, Design and Optimization Symposium (IPDO-2013), Book of Abstracts, Albi, 26-28 June 2013.

[10] Yu. L. Menshikov, “Synthesis of Adequate Mathematical Description as Solution of Special Inverse Problems,” European Journal of Mathematical Sciences, Vol. 2, No 3, 2013, pp. 256-271.

[11] Yu. L. Menshikov, “The Reduction of Initial Date Inaccuracy in Ill-Posed Problems,” Proceedings of 15th IMACS World Congress on Scientific Computation, Modelling and Applied Mathematics, Vol. VI, Berlin, 24-29 August 1997, pp. 577-582.

[12] Yu. L. Menshikov, “Identification of Optimal Mathematical Model of External Impacts,” Proceedings of MATHMOD 03, Vienna, 5-7 February 2003.

[13] Yu. L. Menshikov, “Some New Statements of Inverse Problems of Recognition,” 10th International Conference on Mathematical Methods in Electromagnetic Theory, Kharkov, 14-17 September 2004, pp. 464-466.

[14] Yu. L. Menshikov, “Recognition of Rotor Machines Unbalance,” Differential Equations and Their Appendices in Physics. The Collection of the Proceedings, Dnepropetrovsk, 1988, pp. 44-56.

[15] Yu. L. Menshikov, “Identification of Rotor Unbalance in Minimax Statement,” 13th International Congress on Sound and Vibration, Vienna, 2-6 July 2006, pp. 345-352.