Araştırma Makalesi
BibTex RIS Kaynak Göster

Lyapunov Üstelleri Kullanılarak Dönen Bir Bıçağın Titreşimlerinin Stabilite Analizi

Yıl 2018, Cilt: 5 Sayı: 1, 11 - 23, 31.01.2018
https://doi.org/10.31202/ecjse.339035

Öz

Bu makalede, dönen bir bıçağın titreşim kararlılığının mil titreşim uyarılmasına bağlı olarak analizi sunulmaktadır. Çalışmada kullanılan temel denklem, birden fazla harmonik olarak değişken katsayı terimine sahip Hill tipi doğrusal ikinci dereceden adi diferansiyel denklemdir. Sistemin diferansiyel denklemi, iki birleşmiş birinci mertebeden adi diferansiyel denklem olarak yeniden yazılmıştır. Kararlı ve kararsız bölgeler, rotor hızına, burulma titreşim uyarılma frekansına ve bıçak doğal frekansına ilişkin parametre uzayındaki Lyapunov karakteristik üstelleri tarafından belirlenir. Sonuçlar, genişletilmiş parametreler yöntemiyle (bir pertürbasyon tekniği) elde edilenlerle karşılaştırıldığında, e küçük burulma titreşim genlikleri için mükemmel bir eşleşme gözlemlenmiştir.       

Kaynakça

  • [1] Srinivasan, A.V., “Vibrations of bladed-disk assemblies: a selected survey”, Journal of Vibration and Acoustics, 106 (1984) 165-168.
  • [2] Crawley, E. F., Mokadam, D. R., “Stagger angle dependence of inertial and elastic coupling in bladed disks”, ASME Journal of Vibration, Acoustics Stress and Reliability in Design, 106 (1984) 181-188.
  • [3] Tang, D., Wang, M., “Coupling technique of rotor-fuselage dynamic analysis”, ASME Journal of Vibration and Acoustics, 106 (1984) 235–238.
  • [4] Okabe, A., Otawara, Y., Kaneko, R., Matsushita, O., Namura, K., ''An equivalent reduced modeling method and its application to shaft-blade coupled torsional vibration analysis of a turbine-generator set", Proceedings of Institute of Mechanical Engineers, 205 (1991) 173-181.
  • [5] Huang, S., Ho, K., “Coupled shaft-torsion and blade-bending vibrations of a rotating shaft-blade unit”, Journal of Engineering for Gas Turbines and Power, 118 (1996) 100-106.
  • [6] Al-Bedoor, B.O., ''Reduced-order nonlinear dynamic model of coupled shaft-torsional and blade-bending vibrations in rotors", Journal of Engineering for Gas Turbines and Power, 123 (2001) 82-89.
  • [7] Al-Nassar, Y. N., Al-Bedoor, B. O.,“On the vibration of a rotating blade vibration on a torsionally flexible shaft”, Journal of Sound and Vibration, 259 (2003) 1237–1242.
  • [8] Al-Nassar, Y. N., Kalyon, M., Pakdemirli, M., Al-Bedoor, B. O., (2007) “Stability Analysis of Rotating Blade Vibration due to Torsional Excitation”, Journal of Vibration and Control, 13 1379-1391.
  • [9] Chen, Z. M., et al., “Computing Lyapunov exponents based on the solution expression of the variational system”, Appl. Math. Comput., 174 (2006) 982-996.
  • [10] Al-Bedoor, B. O., Al-Qaisia, A. A., “Stability analysis of rotating blade bending vibration due to torsional excitation”, Journal of Sound and Vibration, 282 (2005) 1065-1083.
  • [11] Sandri, M., “Numerical calculation of Lyapunov Exponents”, The Mathematica Journal, 3 (1996) 78-84.
  • [12] Wolf, A., Swift, J. B., Swinney, H. L., Vastano, J. A., “Determining Lyapunov exponents from a time series”, Physica D, 16 (1985) 285-317.
  • [13] Geist, K., Parlitz, U., Lauterborn, W., “Comparison of different methods for computing Lyapunov exponents”, Prog. Theor. Phys., 83 (1990) 875-892.
  • [14] Oseledec, V. I., “A multiplicative ergodic theorem. characteristic Lyapunov exponents of dynamical systems”, Trans. Moscow Math. Soc., 19 (1968) 197-231.
  • [15] Benettin, G., Galgani, L., Giorgilli, A., Strelcyn J. M., “Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; a method for computing all of them”, Part I and Part II, Meccanica 15 (1980) 9-30.
  • [16] Christiansen, F., Rugh, H. H., “Computing Lyapunov spectra with continuous Gram–Schmidt orthonormalization”, Nonlinearity 10 (1997) 1063-1072.
  • [17] Lu, J., Tang, G., Oh, H., Luo, A. C. J., “Computing Lyapunov exponents of continuous dynamical systems: method of Lyapunov vectors”, Chaos Solitons Fract. 23 (2005) 1879-1892.
  • [18] Moler, C., Van Loan, C., “Nineteen bubious ways to compute the exponential of a matrix, twenty-five years later”, SIAM Rev., 45 (2003) 3-49.
  • [19] Lambert, J. D., “Numerical Methods for Ordinary Differential Systems”, Wiley, Chichester 1991.
  • [20] Nayfeh, A. H., “Introduction to Perturbation Techniques”, Wiley, New York, 1981.
  • [21] Haken, H., “At least one Lyapunov exponent vanishes if the trajectory of an attractor does not contain a fixed point”, Phys. Lett. A, 94 (1983) 71.

Stability Analysis of Rotating Blade Vibrations Using Lyapunov Exponents

Yıl 2018, Cilt: 5 Sayı: 1, 11 - 23, 31.01.2018
https://doi.org/10.31202/ecjse.339035

Öz

This paper presents an analysis of the vibration stability of a rotating blade due to shaft torsional vibration excitation. The governing equation adopted in the study is a Hill’s type linear second order ordinary differential equation with multiple harmonically variable coefficient terms. The differential equation of the system is rewritten as two coupled first order ordinary differential equations. The stable and unstable regions are determined by the Lyapunov characteristics exponents on parameter space (grid) relating to the rotor speed, the torsional vibration excitation frequency and the blade natural frequency. The results are contrasted to those obtained by the strained parameter method (a perturbation technique), an excellent match is observed for small torsional vibration amplitudes .

Kaynakça

  • [1] Srinivasan, A.V., “Vibrations of bladed-disk assemblies: a selected survey”, Journal of Vibration and Acoustics, 106 (1984) 165-168.
  • [2] Crawley, E. F., Mokadam, D. R., “Stagger angle dependence of inertial and elastic coupling in bladed disks”, ASME Journal of Vibration, Acoustics Stress and Reliability in Design, 106 (1984) 181-188.
  • [3] Tang, D., Wang, M., “Coupling technique of rotor-fuselage dynamic analysis”, ASME Journal of Vibration and Acoustics, 106 (1984) 235–238.
  • [4] Okabe, A., Otawara, Y., Kaneko, R., Matsushita, O., Namura, K., ''An equivalent reduced modeling method and its application to shaft-blade coupled torsional vibration analysis of a turbine-generator set", Proceedings of Institute of Mechanical Engineers, 205 (1991) 173-181.
  • [5] Huang, S., Ho, K., “Coupled shaft-torsion and blade-bending vibrations of a rotating shaft-blade unit”, Journal of Engineering for Gas Turbines and Power, 118 (1996) 100-106.
  • [6] Al-Bedoor, B.O., ''Reduced-order nonlinear dynamic model of coupled shaft-torsional and blade-bending vibrations in rotors", Journal of Engineering for Gas Turbines and Power, 123 (2001) 82-89.
  • [7] Al-Nassar, Y. N., Al-Bedoor, B. O.,“On the vibration of a rotating blade vibration on a torsionally flexible shaft”, Journal of Sound and Vibration, 259 (2003) 1237–1242.
  • [8] Al-Nassar, Y. N., Kalyon, M., Pakdemirli, M., Al-Bedoor, B. O., (2007) “Stability Analysis of Rotating Blade Vibration due to Torsional Excitation”, Journal of Vibration and Control, 13 1379-1391.
  • [9] Chen, Z. M., et al., “Computing Lyapunov exponents based on the solution expression of the variational system”, Appl. Math. Comput., 174 (2006) 982-996.
  • [10] Al-Bedoor, B. O., Al-Qaisia, A. A., “Stability analysis of rotating blade bending vibration due to torsional excitation”, Journal of Sound and Vibration, 282 (2005) 1065-1083.
  • [11] Sandri, M., “Numerical calculation of Lyapunov Exponents”, The Mathematica Journal, 3 (1996) 78-84.
  • [12] Wolf, A., Swift, J. B., Swinney, H. L., Vastano, J. A., “Determining Lyapunov exponents from a time series”, Physica D, 16 (1985) 285-317.
  • [13] Geist, K., Parlitz, U., Lauterborn, W., “Comparison of different methods for computing Lyapunov exponents”, Prog. Theor. Phys., 83 (1990) 875-892.
  • [14] Oseledec, V. I., “A multiplicative ergodic theorem. characteristic Lyapunov exponents of dynamical systems”, Trans. Moscow Math. Soc., 19 (1968) 197-231.
  • [15] Benettin, G., Galgani, L., Giorgilli, A., Strelcyn J. M., “Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; a method for computing all of them”, Part I and Part II, Meccanica 15 (1980) 9-30.
  • [16] Christiansen, F., Rugh, H. H., “Computing Lyapunov spectra with continuous Gram–Schmidt orthonormalization”, Nonlinearity 10 (1997) 1063-1072.
  • [17] Lu, J., Tang, G., Oh, H., Luo, A. C. J., “Computing Lyapunov exponents of continuous dynamical systems: method of Lyapunov vectors”, Chaos Solitons Fract. 23 (2005) 1879-1892.
  • [18] Moler, C., Van Loan, C., “Nineteen bubious ways to compute the exponential of a matrix, twenty-five years later”, SIAM Rev., 45 (2003) 3-49.
  • [19] Lambert, J. D., “Numerical Methods for Ordinary Differential Systems”, Wiley, Chichester 1991.
  • [20] Nayfeh, A. H., “Introduction to Perturbation Techniques”, Wiley, New York, 1981.
  • [21] Haken, H., “At least one Lyapunov exponent vanishes if the trajectory of an attractor does not contain a fixed point”, Phys. Lett. A, 94 (1983) 71.
Toplam 21 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Mühendislik
Bölüm Makaleler
Yazarlar

Tarık Kunduracı

Yayımlanma Tarihi 31 Ocak 2018
Gönderilme Tarihi 20 Eylül 2017
Kabul Tarihi 31 Ekim 2017
Yayımlandığı Sayı Yıl 2018 Cilt: 5 Sayı: 1

Kaynak Göster

IEEE T. Kunduracı, “Lyapunov Üstelleri Kullanılarak Dönen Bir Bıçağın Titreşimlerinin Stabilite Analizi”, ECJSE, c. 5, sy. 1, ss. 11–23, 2018, doi: 10.31202/ecjse.339035.