Kompleks Koordinat Formülasyonuna Dayalı İzotropik ve Anizotropik Mesnetli Rotorların İç Sönümleme Kararsızlığı

Abstract

Bu çalışma, rotordinamik sistemlerin iç sönümleme kararlılığı için kompleks koordinat formülasyonunun avantaj ve dezavantajlarını araştırmaktadır. Rotor yapısına özgü sönümleme mekanizmaları, sabit sönümleme kaynaklarına kıyasla titreşimler üzerinde farklı etkilere sahiptir. İç ve dış sönümleme kaynakları, sabit referans çerçevesine göre farklı titreşim frekanslarına sahiptir. Bu nedenle, harici sönümlemenin aksine, dahili sönümleme titreşimleri her zaman kararlı hale getirmez. Bu nedenle, titreşim özelliklerini doğru bir şekilde tahmin etmek için sönümleme kuvvetlerinin modele doğru bir şekilde dahil edilmesi araştırılmıştır. Rotor parçaları arasındaki sürtünme bağlantılarının ve yapısal sönümlemenin neden olduğu iç sönümleme nedeniyle rotordinamik kararlılığı incelemek için birleşik bir sonlu eleman modeli geliştirilmiştir. İç sürtünme bağlantılarını modellemek için döner yatak elemanları kullanılmış ve histerezis sönümlemeli dinamik denklemi, izotropik ve anizotropik bağlantılar üzerindeki rotorlar için karmaşık vektör gösterimi kullanılarak sağlanmıştır. Kompleks koordinat formülasyonu, dönen ve sabit referans çerçeveleri arasındaki vektörlerin dönüşümünde matematiksel avantajlar sağlamaktadır. İzotropik mesnetler söz konusu olduğunda, kompleks koordinat formülasyonunun kullanılması düşük boyutlu bir model ortaya çıkarmakta ve modelin verimliliğini artırmaktadır. Ancak, anizotropik mesnetler söz konusu olduğunda, modelin mertebesinin azaltılması mümkün olmamakta ve sistemin kinematiği nedeniyle hareket denklemi doğrusal değildir. Bu durum, özdeğer problemini çözmek için iteratif bir yöntem gerektirmektedir. Doğrulama için, geliştirilen modellerin sonuçları ticari bir sonlu elemanlar yazılımının sonuçları ile karşılaştırılmıştır. Sonuç olarak, farklı iç sönümleme kaynaklarının genel rotordinamik kararlılık üzerindeki etkisi gösterilmiştir.

Keywords

• Rotordinamik
• iç sönüm kararlılığı
• kompleks koordinat formülasyonu
• sürtünmeli mafsallar
• yapısal sönümleme

Internal Damping Instability of Rotors with Isotropic and Anisotropic Supports based on Complex Coordinates Formulation

Abstract

This study investigates the advantages and disadvantages of complex coordinates formulation for internal damping stability of rotordynamic systems. Damping mechanisms inherent to the rotor structure have different effects on vibrations when compared to stationary damping sources. The internal and external damping sources experience different vibration frequencies with respect to the stationary reference frame. Thus, in contrast to external damping, internal damping does not always stabilize vibrations. Therefore, the correct incorporation of damping forces into the model is investigated to predict vibration characteristics accurately. A unified finite element model is developed to study rotordynamic stability due to internal damping caused by frictional joints between rotor parts and structural damping. Rotating bearing elements are used to model internal frictional joints and the governing equation with hysteresis damping is provided using complex vector notation for rotors on isotropic and anisotropic mounts. Complex coordinates formulation provides mathematical advantages in transformation of vectors between rotating and stationary reference frames. In the case of isotropic supports, the use of complex coordinates formulation yields a low-dimensional model and increases the efficiency of the model. However, in the case of anisotropic supports, reduction in the order of the model is not possible and the equation of motion is nonlinear due to kinematics of the system. This requires an iterative method to solve the eigenvalue problem. For verifications, the results of the developed models are compared to those of a commercial finite element software. Consequently, the effect of different internal damping sources on the overall rotordynamic stability is demonstrated.

Keywords

• Rotordynamics
• internal damping stability
• complex coordinates formulation
• frictional joints
• structural damping

References

1. [1] Wu, J., Rezgui, D., Titurus, B. 2023. Model and experimental analysis of a rotor rig dynamics with time-varying characteristics, Journal of Sound and Vibration, Vol. 557, pp. 117683.
2. [2] Lotfan, S., Salehpour, N., Adiban, H., Mashroutechi, A., 2015. Bearing fault detection using fuzzy C-means and hybrid C-means-subtractive algorithms. IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), August, Istanbul, pp.1–7.
3. [3] Pattnayak, M., Dutt, J., Pandey, R., 2022. Rotordynamics of an accelerating rotor supported on aerodynamic journal bearings. Tribology International, Vol.176, pp.107883.
4. [4] Lotfan, S., Bediz, B., 2022. Free vibrations of rotating pre-twisted blades including geometrically nonlinear pre-stressed analysis. Journal of Sound and Vibration, Vol.535, pp.117109.
5. [5] Chipato, E.T., Shaw, A.D., Friswell, M.I., 2021. Nonlinear rotordynamics of a MDOF rotor–stator contact system subjected to frictional and gravitational effects. Mechanical Systems and Signal Processing, Vol.159, pp.107776.
6. [6] De Felice, A., Sorrentino, S., 2021. Damping and gyroscopic effects on the stability of parametrically excited continuous rotor systems. Nonlinear Dynamics, Vol.103, pp.3529–3555.
7. [7] Lotfan, S., Anamagh, M.R., Bediz, B., Cigeroglu, E., 2022. Nonlinear resonances of axially functionally graded beams rotating with varying speed including Coriolis effects. Nonlinear Dynamics, Vol.107, pp.533–558.
8. [8] Krack, M., Salles, L., Thouverez, F., 2017. Vibration prediction of bladed disks coupled by friction joints. Archives of Computational Methods in Engineering, Vol.24, pp.589–636.

9. [9] Lund, J.W., 1967. Destabilisation of rotors from friction in internal joints. In: Vibration and Wear in High Speed Rotating Machinery, pp.617–629.
10. [10] Walton, J., Artiles, A., Lund, J., Dill, J., Zorzi, E., 1990. Internal rotor friction instability. NASA Report, Vol. NAS 1.26: 183942.
11. [11] Wang, L., Wang, A., Jin, M., Yin, Y., Heng, X., Ma, P., 2021. Nonlinear dynamic response and stability of a rod fastening rotor with internal damping effect. Archive of Applied Mechanics, Vol.91, pp.3851–3867.
12. [12] Dai, Z., Jing, J., Chen, C., Cong, J., 2018. Extensive experimental study on the stability of rotor system with spline coupling. In: Turbo Expo: Power for Land, Sea, and Air, Vol.51135, pp.V07AT33A021.
13. [13] Melanson, J., Zu, J., 1998. Free vibration and stability analysis of internally damped rotating shafts with general boundary conditions. ASME Journal of Vibration and Acoustics, Vol.120, pp.776–783.
14. [14] Genta, G., Brusa, E., 2000. On the role of nonsynchronous rotating damping in rotordynamics. International Journal of Rotating Machinery, Vol.6, pp.467–475.
15. [15] Forrai, L., 2000. A finite element model for stability analysis of symmetrical rotor systems with internal damping. Journal of Computational and Applied Mechanics, Vol.1, pp.37–47.
16. [16] Cerminaro, A.M., Nelson, F.C., 2000. The effect of viscous and hysteretic damping on rotor stability. In: Turbo Expo: Power for Land, Sea, and Air, Vol.78576, pp.V004T03A043.
17. [17] Ku, D.-M., 1998. Finite element analysis of whirl speeds for rotor-bearing systems with internal damping. Mechanical Systems and Signal Processing, Vol.12, pp.599–610.
18. [18] Genta, G., 2004. On a persistent misunderstanding of the role of hysteretic damping in rotordynamics. ASME Journal of Vibration and Acoustics, Vol.126, pp.459–461.
19. [19] Genta, G., Amati, N., 2010. Hysteretic damping in rotordynamics: An equivalent formulation. Journal of Sound and Vibration, Vol.329, pp.4772–4784.
20. [20] Genta, G., 2015. On the stabilizing effect of support asymmetry in rotordynamics. In: Proceedings of the 9th IFToMM International Conference on Rotor Dynamics, Springer, pp.2045–2057.
21. [21] d'Alessandro, F., Festjens, H., Chevallier, G., Cogan, S., 2021. Hysteretic damping in rotordynamics: A focus on damping induced instability. IMAC XL, SEM, February, Orlando, FL, USA.
22. [22] Chandra, N.H., Sekhar, A., 2016. Nonlinear damping identification in rotors using wavelet transform. Mechanism and Machine Theory, Vol.100, pp.170–183.
23. [23] Kozioł, M., Cupiał, P., 2022. The influence of the active control of internal damping on the stability of a cantilever rotor with a disc. Mechanics Based Design of Structures and Machines, Vol.50, pp.288–301.
24. [24] Mori, H., Sueda, M., Shiroshita, K., Kondou, T., 2024. Effect of damping and rotor moment of inertia on stability of self-synchronization for two unbalanced rotors. Journal of Sound and Vibration, Vol.570, pp.118103.
25. [25] Arab, S.B., Rodrigues, J.D., Bouaziz, S., Haddar, M., 2018. Stability analysis of internally damped rotating composite shafts using a finite element formulation. Comptes Rendus Mécanique, Vol.346, pp.291–307.
26. [26] Friswell, M.I., 2010. Dynamics of Rotating Machines. Cambridge University Press.
27. [27] Roy, D.K., Tiwari, R., 2019. Development of identification procedure for the internal and external damping in a cracked rotor system undergoing forward and backward whirls. Archive of Mechanical Engineering, Vol.66, pp.229–255.
28. [28] Mendonça, W.R.D.P., De Medeiros, E.C., Pereira, A.L.R., Mathias, M.H., 2017. The dynamic analysis of rotors mounted on composite shafts with internal damping. Composite Structures, Vol.167, pp.50–62.
29. [29] Zorzi, E., Nelson, H., 1977. Finite element simulation of rotor-bearing systems with internal damping. ASME Journal of Engineering for Power, Vol.99, pp.71–76.
30. [30] Nelson, H., 1980. A finite rotating shaft element using Timoshenko beam theory. ASME Journal of Mechanical Design, Vol.102(4), pp.793–803.
31. [31] Genta, G., 2005. Dynamics of Rotating Systems. Springer Science & Business Media.
32. [32] Rezaee, M., Lotfan, S., 2015. Non-linear nonlocal vibration and stability analysis of axially moving nanoscale beams with time-dependent velocity. International Journal of Mechanical Sciences, Vol.96, pp.36–46.
33. [33] Farshbaf Zinati, R., Rezaee, M., Lotfan, S., 2020. Nonlinear vibration and stability analysis of viscoelastic Rayleigh beams axially moving on a flexible intermediate support. Iranian Journal of Science and Technology, Transactions of Mechanical Engineering, Vol.44, pp.865–879.
34. [34] Lotfan, S., Sadeghi, M.H., 2017. Large amplitude free vibration of a viscoelastic beam carrying a lumped mass–spring–damper. Nonlinear Dynamics, Vol.90, pp.1053–1075.