Nonlinear Dynamics

, Volume 88, Issue 4, pp 2531–2551

# Application of the HB–AFT method to the primary resonance analysis of a dual-rotor system

• Lei Hou
• Yushu Chen
• Yiqiang Fu
• Huizheng Chen
• Zhenyong Lu
• Zhansheng Liu
Original Paper

## Abstract

This paper focuses on the primary resonance analysis of a dual-rotor system having two rotor unbalance excitations of different rotating speeds and being connected by an inter-shaft ball bearing. Due to the complex excitation condition and the complicated nonlinear bearing forces of the inter-shaft bearing, the general analytical methods, e.g., the multiple scales method or the harmonic balance method, are failed to give the theoretical solutions. Thus, the harmonic balance–alternating frequency/time domain (HB–AFT) method is formulated to deal with this problem. The basic idea of the method is using the inverse discrete Fourier transform and the discrete Fourier transform, instead of the direct analytical relationship between the supposed solutions of the system and the nonlinear forces, to construct the harmonic expressions of the nonlinear forces, which is the so-called alternating frequency/time domain technique. By using the HB–AFT method, therefore, a Newton– Raphson iteration procedure can be performed to demonstrate the approximate solutions of the system. Accordingly, the frequency responses of the system affected by some critical parameters, such as rotating speed ratio, unbalances of both the inner and outer rotors, and clearance of the inter-shaft bearing, are analyzed, respectively. A variety of phenomena including double resonance peaks, biperiodic and quasi-periodic behaviors, and resonance hysteresis phenomenon are obtained, which are discussed in detail through diagrams for separated frequency responses with different frequency components and rotors’ orbits for different combinations of system parameters. Most prominently, for a relatively small unbalance of rotor as well as a relatively large clearance of the inter-shaft bearing, the resonance hysteresis phenomena are more obvious. The results obtained are also compared with the direct numerical simulation results, and the comparisons show good agreements. In addition, the methodology formulated in this paper is a general approach, which can be applied to other engineering systems with multi-frequency excitations.

## Keywords

Primary resonance analysis Dual-rotor system; HB–AFT method Inter-shaft ball bearing Resonance hysteresis phenomenon

## Notes

### Acknowledgements

The authors would like to acknowledge the financial supports from China Postdoctoral Science Foundation (Grant No. 2016M590277), the National Natural Science Foundation of China (Grant No. 11602070), and the National Basic Research Program (973 Program) of China (Grant No. 2015CB057400).

## References

1. 1.
Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillation. Wiley, New York (1979)
2. 2.
Cameron, T.M., Griffin, J.H.: An alternating frequency/time domain method for calculating the steady state response of nonlinear dynamic systems. ASME J. Appl. Mech. 56, 149–154 (1989)
3. 3.
Kim, Y.B., Noah, S.T.: Stability and bifurcation analysis of oscillators with piecewise-linear characteristics: a general approach. ASME J. Appl. Mech. 58, 545–553 (1991)
4. 4.
Kim, Y.B., Noah, S.T.: Periodic response and crisis behavior for a system with piecewise-smooth non-linearities. Int. J. Nonlinear Mech. 27, 833–843 (1992)
5. 5.
Kim, Y.B., Noah, S.T.: Bifurcation analysis for a modified Jeffcott rotor with a bearing clearance. Nonlinear Dyn. 1, 221–241 (1991)
6. 6.
Kim, Y.B., Noah, S.T.: Response and bifurcation analysis of a MDOF rotor system with a strong nonlinearity. Nonlinear Dyn. 2, 215–234 (1991)
7. 7.
Kim, Y.B., Noah, S.T.: Quasi-periodic response and stability analysis for a non-linear Jeffcott rotor. J. Sound Vib. 190, 239–253 (1996)
8. 8.
Kim, Y.B., Choi, S.K.: A multiple harmonic balance method for the internal resonant vibration of a non-linear Jeffcott rotor. J. Sound Vib. 208, 745–761 (1997)
9. 9.
Xie, G., Lou, J.Y.K.: Alternating frequency/coefficient (AFC) technique in the harmonic balance method. Int. J. Nonlinear Mech. 31, 517–529 (1996)
10. 10.
Ma, Q.L., Kahraman, A.: Period-one motions of a mechanical oscillator with periodically time-varying, piecewise-nonlinear stiffness. J. Sound Vib. 284, 893–914 (2005)
11. 11.
Legrand, M., Roques, S., Pierre, C., Peseux, B., Cartraud, P.: n-dimensional Harmonic Balance Method extended to non-explicit nonlinearities. Eur. J. Comput. Mech. 15, 269–280 (2006)
12. 12.
Guskov, M., Sinou, J.J., Thouverez, F.: Multi-dimensional harmonic balance applied to rotor dynamics. Mech. Res. Commun. 35, 537–545 (2008)
13. 13.
Groll, G.V., Ewins, D.J.: The harmonic balance method with arc-length continuation in rotor/stator contact problems. J. Sound Vib. 241, 223–233 (2001)
14. 14.
Rook, T.: An alternate method to the alternating time-frequency method. Nonlinear Dyn. 27, 327–339 (2002)
15. 15.
Dimitriadis, G.: Continuation of higher-order harmonic balance solutions for nonlinear aeroelastic systems. J. Aircr. 45, 523–537 (2008)
16. 16.
Jaumouille, V., Sinou, J.J., Petitjean, B.: An adaptive harmonic balance method for predicting the nonlinear dynamic responses of mechanical systems—application to bolted structures. J. Sound Vib. 329, 4048–4067 (2010)
17. 17.
Grolet, A., Thouverez, F.: On a new harmonic selection technique for harmonic balance method. Mech. Syst. Signal Process. 30, 43–60 (2012)
18. 18.
Didier, J., Sinou, J.J., Faverjon, B.: Nonlinear vibrations of a mechanical system with non-regular nonlinearities and uncertainties. Commun. Nonlinear Sci. Numer. Simul. 18, 3250–3270 (2013)
19. 19.
Valipour, M., Banihabib, M.E., Behbahani, S.M.R.: Comparison of the ARMA, ARIMA, and the autoregressive artificial neural network models in forecasting the monthly inflow of Dez dam reservoir. J. Hydrol. 476, 433–441 (2013)
20. 20.
Valipour, M., Sefidkouhi, M.A.G., Eslamian, S.: Surface irrigation simulation models: a review. Int. J. Hydrol. Sci. Technol. 5, 51–70 (2015)
21. 21.
Woo, K.C., Rodger, A.A., Neilson, R.D., Wiercigroch, M.: Application of the harmonic balance method to ground moling machines operating in periodic regimes. Chaos Solitons Fractals 11, 2515–2525 (2000)
22. 22.
Chen, J.J., Yang, B.D., Meng, C.H.: Periodic forced response of structures having three-dimensional frictional constraints. J. Sound Vib. 229, 775–792 (2000)
23. 23.
Sinou, J.J., Thouverez, F., Jezequel, L.: Non-linear stability analysis of a complex rotor/stator contact system. J. Sound Vib. 278, 1095–1129 (2004)
24. 24.
Sinou, J.J., Lees, A.W.: The influence of cracks in rotating shafts. J. Sound Vib. 285, 1015–1037 (2005)
25. 25.
Sinou, J.J., Lees, A.W.: A non-linear study of a cracked rotor. Eur. J. Mech. A Solids 26, 152–170 (2007)
26. 26.
Laxalde, D., Thouverez, F., Sinou, J.J., Lombard, J.P.: Qualitative analysis of forced response of blisks with friction ring dampers. Eur. J. Mech. A Solids 26, 676–687 (2007)
27. 27.
Saito, A., Castanier, M.P., Pierre, C., Poudou, O.: Efficient nonlinear vibration analysis of the forced response of rotating cracked blades. ASME J. Comput. Nonlinear Dyn. 4, 011005 (2008)
28. 28.
Sinou, J.J.: Non-linear dynamics and contacts of an unbalanced flexible rotor supported on ball bearings. Mech. Mach. Theory 44, 1713–1732 (2009)
29. 29.
Villa, C., Sinou, J.J., Thouverez, F.: Investigation of a rotor-bearing system with bearing clearances and hertz contact by using a harmonic balance method. J. Braz. Soc. Mech. Sci. Eng. 29, 14–20 (2007)
30. 30.
Villa, C., Sinou, J.J., Thouverez, F.: Stability and vibration analysis of a complex flexible rotor bearing system. Commun. Nonlinear Sci. Numer. Simul. 13, 804–821 (2008)
31. 31.
Zhang, Z.Y., Chen, Y.S.: Harmonic balance method with alternating frequency/time domain technique for nonlinear dynamical system with fractional exponential. Appl. Math. Mech. Engl. 35, 423–436 (2014)
32. 32.
Fu, Y.Q., Chen, Y.S., Hou, L., Li, Z.G.: A counter-rotating dual-rotor system’s hysteretic characteristics. J. Vib. Shock 34, 23–27 (2015)Google Scholar
33. 33.
Luo, G.H., Hu, X., Yang, X.G.: Nonlinear dynamic performance analysis of counter-rotating dual-rotor system. J. Vib. Eng. 22, 268–273 (2009)Google Scholar
34. 34.
Chen, S.T., Wu, Z.Q.: Rubbing vibration analysis for a counter-rotating dual-rotor system. J. Vib. Shock 31, 142–147 (2012)Google Scholar
35. 35.
Hou, L., Chen, Y.S.: Dynamical simulation and load control of a Jeffcott rotor system in Herbst maneuvering flight. J. Vib. Control 22, 412–425 (2016)
36. 36.
Han, Q.K., Chu, F.L.: Parametric instability of a rotor-bearing system with two breathing transverse cracks. Eur. J. Mech. A Solids 36, 180–190 (2012)
37. 37.
Chen, G.: Vibration modeling and analysis for dual-rotor aero-engine. J. Vib. Eng. 24, 619–632 (2011)Google Scholar
38. 38.
Chen, Y.S.: Nonlinear Vibrations. Higher Education Press, Beijing (2002)Google Scholar
39. 39.
Zhou, J.Q., Zhu, Y.Y.: Nonlinear Vibrations. Xi’an Jiao Tong University Press, Xi’an (1998)Google Scholar
40. 40.
Friedmann, P., Hammond, C.E.: Efficient numerical treatment of periodic systems with application to stability problems. Int. J. Numer. Methods Eng. 11, 1117–1136 (1977)
41. 41.
Hou, L., Chen, Y.S., Cao, Q.J., Zhang, Z.Y.: Turning maneuver caused response in an aircraft rotor-ball bearing system. Nonlinear Dyn. 79, 229–240 (2015)Google Scholar
42. 42.
Hou, L., Chen, Y.S., Fu, Y.Q., Li, Z.G.: Nonlinear response and bifurcation analysis of a Duffing type rotor model under sine maneuver load. Int. J. Nonlinear Mech. 78, 133–141 (2015)

## Authors and Affiliations

• Lei Hou
• 1
• 2
• Yushu Chen
• 1
• Yiqiang Fu
• 1
• Huizheng Chen
• 1
• Zhenyong Lu
• 1
• Zhansheng Liu
• 2
1. 1.School of AstronauticsHarbin Institute of TechnologyHarbinPeople’s Republic of China
2. 2.School of Energy Science and EngineeringHarbin Institute of TechnologyHarbinPeople’s Republic of China