Bifurcation analysis for 2:1 and 3:1 super-harmonic resonances of an aircraft cracked rotor system due to maneuver load

This paper focuses on the local bifurcation characteristics of an aircraft cracked rotor system mainly for the 2:1 and 3:1 super-harmonic resonances induced by the maneuver load. The motion equations of the system are formulated with the consideration of the nonlinear stiffness of the Duffing type and the breathing of a transverse crack on the shaft, as well as the maneuver load induced by the climbing and diving flight of the aircraft. By using the multiple scales method, the motion equations are analytically solved to obtain the bifurcation equations for 2:1 and 3:1 super-harmonic resonances, respectively. Furthermore, the two-state variable singularity method is employed to analyze the local bifurcation characteristics of the system affected by crack coefficient and maneuver load. For each case, two curves of hysteresis set dividing $$K-G$$K-G parameter plane into three regions are demonstrated. Accordingly, bifurcation modes for different parameter combinations from the three regions and the two curves are obtained. The approach in this paper will provide an effective and convenient way to analyze the bifurcation characteristics of dynamical systems. The results in this paper will contribute to a better understanding of the effect of the maneuver load on the response and bifurcation characteristics of aircraft cracked rotor systems.

[1]  Oh Sung Jun.  Dynamic behavior analysis of cracked rotor based on harmonic motion , 2012 .

[2]  Chong-Won Lee,et al.  MODELLING AND VIBRATION ANALYSIS OF A SIMPLE ROTOR WITH A BREATHING CRACK , 1992 .

[3]  M. Golubitsky,et al.  Singularities and groups in bifurcation theory , 1985 .

[4]  Liming Dai,et al.  Nonlinear response and dynamic stability of a cracked rotor , 2007 .

[5]  C. Pierre,et al.  Non-linear normal forced vibration modes in systems with internal resonance , 2013 .

[6]  William L. Garrard,et al.  Nonlinear control of a supermaneuverable aircraft , 1989 .

[7]  Andrew D. Dimarogonas,et al.  Vibration of cracked structures: A state of the art review , 1996 .

[8]  Livija Cveticanin.  Analytic approach for the solution of the complex-valued strong non-linear differential equation of Duffing type , 2001 .

[9]  Meng Guang,et al.  Nonlinear dynamics of a cracked rotor in a maneuvering aircraft , 2004 .

[10]  Liming Dai,et al.  Dynamic Stability Analysis of a Cracked Nonlinear Rotor System Subjected to Periodic Excitations in Machining , 2007 .

[11]  Robert Gasch,et al.  Dynamic behaviour of the Laval rotor with a transverse crack , 2008 .

[12]  A. W. Lees,et al.  The influence of cracks in rotating shafts , 2005 .

[13]  G. Genta,et al.  Conditions for noncircular whirling of nonlinear isotropic rotors , 1993, Nonlinear Dynamics.

[14]  Zhengjia He,et al.  The influence of crack breathing and imbalance orientation angle on the characteristics of the critical speed of a cracked rotor , 2011 .

[15]  Eric A. Butcher,et al.  General harmonic balance solution of a cracked rotor-bearing-disk system for harmonic and sub-harmonic analysis: Analytical and experimental approach , 2010 .

[16]  Fulei Chu,et al.  Numerical and experimental investigations of flexural vibrations of a rotor system with transverse or slant crack , 2009 .

[17]  Paolo Pennacchi,et al.  A model based identification method of transverse cracks in rotating shafts suitable for industrial machines , 2006 .

[18]  Mohamed S. Soliman.  Suppression of steady state bifurcations and premature fractal basin erosion in nonlinear systems subjected to combined external and parametric excitations , 1994 .

[19]  Amiya R Mohanty,et al.  Transient lateral analysis of a slant-cracked rotor passing through its flexural critical speed , 2002 .

[20]  Lei Hou,et al.  Nonlinear vibration phenomenon of an aircraft rub-impact rotor system due to hovering flight , 2014, Commun. Nonlinear Sci. Numer. Simul..

[21]  M. Ruijgrok,et al.  Bifurcations in an autoparametric system in 1:1 internal resonance with parametric excitation , 2002 .

[22]  Fulei Chu,et al.  Parametric instability of a rotor-bearing system with two breathing transverse cracks , 2012 .

[23]  Arthur W. Lees,et al.  A non-linear study of a cracked rotor , 2007 .

[24]  Lei Hou,et al.  Analysis of 1/2 sub-harmonic resonance in a maneuvering rotor system , 2014 .

[25]  鈴木 増雄.  A. H. Nayfeh and D. T. Mook: Nonlinear Oscillations, John Wiley, New York and Chichester, 1979, xiv+704ページ, 23.5×16.5cm, 10,150円. , 1980 .

[26]  Jörg Wauer,et al.  On the Dynamics of Cracked Rotors: A Literature Survey , 1990 .

[27]  Jean-Jacques Sinou,et al.  Detection of cracks in rotor based on the 2× and 3× super-harmonic frequency components and the crack–unbalance interactions , 2008 .

[28]  R. Gasch,et al.  A Survey Of The Dynamic Behaviour Of A Simple Rotating Shaft With A Transverse Crack , 1993 .

[29]  P. N. Saavedra,et al.  Vibration Analysis of Rotor for Crack Identification , 2002 .

[30]  Fulei Chu,et al.  Dynamic response of cracked rotor-bearing system under time-dependent base movements , 2013 .

[31]  A. H. Nayfeh,et al.  GLOBAL BEHAVIOR OF A BIASED NON-LINEAR OSCILLATOR UNDER EXTERNAL AND PARAMETRIC EXCITATIONS , 1997 .

[32]  Koji Kimura,et al.  Dynamics of a Weakly Nonlinear System Subjected to Combined Parametric and External Excitation , 1990 .

[33]  Vibration of a Non-Linear Self-Excited System with Two Degrees of Freedom under External and Parametric Excitation , 1997 .

[34]  Yukio Ishida.  Cracked rotors: Industrial machine case histories and nonlinear effects shown by simple Jeffcott rotor , 2008 .

[35]  Yushu Chen,et al.  Singular analysis of bifurcation systems with two parameters , 2010 .

[36]  Raymond H. Plaut,et al.  Non-linear structural vibrations under combined parametric and external excitations , 1987 .

[37]  Xingmin Ren,et al.  Analysis on the nonlinear response of cracked rotor in hover flight , 2010 .

[38]  Paolo Pennacchi,et al.  Discussion of the dynamic stability of a multi-degree-of-freedom rotor system affected by a transverse crack , 2012 .

[39]  Jean-Jacques Sinou,et al.  Effects of a crack on the stability of a non-linear rotor system , 2007 .

[40]  W. B. Herbst,et al.  Dynamics of Air Combat , 1983 .

[41]  Mohamed S. Gadala,et al.  Dynamic behavior analysis of cracked rotor , 2008 .

[42]  Mohamed Belhaq,et al.  Quasi-Periodic Oscillations, Chaos and Suppression of Chaos in a Nonlinear Oscillator Driven by Parametric and External Excitations , 1999 .