This article combines the three-order harmonic balance method and eight-order Taylor expansion technique to give the qualitative analysis of responses for the spindle-ball bearing system. The effects of the bearing clearance and initial position of rotor on the frequency–response curves are investigated. As the number of contact balls alters due to the combining actions of the bearing clearance and initial position of rotor, the grazing phenomenon occurs and the frequency–response curve undergoes a major change. The system’s characteristic coefficients are also found to sharply grow at the point the number of contact balls alters. They therefore can be treated as the signs to indicate the occurrence of the grazing phenomenon, which is helpful for the researchers to study the non-linear and non-smooth properties of the spindle-ball bearing system.
SchmidtFLamarqueCH. How to improve confidence in calculations for non-smooth dynamical systems. J Vib Control2008; 14(1/2): 231–253.
2.
SieberJKowalczykP. Small-scale instabilities in dynamical systems with sliding. Physica D2010; 239: 44–57.
3.
HuHY. Detection of grazing orbits and incident bifurcations of a forced continuous, piecewise-linear oscillator. J Sound Vib1995; 187(3): 485–493.
4.
BishopSRXuD. The use of control to eliminate subharmonic and chaotic impacting motions of a driven beam. J Sound Vib1997; 205(2): 223–234.
5.
IvanovAP. Bifurcations in impact systems. Chaos Soliton Fract1996; 7(10): 1615–1634.
6.
LuoGW. Periodic motions and transition phenomena in a two-degrees-of-freedom system with perfectly plastic impact. Phys Lett A1999; 263: 83–90.
7.
QiuJFengZC. Parameter dependence of the impact dynamics of thin plates. Comp Struct2000; 75: 491–506.
8.
RangoliSSoumitroB. Character of the map for switched dynamical systems for observations on the switching manifold. Phys Lett A2008; 372: 4234–4240.
9.
TanakaGTsujiSAiharaK. Grazing-induced crises in hybrid dynamical systems. Phys Lett A2009; 373: 3134–3139.
10.
ConeKMZadoksRI. A numerical study of an impact oscillator with the addition of dry friction. J Sound Vib1995; 188(5): 659–683.
11.
BishopSRWaggDJXuD. Use of control to maintain period-l motions during wind-up or wind-down operations of an impacting driven beam. Chaos Soliton Fract1998; 9(112): 261–269.
12.
BernardoMDBuddbCJChampneysAR. Normal form maps for grazing bifurcations in n-dimensional piecewise-smooth dynamical systems. Physica D2001; 160: 222–254.
13.
QinWYChenGRMengG. Nonlinear responses of a rub-impact overhung rotor. Chaos Soliton Fract2004; 19: 1161–1172.
14.
QinWYChenGRRenXM. Grazing bifurcation in the response of cracked jeffcott rotor. Nonlinear Dyn2004; 35: 147–157.
15.
ZhaoXReddyCKNayfehAH. Nonlinear dynamics of an electrically driven impact microactuator. Nonlinear Dyn2005; 40: 227–239.
16.
LuoACJChenLD. Periodic motions and grazing in a harmonically forced, piecewise, linear oscillator with impacts. Chaos Soliton Fract2005; 24: 567–578.
17.
LuoACJGeggBC. Stick and non-stick periodic motions in periodically forced oscillators with dry friction. J Sound Vib2006; 291: 132–168.
18.
LuoACJ. Periodic motions and stability in a semi-active suspension system with MR damping. J Vib Control2007; 13(5): 687–709.
19.
LuoACJThapaS. Periodic motions in a simplified brake system with a periodic excitation. Commun Nonlinear Sci Numer Simulat2009; 14: 2389–2414.
20.
SouzaSLTWiercigrochMCaldasIL. Suppressing grazing chaos in impacting system by structural nonlinearity. Chaos Soliton Fract2008; 38: 864–869.
21.
GeggBCLuoACJSuhSC. Grazing bifurcations of a harmonically excited oscillator moving on a time-varying translation belt. Nonlinear Anal: Real World Appl2008; 9: 2156–2174.
22.
LundbergGPalmgrenA. Dynamic capacity of rolling bearings. J Acta Politech Mech Eng Ser1952; 2(4): 1–50.
23.
PalmgrenA. Ball and roller bearing engineering, Philadelphia, PA: SH Burbank, 1959.
24.
GaoSHLongXHMengG. Nonlinear response and nonsmooth bifurcations of an unbalanced machine-tool spindle-bearing system. Nonlinear Dyn2008; 54: 365–377.
25.
GaoSHMengGLiHG. Non-linear dynamics of an unbalanced machine-tool spindle system supported by ball bearings. Proc IMechE Part K: J Multi-body Dynamics2008; 222: 201–214.
26.
GaoSHLongXHMengG. Multiple resonances and nonlinear behaviors of a spindle ball bearings system with 3/2 nonlinear term. Int J Nonlinear Sci Numer Simul2009; 10(7): 945–957.
27.
GaoSHLongXHMengG. Nonlinear response and stability of a spindle system supported by ball bearings. Arch Appl Mech2010; 80: 1069–1081.
28.
WardleFPPoonSY. Rolling bearing noise—cause and cure. Chartered Mech Eng1983; 30(7–8): 36–40.
29.
WardleFPLaceySJPoonSY. Dynamic and static characteristics of a wide speed range machine tool spindle. Prec Eng1983; 5(4): 175–183.
30.
HarrisTA. Rolling bearing analysis, 1st ed. New York: John Wiley and Sons, 1966.
31.
TamuraHShimizuH. Vibration of rotor based on ball bearings, 3rd report: static stiffness of ball bearings containing a large number of balls. Bull JSME1968; 34(258): 298–307.
32.
RahnejatHGoharR. The vibrations of radial ballbearings. Proc IMechE Part C: J Mechanical Engineering Science1985; 199: 181–193.
33.
WardleFP. Vibration forces produced by waviness of the rolling surfaces of thrust loaded ball bearings. PartI: theory. Proc IMechE Part C: J Mechanical Engineering Science1988; 202: 305–312.
34.
WardleFP. Vibration forces produced by waviness of the rolling surfaces of thrust loaded ball bearings. PartII: experimental validation. Proc IMechE Part C: J Mechanical Engineering Science1988; 202: 313–319.
35.
YhlandEA. linear theory of vibrations caused by ball bearings with form errors operating at moderate speed. ASME J Tribol1992; 114: 348–359.
36.
LynaghNRahnejatHEbrahimiM. Bearing induced vibration in precision high speed routing spindles. Int J Mach Tools Manufac2000; 40: 561–577.
37.
VafaeiSRahnejatH. Indicated repeatable runout with wavelet decomposition (IRR-WD) for effective determination of bearing-induced vibration. J Sound Vib2003; 260: 67–82.
38.
AktürkNGoharR. The effect of ball size variation on vibrations associated with ball-bearings. Proc IMechE Part J: J Engineering Tribology1998; 212: 101–110.
39.
AiniRRahnejatHGoharR. Vibration modelling of rotating spindles supported by lubricated bearings. ASME J Tribol2002; 124(1): 158–165.
40.
WalfordTLHStoneBJ. The sources of damping in rolling element bearings under oscillating conditions. Proc IMechE Part C: J Mechanical Engineering Science1983; 197: 225–232.
41.
AiniRRahnejatHGoharR. A five degrees of freedom analysis of vibrations in precision spindles. Int J Mach Tools Manufac1990; 30(1): 1–18.
42.
AiniRRahnejatHGoharR. An experimental investigation into bearing-induced spindle vibration. Proc IMechE Part C: J Mechanical Engineering Science1995; 209: 107–114.
43.
RahmanAKAiniRGoharR. On the performance of multi-support spindle–bearing assemblies. Proc IMechE Part K: J Multi-body Dynamics2002; 216: 117–132.
44.
RahmanAKAiniRGoharR. On the effects of bearing settings on the dynamic performance of a three-bearing machine tool spindle assembly. Proc IMechE Part K: J Multi-body Dynamics2002; 216: 133–141.
45.
RahnejatH. Computational modelling of problems in contact dynamics. Eng Anal1985; 2(4): 192–197.
46.
AktürkNUneebMGoharR. The effect of number of balls and preload on vibrations associated with ball bearings. ASME J Tribol1997; 119: 747–753.
47.
HarshaSPKankarPK. Stability analysis of a rotor bearing system due to surface waviness and number of balls. Int J Mech Sci2004; 46: 1057–1081.
48.
HarshaSP. Non-linear dynamic response of a balanced rotor supported on rolling element bearings. Mech Syst Signal Process2005; 19: 551–578.
49.
HarshaSP. Nonlinear dynamic response of a balanced rotor supported by rolling element bearings due to radial internal clearance effect. Mech Mach Theor2006; 41: 688–706.
50.
HarshaSPSandeepKPrakashR. The effect of speed of balanced rotor on nonlinear vibrations associated with ball bearings. Int J Mech Sci2003; 45: 725–740.
51.
WangCC. Bifurcation analysis of high speed spindle air bearings. J Vib Control2011; 17(1): 103–114.
ChuFZhangZ. Bifurcation and chaos in a rub-impact jeffcott rotor system. J Sound Vib1998; 210(1): 1–18.
54.
ThotaPDankowiczH. Analysis of grazing bifurcations of quasiperiodic system attractors. Physica D2006; 220: 163–174.
55.
ThotaPDankowiczH. Continuous and discontinuous grazing bifurcations in impacting oscillators. Physica D2006; 214: 187–197.
56.
PiiroinenPTVirginLNChampneysAR. Chaos and period-adding: experimental and numerical verification of the grazing bifurcation. J Nonlinear Sci2004; 14: 383–404.
57.
LongXHLinGBalachandranB. Grazing bifurcations in an elastic structure excited by harmonic impactor motions. Physica D2008; 237: 1129–1138.
58.
IngJPavlovskaiaEWiercigrochM. Bifurcation analysis of an impact oscillator with a one-sided elastic constraint near grazing. Physica D2010; 239: 312–321.
59.
MehranKZahawiBGiaourisD. Investigation of the near-grazing behavior in hard-impact oscillators using model-based TS fuzzy approach. Nonlinear Dyn2012; 69: 1293–1309.
60.
GaoSHLongXHMengG. Nonlinear response and stability of a spindle system supported by ball bearings. Arch Appl Mech2010; 80: 1069–1081.
61.
GaoSHNongSLMengG. Frequency response curves of a spindle bearings system with non-linear and non-smooth terms. Proc IMechE Part K: J Multi-body Dynamics2013; 227: 61–75.
62.
MehdigoliHRahnejatHGoharR. Vibration response of wavy surfaced disc in elastohydrodynamic rolling contact. Wear1990; 139: 1–15.
