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Vibration characteristics of a rotor-bearing system with pedestal looseness areinvestigated. A non-linearmathematical model containing sti!ness and damping forces withtri-linear forms is considered. The shooting method is used to obtain the periodic solutionsof the system. Stability of these periodic solutions is analyzed by using the Floquet theory.Period-doubling bifurcation and Naimark}Sacker bifurcation are found. Finally, thegoverning equations are integrated using the fourth order Runge}Kutta method. Di!erentforms of periodic, quasi-periodic and chaotic vibrations are observed by taking the rotatingspeed and imbalance as the control parameter. Three kinds of routes to or out of chaos, thatis, period-to-chaos, quasi-periodic route and intermittence, are found.( 2001 Academic Press1. INTRODUCTIONIn diagnosing mechanical faults of rotating machinery, it is very important to know thevibration feature of the machine with various forms of fault. A rotor system with fault isgenerally a complicated non-linear vibrating system.43861
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Its vibration is in a very complex form.A comprehensive investigation on periodic, quasi-periodic and chaotic vibrations of therotating system with faults will be very bene"cial to the e!ective fault diagnostics of rotatingmachinery. Pedestal looseness is one of the common faults that occur in rotatingmachinery.It is usually caused by the poor quality of installation or long period of vibration of themachine. Under the action of the imbalance force, the rotor system with pedestal loosenesswill have a periodic beating. This will generally lead to a change in sti!ness of the systemand the impact e!ect. Therefore, the system will often show very complicated vibrationphenomenon.There have been very few publications on this topic. Goldman and Muszynska [1]performed experimental, analytical and numerical investigations on the unbalance responseof a rotating machine with one loose pedestal. The model was simpli"ed as a vibratingsystem with bi-linear form. Synchronous and subsynchronous fractional components of theresponse were found. In a subsequent paper [2], they discussed the chaotic behavior of thesystem based on the bi-linear model.In this paper, a simple rotor system with a disk in the middle span and with the pedestallooseness in one support is investigated theoretically. We think that the sti!ness and damping of the foundation to the pedestal can actually be pided into three parts while thesystem is vibrating. Therefore, the system is simpli"ed as a model of di!erential equationswith tri-linear forms of sti!ness and damping. The shooting method is used to calculate thestable periodic solutions of the system. The Floquet theory is used to analyze stability andbifurcation of the periodic solutions.
Finally, the equations are integrated by using thefourth order Runge}Kutta method to discuss the periodic, quasi-periodic and chaoticvibrations of the system, and the relevant phenomena. The rotating speed and imbalanceare used as control parameters to investigate the bifurcation characteristics.2. FORMULATIONThemodel discussed is a simple rotor systemas shown in Figure 1. The rotor is supportedon identical oil "lm bearings at both sides. The equivalent lumped mass in the position ofthe disk is 2m. The shaft sections between disk and bearings are considered massless andelastic. It is assumed that the left support has pedestal looseness, the maximum static gap ofthe looseness is d, and mass of the pedestal involving looseness is M. Values for theparameters of the system used in the analysis and the subsequent simulation are as follows:2m"5 kg, c"0)8]103 Ns/m, k"0)34]106 N/m,u"0)5]10~4 m, d"0)8 mm, M"8 kg,cf1"0)2]104 Ns/m, cf2"0)2]104 Ns/m, cf3"0)2]104 Ns/m,kf1"0)4]108 N/m, kf2"0)1]103 N/m, kf3"0)8]106 N/m,k"0)015 Ns/m2, R"50 mm, ¸"10 mm, cl"0)1 mm, g"9)81 N/kg.Based on these values the "rst undamped natural frequency of the rotor system is obtainedas n1"Jk/m"3521)6 r.p.m., or u1"368)81/s.2.1. OIL FILM FORCES OF THE BEARINGFor the plain bearing, the housing is constrained from rotating. The Reynolds equationfor the short-bearing approximation is given in both "xed co-ordinates [3] byLLz Ch3kLpLzD"6uLhLh#12LhLt, where h is the "lm thickness and is given by h"cl!x cos h!y sin h, as shown in Figure 2,p is the oil "lmpressure, z, the axial co-ordinate, k, the oil viscosity, u, the rotating speed, cl,the bearing clearance, and t, the time.Integrating the equation for z yieldsp(h, z)"3kh3[z2!¸z]Cu LhLh#2LhLtD,where ¸ is the bearing length. The total force components in the x and y directions can beobtained by integrating the pressure over the entire journal surface as follows:GPx(x, y, x, y)Py(x, y, x, y) H"P2n0PL@2~L@2p(h, z)Gcos hsin h HRdz dh.Finally, the two forces can be written asPx(x, y, x, y)"!knR¸3Cuy#2x2(c2l!x2!y2)3@2# 3x(xx#yy)(c2l!x2!y2)5@2D,(1)Py(x, y, x, y)"!knR¸3C2y!ux2(c2l!x2!y2)3@2# 3y(xx#yy)(c2l!x2!y2)5@2D.2.2. GOVERNING EQUATIONSIt is assumed that the radial displacements in the right-bearing position are x1, y1, in thedisk position x2, y2, and in the left-bearing position x3, y3. The small movement of the leftpedestal in the horizontal direction is considered negligible and its displacement in thevertical direction is assumed as y4. The di!erential equations for the system can then bewritten asc(x1!x2)#k(x1!x2)"Px1(x1, y1, x1, y1),c(y1!y2)#k(y1!y2)"Py1(x1, y1, x1, y1), 4 f 4 f 4 y3 3 3 4 3 3 4(2)where the change in the unbalance of the rotating disk and shaft caused by the looseness gapis neglected. In the above equation c is the damping coe$cient of shaft, k the sti!nesscoe$cient, u the unbalance, Px1, Py1the oil "lmforce components of the right bearing in thex and y directions, Px3, Py3of the left bearing, and cf, kfare the damping and sti!nesscoe$cients of the foundation or the joint to the pedestal. When the looseness occurs, thesetwo coe$cients can be expressed bycf"igjgkcf1, y4(0,cf2,0)y4)d,cf3, y4'd,kf"igjgkkf1, y4(0,kf2,0)y4)d,kf2#kf3!kf3dy4, y4'd,(3)where the sti!ness and damping actions are considered in three parts. When y4"0, thepedestal is in contact with the foundation. y4(0 means that the pedestal and thefoundation are in compression state and the impact is considered elastic. y4'd describesthe extension of the joint and also the deformation of the joint is assumed as elastic.Equation (2) including equation (3) is a non-linear vibrating system with piecewise-linearsti!ness and damping.If one assumes that In equation (6), some of the right-hand terms contain derivatives. In performing numericalintegration, one can compute s@1, s@2, s@7, s@8"rst, then use the obtained results to calculateother derivatives. In this way, the problemS@"f (S@,S, q) can be processed by using the samemethod as in the form of S@"g(S, q).3. BIFURCATION AND STABILITY ANALYSISEquation (6) is a non-linear vibrating system with seven degrees of freedom and withpiecewise-linear form. Because of these features, when performing a theoretically qualitativeanalysis it is very di$cult to discuss the equations of motion in an analytical way andimpossible to obtain the solutions in a closed form. Therefore, numerical methods have tobe resorted.There have been several methods for determining the periodic response of the non-linearrotor systems, including the series expansion [4] and the harmonic balance method as usedin references [5, 6]. However, for a multi-degree of freedom, these methods often su!er theproblem of convergence to some extent when iteration is performed. In this aspect, theshooting method has shown good ability of convergence. The shooting method has beenpreviously used to obtain periodic solutions of non-linear di!erential equations. The algorithm is based on the utilization of the Poincare'smap in which the #ow of an nth ordercontinuous-time system is replaced with an (n!1)th order discrete-time system, thustransforming the problem of "nding a periodic solution to that of "nding a "xed point.Kaas}Petersen [7] discussed the method and extended it to "nd quasi-periodic solutions. Inthis section, the shooting method to calculate periodic solutions and to analyze stability ispresented followed by some of the numerical results.Consider a system described by the following equation:s"f (s, u, t)3Rn, (7)where f is periodic in t with period ¹"2n/u, f (s, u, t#¹)"f (s, u, t) ands"(s1, s2,2, sn)3Rn. For s03Rn, let q(t, s0) be the solution of equation (7) with initialvalue g(0, s0)"s0. Then the Poincare's map for the system (7) isP :RnPRn, P(s)"q(¹, s). (8)A ¹-periodic orbit q(t, s0) of equation (7) obviously corresponds to a "xed point of thePoincare's map (8), P(s0)"s0.The map P can be used to de"ne a map Q as follows:Q"P!I, (9)where I is the identity. Then a periodic solution of equation (7) corresponds to a zero of Q.Newton}Raphson method is very e$cient for the purpose of "nding zeros of Q. The valuesof Q(s) and the derivative DQ(s) needed for the iteration procedure can be computednumerically. The iteration formula can then be obtained assk`1"sk![DQ(sk)]~1Q(sk), k"0, 1,2. (10)The iteration process is repeated until EQ(sk)E(e for some preassigned e. If too manyiterations are performed, then the process is stopped, which could be an indication of a toopoor initial guess. The convergent result of MskN as a "xed point of the map P is justa periodic solution to the system (7).For the stability analysis, based on the iteration result, stability of a periodic solutioncould be determined as the stability of the "xed point. Now let s be the "xed point of themap P,sos"P(s).If eu is any disturbance, then by Taylor's theoremP(s#eu)"P(s)#DP(s)eu#O( DeD2).Let s#eu be mapped into s#/, thens#/"P(s#eu)"P(s)#DP(s)eu#O( DeD2).Retaining only the lowest-order terms gives/"DP(s)eu. (11) TABLE 1Characteristics of the rotor system at di+erent rotating speedsu/u1Floquet multipliers j1Conclusions0)64 !1)08279#i0)0;!1)03667#i0)0 Period-doubling bifurcation!0)65323#i0)12867;!0)65323!i0)12867!0)11665#i0)06240;!0)11665!i0)062400)01217#i0)0;!0)01101#i0)0!0)01047#i0)00732;!0)01047!i0)007321)98 !1)28657#i0)36588;!1)28657!i0)36588 Naimark}Sacker