Dynamic characteristics of self-acting gas bearing flexible rotor coupling system based on the forecasting orbit method

Transcription

1 Nonlinear Dyn (2012) 69: DOI /s z Dynamic characteristics of self-acting gas bearing flexible rotor coupling system based on the forecasting orbit method Guang-hui Zhang Yi Sun Zhan-sheng Liu Min Zhang Jia-jia Yan Received: 26 November 2010 / Accepted: 4 November 2011 / Published online: 3 December 2011 Springer Science+Business Media B.V Abstract This paper studies the nonlinear dynamic characteristics of a flexible rotor supported by selfacting gas bearings theoretically. The multiple degree freedom model of flexible rotor is established by the finite element method and analyzed coupled with the transient gas lubricated Reynolds equation by employing the forecasting orbit method. The Reynolds equation is solved by the alternating direction implicit method and the dynamic response of the rotor is calculated by the Newmark integral method. To settle the problem that the two kinds of transient solving processes (transient Reynolds equation for bearing and transient equation of motion for rotor) cannot be solved simultaneously, which arises from the fact that they need each other s results as their initial values, the multi-field coupling algorithm based on the forecasting method is proposed and applied in this paper. By employing the numerical method, the rotor trajectory diagram, phase diagram, frequency spectrum, power spectrum, bifurcation diagram, and vibration mode diagram were obtained. It is to note that the dynamic characteristics of self-acting gas G.-h. Zhang ( ) Z.-s. Liu M. Zhang J.-j. Yan School of Energy Science and Engineering, Harbin Institute of Technology, Harbin, China qq @gmail.com Y. Sun School of Astronautic, Harbin Institute of Technology, Harbin, China bearing rotor system and whirling instability of the system could be depicted successfully. This would establish the foundation for contributing to a further understanding of the gas bearing flexible rotor system. Keywords Gas bearing rotor system Stability Nonlinear dynamic Bifurcation Forecasting orbit method Nomenclature c average air film thickness e eccentricity e x,e y eccentricity in x, y direction h air film thickness l T the length of the rotor element p air film pressure p a atmospheric pressure t time x,y,z coordinates A the section area of the rotor F x,f y the gas film forces act on the rotor F n,f n+1 the gas film forces at time n and n + 1 H dimensionless air film thickness H n gas film thickness function at time n H n+1 forecast gas film thickness function at time n + 1by forecasting I y moment of the inertia for rotor L the length of the bearing NX the mesh number of circumferential direction

2 342 G.-h. Zhang et al. NZ the mesh number of bearing length direction P dimensionless pressure P n+1 dimensionless pressure at time n + 1 Q the square of the P ; Q = P 2 R radius of the bearing W load capacity ε the eccentricity ratio η the dynamic viscosity of air θ the angular coordinate θ 0 attitude angle ξ the dimensionless coordinates in length direction ρ the air density under arbitrary pressure ρ a the air density under atmospheric pressure ρ r the density of the rotor τ the dimensionless time ϕ the attitude angle ω rotating angular speed of the shaft Λ the bearing number shear factor for the rotor Φ s 1 Introduction The gas bearings are more and more widely used in aerospace and mechanical engineering fields for the benefit of high rotating speeds, low friction, no wear, no contamination and smooth functioning, which are generated from the features of gas-lubricated film. However, the self-excited whirling phenomenon exists in the gas bearing which is similar to that of liquid bearing. The whirling speed of gas bearing is 0.45 times of rotating speeds [1], which will cause the instability of gas bearing rotor system and limit the application of gas bearing. So for the proper design of such systems, the dynamic of gas bearing rotor system should be identified accurately and effectively. Although abundant literature for predicting the nonlinear whirling behavior of gas bearing can be found, most of it has limitations: some ignore the partial derivative of pressure or film thickness with respect to time for gas film force; some simplify the rotor as the rigid kinematic model; and nearly none takes into account the coupling mechanism between the gas film forces and the rotor motion. Those factors should be considered simultaneously to obtain the dynamic characteristics of gas bearing rotor system accurately and effectively. Usually there are two ways to analyze the dynamic characteristics of gas bearing rotor system: one is the perturbation method based on the steady-state operations, the other one is the orbit method. In the perturbation method, the load on perturbation state is calculated by applying small displacement or velocity on the steady-state operation; the variation of load is decomposed by perturbation method and then the stiffness and damping coefficients are obtained. This method was employed by Sternlicht [2], Rentzepis and Sternlicht [3], Ausman [4], and Lund [5] and has the advantage of high maneuverability and accuracy on analyzing the smaller whirling orbit (for example, the lower rotating angular velocity and higher load); however, for the case of high rotating velocity and light rotor weight, where the whirling orbit of the rotor is relatively bigger, the difficult-to-estimate numerical error will be generated by applying the perturbation method. Another way to solve the stability problem of gas bearing rotor system is the orbit method, which was proposed by Castelli in Ref. [6]. In the orbit method, the complete nonlinear equations are included, the calculation procedures highly coincide with the assumed governing equations, and the trajectory orbit of the rotor can be obtained. So the orbit method enables both establishment of the stability threshold and prediction of the behavior of the bearing into the instability region. Because of the above-mentioned advantages, the orbit method is being more and more widely employed. Castelli discussed the numerical method for solving the transient Reynolds equation in Ref. [7]. The proposed method comprises the satisfactory numerical stability of implicit arithmetic and the fast convergence of the explicit method. In Ref. [8] Dimofte modified the alternating direction implicit method and developed the code for analyzing the steady and dynamic characteristics. Bou-Sa solved the nonlinear Reynolds equation with time terms to simulate the dynamic characteristics of the rotor in Ref. [9]. The proposed model was compared with conventional linear journal bearing model which employs the stiffness and damping coefficients. It is to note that the linear model would lead to the incorrect solutions; the nonlinear model could forecast the rotor dynamic characteristics better. In Refs. [10 12], the gas-lubricated Reynolds equation with time term was solved by employing the finite difference method and over-relaxation method. The dynamic behavior in horizontal and vertical directions of the journal was analyzed by locus diagram, Poincaré map, power spectrum and bifurcation diagram. The concept of whirling ratio was imported into

3 Dynamic characteristics of self-acting gas bearing flexible rotor coupling system 343 the solving procedure. Some limits existed in the research for journal gas bearing rotor system mentioned above. Most of the analysis was particular for bearing without considering the effect of the rotor; some assumed the rotor model as rigid rotor with little number of freedoms, but nearly no-one modeled the multiple degrees of freedom for gas bearing rotor system. Also most of the papers employed the last-step results to solve the next step, without considering the coupling between the transient Reynolds equation for bearing and transient equation of motion for rotor. The essence of the rotor gas bearing coupling dynamic system is the fluid structure interaction problem. Because of the characteristics of different physical fields, the numerical methods for solving them are distinct. Many different coupling algorithms are proposed. The full coupling algorithm which is called the monolithical approach is proposed to solve the fluid structure interaction problem in Refs. [13 16]. The full coupling algorithm has the advantages of good stabilization and introducing no energy error. But this approach cannot employ the originally developed flow domain and structure domain simulation code, which limits the application of the monolithical approach. With the status of the sophisticated flow domain and structure domain calculation, it is better to develop the proper multi-field algorithm based on the interactive relation. Piperno and Farhat in Refs. [17, 18] developed the partitioned procedures for the twodimensional and three-dimension aeroelastic problems. Farhat in Ref. [19] proposed the structure forecasting predictor that is based on the energy conservation law to simulate the aeroelastic problems. The choice of the structures predictor has to do with the calculation form of the flow domain. Usually it is difficult to choose the structures predictor. Also, for the realistic cases, the exactly energy conservation is impossible. Piperno [20] analyzed the predictor, and obtained the same conclusion. Since there are few references regarding the coupling between the gas bearing flow and the rotor motion this topic needs to be studied in depth. The self-acting gas bearing rotor system with multiple degrees of freedom is modeled by finite element method in this paper. The ADI (alternating direction implicit) is employ to solve the transient gas lubricated Reynolds equation and the Newmark method is used to solve the equations of motion for rotor. The forecasting orbit method is proposed to couple the gaslubricated Reynolds equation with the equation of motion for rotor. The nonlinear dynamic characteristics of journal gas bearing rotor system are analyzed, and the phenomenon and mechanism of the whirling in the gas bearing are studied. The effects of the imbalance mass on the dynamic behavior are investigated in this paper. 2 Mathematical modeling 2.1 Modeling for transient gas bearing forces The mathematical model of gas bearings is much more complicated to deal with than that of the liquid film bearing because of the compressibility of the gas. So we need the following assumptions: the variation of viscosity and density across the film direction is neglected, the flow is isothermal, the mass flow loss caused by the end leakage of the bearing is neglectable, and the air is an ideal gas. The well-known transient Reynolds equation of an isothermal gas journal bearing is as follows: ( ρh 3 ( ρh 3 ) z ) + 12 t (ρh) p p R θ η R θ η z + 6ω (ρh) = 0 (1) θ The film thickness is as follows: h = c ( 1 + ε cos(θ ϕ) ) (2) ε = 1 (e x ) c 2 + (e y ) 2 ; ϕ = arctg e y (3) e x By employing the dimensionless parameters, ξ = z R, P = p p a, H = h c, Λ = 6ηωR2 p a c 2, τ = ωt 2Λ The dimensionless form of transient Reynolds equation is derived by introducing the isothermal gas condition p/ρ = p a /ρ a into (1): ( PH 3 P ) ( PH 3 P ) θ θ ξ ξ + Λ (PH) (P H ) + = 0 (4) θ τ Equation (4) is the nonlinear partial differential equation, so the iteration method is employed to solve the equation.

4 344 G.-h. Zhang et al. Fig. 1 The model of self-acting gas journal bearings Fig. 2 Calculation domain of journal gas bearings (self-acting type) The structure schematic of self-acting gas journal bearing is shown in Fig. 1. The symmetry boundary condition, periodic boundary condition, mass flow boundary condition and atmospheric boundary condition are defined in Fig. 2. The computational domain is θ(0 : NX+1) and ξ(0 : NZ+1). The solution domain is θ(1 : NX) and ξ(1 : NZ). The atmospheric boundary condition is p(ξ = 0) = p(ξ = NZ)= p a ; the periodic boundary condition is p(θ = 0) = p(θ = NX) and p(θ = 1) = p(θ = NX+ 1). By submitting Q = P 2, the central difference method is employed to separate the variables in the θ and ξ direction, and the ADI (alternating direction implicit) method is used to separate the variables in the form of the time τ. According to the rules governing the scheme, the computations are carried out for three discrete time levels, n, n + 1 and n + 2, which are separated from each other by segments of the dimensionless time τ. One iterative cycle consists of solving procedure for the levels n + 1 and n + 2. In the beginning of the computation, the pressure distribution in the level n is assumed to be known. In the first stage of the iterative cycle, for the level n + 1, the derivatives with respect to θ are written as unknowns, and the derivatives with respect to ξ are considered to be the values known from the level n. In the second stage (level n + 2), the derivatives with respect to ξ are unknown, and the derivatives with respect to θ are treated as the known values from the level n + 1. On finishing the iterative cycle, the level n + 1 is treated as a new level n and the next cycle is started. In this scheme the following formulas are applied: h n p n Q n+1 Q n + 2 p n τ + 2Λ Qn+1 p n h n h n +1 h n 1 2 θ h n+1 h n τ Q n Λ Qn+1 1 p n 2 θ 3 ( h n ) 2 ( h n +1 h n 1 )(Qn+1 +1 Qn+1 1 ) 4 θ 2 3 ( h n ) 2 ( h n i+1,j h n i, 1j )(Qn i+1,j Qn i 1,j ) 4 ξ 2 ( h n ) 3 Q n Qn+1 + Q n+1 1 θ 2 ( h n ) 3 Q n i+1,j 2Qn + Qn i 1,j ξ 2 = 0 (5), Qn+1 1 are the unknown param- where Q n+1 eters for step 1. +1, Qn+1 h n+1 Q n+2 Q n+1 p n+1 τ + 2Λ Qn+1 p n p n+1 h n+1 +1 h n θ h n+2 h n+1 τ h n+1 Q n Λ Qn+1 1 p n+1 2 θ 3 ( h n+1 ) 2 ( h n+1 +1 h n+1 1 )(Qn+1 +1 Qn+1 1 ) 4 θ 2 3 ( h n+1 ) 2 ( h n+1 i+1,j h n+1 i, 1j )(Qn+2 i+1,j Qn+2 i 1,j ) 4 ξ 2 ( h n+1 ) 3 Q n Qn+1 + Q n+1 1 θ 2 ( h n+1 ) 3 Q n+2 i+1,j 2Qn+2 + Q n+2 i 1,j ξ 2 = 0 (6)

5 Dynamic characteristics of self-acting gas bearing flexible rotor coupling system 345 EI y K = lt 3 (1 + Φ s) 12 6l T 12 6l T (4 + Φ s )lt 2 6l T (2 Φ s )l 2 T 12 6l T symm. (4 + Φ s )lt 2 Fig. 3 A model of gas journal bearing system where Q n+2 i+1,j, Qn+2, Qn+2 i 1,j are the unknown parameters for step 2. Equations (5) and (6) are the nonlinear equations, so the ADI method is not unconditionally stable. To keep the stability of the calculation scheme, the condition following should be considered: the step time τ should be less than During the solving procedure, the over-relaxation method is employed to form and update the triangular matrix and the discrete equations are solved. By integrating the transient gas film pressure p(θ, ξ,τ) acting on the rotor, the transient gas film forces can be obtained as follows: 2π L/R F x = P a R 2 p(θ,ξ,τ)cos θdξdθ; (7) 0 F y = P a R 2 2π 0 0 L/R 0 p(θ,ξ,τ)sin θdξdθ (8) 2.2 Modeling for motion equation of elastic rotor The model of journal gas bearing system is presented in Fig. 3. The rotor system is formed by discrete disk, elastic shaft segment with distributed mass. The numbering scheme is shown in Fig. 3, where the journal gas bearing locates on nodes 2, 6 and the imbalance mass locates on node 4. The finite element method is employed to form the stiffness and mass matrixes. The Timoshenko beam element is used. Then the equation of motion is as follows: M{ü}+C{ u}+k{u}={f } (9) where {u} is the displacement and angular vector of each node for the rotor system is {x 1,θ y1,y 1, θ x1, x 2,θ y2,y 2 θ x2,...,x N,θ yn,y N, θ xn }. N is the total number of nodes of the rotor system. K is the stiffness matrix, and M is the mass matrix. By employing the strain energy and kinematic energy expression, the above matrix is as follows: M = M T + M R ρal T M T = 420(1 + Φ s ) 2 m 1 l T m 2 m 3 l T m 4 l T 2 m 5 l T m 4 lt 2 m 6 m 1 l T m 2 symm. lt 2 m 5 ρi y M R = 30l T (1 + Φ s ) 2 m 7 l T m 8 m 7 l T m 8 l T 2 m 9 l T m 8 lt 2 m 10 m 7 l T m 8 symm. lt 2 m 9 m 1 = Φ s + 140Φ 2 s ; m 2 = Φ s Φ 2 s ; m 3 = Φ s + 70Φ 2 s ; m 4 = Φ s Φ 2 s ; m 5 = 4 + 7Φ s + 3.5Φ 2 s ; m 6 = 3 + 7Φ s + 3.5Φ 2 s ; m 7 = 36; m 8 = 3 15Φ s ; m 9 = 4 + 5Φ s + 10Φ 2 s ; m 10 = 1 + 5Φ s 5Φ 2 s The gyroscopic matrix is equal to 2M R ; C is the damping matrix, which includes the gyroscopic matrix; {F } are the forces acting on the rotor. In this paper, {F } include the transient gas film forces F x and F y which act on the elastic rotor. Equation (9) is the nonlinear ordinary differential equation; the step-by-step integration method is applied to carry out the numerical simulation. The Newmark integration method is employed to calculate the displacement, velocity and acceleration for each node of the rotor.

6 346 G.-h. Zhang et al. Fig. 4 The flow chart for weak coupling orbit method 2.3 Solving process of forecasting orbit method The orbit method based on weak coupling process The weak coupling process between the rotor dynamic response algorithm and transient gas film forces algorithm is depicted as follows: as shown in Fig. 4, the initial condition for the loop is the rotor initial position in the bearing at time n, expressed by the initial gas film thickness function H n. In weak coupling algorithm, the gas film thickness function H n+1 forecast at time n + 1 is equal to the gas film thickness function at time n, and then it will be taken into the ADI algorithm as the initial condition to solve the transient gas lubricated Reynolds equation at time n + 1, where the transient pressure distribution will be obtained. By integrating the gas pressure distribution P n+1, the transient gas film forces acting on the rotor will be obtained. With the obtained forces, the Newmark method will be employed to calculate the dynamic response of the rotor system. The orbit trajectory of the rotor at time n + 1 expressed by the gas film thickness function H n+1 will be obtained and then taken as the initial value for the next loop. From the above weak coupling process, the rotor dynamic equation is the ordinary differential equation and the gas-lubricated Reynolds equation is the second-order partial differential equation, which comprise the two analysis steps for one step of the bearing rotor system solving procedure. Solving results of the two types of analyses are needed as each other s initial value, which cannot be realized simultaneously in the coupling process. For this problem, the dynamic orbit method based on the weak coupling process is being proposed. In this procedure, the gas film thickness function from the rotor dynamic response calculation at last time step will be used as initial value for transient gas lubricated Reynolds equation for the next time step. Because of the delay of the time step of the two kinds of equation, this method is called the weak coupling process The orbit method based on forecasting coupling process The forecasting coupling process between the rotor dynamic response algorithm and transient gas film forces algorithm is depicted as follows: as shown in Fig. 5, based on the weak coupling process, the initial condition for the loop is also the rotor initial position in the bearing at time n, expressed by the initial gas film thickness function H n. Differently from the weak coupling algorithm, the gas film thickness function H n+1 forecast at time n + 1 will be calculated by forecasting method in forecasting coupling procedure. Then it will be taken into the ADI algorithm as the initial condition to solve the transient gas lubricated Reynolds equation at time n + 1, where the transient pressure distribution will be obtained. By integrating the gas pressure distribution P n+1, the transient gas film forces acting on the rotor will be obtained. With the obtained forces, the Newmark method will be employed to calculate the dynamic response of the rotor system. The orbit trajectory of the rotor at time n + 1 expressed by the gas film thickness function H n+1 will be obtained and then taken as the initial value for the next loop.

7 Dynamic characteristics of self-acting gas bearing flexible rotor coupling system 347 Fig. 5 The flow chart for forecasting coupling orbit method There are many ways to predict the gas film thickness function H n+1 forecast at time n + 1. Different methods need a varied number of the gas film thickness at previous time. The linear interpolation method is employed in this study to forecast he gas film thickness function at the next time step. It assumes that the trend of the function varying with time is linear, and then the following equation will be listed: H n+1 H n τ n = H n H n 1 τ n 1 and then ( H H n+1 n H n 1 = τ n 1 + H n ) τ n τ n So it is obvious that the gas film thickness function of the two previous time steps is needed for the linear forecasting procedure. 3 Numerical simulation and dynamic analysis The simulation model for numerical analysis is as follows: the length of the bearing L = 0.05 m, the diameter of the bearing D = m, air density ρ a = kg/m 3, air viscosity μ = 1.82E 5 N s/m 2, specific heat ratio k = 1.401, atmospheric pressure p a = Pa, the mesh grid size: in z direction (axial direction) nz = 20, in x direction (circumferential direction) nx = 40, the length of the rotor: 0.3 m, the diameter of the rotor: 0.05 m, the elastic modulus of the rotor 2.0E11 Pa, the density of the rotor: 7825 kg/m 3. The convergence condition is Max p n+1 p n Table 1 Comparison of calculated results Λ ε W θ 0 FEM DQM FEM DQM Verification of the proposed method To verify the accuracy of the orbit method employed in this study, the comparison is presented with the results from Ref. [21]bymodelingthe bearingof length diameter equal to 1. By ignoring the effects of rotor and the transient term of the gas lubricated Reynolds equation, the load capacity W and attitude angle θ 0 are obtained as shown in Table 1. The columns with are the results of the proposed method. It indicates that the numerical results of this paper are in good agreement with the reference results, which makes the foundation for the analysis of the gas bearing rotor system. 3.2 Comparison between the weak coupling and forecasting coupling methods The comparison between the weak coupling orbit method and the forecasting coupling orbit method is

8 348 G.-h. Zhang et al. Fig. 6 Shaft locus of different rotating speeds for different coupling algorithms without imbalance mass presented in this paper. The time step in this analysis is s. The effect of the shaft locus obtained by weak coupling and forecasting coupling methods are observed in Fig. 6, where the imbalance mass is ignored. As shown in Fig. 6(a,b), when the rotating speed is 5000 r/min, the difference between the two method is neglectable. As the rotating speed increases to 11,000 r/min, it indicates that the shaft locus amplitude of the forecasting method is a bit smaller than that of the weak coupling method in Fig. 6(c,d), but the same trend exists. As the rotating speed reaches 14,000 r/min, the regularity is the same as for rotating speed of 11,000 r/min. It is observed that in the case of ignoring the imbalance mass, the tendency for the weak coupling and forecasting coupling methods is the same, but the amplitude of the shaft locus for weak coupling is bigger. The effect of the shaft locus obtained by weak coupling and forecasting coupling method is observed in Fig. 7, where the imbalance mass is considered. AsshowninFig.7(a,b), when the rotating speed is 5000 r/min, the difference between the two methods is small. As the rotating speed increases to 11,000 r/min, it indicates that the shaft locus amplitude of the forecasting method is a bit smaller than that of the weak coupling method in Fig. 7(c,d), but the same trend exists. As the rotating speed reaches 14,000 r/min, the regularity is the same as for rotating speed of 11,000 r/min. It is observed that in the case of considering the imbalance mass, the tendency for the weak coupling and forecasting coupling methods is the same, but the amplitude of the shaft locus for weak coupling is bigger. The case which considers the effect of gravity forces and ignores the imbalancing mass is presented in Fig. 8. It indicates that for different rotating speeds (5000, 11,000 and 14,000 r/min), the tendency of the shaft locus for the two coupling algorithms is same. The static balance positions for the two methods are also same. In is observed that the weak coupling method and forecasting coupling method have little influence on the dynamic response when the gravity forces are considered. The variation of iterative number for different time steps which reach the stable condition is shown in Fig. 9, where the step times , , , , and s are employed. The calculated number of steps de-

9 Dynamic characteristics of self-acting gas bearing flexible rotor coupling system 349 Fig. 7 Shaft locus of different rotating speeds for different coupling algorithms with imbalance mass Fig. 8 Shaft locus of different rotating speeds for different coupling algorithms with imbalance under gravity force

10 350 G.-h. Zhang et al. creases as the time step increases, but the total time of reaching the stable condition increases. The reason for this phenomenon is that the essential asynchronous characteristics of weak coupling method introduce the delay error. For the cases of forecasting coupling method, the tendency for decrement of calculated step is more intensive than that of weak coupling method. It is observed that as the time step increases, the iterative number decreases, and the prod- uct of the two nearly keeps unchanged. The total time of reaching the stable condition for forecasting coupling method does change with the variation of the time step. So the forecasting coupling method can eliminate the delay of the weak coupling algorithms effectively. In the following study, the forecasting coupling method will be used to analyze the dynamics of the gas bearing rotor system. 3.3 The dynamic analysis of self-acting gas bearing rotor system The dynamic analysis without imbalance mass Fig. 9 The variation of iterative number with time steps The shaft locus and time response spectra diagrams are presented in Fig. 10 by analyzing the rotor bearing system without imbalance mass. As shown in Fig. 10(a), when the rotating speed is 5000 r/min, the dynamic characteristics of the rotor bearing system is point stabilization, where the rotor is running near some stable position after initial motion. When the rotating speed reaches 11,500 r/min (Fig. 10(b)), the Fig. 10 The shaft locus, time response and frequency spectrum diagram for different rotating speeds with no imbalance mass

11 Dynamic characteristics of self-acting gas bearing flexible rotor coupling system 351 Fig. 11 The power spectrum, phase diagram and Poincaré maps for different rotating speeds with no imbalance mass stable limit cycle motion appears. With the increment of the rotating speed as in Fig. 10(c), the motion of the rotor does not attenuate, and keeps the stable limit cycle motion status. The phase diagram in X direction in Fig. 11 also indicates the existence of the limit cycle motion. The frequency analysis is performed and the results are presented in Fig. 10(d,e,f) and Fig. 11(a,b,c) by ignoring the imbalance mass effect. It indicates that as the rotating speed increases, the dynamic response amplitude for the cases of 11,500 and 14,000 r/min changes little. The lower frequency component that is smaller than 0.5 frequency multiplication appears generally. The range of lower frequency whirl for gas bearing is smaller than that of oil bearing because of the lower viscosity of the gas. For frequency analysis, the data of stable motion are employed, but for the shaft locus the initial data are used. The power spectral diagram is presented in Fig. 11(d,e,f). It is observed that the frequency components of the gas bearing rotor system are complicated, where two lower frequency components appear. The difference between the two lower frequencies is small for the point stable condition (Fig. 4(a)). For the cases of the limit cycle, the frequency components are complicated, and the amplitude of different fre-

12 352 G.-h. Zhang et al. quency components varies with the rotating speeds in power spectrum (Fig. 11(b,c)), which is single frequency in frequency spectrum diagram. The Poincaré maps for different rotating speeds are presented in Fig. 11(g,h,i). In non-autonomous system, a point on the Poincaré section is referred to as the return point of the time series at a constant interval of T, where T is the period of the rotating speed. As the rotating speed is 5000 r/min (Fig. 11(g)), the T period motion is stable. As the rotating speed increased to 11,500 r/min (Fig. 11(h)), the T period motion loses the stability and is replaced by 8T period motion. When the rotating speed reaches 14,000 r/min (Fig. 11(i)), the system continues to exhibit the 8T period motion. The bifurcation diagram of the gas bearing rotor system is shown in Fig. 12, which takes the rotating speed as the bifurcation parameter. It indicates that as the rotating speed increases to 8000 r/min, the Hopf bifurcation occurs. The motion of the system changes from the balance points of stabilization to the limit cycle condition The dynamic analysis with imbalance mass Fig. 12 The x direction displacement bifurcation with no imbalance mass The cases that consider the imbalance mass are carried out. The imbalance mass acts on the node 4. The mass weight is 0.2 kg, and the eccentricity is m. Fig. 13 The shaft locus, time response and frequency spectrum diagram for different rotating speeds with imbalance mass

13 Dynamic characteristics of self-acting gas bearing flexible rotor coupling system 353 Fig. 14 The power spectrum, phase diagram and Poincaré maps for different rotating speeds with imbalance mass When the rotating speeds are 5000 and 10,000 r/m, the rotor runs with the status of stable periodicity which is shown in Fig. 13(a,b). The amplitude of the motion increases as the rotating speed increases. When the rotating speed reaches 3000 r/min, the stable periodic motion destabilizes. From the phase diagram of Fig. 14(d,e,f), it indicates that the behavior of the motion becomes more complicated. Then the frequency analysis for the gas bearing rotor system with imbalance mass is carried out. When the rotating speed is 5000 r/min, the basic frequency generated from the imbalance mass is the main component, as shown in Fig. 13(d). As the rotating speed increases to 10,000 r/min, the 0.35 frequency multiplication that the amplitude is small appears in Fig. 13(e). As the rotating speed reaches 30,000 r/min, the am-

14 354 G.-h. Zhang et al. plitude of the 0.35 frequency multiplication increases and then exceeds the value of the basic frequency component. The lower frequency whirling phenomenon is obvious. As the rotating speed increases, the lower frequency components that are smaller than the 0.35 frequency multiplication appear based on the original 0.35 frequency multiplication in Fig. 13(e). But the amplitude of 0.35 frequency multiplication is the biggest in the system. The power spectral analysis is performed for different rotating speeds, which is shown in Fig. 14(a,b,c). It is observed that the frequency component of the power spectrum is more complicated than that of the no imbalance mass. With the increment of the rotating speed, the power of the basic frequency is becoming smaller. On the contrary, the lower frequency component is the main part of the whole frequency component. And also some certain frequency components that exist between the basic and second frequency components appear. The Poincaré maps for different rotating speeds with imbalance mass are presented in Fig. 14(g,h,i). As the rotating speed is 5000 r/min (Fig. 14(g)), the T period motion is stable. As the rotating speed increases to 10,000 r/min (Fig. 14(h)), the system continues to exhibit the T period motion. When the rotating speed reaches 30,000 r/min (Fig. 14(i)), the T period motion loses the stability and is replaced by 8T period motion. AsshowninFig.15, the rotating speed is taken as the bifurcation parameter. It is observed that as the rotating speed reaches 11,000 r/min, the stable periodic motion destabilizes. It indicates that the motion changes into the multiply periodic. Compared with the Fig. 12, the imbalance mass can be employed to increase the whirling speed by delaying the occurrence of whirling. Similar conclusion is given in Ref. [22]: The tests previously described indicated that large amplitude self-excited whirl cannot exist if the magnitude of the eccentricity ratio is kept above a certain critical level. By plotting the deformation of the nodes, the modal diagram of the rotor system with rotating speed of 47,000 r/min is presented in Fig. 16. It indicates that the rotor is running with the first bending natural frequency modes. 4 Conclusion This paper has studied the nonlinear dynamic characteristic of self-acting gas bearing flexible rotor sys- Fig. 15 The x direction displacement bifurcation with imbalance mass Fig. 16 The rotor modal diagram with rotating speeds of r/min with no imbalance mass tem by forecasting coupling orbit method. To settle the problem that the two kinds of transient solving processes (transient Reynolds equation for bearing and transient equation of motion for rotor) cannot be solved simultaneously, which arises from the fact that they need each other s results as their initial values, the multi-field coupling algorithm based on the forecasting method is proposed and applied in this paper. By employing the numerical method, the nonlinear gas film forces, the trajectory diagram, phase diagram, frequency spectrum, power spectrum, bifurcation diagram, vibration mode diagram and the equilibrium position diagram were obtained. It is to be noted that the dynamic characteristics of journal gas bearings rotor system and whirling instability of the gas film could be depicted successfully. The results reveal that the presence of a complex dynamic behavior is comprised of periodic, subharmonic responses of the rotor. By comparing the weak coupling and forecasting coupling method, the forecasting coupling method can

15 Dynamic characteristics of self-acting gas bearing flexible rotor coupling system 355 eliminate the delay of the weak coupling algorithms effectively. Regarding the behavior of the rotor bearing system with respect to variation in rotating speed and imbalance mass, the results indicate that the imbalance mass can be employed to increase the whirling speeds by delaying the occurrence of whirling. The method presented in this study provides the foundation for the design of gas bearing flexible rotor system. The results indicate suitable operating conditions that suppress subharmonic motion and prevent instable behavior. Acknowledgements The research presented here was supported by the National Natural Science Foundation of China (Grant No ) and the Fundamental Research Funds for the Central Universities (Grant No. HIT. NSRIF ). The authors are grateful for the support. References 1. Yang, P., Zhu, K.Q., et al.: On the nonlinear stability of self-acting gas journal bearings. Tribol. Int. 42(1), (2009) 2. Sternlicht, B.: Elastic and damping properties of cylindrical journal bearings. J. Basic Eng. 81, (1959) 3. Rentzepis, G.M., Sternlicht, B.: On the stability of rotors in cylindrical journal bearings. J. Basic Eng. 84(3), (1961) 4. Ausman, J.S.: An improved analytical solution for selfacting, gas-lubricated journal bearings of finite length. J. Basic Eng. 83(2), (1961) 5. Lund, J.W.: The stability of an elastic rotor in journal bearings with flexible, damped supports. J. Appl. Mech. 87, (1965) 6. Castelli, V., Elrod, H.G.: Solution of the stability problem for 360 degree self-acting, gas-lubricated bearing. J. Basic Eng. 87(1), (1961) 7. Castelli, V., Stevenson, C.H.: A Semi-implicit Numerical Method for Treating the Time Transient Gas Lubrication Equation. Mechanical Technology Inc, New York, TID (1967) 8. Dimofte, F.: Fast methods to numerically integrate the Reynolds equation for gas fluid films. New York, NASA Technical Memorandum, (1992) 9. Bou-sa, B., Grau, G., Iordanoff, I.: On nonlinear rotor dynamic effects of aerodynamic bearings with simple flexible rotors. J. Eng. Gas Turbine Power 130(1), (2008) 10. Wang, C., Jang, M., Yeh, Y.: Bifurcation and nonlinear dynamic analysis of a flexible rotor supported by relative short gas journal bearings. Chaos Solitons Fractals 32(2), (2007) 11. Wang, C.C.: Application of a hybrid method to the nonlinear dynamic analysis of a flexible rotor supported by a spherical gas-lubricated bearing system. Nonlinear Anal. 70(5), (2009) 12. Wang, C.C.: Bifurcation and nonlinear analysis of a flexible rotor supported by a relative short spherical gas bearing system. Commun. Nonlinear Sci. Numer. Simul. 15(9), (2010) 13. Frederic, J.B.: A monolithical fluid-structure interaction algorithm applied to the piston problem. Comput. Methods Appl. Mech. Eng. 167, (1998) 14. Michler, C., Hulshoff, S.J., van Brummelen, E.H., et al.: A monolithic approach to fluid-structure interaction. Comput. Fluids 33, (2004) 15. Longatte, E., Verreman, V., Bendjeddou, Z., et al.: Comparison of strong and partitioned fluid structure code coupling methods. In: ASME, Pressure Vessels and Piping Division (Publication) PVP, v 4, Proceedings of the ASME Pressure Vessels and Piping Conference 2005 Fluid-Structure Interaction, PVP2005, pp (2005) 16. Etienne, S., Pelletier, D., Garon, A.: An updated lagrangian monolithic formulation for steady-state fluidstructure interaction problems. In: 43rd AIAA Aerospace Sciences Meeting and Exhibit Meeting Papers, pp (2005) 17. Piperno, S., Farhat, C., Larrouturou, B.: Partitioned procedures for the transient solution of coupled aeroelastic problems Part I: Model problem, theory and twodimensional applications. Comput. Methods Appl. Mech. Eng. 124, (1995) 18. Piperno, S.: Explicit/implicit fluid/structure staggered procedures with a structural predictor and fluid subcycling for 2d inviscid aeroelastic simulations. Int. J. Numer. Methods Fluids 25, (1997) 19. Farhat, C., Lesoinne, C., Letallec, P.: Load and motion transfer algorithms for fluid/structure interaction problems with non-matching discrete interfaces: momentum and energy conservation, optimal discrimination and application to aeroelasticity. Comput. Methods Appl. Mech. Eng. 157(1 2), (1998) 20. Piperno, S.: Partitioned procedures for the transient solution of coupled aeroelastic problems Part II: Energy transfer analysis and three-dimensional applications. Comput. Methods Appl. Mech. Eng. 190(24 25), (2001) 21. Malik, M., Bert, C.W.: Differential quadrature solutions for steady-state incompressible and compressible lubrication problems. J. Tribol. 116(2), (1994) 22. Reynolds, D.B., Gross, W.A.: Experimental investigation of whirl in self-acting air-lubricated journal bearings. Tribol. Trans. 5(2), (1962)