Effects of Surface Waviness on the Nonlinear Vibration of Gas Lubricated Bearing-Rotor System

-is paper presents the effects of the surface waviness on the nonlinear dynamic performance of a gas bearing-rotor system. -e coupled vibration with the elastomer is taken into consideration to fit the actual engineering application. -e effects of the directions, the amplitudes, and the numbers of waves to the nonlinear dynamic performance are investigated. -e results show that the existence of the surface waviness in the circumferential direction can improve the stability of the system obviously. But the surface waviness in the axial direction is damage to the system.-e nonlinear dynamic performance of the system is insensitive to the number of surface waviness. -e increase of the amplitude of the waviness in circumferential direction can improve the stability of the system.


Introduction
Bearing is an essential component of most rotating machines.ey provide the load capacity and stiffness to support shafts against static loads and dynamic forces.Gas lubricating journal bearings have many advantages of less heat generation, more environment friendly, higher accuracy, and longer lives by comparing with traditional oil bearings.us, the gas lubricating bearing-rotor system is widely used in engineering applications such as turboexpanders in large-scale cryogenic systems, superprecision machine tools, and highspeed machines [1,2].However, the inherent characteristic of compressibility and low viscidity of gas may cause instability to the gas bearing-rotor system which limits the range of the application of gas bearings.Furthermore, the low viscosity of gas makes a thin gas film generally lower than 50 micrometers.As the result, the assumption of the journal bearing is that the smooth surface is far from bearing practical application especially for bearing-rotor system operating under a relatively small film thickness [3].
Manufacturing error may cause some waves on the surfaces of bearing.ese waves can change the thickness distribution of the gas film which can significantly influence the static and dynamic performance of the gas bearing-rotor system.During the past years, authors have published many articles about the effects of the surface waviness on the performance of the bearing.Dimofte [4] compared the three-waved journal bearing with both the three-wavegroove bearing and three-lobe bearing [5], they concluded that the load capacity of the wave bearing is better than other types, and the performance of the wave bearing is dependent on the parameters and position of waves.Rasheed [3] studied the influence of circumferential, axial, and combined surface waviness on the performance of plain cylindrical bearings.ey noted that the circumferential waviness can increase the load capacity and decrease the friction variable when the wave number is below 9, and the axial waviness has the adverse effects on load capacity and friction.Kwan and Post [6] studied the influence of manufacturing error of rectangular thrust bearing on the static load and stiffness performance.
ey concluded that the load capacity and stiffness are sensitive to the manufacturing errors.Nagaraju et al. [7] studied the combined influence of bearing flexibility and surface roughness on the characteristics of the journal bearing system.e results indicated that the longitudinal roughness pattern of the bearing can improve the load capacity.Ene et al. [8] used both the transient and small perturbation methods to investigate the dynamic behavior of three-wave journal bearing system.e results showed the influence of the wave amplitude and oil supply pressure on the threshold of stability.Sharma and Kushare [9] investigated the influence of surface roughness on the performance of two-lobe journal bearing.e results indicated that the performance of the bearing system is significantly affected by the orientation of roughness.And thus, the performance of bearing can be enhanced by the proper selection of the roughness pattern parameter.Wang et al. [10] studied the influence of the waves of bearing surface on the static and dynamic performance of the aerostatic journal bearing.e results showed that the wave amplitudes significantly affect the static and dynamic performance of the bearing.
e stability is the most important character for the bearing-rotor system.Large number of articles has been published to investigate the stability of gas-lubricant bearing-rotor system.Among these papers, two different approaches were published to analyze the dynamic stability of the rotor-bearing system: one is the linear small perturbation analysis and another is the nonlinear transient analysis.
Linear small perturbation theory assumes that the journal obit whirls around its equilibrium position with a small perturbation.is assumption makes the dynamic stiffness, and dynamic damping coefficients can be computed.And then, the whirl frequency and critical mass can be computed by these dynamic coefficients.is method can provide the instability boundary.us, the stability of the system can be estimated by the threshold.However, this method cannot present the characters of the motion.
Gross and Zachmanoglou [11] studied the performance of the self-acting journal bearing with infinite length by using the small perturbation method.e calculation results are valid for other types of bearings with high accuracy.Belforte et al. [12] investigated dynamic coefficients of selfacting gas bearing, they used the developed mathematical model, and the results were in good agreement with the results using the nonlinear distribution parameter model.Yang et al. [13] studied the dynamic coefficients of selfacting tilting pad gas bearing, and the results showed the relationship between dynamic coefficients and tilting pad perturbation.Yu et al. [14] used numerical simulations and experiments to investigate the influence of perturbations on the dynamic stiffness for aerostatic bearings.
e results revealed that the dynamic stiffness coefficients are more sensitive to the frequency of perturbation than the amplitude; thus, they concluded that the dynamic stiffness can be enhanced through active control which can generate perturbations.
In nonlinear transient analysis, the time-dependent Reynolds equation is solved to get the transient forces, and then, the transient journal center obits are obtained by solving the motion equations.e stability of the system can be judged by the shape of obits.
e nonlinear dynamic characters are available by using this method which is different from the perturbation analysis.
Kim and Noah [15] coupled harmonic balance with alternating frequency technique to obtain synchronous and subsynchronous whirling motions of a Jeffcott rotor-bearing system.e results showed that as the parameters of the system changed the boundary of flip bifurcation were obtained.Wang et al. [16][17][18][19][20] investigated the bifurcation performance of rigid rotor and flexible rotor supported by a pair of gas bearing, and the results presented a better understanding to the nonlinear dynamic characters of the gas bearing-rotor system.From the calculation results, we can also find that the dynamic behavior of the rotor center is very complex which includes periodic and subharmonic response.Rashidi et al. [1,2] presented the effect of operating parameters on nonlinear dynamic performance for noncircular journal gas-lubricated bearing-rotor system.e complex nonlinear dynamic behaviors such as periodic, KTperiodic, and quasiperiodic responses were analyzed by using dynamic trajectory, the Poincare maps, and bifurcation diagrams.Hassini and Arghir [21] used the dynamic coefficients to analyze the nonlinear characteristics, and the results were in good agreement with the results calculated by nonlinear transient Reynolds equations.Abbasi et al. [22] studied the vibration control of the rotating machine with a high-static low-dynamic stiffness suspension.In their study, the complex nonlinear dynamic behavior was showed, and the results present that the method can suppress the nonperiodic behavior in the considered speed range.
Generally speaking, the vibrations of most manufacture machines are inevitable which will cause the surface to be unsmooth.e effects of waves on static performances and dynamic coefficients of bearing are investigated in most previous published articles.However, few papers have mentioned the effects of waves on the stability and nonlinear dynamic performance of the gas bearing.us, in this paper, the effects of surface waviness in circumferential and axial directions on the nonlinear dynamic characteristics of the rotor-gas bearing system have been investigated.e method used in this paper is nonlinear transient approach which is similar to the previous articles [18,23], but our focuses are different.In practical engineering, the elastomers are mounted between bearing and shaft; thus, the elastomer suspension is taken into consideration.e results in this paper can provide guidance on the design and manufacture of the gas bearing system.

Numerical Solution of Gas
Pressure Distribution e model in this paper is a massless shaft with a disk supported by a pair of self-acting gas journal bearings.Generally speaking, the manufacture precision of rotors is always better than that of bearings, so, the waves on the surface of rotors can be neglected [22].In this paper, the journal bearings used for calculation include axial and 2 Shock and Vibration circumferential surface waviness which is shown in Figure 1.Before numerical simulation, some design assumptions are given as follows [16][17][18]23]: (1) Isothermal flow: due to the heat conductivity ability of the bearing material (copper) is much bigger than that of gas, the heat generation in the gas film will conduct away quickly.us, the temperature of gas is constant.
(2) Curvature insensitive: the gas film thickness is much smaller than the diameter of the shaft.So, the change of velocity caused by the curvature is neglected.(3) Constant viscosity: due to the viscosity of gas is insensitive to the changes of pressure, we assume it as a virtually constant.(4) e side flow of the bearing can be neglected.
To simplify the model of surface waves, we assume that the form of waves fits the function of sinusoidal; thus, the waviness in circumferential direction can be described as the following form [10,24]: e waves in axial direction of the bearing can be denoted as Based on the assumption above, the pressure distribution of the gas film can be described by the Reynolds equation: where h is the gas film thickness and can be denoted as Some dimensionless parameters are defined as follows to transform Equation (3) into polar coordinate and nondimensionalized: ( Taking the dimensionless parameters above into Equation (3), the dimensionless Reynolds equation is obtained and can be written as e finite difference method is used to get the pressure distribution of the gas film.e discretized scheme of the Reynolds equation was derived by using the central difference method in circumferential and axial directions and the implicit backwards method in time τ. e methods used in this paper are similar to the previous articles [18,23], and the discretized version can be described as where S � P 2 ; i � 1, 2, . .., N; j � 1, 2, . .., M. e film thickness distribution is depended by the position of the rotor.For parameter P n+1 i,j , the equation is a quadratic equation with one unknown and determined by the pressure of the surrounding four nodes (i − 1, j), (i + 1, j), (i, j + 1), and (i, j − 1).
e pressure distribution of the gas film between the rotor and bearing enables the estimation of the supporting force which is acting on the rotor.e supporting force can be described as

Shock and Vibration
where P(θ, η) is the pressure distribution in the dimensionless coordinate system.

Rotor Dynamic Equations and Obits Calculation Procedure
In this paper, we use the nonlinear transient approach to investigate the stability of the rotor-bearing system.e model for calculation is a massless rotor with a disk, and it was supported by a pair of self-acting gas bearings.In engineering reality, the elastic dampers such as O-rings are mounted between bearings and shell to improve its stability, shown in Figure 2. e damping force of the elastomer obeys the viscous damping law which is one of the simplest and most widely used. is model is expressed as [25][26][27] where c is the damping exponent; sign() is the signum function.e case c � 1 is chose which indicates the linear viscous damping model and used for calculation.us, the equation of motion of the rotor and bearing in transient state can be written as e nondimensional groups can be defined as follows: Substituting Equation (11) into Equation ( 10), we will get the dimensionless dynamic equations as follows:  Shock and Vibration As we get acceleration of the rotor and bearing in horizontal and vertical directions, the center obits of rotor and bearing can be acceptable by using the following equation: Computation of the center trajectory of rotor and bearing is an iterative procedure.e procedure begins with the initial equilibrium state.e initial position of the rotor (X 0 , Y 0 ) is a static balance position which is defined by the external loading and rotational speed.To estimate the influence of the initial condition, the first 600 periodic calculation data can be ignored and the nonlinear behavior analysis is based on the data after the vibration is stable.e initial velocities of the rotor in two different directions are assumed to be zero.

Results and Discussion
In order to verify the correctness of the program, the attitude angle φ and supporting force F of a self-acting journal gas bearing with smooth surface are calculated and compared

Shock and Vibration
with previous investigations [23,28], as is shown in Figure 3.
e results in this paper are in good agreement with the previous investigations; thus, the validity of the program is verified.
e rotor unbalance mass is an important parameter for rotating machine which influence the response of the rotorbearing system [28,29].In this paper, what we focused on is the influence of the surface waviness to the nonlinear dynamic response.So, the single rotor unbalance value (0.15 μm) is selected in this study.e rotor-bearing system is used in the turbomachine with a superhigh speed.us, the mass of the rotor is usually quite small, less than 0.5 kg.
In the numerical integration, the aspect ratio (L/D) of self-acting journal gas bearing for calculation is chosen as 1.44.e lubricant in the film is air.e parameters of elastomer are set as follows: k � 4e6N/m, ξ � 0.005, and c � 1.To show the influence of the surface waviness to the nonlinear dynamic response clearly, the bifurcation diagrams of the rotor-bearing system with flat surface are plotted, which is presented in Figure 4. From the results, we can conclude that when the half mass of the rotor is equal or less than 0.18 kg, the rotor center presents a regular periodic motion.But, motion loses its regularity and becomes irregular when the half mass of the rotor increased to 0.19 kg.
Figure 5 shows the bifurcation diagrams of the rotating system with circumferential surface waviness.In this calculation, we assume that the surface waviness is only included in the circumferential direction of the bearing, and the number of waves is equal to 5, and the amplitude of the waves is 1 micrometer.Compared with the results in Figure 4, the discrete point of the system with circumferential waviness is much bigger than the system with flat surface which indicated that the waviness in circumferential direction can increase the stability of the system obviously.
In order to show nonlinear dynamic performance of the system clearly, a set of maps such as the rotor center obits, phase portraits, logarithmic spectra maps in two directions, and Poincare maps are used to analyze the nonlinear dynamic performance in different conditions.From Figures 6-8, when the half mass of the rotor is less than 0.75 kg, the vibration is mainly caused by the mass unbalance, the frequency of rotating is much bigger than subharmonic frequency; thus, the rotor center obits show a regular periodic motion.When the half mass of the rotor is in the range of 0.75 kg and 1 kg, the amplitude of subharmonic vibration increased and cannot be ignored.e motion losses its periodic motion, and subharmonic periodic motion appears.However, the amplitude of motion is also small.When the half mass of the rotor is bigger than 1.0 kg, shown in Figures 8 and 9, the amplitude of vibration becomes much bigger and a closed curve occurs in the Poincare map, which indicated that quasiperiodic motion appears.
e effects of the number of waviness in the circumferential direction on the nonlinear dynamic performance of the system are analyzed in which different numbers of waviness (N � 3, 5, and 7) with m s � 1.0 kg, ξ � 0.005, k � 4e6N/m are used for calculation.Figures 8-11 show the obits, logarithmic spectrum maps, and Poincare map with surface waviness (N � 3, 5, and 7), respectively.e comparison between different numbers of waviness in circumferential direction shows that the number of waviness do not change the nonlinear dynamic performance of the system, but the amplitude of the vibration declines slightly with the increase of the number of waviness (shown in the Reference [23] Reference [28] This paper Reference [23] Reference [28] This paper 6 Shock and Vibration logarithmic spectrum maps).e increase of the number of waves can only improve the static and dynamic performance of the bearing slightly which is not enough to change the nonlinear dynamic behavior of the system.Figures 8, 12, and 13 display the obits, logarithm amplitude spectrum maps, and Poincare maps of the rotor center with different surface waviness amplitudes in circumferential direction.In this solution, three cases (A � 1, 2, and 3) with other parameters such as m s � 1 kg, ξ � 0.005, and k � 4e6N/m are used and compared.e nonlinear dynamic performance changes from quasiperiodic motion (A � 1) to regular periodic motion (A � 4). is means the large amplitude of the waviness can increase the stability of the system, but the amplitude of the waviness is restricted by the gas clearance.From the obits and logarithm spectrum maps in these pictures above, we can find that the increase of the amplitude of the surface waviness in circumferential direction can decline the amplitude of the subharmonic frequency.at is because the increase of the amplitude of the waviness can increase the static and dynamic performance.-0.14 -0.16 -0.15 -0.17 -0.13 -0.18 -0.12 X s -0.13 -0.14 -0.15 -0.12 -0.18 -0.17 -0.16  -0.13 -0.11 -0.12 -0.14 -0.16 -0.15    Shock and Vibration Figure 14 displays the bifurcation maps of the surface waviness in the axial direction.In this solution, m s � 1 kg, ξ � 0.005, k � 4e6N/m, and A � 1μm are used.By comparing with the flat surface bearing system, we can conclude that the nonlinear dynamic performance of the system with axial surface waviness is similar to that of the system with flat surface.e discrete point of the system with axial surface waviness is 0.2 kg, a little bigger than that of the system with flat surface (0.19 kg).But the existence of surface waviness results in the decrease of the minimum gas film which easily causes the collision friction between rotor and bearing.

Conclusions
In this paper, the effects of surface waviness in circumferential and axial direction to the nonlinear dynamic characteristics of the rotor-bearing system with elastomer suspension have been investigated.e nonlinear dynamic behaviors under different conditions have been investigated by inspecting the rotor center obits, logarithm spectrum maps, and Poincare maps.e results of this study are more applied to the practical engineering and can provide a forward guidance of the manufacture for the bearing.
By comparing with the rotor-bearing system with flat surface, the surface waviness in circumferential direction does raise the stability of the system.For rotor-bearings system with surface waviness in circumferential direction (N � 5 and A � 1μm), the discrete point in the bifurcation diagram (0.75 kg) is much bigger than that of flat surface system (0.18 kg).
ree values (N � 3, 5, and 7) of the number of surface waviness are investigated and compared.
e nonlinear dynamic performance is insensitive to the number of the waviness in circumferential direction.With the increase of the number of the waviness, the amplitude of the vibration declines slightly.
-0.16 -0.14 -0.12 -0.1 -0.08 1.16    Shock and Vibration e increase of the amplitude of the surface waviness in circumferential can improve the static and dynamic performance of the bearing obviously.With the increase of the amplitude, the motion of the system is changed from quasiperiodic motion to regular periodic motion.us, the large amplitude can increase the stability of the system.e effect of waviness in the axial direction is investigated by comparing with the flat surface.
e axial surface waviness cannot change the nonlinear dynamic performance of the system.But the existence of the waviness can reduce the minimum gas clearance which is damage for the bearing.14

Shock and Vibration
V sx , V sy : Dimensionless horizontal and vertical velocity of rotor V bx , V by : Dimensionless horizontal and vertical velocity of bearing ϖ: Angular speed of rotor, rad/s x, y: Horizontal and vertical coordinates, m X s , Y s : Dimensionless horizontal and vertical displacement of rotor X b , Y b : Dimensionless horizontal and vertical displacement of bearing X j , Y j : Dimensionless displacement of rotor relative to the bearing in horizontal and vertical directions Bearing number ε: Eccentricity ratio of the system δ: Height distribution of waviness, m λ: Wavelength of surface waviness, m θ, η: Dimensionless coordinate in circumferential and axial directions n: Time level i, j: Grid location in circumferential and axial directions b: Parameters of bearing s: Parameters of rotor 0: Initial parameter.

Figure 1 :
Figure 1: Model of self-acting gas journal bearing with surface waviness.

Figure 2 :
Figure 2: Schematic of a rigid rotor (massless shaft and disk) in journal bearing with elastic damper.

Figure 3 :
Figure 3: Comparison of steady state calculation results for journal gas bearing, D � L.

Figure 4 :Figure 5 :
Figure 4: Bifurcation diagrams of rotor: (a)X s (nT) and (b)Y s (nT) versus half mass of rotor.
Amplitude of the waves on the surface, m A sx , A sy : Dimensionless horizontal and vertical acceleration of rotor A bx , A by : Dimensionless horizontal and vertical acceleration of bearing Cr: Average gas film clearance, m d: Diameter of the rotor, m e ub : Mass eccentricity of the rotor, m f gx, f gy : Supporting forces in horizontal and vertical directions, N f ex , f ey : External forces in horizontal and vertical directions, N f d : Damping force of elastomer, N F gx , F gy : Dimensionless supporting forces in horizontal and vertical directions F: Defined as tan −1 (F gx /F gy ) F ex , F ey : Dimensionless external forces in horizontal and vertical directions H: Dimensionless film thickness between shaft and bearing h: Film thickness, m k, c: Stiffness(N/m) and damping(N•s/m)of elastic damper c: Damping exponent K, C: Dimensionless stiffness and damping of damper ξ: Dimensionless damping coefficient of elastomer, ξ � c/2 ���� km b  L: Length of journal bearing, m m s : One-half mass of rotor, Kg m b : Mass of bearing, Kg M s , M b : Dimensionless mass of rotor and bearing N: Number of waviness p: Pressure, N/m 2 P: Dimensionless pressure P a : Atmosphere pressure, N/m 2 R: Radius of shaft, m t: Time, s τ: Dimensionless time, ϖt U: Peripheral speed of rotor in dynamical state, m/s μ: Fluid kinematic viscosity, Pa•s

Figure 14 :
Figure 14: Bifurcation diagrams of rotor center with axial surface waviness: (a) X s (nT) and (b) Y s (nT) versus half mass of rotor.