A new model of water-lubricated rubber bearings for vibration analysis of flexible multistage rotor systems

Journal of Sound and Vibration ] (]]]]) ]]]–]]]

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A new model of water-lubricated rubber bearings for vibration analysis of flexible multistage rotor systems Shibing Liu, Bingen Yang n Department of Aerospace and Mechanical Engineering, University of Southern California, Los Angeles, CA 90089, USA

a r t i c l e i n f o

abstract

Article history: Received 25 July 2014 Received in revised form 23 March 2015 Accepted 23 March 2015 Handling Editor: H. Ouyang

Flexible multistage rotating systems that are supported or guided by long waterlubricated rubber bearings (WLRBs) have a variety of engineering applications. Vibration analysis of this type of machinery for performance and duality requires accurate modeling of WLRBs and related rotor–bearing assemblies. This work presents a new model of WLRBs, with attention given to the determination of bearing dynamic coefficients. Due to its large length-to-diameter ratio, a WLRB cannot be described by conventional pointwise bearing models with good fidelity. The bearing model proposed in this paper considers spatially distributed bearing forces. For the first time in the literature, the current study addresses the issue of mixed lubrication in the operation of WLRBs, which involves interactions of shaft vibration, elastic deformation of rubber material and fluid film pressure, and validates the WLRB model in experiments. Additionally, with the new bearing model, vibration analysis of WLRB-supported flexible multistage rotating systems is performed through use of a distributed transfer function method, which delivers accurate and closed-form analytical solutions of steady-state responses without discretization. & 2015 Published by Elsevier Ltd.

1. Introduction Water-lubricated rubber bearings (WLRBs) are widely used to support or guide flexible multistage rotor systems in a variety of engineering applications. Take water-circulated vertical pumps as an example, which, as large and expensive industrial equipment, are commonly found in power generation, mining dewatering, water waste treatment, oil and gas industries and industrial processes. In a vertical pump, WLRBs guide the rotating shaft that carries multiple rigid bodies (impellers). Due to high flexibility of the shaft, dynamic modeling and vibration analysis is essentially important to optimal design and reliable operation of this kind of rotor systems. A typical WLRB is a long cylindrical metal shell that hosts multiple rubber staves separated by axial grooves; see Fig. 1. The usage of the bearing has three major advantages: (i) pumped water going through the bearing is conveniently used as a lubricant, which reduces pump operation cost; (ii) water flow takes away heat and fine particles through the bearing grooves; and (iii) the natural resilience of rubber gives the bearing good properties for shock and vibration absorption and wear resistance. The knowledge on the dynamic coefficients (stiffness and damping coefficients) of WLRBs is necessary in vibration analysis and optimal design of flexible rotor–bearing systems.

n

Corresponding author. Tel.: þ1 213 740 7082. E-mail addresses: [email protected] (S. Liu), [email protected] (B. Yang).

http://dx.doi.org/10.1016/j.jsv.2015.03.052 0022-460X/& 2015 Published by Elsevier Ltd.

Please cite this article as: S. Liu, & B. Yang, A new model of water-lubricated rubber bearings for vibration analysis of flexible multistage rotor systems, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.03.052i

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Nomenclature C D e E h hðpÞ

radial clearance of the WLRB inside diameter of the WLRB eccentricity Young's modulus of the shaft fluid film thickness of the WLRB non-dimensional rubber deformation caused by fluid pressure inside the WLRB H ðpÞ rubber deformation caused by fluid pressure inside the WLRB I moment of inertial of the shaft L length of the WLRB p pressure distribution inside the WLRB R inside radius of the WLRB t time u; v experimental vibration amplitudes of the shaft in x and y directions U; V theoretical vibration amplitudes of the shaft in x and y directions x; Rθ circumferential coordinate in the direction of rotation z coordinate along the bearing length α; αexp non-dimensional number representing the percentage contribution of the stiffness of rubber staves to the overall rubber bearing stiffness Γ compliance operator ε; ε0 non-dimensional eccentricity η non-dimensional compliance operator μ fluid viscosity υ Poisson's ratio of the rubber material ξ; λ; Λ non-dimensional parameter of the WLRB ϕ; ϕ0 attitude angle Ω; ω shaft rotation speed a1 ; a2 ; b1 ; b2 coefficients obtained from the complex variable filtering method A1 ; A2 ; B1 ; B2 coefficients obtained from the distributed transfer function method AL area of the land region Cr constant used in shape factor method C h;ij non-dimensional elastohydrodynamic damping coefficients used in the distributed model of long WLRBs C rb;ij non-dimensional overall rubber bearing damping coefficients using the distributed model of long WLRBs C rr ; C ϕr ; C rϕ ; C ϕϕ non-dimensional elastohydrodynamic damping coefficients in r  θ coordinate

C xx ; C xy ; C yx ; C yy non-dimensional elastohydrodynamic damping coefficients in x  y coordinate er applied strain Er linear Young's modulus of the rubber stave Erh homogenous compression modulus of the rubber block f x; f y equivalent spring force generated at the top boundary in the experiment Fx; Fy non-dimensional load components in the x-, y-direction h0 steady-state fluid film thickness Kr equivalent stiffness of the rubber stave K r ; K r;d non-dimensional stiffness of rubber staves used in the distributed model of long WRLBs kuu ; kuψ ; kvθ ; kvv equivalent spring stiffness of the upper shaft at the top boundary in the experiment K rr ; K ϕr ; K rϕ ; K ϕϕ non-dimensional elastohydrodynamic stiffness in r–θ coordinate K xx ; K xy ; K yx ; K yy elastohydrodynamic stiffness in the x  y coordinate K xx ; K xy ; K yx ; K yy non-dimensional elastohydrodynamic stiffness in the x  y coordinate K h;d ; K h ; K h;ij non-dimensional elastohydrodynamic direct stiffness used in the distributed model of long WRLBs K rb;d ; K rb;exp non-dimensional overall rubber bearing stiffness using the distributed model of long WLRBs lbc effective length of the shaft at the top boundary in the experiment Ls equivalent length of the rubber block p1 ; p2 perturbed pressure p0 steady-state fluid pressure distribution ps supply pressure of the WLRB Sr0 original shape factor of the rubber block Ts equivalent thickness of the rubber block Ws total width of the rubber stave W r ; W t load components generated by fluid pressure W r ; W t non-dimensional load components generated by fluid pressure αr ; β r constants in the function of compliance operator αw ratio of the width of the rectangular rubber block to the width of the rubber stave ΔT r deformation displacement of the stave surface ε1 ; ϕ1 magnitude of infinitesimal perturbation ωp whirling frequency of the rotating shaft

Modeling and analysis of bearings have been long studied by many researchers; for instance see [1–7] and the references cited therein. Experimental methods to identify dynamic coefficients of bearings were reviewed by Tiwari [8] and Dimond [9]. Although the literature on the subject is rich, no experimental investigation on the dynamics stiffness coefficients of water-lubricated rubber bearings has been reported. Some recent experiments on WLRBs have mainly been focused on tribological properties [10–11] and circumferential pressure distribution [12–13]. By measuring the circumferential pressure distribution in a WLRB and comparing the experimental data with the CFD simulation results, Cabrera et al. [13] pointed out that water-lubricated rubber bearings operate under mixed lubrication. This indicates that both the fluid film lubrication and the contact between the rotating Please cite this article as: S. Liu, & B. Yang, A new model of water-lubricated rubber bearings for vibration analysis of flexible multistage rotor systems, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.03.052i

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Fig. 1. A typical water-lubricated rubber bearing: (a) 3D view and (b) cross-section view.

shaft and rubber bearing liners must be taken into consideration in modeling of WLRBs. Nevertheless, in the literature, no model of WLRBs has addressed the issue of mixed lubrication. Pai et al. [14–17] derived a model for water-lubricated journal bearings with multiple axial grooves. This model, however, does not consider the elastic deformation of rubber. On the other hand, a thin elastic liner model was used to examine the static and dynamic rubber deformation [18–20], but this model is only valid for compressible materials [21–22]. The rubber in WLRBs is an incompressible material with Poisson's ratio almost equal to 0.5. Hence, the previous results [14–20] cannot be directly used for WLRB modeling in the current investigation. A WLRB is different from other types of bearings in its large length-to-diameter ratio (often larger than two). Most previous investigations treat the bearing as pointwise springs and dampers. Obviously, these pointwise models cannot fully describe the spatial distributions of stiffness, damping and fluid pressure along the axis of a WLRB. In summary of previous studies, three open issues regarding modeling WLRBs are as follows. First, there exists a big gap between theoretical and experimental results. In fact, no theoretical model of WLRBs has ever been well validated by experiments, considering the mixed lubrication phenomenon of WLRBs. Second, most of previous works treat bearings as pointwise springs and dampers. WLRBs in flexible multistage rotating systems usually have a large length-to-diameter ratio, indicating a large area of application of fluid-film pressure on a rotating shaft. Therefore, standard pointwise bearing models are not sufficient enough to capture the physics of WLRB-supported rotors in vibration. Third, the existing models accounting for elastic deformation of bearings are only valid for compressible materials, and they are not applicable to incompressible materials like rubber used in WLRBs. In the current work, a new model of WLRBs is proposed to address the above-mentioned issues. Different from previous research, this bearing model describes mixed lubrication through combination of a fluid lubrication model and an elastic rubber model, and treats spatial distributions of stiffness and pressure of WLRBs through use of a viscoelastic foundation. For the first time in the literature, the WLRB model is validated in experiments. To this end, a distributed transfer function method (DTFM) is developed to model the vibration of flexible multistage rotor systems. With the DTFM, the transfer function between the unbalanced mass force and vibration displacement is used in this work to deliver accurate and closed form analytical solution of steady-state responses and eigensolutions of the rotor system. The predictions by the bearing model and DTFM formulation are in good agreement with the experimental results. The remainder of the paper is organized as follows. In Section 2, an experimental device is introduced to identify the dynamic stiffness coefficients of WLRBs. In Section 3, the theoretical model of WLRBs is presented. Numerical studies of the dynamic coefficients of WLRBs are presented in Section 4. And finally, the main results and contributions from this investigation are summarized in Section 5.

2. Experimental identification of dynamic stiffness of WLRBs The experimental identification of the dynamic stiffness coefficients of WLRBs is performed on a water-lubricated rubber bearing test device [23,24]. Fig. 2 shows two photos of the test device and Fig. 3 gives a schematic of the test device. The test device can host one water-lubricated rubber bearing with diameter ranging from 50.8 mm (2 in.) to 152.4 mm (6 in.) and for a length-to-diameter ratio of 2. In experiments, bearings with diameters 50.8 mm (2 in.) and 101.6 mm (4 in.) were tested. For these rubber bearings, the liner material is Nitrile; the rotating journal sleeves are made of polished stainless steel and they are mounted on a shaft that is adjustable to be concentric with the centerline of the rubber bearing. The lower end of the shaft is connected to a disk with four symmetric holes, which is used to introduce unbalanced mass to the rotor. To lubricate the WLRB, city water flows through the running clearance by injection to the water inlet and collection from the water outlet; see Fig. 3. Because the water inlet is lower than the water outlet, the bearing housing is always completely filled with water, making water fully functioning as a lubricant during the operation of the test device. Please cite this article as: S. Liu, & B. Yang, A new model of water-lubricated rubber bearings for vibration analysis of flexible multistage rotor systems, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.03.052i

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Fig. 2. Water-lubricated rubber bearing test device.

Fig. 3. Schematic of the water-lubricated rubber bearing test device.

The bearing housing is sealed by two face-type mechanical seals (modified type 5610Q from John Crane, see Fig. 2), whose radial forces applied onto the shaft are negligible. Four proximity probes (3300XL NSv proximity transducer system) that are mounted above and below the bearing housing are used to measure the transverse displacement of the rotating shaft in x and y directions. A data acquisition system records pressure and vibration measurements. Please cite this article as: S. Liu, & B. Yang, A new model of water-lubricated rubber bearings for vibration analysis of flexible multistage rotor systems, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.03.052i

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2.1. Approach In this research, the dynamic stiffness coefficients of the WLRB shown in Fig. 2 are obtained by matching unbalanced mass responses of the rotor system (test device) that are measured experimentally and predicted theoretically. In experiments, unbalanced responses of the rotor–bearing system in both time and frequency domains were recorded by the four proximity probes shown in Fig. 3. As mentioned previously, unbalanced mass is introduced by mounting a pair of bolt and nut in one of the holes of the disk (see Fig. 3). Theoretically, unbalanced mass response of a rotor system only consists of the frequency that equals to 1  running speed. However, from the experimental results, the recorded frequency response always shows higher harmonic frequencies, which may result from shaft misalignment. But, the amplitude of the higher harmonic response is significantly small, compared with that at the frequency equal to 1  running speed. Because the lower two proximity probes are closer to the unbalanced mass, the signal recorded by the lower proximity probes is much cleaner. So, the unbalanced mass response recorded by the lower proximity probes is used in the analysis. To obtain the pure 1  running speed frequency from the recorded response, the complex variable filtering method explained by Muszynska [25] was applied. The purpose of using complex variable filtering is to separate the frequency components contained in a rotor orbit into circular, forward and reverse frequency components. Following the procedure of the complex variable filtering method, the experimental 1  -filtered orbit can be expressed by
 u v

" ¼

a1

b1

a2

b2

#


cos Ωt

 (1)

sin Ωt

where u and v are the experimental vibration amplitudes in the x and y directions; a1 ; b1 ; a2 and b2 are the coefficients obtained from the complex variable filtering method; and Ω is the rotation speed of the shaft. In this effort, theoretical unbalanced mass response is derived through use of a distributed transfer function method (DTFM); see Appendix A. The DTFM is an analytical method for dynamic analysis and feedback control of multibody flexible dynamic systems. One obvious advantage of the DTFM over traditional finite element methods (FEM) is that it can deliver highly accurate solutions of dynamic problems without the need for discretization [26]. The DTFM has been successfully applied to complex flexible rotor systems before [27]. The DTFM formulation presented in this paper is an extension of the method for WLRB-supported rotor systems. In modeling long WLRBs, the deformation of the bearing rubber surface must be considered. Conventional pointwise bearing models cannot precisely represent the characteristics of rubber bearings of large length-to-diameter ratio. In the DTFM formulation, a shaft segment guided by a long WLRB is treated as the rotating beam segment supported by a viscoelastic foundation (Fig. 4), which yields a distributed bearing model. By the DTFM as described in Appendix A, the unbalanced mass response of a flexible rotor–bearing system can be written as


U V

"

 ¼

A1 A2

B1 B2

#


cos Ωt



sin Ωt

(2)

where U and V are the transverse vibration amplitudes of the shaft in x and y directions; A1 ; B1 ; A2 and B2 are coefficients computed by the DTFM. Finally, comparison of Eq. (1) and Eq. (2) yields a1 ¼ A1 ;

b1 ¼ B1 ;

a2 ¼ A2 ;

b2 ¼ B2

(3)

Solution of these equations eventually gives the stiffness coefficients of the bearing, namely, K xx ; K xy ; K yx and K yy . A flow chart of the related data processing procedure is shown in Fig. 5. It should be pointed out that damping also plays an important role in vibration of rotor systems. However, damping in a rotor system has little effect on the values of its whirl frequencies even though it significantly affects the decay rate of system response [28]. Because this effort is focused on dynamic stiffness of WLBRs under the influence of mixed lubrication, identification of damping coefficients of this type of bearings will be left as a topic of future investigation.

Fig. 4. Distributed model of long WLRB.

Please cite this article as: S. Liu, & B. Yang, A new model of water-lubricated rubber bearings for vibration analysis of flexible multistage rotor systems, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.03.052i

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Fig. 5. Experimental data processing procedure.

Fig. 6. Top end of the rotating shaft and its spring constraint model.

Fig. 7. Unbalanced mass response for the 101.6 mm rubber bearing.

2.2. End conditions of rotating shaft To obtain accurate results, the end conditions of the rotating shaft in the test device (see Fig. 3) need to be properly prescribed. At the bottom end of the shaft, the boundary conditions are those as a free end with a mounted disk. At the top Please cite this article as: S. Liu, & B. Yang, A new model of water-lubricated rubber bearings for vibration analysis of flexible multistage rotor systems, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.03.052i

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end, the shaft is connected to the motor by a nut and a key, as shown in Fig. 6. Due to the elastic properties of the nut and key, the shaft end is neither simply supported nor clamped. In this work, the top end of the rotating shaft is modeled as a hinged end that is constrained by translational and torsional springs. The constraint forces of the springs are of the form [29] f x ¼ kuu U þ kuψ

∂U ; ∂z

f y ¼ kvv V kvθ

∂V ∂z

(4)

where kuu ; kuψ ; kvv and kvθ are the equivalent spring coefficients. By Euler–Bernoulli beam theory, these equivalent spring coefficients are found as kuu ¼ kvv ¼

12EI lbc

3

;

kuψ ¼  kvθ ¼ 

6EI lbc

2

(5)

where E and I are the Young's modulus and moment of inertial of the shaft, respectively; lbc is the effective length of the shaft at the top boundary (see Fig. 6). For given geometric and material parameters of the nut and key, the lbc can be computed by the finite element method. The aforementioned end conditions of the rotating shaft will be implemented in the DTFM formulation for the WLRBsupported rotor system.

Fig. 8. Dynamic stiffness of the 101.6 mm rubber bearing (a) direct stiffness and (b) cross-coupled stiffness.

Please cite this article as: S. Liu, & B. Yang, A new model of water-lubricated rubber bearings for vibration analysis of flexible multistage rotor systems, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.03.052i

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2.3. Results As stated previously, WLRBs of two different sizes were used in experiments: a 101.6 mm rubber bearing and a 50.8 mm rubber bearing. The parameters of these bearings are given in Appendix B. In experiments, a constant supply pressure is maintained. The unbalanced mass response of the flexible rotor–bearing system is recorded in three disk-mounting cases: the rotating shaft carrying one disk, two disks, and three disks, with unbalanced mass of 36.3 g (0.08 lb), 43.1 g (0.095 lb) and 45.4 g (0.1 lb), respectively. The shaft rotation speed varies from 600 rev/min to 1600 rev/min. Due to the limitation of the data acquisition system, the maximum vibration amplitude that can be recorded is 0.4 mm (0.015 in.) and experimental data can only be obtained with the shaft rotation speed up to 1350 rev/min in the two-disk case, and 1150 rev/min in the three-disk case. The Reynolds number (Re ¼ ðρΩRC=μÞ, with ρ being fluid density) at rotation speed 600 rev/min and 1600 rev/min for the 101.6 mm rubber bearing is 767.8 and 2047.5, respectively. For 50.8 mm rubber bearing at rotation speed 600 rev/min and 1500 rev/min, the Reynolds number is 170.7 and 426.8, respectively. The experimental results on the larger (101.6 mm) rubber bearing are first presented. The unbalanced mass response of the rotor system in the three disk-mounting cases is shown in Fig. 7. where the 1  running speed and 2  running speed components of vibration are obtained separately through use of complex variable filtering method. As expected, the 1  running speed response increases as the rotation speed increases. The response at the 2  running speed, however, is seen to have peak amplitudes at 1050 rev/min, 950 rev/min and 850 rev/min for one-, two- and three-disk cases, respectively. Indeed, resonant vibration was observed at these speeds during the tests, which indicates that the natural frequencies of the rotor system equal the 2  running speeds at the peak response. It follows that the natural frequencies of the rotor system with one, two and three disks, are 35.00 Hz, 31.67 Hz and 28.33 Hz, respectively. Because the amplitude of the 1  running Table 1 Theoretical and experimental natural frequency for 101.6 mm rubber bearing Cases

Theoretical natural frequency (f theo , Hz)

Experimental natural frequency (f exp , Hz)

  f  f  Error  expf theo  (%) exp

One disk at 1050 rev/min Two disks at 950 rev/min Three disks at 850 rev/min

32.77 30.70 29.27

35.00 31.67 28.33

6.3 3.1 3.3

Fig. 9. Comparison of direct stiffness for 50.8 mm and 101.6 mm rubber bearing.

Table 2 Theoretical and experimental natural frequency for the 50.8 mm rubber bearing Cases

Theoretical natural frequency (f theo , Hz)

Experimental natural frequency (f exp , Hz)

  f  f  Error  expf theo  (%) exp

One disk at 1050 rev/min

32.24

35.00

7.9

Please cite this article as: S. Liu, & B. Yang, A new model of water-lubricated rubber bearings for vibration analysis of flexible multistage rotor systems, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.03.052i

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speed response is much larger than that of the 2  running speed response, the unbalanced mass response of the rotor system is almost the same as the 1  running speed response. The dynamic stiffness coefficients of the rubber bearing are identified by Eqs. (1)–(3). Plotted in Fig. 8 are the dynamic stiffness coefficients of the 101.6 mm rubber bearing versus the shaft rotation speed. As can be seen from the figure, K xx ¼ K yy and K xy   K yx at most points of measurement. Fig. 8a shows that the direct stiffness coefficients change in an approximately parabolic manner as the rotation speed increases. Furthermore, Fig. 8a and Fig. 7 suggest that, as the rotation speed of the shaft increases, so does its vibration amplitude, which causes more contacts between the rotating shaft and the rubber staves. However, Fig. 8b does not show obvious correlation between the cross-coupled stiffness and the rotation speed. As stated by Vance et al. [30], the cross-coupled dynamic stiffness coefficients that produce forces tangential to the whirl orbit have little effect on system natural frequencies, but they affect stability and amplification factors at a critical speed. As such, the natural frequencies of the rotor system can be predicted with or without the cross-coupled stiffness coefficients. The natural frequencies of the rotor system can be predicted by using the determined dynamic stiffness coefficients with the DTFM. In Table 1, the theoretical predictions are compared with the experimentally-determined natural frequencies in the three disk-mounting cases. It is seen that the theoretical and experimental natural frequencies are in good agreement. Now the results on the smaller (50.8 mm) rubber bearing are presented. The unbalanced mass response of the rotor system with this bearing is only recorded in the case of the shaft carrying one disk. All other conditions are the same as for the larger (101.6 mm) bearing. Fig. 9 compares the direct stiffness parameters of the two bearings, which show the similar parabolic patterns as the shaft rotation speed increases. Also, Table 2 lists the results on the first natural frequency of the rotor system with the 50.8 mm rubber bearing. It is seen from the above-mentioned results on the 50.8 mm and 101.6 mm bearings that the proposed experimental method can give fairly accurate prediction of dynamic stiffness of WLRBs.

3. A model of water-lubricated rubber bearings in mixed lubrication In a post-test examination, several polished areas on the rotating shaft were found in the regions that are covered by the inner surface of the WLRB; see Fig. 10. These polished areas indicate that during the experiments on the test stand, the rotating shaft and the rubber bearing were in contact with each other from time to time, which is in line with the prediction by Cabrera et al. [13]. It is therefore reasonable to assume that the WLRB operates under the influence of mixed lubrication, which involves the interactions among the shaft vibration, the elastic deformation of rubber material and the fluid film pressure by the lubricant (water in the current case). In other words, under mixed lubrication, the contact between the rotating shaft and the surface of the bearing causes elastic deformation of the rubber staves. Although mixed lubrication occurs in a WLRB, it is difficult to know when and where the shaft is in touch with the rubber staves of the bearing because the vibration of the shaft is unknown in the first place. Also, the shaft and bearing do not engage in full contact all the time, indicating that the shaft–bearing interaction is nonlinear in nature. In this work, investigation on mixed lubrication in WLRBs takes two steps (i) study the effects of boundary lubrication and elastohydrodynamic lubrication separately and (ii) study the mixed lubrication through combination of the effects of boundary lubrication and elastohydrodynamic lubrication.

Fig. 10. The rotating shaft with polished areas (in squares).

Please cite this article as: S. Liu, & B. Yang, A new model of water-lubricated rubber bearings for vibration analysis of flexible multistage rotor systems, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.03.052i

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3.1. Boundary lubrication In boundary lubrication, it is assumed the rotating shaft is in full contact with the surface of the rubber bearing. Because small amount of fluid always exists between the asperities of the contact surfaces, the shaft rotation speed has little effect on the stiffness in boundary lubrication. As such, the dynamic stiffness of the WLBR bearing in boundary lubrication is only related to the stiffness of rubber staves of the bearing that are in elastic deformation. To derive a simple analytical expression for the above-mentioned stiffness, a rubber stave is approximated as a rectangular rubber block as shown in Fig. 11, where the length and height of the rectangular block are the same as those of the stave. Also, it is assumed that the shaft is only in touch with the land region of the rubber block (the slightly-curved top area of the stave in Fig. 11a), which is 38% of the total width W s of the stave. Matching the stiffness of the stave computed by FEM to that of the rectangular rubber block gives the effective width of the block as half of W s ; see Fig. 11b. Under the assumptions, the shape factor method, which was first developed by Gent [31] and Lindley [32] and later on documented by Hill [33], is applied to obtain the stiffness K r of the rectangular rubber block as follows: ! ðαw W s ÞLs Er C r S2r0 Erh (6) þ Kr ¼ Ts ð1 er Þ3 ð1  er Þ In the previous equation, αw is the percentage width of the rectangular block compared to the rubber stave (namely, αw ¼ 0:5 for the tested bearings); Ls is the length of the rubber block; Er is the linear Young's modulus; and C r is a constant given by
 4 ð αw W s Þ 11 αw W s Cr ¼ þ 2 (7) 3 Ls 10 Ls Sr0 is the original shape factor of the rubber stave, which is Sr0 ¼

Ls ðαw W s Þ 2ðαw W s þ Ls ÞT s

Erh is the ‘homogenous’ compression modulus defined by 8 !2 9 < 1 L2s  ðαw W s Þ2 = Erh ¼ Er 1 þ : 3 L2s þ ðαw W s Þ2 ;

(8)

(9)

and er is the applied strain, which is er ¼

ΔT s Ts

(10)

with T s being the thickness of the stave and ΔT s the deformation displacement of the stave surface. For small applied strain (say er o2%), er in Eq. (6) is negligible, rubber thus deforms linearly [34], and the stiffness of rubber staves is expressed by o ðαw W s ÞLs n (11) Kr ¼ Erh þ Er C r S2r0 Ts The stiffness of rubber staves given by Eq. (6) is compared with that computed by the nonlinear FEM software Abaqus. To this end, an FEM model for use in Abaqus is built; see Fig. 12, where the entire rubber body of the bearing is considered. In simulation, Neo-Hookean model for rubber material is used, a constant pressure is applied on the half shaft surface, and the bottom of the shaft is in full contact with the rubber stave. The stiffness of rubber staves versus applied strain for the two bearings used in the experiments is plotted in Fig. 13, where the solid lines represent the Abaqus results and the dashed lines are for the predictions by Eq. (6). It is seen from the figure that the stiffness by Abaqus is always larger than that given by Eq. (6). This is due to the fact that the entire body of the rubber bearing is engaged in elastic deformation in the FEM model. If the FEM results are used as reference solutions, the maximum error in the stiffness predictions by Eq. (6) is around

Fig. 11. Rubber bearing in boundary lubrication: (a) a stave of WLRBs and (b) rectangular rubber block as approximation.

Please cite this article as: S. Liu, & B. Yang, A new model of water-lubricated rubber bearings for vibration analysis of flexible multistage rotor systems, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.03.052i

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Fig. 12. Model used in Abaqus–Contact.

Fig. 13. Stiffness of rubber staves in boundary lubrication.

3.5%, for both the bearings. As such, the analytical expression given in Eq. (6) can be used to accurately predict the stiffness of rubber staves of WLRBs in boundary lubrication.

3.2. Elastohydrodynamic lubrication In elastohydrodynamic lubrication, the rotating shaft is always separated from the rubber surface by the fluid. In this research, the following assumptions about elastohydrodynamic lubrication are made: (i) the fluid is Newtonian fluid with constant viscosity; (ii) inertial and body force terms are negligible compared to the viscous term; (iii) the variation of pressure across the film thickness is negligibly small; and (iv) the flow is laminar. Under these assumptions, the elastohydrodynamic lubrication can be described by Reynolds equation [15]
 2 ∂ ∂h ∂h 3 ∂p 3∂ p h þh 2 ¼ 6μωR þ12μ ∂x ∂x ∂x ∂t ∂z

(12)

where p is the distribution of the fluid pressure; x ¼ Rθ is the circumferential coordinate   in the direction of rotation as shown in Fig. 14a; ω is the shaft rotation speed; μ is the fluid viscosity; and h ¼ C þe cos θ þ HðpÞ is the fluid film thickness, with C being the radial clearance, e the eccentricity, and HðpÞ the rubber deformation caused by the fluid pressure. Please cite this article as: S. Liu, & B. Yang, A new model of water-lubricated rubber bearings for vibration analysis of flexible multistage rotor systems, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.03.052i

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Fig. 14. Coordinates in elastohydrodynamic lubrication.

By introducing dimensionless parameters x R

p ps

h C

z L

θ ¼ ; p0 ¼ ; τ ¼ ωp t; h0 ¼ ; z0 ¼ ; ξ ¼

D L

where ps and ωp are the supply pressure and whirling frequency, and L and D are the length and diameter of the bearing, Eq. (12) is reduced to 0 0

0 0 ∂ 1 2 03 ∂2 p ∂h ∂h 03 ∂p þ ξ h ¼ Λ þ 2Λλ h (13) 02 4 ∂τ ∂z ∂θ ∂θ ∂θ Furthermore, dropping the prime in Eq. (13) yields the dimensionless Reynolds equation

 ∂ 1 2 3 ∂2 p ∂h ∂h 3 ∂p h þ ξ h ¼ Λ þ 2Λλ 2 4 ∂τ ∂z ∂θ ∂θ ∂θ

(14)

where bearing number   ω 6μω Λ ¼ h  2 i; λ ¼ p ; h ¼ 1 þ ε cos θ þ hðpÞ C ω ps

R

with hðpÞ ¼ ðH ðpÞ=CÞ. With the same token, non-dimensional boundary conditions for the rubber bearing can be written as (i) at z ¼ 0 andz ¼ 1; p ¼ 0 at staves; (ii) in the region 0 r z r1; p ¼ 1 at grooves; and (iii) in the cavitation zone, Reynolds boundary conditions are used and they are p ¼ 0 and ∂∂pθ ¼ 0. 3.2.1. Determination of hðpÞ by compliance operator method To solve Eq. (14), the elastic deformation hðpÞ of the rubber caused by the fluid pressure must be known in advance. As stated in Section 1, the available models in the literature do not deal with hðpÞ for rubber material. In this research, a model of the pressure-induced rubber deformation is proposed. Two methods are used to obtain hðpÞ: the compliance operator method and the finite element method (FEM). In the compliance operator method, a compliance operator (η) is used to relate the rubber deformation to the fluid pressure, namely hðpÞ ¼ ηp

(15)

where η ¼ ðΓ ps =CÞ in Eq. (14) with Γ being the dimensional representation of η. Because a compliance operator can be easily implemented in Reynolds equation, the compliance operator method has been popular in studies on relevant topics [18–21]. In this investigation, the compliance operator method is applied to determine the rubber deformation. A relationship between rubber deformation and applied strain is first determined. To this end an Abaqus model of the rubber bearing body is built; see Fig. 15, where a varied pressure (p) is applied at one land region AL and a constant supplied pressure (ps ) is applied elsewhere in the bearing. The FEM simulation for bearings with different diameters, different length and thickness results in plots of the compliance operator η against the strain er of the land region of the bearing; see the solid lines in Fig. 16. In the figure, the compliance operator, which is determined by Eq. (15), is equal to ðH ðpÞ=T r Þ, and the strain equals to ðhðpÞ=pÞ. Please cite this article as: S. Liu, & B. Yang, A new model of water-lubricated rubber bearings for vibration analysis of flexible multistage rotor systems, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.03.052i

S. Liu, B. Yang / Journal of Sound and Vibration ] (]]]]) ]]]–]]]

13

Fig. 15. Bearing model used in Abaqus – compliance operator.

Fig. 16. Compliance operator vs. applied strain.

Two observations can be made from the FEM simulation results: (i) operator η only depends on the strain er of the land region of the rubber bearing and (ii) operator η can be approximately expressed by an exponential function   (16) η ¼ αr 1 e  βr er A typical range of strain in WLRB applications is 0:2%r er r0:5%. To obtain an accurate formula of η in this strain range, parameters αr and βr in Eq. (16) are determined in the following two cases: Case I: At er ¼ 0:2%, η can be approximated by Winkler0 s hypothesis [20,22] as   2 ps T r 1  12υ υ (17) η¼ Er C where υ is Poisson0 s ratio of the rubber material. Case II: At er ¼ 0:6%, the compliance operator is approximated by

η¼

ps AL KrC

(18)

where AL is the area of the land region shown in Fig. 15, Cis the radial clearance of the rubber bearing, and K r is the stiffness of rubber staves given by Eq. (11). Eq. (18) describes the rubber stave as a linear spring of coefficient K r . The dimensional pA  L compliance operator is Γ ¼ HpðpÞ ¼ Kpr ¼ KALr : The dimensionless representation of η ¼ ðΓ ps =CÞ thus yields Eq. (18). Hence, matching the compliance operator by Eq. (18) to that obtained by Abaqus gives er ¼ 0:6%. With Eqs. (17) and (18), parameters αr and β r in the exponential expression in Eq. (16) can be obtained. The abovedescribed approach to the determination of η shall be called the two-point method. The dashed lines in Fig. 16 are the Please cite this article as: S. Liu, & B. Yang, A new model of water-lubricated rubber bearings for vibration analysis of flexible multistage rotor systems, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.03.052i

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Fig. 17. Compliance operator by multi-region exponential approximation.

results predicted by the two-point method. It is seen from the figure that the two-point method has a maximum deviation of 5% from the Abaqus results for 0% rer r 0:5%, and a maximum deviation of 10% for 0:5% o er r2%. Because a typical range of strain for WLRBs is 0:2% r er r0:5%, the two-point method proposed herein gives fairly accurate results without having to rely on computationally intensive numerical methods. The accuracy of analytical estimation of η can be further improved by a method of multi-region exponential approximation. According to the FEM (Abaqus) simulation results, the compliance operator can be described in multiple regions of strain. Take the 101.6-mm (4-inch) rubber bearing with length 203.2 mm (8 in.) for example. Select five pairs of parameters ðη; er ) from the Abaqus data: (0.2%, 0.078), (0.4%, 0.11), (0.8%, 0.136), (1.6%, 0.155) and (3.2%, 0.162). Application of the two-point method to these pairs of parameters yields the following multi-region function for η:   8 er > for 0%o er r 0:4% 0:1323 1  e0:002 lnð0:4103Þ > > >   > > er > ð Þ ln 0:2364 > for 0:4%o er r 0:8% < 0:1441 1  e0:004   η¼ er > > 0:1581 1  e0:008 lnð0:1397Þ for 0:8%o er r 1:6% > > >   > > er > : for e 4 1:6% 0:1623 1  e0:016 lnð0:0452Þ

(19)

r

As shown in Fig. 17, the multi-region exponential approximation has a maximum deviation of 2% from the Abaqus results for er o 9%. For the strain range of 0:2% rer r 0:5% in WLRB applications, the multi-region exponential approximation has a maximum deviation of 1.5%. It should be pointed out that, in the two-point method, Winkler0 s model is not valid for υ ¼0.5. The aforementioned multi-region exponential approximation, however, is applicable to any values of Poisson0 s ratio. In fact, even the two-point method does not have to rely on Winkler0 s hypothesis if Eq. (17) is replaced by a relation that is obtained by matching the FEM result at er ¼ 0:2%. Therefore, the proposed formulas for estimation of η are accurate enough for WLRB applications, and are applicable to both compressible and incompressible materials. In the above analysis, uniform pressure distribution is applied at the land region to determine the compliance operator shown in Fig. 16. However, for a WLRB, the pressure distribution on the land region is typically parabolic or alike. To check the accuracy of η predicted by the two-point method, a parabolic pressure distribution shown in Fig. 18(a), is applied at one of the land regions and the fourth land region (θ7 r θ r θ8 in Fig. 14(b)). With the parabolic pressure distribution, the dimensionless elastic deformation of the rubber bearing by the two-point method is obtained by hðpÞ ¼ ηp, where η is given in Eq. (16). Fig. 18(b) compares the results on the distribution of the elastic deformation of the rubber bearing along the annual coordinate θ (see Fig. 14), where the solid line is the prediction by the two-point method with a uniform distribution of equivalent resultant force, and the dashed line is the result by the FEM (Abaqus) with the parabolic pressure distribution given in Fig. 18(a). According to the figure, the maximum deviation of the result by the two-point method from the FEM result is 3.5%. This comparison result indicates that the two-point method is accurate enough to describe the actual rubber deformation. The accuracy can be understood by the fact that the pressure variation is small due to the small angle θL in Fig. 14. Please cite this article as: S. Liu, & B. Yang, A new model of water-lubricated rubber bearings for vibration analysis of flexible multistage rotor systems, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.03.052i

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15

Fig. 18. Rubber deformation for parabolic-shape pressure distribution (a) parabolic distribution of pressure and (b) comparison of rubber deformation.

3.2.2. Steady state and dynamic characteristics With the hðpÞ obtained in the previous section, the steady-state solution of the elastohydrodynamic lubrication can be determined. Drop the time-dependent term in Eq. (14), to have the steady-state characteristic equation

 ∂ 1 2 3 ∂2 p ∂h 3 ∂p h (20) þ ξ h ¼Λ 2 4 ∂z ∂θ ∂θ ∂θ By solving Eq. (20) for the steady-state pressure p0 , the elastohydrodynamic forces are determined by the following integrals:   X Z 1 Z θi þ 1 Wr n Wr ¼ ¼ p0 cos θ  ϕ0 dθdz LDps i ¼ 1;3;:::15 0 θi   X Z 1 Z θi þ 1 Wt n Wt ¼ ¼ p0 sin θ  ϕ0 dθdz (21) LDps i ¼ 1;3;:::15 0 θi where W r and W t are the elastohydrodynamic forces in radial and circumferential directions, respectively; ϕ0 is the attitude angle at steady state (Fig. 14a); and θi is shown in Fig. 14b. The load capacity and steady-state attitude angle ϕ0 are thus given by ! qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Wt 2 2 (22) W ¼ W r þ W t ; ϕ0 ¼ tan  1  Wr In this work, both infinitesimal perturbation method [15,35] and finite perturbation method [35–37] are used to study the elastohydrodynamic coefficients (stiffness (K h ) and damping coefficients (C h )) of water-lubricated rubber bearings, which are produced by the elastohydrodynamic lubrication. 3.2.2.1. Infinitesimal perturbation method (IPM). For small amplitude whirl motions of frequency ωp about the steady-state position (ε0 ; ϕ0 ) of the rotating shaft, the pressure and film thickness can be written as p ¼ p0 þ ε1 eiτ p1 þ ε0 ϕ1 eiτ p2

(23)

  h ¼ Reðh0 þeiτ ε1 cos θ þ ε0 ϕ1 sin θ þ ηðε1 eiτ p1 þ ε0 ϕ1 eiτ p2 ÞÞ (24)       where jε1 j 5 ε0 ; ϕ1 5 ϕ0 ; h0 ¼ 1 þ ε0 cos θ þ ηp0 ; with h0 and p0 being the steady-state fluid film thickness and pressure distribution, respectively. The steady-state eccentricity (ε0 ) is equivalent to the steady-state vibration amplitude at WLRBs of the rotor system (test device) with an unbalanced mass, which can be determined by the DTFM method with the experimental dynamic coefficients of WLRBs in Section 2. Substitute Eqs. (23) and (24) into (14) and retaining up to first linear terms, to obtain the following equations:

ε0 : ε1 eiτ : h30

∂2 p1 ∂θ

2

2 3 ∂ p0 2 ∂h0 þ 3h0 2 ∂

h0

∂θ

∂p0 dh0 1 2 3 ∂2 p0 Λ þ ξ h0 ¼0 ∂z2 θ ∂θ dθ 4

(25)

1 2 3 ∂ 2 p1 ∂p ∂2 p 2 þ ξ h0  ηΛ 1  2iηΛλp1 þ 3h0 cos θ 20 2 ∂z ∂θ ∂θ 4 ∂θ ∂θ

2 ∂h0 ∂p1

þ 3h0

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16

 3h0 sin θ 2

∂p0 ∂h0 ∂p0 3 2 2 ∂2 p þ 6h0 cos θ þ ξ h0 cos θ 20 þ Λ sin θ 2iΛλ cos θ ¼ 0 ∂θ ∂θ ∂θ 4 ∂z

∂p2 1 2 3 ∂2 p2 ∂p ∂2 p 2 þ ξ h0  ηΛ 2 2iηΛλp2 þ3h0 sin θ 20 2 4 ∂z ∂ θ ∂ θ ∂ θ ∂θ ∂θ ∂p0 ∂h0 ∂p0 3 2 2 ∂ 2 p0 2 þ 3h0 cos θ þ6h0 cos θ þ ξ h0 sin θ 2  Λ cos θ 2iΛλ sin θ ¼ 0 ∂z ∂θ ∂θ ∂θ 4

ε0 ϕ1 eiτ : h30

∂2 p2 2

(26)

2 ∂h0

þ 3h0

(27)

Eqs. (25)–(27) are solved via a finite difference method. To this end, the previous differential equations are firstly written in the standard finite difference form, in which a rectangular grid with 20 elements between θi and θj (see Fig. 14b) and 30 elements in the z direction is applied to each land region as shown in Fig. 15. The resulting finite difference equations, which must satisfy the boundary conditions, are then determined numerically via the successive over relaxation (SOR) scheme in the rectangular grid. To use this method, the pressure distribution is determined via an iterative process, with the initial pressure at all grid points set to zero. In each iteration step, the pressure at every grid point is modified by an over relaxation factor (say, the value of the factor can be chosen as 1.7). The solution is reached when the convergence criteria P  P P    pni;jþ 1  pni;j = pni;jþ 1  o 10  6 is met. More detailed description of the solution procedure for bearings with multiple grooves is seen in Ref. [15]. With the perturbed pressure solutions p1 and p2 , the stiffness and damping coefficients of the rubber bearing are expressed as follows: ! ! Z 1 Z 2π Z 1 Z 2π K rr ¼  Re p1 cos θdθdz ; K ϕr ¼  Re p1 sin θdθdz 0

0

Z

K rϕ ¼ Re Z

0

Z 2π 0

0

Z

C rϕ ¼  Im

Z 2π

0 1

C rr ¼ Im

1

0

1

!

0

Z

p2 cos θdθdz ; K ϕϕ ¼  Re

p1 cos θdθdz =λ; C ϕr ¼  Im

Z 2π

0

0

!

1

Z 2π 0

0

Z

1

Z 2π 0

0

! p2 cos θdθdz =λ; C ϕϕ ¼  Im

Z

1 0

(28)

! p1 sin θdθdz =λ

Z 2π 0

! p2 sin θdθdz

! p2 sin θdθdz =λ

(29)

where K ij ¼ ðK ij C=LDps Þ and C ij ¼ ðC ij C ω=LDps Þ. By coordinate transformation, the stiffness and damping coefficients of the bearing in the xy coordinate system are given by [6] 1 ! !0 ! K rr K rϕ K xx K xy cos ϕ cos ϕ  sin ϕ sin ϕ @ A ¼ sin ϕ cos ϕ  sin ϕ cos ϕ K yx K yy K ϕr K ϕϕ 1 ! !0 ! C rr C rϕ C xx C xy cos ϕ sin ϕ cos ϕ  sin ϕ @ A ¼ (30) sin ϕ cos ϕ  sin ϕ cos ϕ C yx C yy C ϕr C ϕϕ where ϕ is the bearing attitude angle as shown in Fig. 14a. 3.2.2.2. Finite perturbation method (FPM). In finite perturbation, the four linear elastohydrodynamic stiffness coefficients of the rubber bearing are evaluated via a small displacement perturbation around the equilibrium position (ε0 ; ϕ0 ) in both positive and negative x and y directions, which is K xx ¼ 

ΔF y ΔF y ΔF x ΔF x ;K ¼  ;K ¼  ;K ¼  Δx yy Δy xy Δy yx Δx

(31)

Based on the results from Ref. [37], the perturbation values are chosen as Δx ¼ Δy ¼ 0:002 C, with C being the bearing radial clearance. The dimensionless displacement perturbations are then Δx ¼ Δy ¼ 0:002. Therefore, using the four points ( 7 Δx; 7 Δy) around the equilibrium point, the dimensionless stiffness coefficients are estimated as below (i) Perturbation in the positive and negative x-direction K xx ¼ 

F x; þ  F x;  ; 2Δx

K yx ¼ 

F y; þ  F y;  2 Δx

(32)

K xy ¼ 

F x; þ  F x;  2 Δy

(33)

(ii) Perturbation in the positive and negative y-direction K yy ¼ 

F y; þ  F y;  ; 2Δy

Please cite this article as: S. Liu, & B. Yang, A new model of water-lubricated rubber bearings for vibration analysis of flexible multistage rotor systems, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.03.052i

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The loads F x ; F y are given by Fx Fy

! ¼

cos ϕ sin ϕ

 sin ϕ cos ϕ

!

Wr

17

!

Wt

(34)

where ϕ is the bearing attitude angle as shown in Fig. 14; and W r and W t are given in Eq. (21). Application of the finite perturbation method takes the following steps: (i) Determine the steady-state hydrodynamic forces (W r ; W t ) and attitude angle (ϕ).   (ii) Determine the eccentricity in x and y directions: εx;0 ¼ ε0 cos ϕ ; εy;0 ¼ ε0 sin ðϕÞ. (iii) Apply a finite perturbation on the eccentricity εx ¼ εx;0 7 Δx; εy ¼ εy;0 7 Δy. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ε (iv) Update eccentricity and attitude angle after perturbation ε ¼ ε2x þ ε2y ; ϕ ¼ tan  1 εyx .

(v) Solve Eq. (20) for the steady-state pressure and compute the hydrodynamic forces (W r ; W t ) by Eq. (21), with updated ε and ϕ. (vi) Evaluate the dynamic stiffness coefficients using Eqs. (32)–(34).

Through use of the aforementioned formulas, the elastohydrodynamic coefficients (stiffness (K ij ) and damping coefficients (C ij )) of the two WLRBs that are used in the experiments are computed. The dynamic stiffness coefficients of the 101.6 mm (4-inch) rubber bearing are computed by both the infinitesimal perturbation method (IPM) and the finite perturbation method (FPM), and they are plotted against the shaft rotation speed in Fig. 19. As can be seen from the figure,

Fig. 19. Elastohydrodynamic coefficients of the 101.6 mm rubber bearing (a) direct stiffness; (b) cross-coupled stiffness; (c) direct damping coefficients; and (d) cross-coupled damping coefficients.

Please cite this article as: S. Liu, & B. Yang, A new model of water-lubricated rubber bearings for vibration analysis of flexible multistage rotor systems, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.03.052i

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Fig. 20. Elastohydrodynamic coefficients of the 50.8 mm rubber bearing (a) direct stiffness; (b) cross-coupled stiffness; (c) direct damping coefficients; and (d) cross-coupled damping coefficients.

the cross-coupled dynamic stiffness coefficients (K xy ; K yx ) obtained by the IPM perfectly match those by the FPM. However, the direct dynamic stiffness coefficients (K xx ; K yy ) obtained by the IPM are larger than those by the FPM, especially at higher rotation speeds. Because the accuracy of FPM is closely related to the perturbed values (Δx; Δy) [37], the results obtained by the IPM seem more trustworthy. It is with this argument that the dynamic coefficients of the 50.8 mm (2-inch) rubber bearing are computed by the IPM; see Fig. 20.

3.2.3. Implementation of finite element method in solution of steady-state pressure As mentioned in Section 3.2.1, two methods can be used to determine the fluid pressure-induced rubber deformation hðpÞ: the finite element method (FEM) and the compliance operator method. In Section 3.2.2, the compliance operator method is implemented in the infinitesimal and finite perturbation method to study the steady and dynamic characteristics of WLRBs. Although computationally intensive and time consuming, the FEM delivers more accurate results. In this section, the finite element method is implemented in the infinitesimal perturbation method (IFM), to obtain steady-state pressure distribution of WLRBs and to determine the fluid pressure-induced rubber deformation. In an iterative solution procedure, the pressure distribution is computed by MATLAB codes and the nonlinear FEM software Abaqus determines the rubber deformation; see Fig. 21. The above-mentioned iterative FEM solution method is used to compute the fluid pressure distribution of the 101.6 mm rubber bearing with ε0 ¼ 0:6 and ω ¼ 800 rev/min; see Fig. 22. The θi 0 s in Fig. 22 are corresponding to the coordinate shown in Fig. 14(b). For comparison, the pressure distribution that is computed by the compliance operator method is also included in the figure, which is in good agreement with the FEM. This confirms that the compliance operator method described in Section 3.2.1 can accurately predict fluid pressure-induced rubber deformation. Please cite this article as: S. Liu, & B. Yang, A new model of water-lubricated rubber bearings for vibration analysis of flexible multistage rotor systems, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.03.052i

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Fig. 21. Procedure to determine steady-state pressure by implementing FEM.

Fig. 22. Circumferential pressure distribution of the 101.6 mm bearing with ε0 ¼ 0:6 at ω ¼ 800 rev/min.

3.3. Mixed lubrication via the α-method The mixed lubrication of a WLRB involves both boundary lubrication (Section 3.1) and elastohydrodynamic lubrication (Section 3.2). Physically, the rotating shaft and the rubber bearing are in direct contact sometimes, but they are separated by the fluid at other times. Obviously, boundary and elastohydrodynamic lubrications cannot coexist at any particular time. However, in steady-state vibration, the relative occurrences of these two types of lubrications are in a relation. Hence, estimation of the overall dynamic coefficients for the WLRB should consider proper combination of the elastohydrodynamic Please cite this article as: S. Liu, & B. Yang, A new model of water-lubricated rubber bearings for vibration analysis of flexible multistage rotor systems, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.03.052i

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coefficients and stiffness of the rubber stave. It with this understanding that the α-method is proposed as follows: K rb;d ¼ α U K r;d þð1  αÞ U K h;d

(35)

where K rb;d is the overall dynamic stiffness of the WLRB, K r;d is the stiffness of rubber staves which can be determined by Eq. (11); K h;d is the fluid film stiffness given by Eq. (30) or Eqs. (32) and (33); and α is a non-dimensional number between 0 and 1 representing the percentage contribution of the stiffness of rubber staves in boundary lubrication to the overall rubber bearing stiffness in steady-state vibration of the rotor system. The α-method can be illustrated by the distributed model of long bearings in Fig. 23, where the overall rubber bearing stiffness is a combination of the stiffness of rubber staves and elastohydrodynamic stiffness. The value of α can be determined experimentally as follows:

αexp ¼

K rb;exp  K h;d K r;d K h;d

(36)

where K rb;exp is the experimental dynamic stiffness of the WLRB as presented in Section 2. Shown in Fig. 24 are the plots of α versus the shaft rotation speed for the two WLRBs used in the experiments, where the experimental value of α is obtained by Eq. (36), and the theoretical value of a is given by the non-dimensional formula  1:8  1:15 6 μω ε0:25 DL ps α¼ C 3:9

(37)

R

Eq. (37) is devised based on following properties of α: (i) α is a non-dimensional value between 0 and 1; and (ii) α depends on bearing parameters (i.e. L; D) and operation parameters (i.e. μ; ω; ps ; ε; C). All the dimensionless terms in Eq. (37), i.e. μω D C ps ; ε; L and R, are chosen from the non-dimensional Reynolds equation, Eq. (14). Based on the experimental results, the values of the indices in Eq. (37) are obtained by curve fitting. Fig. 24 shows the plots of α versus the shaft rotation speed, which are obtained by Eqs. (36) and (37), respectively. The maximum difference between the theoretical and experimental plots is 8%. With the theoretical α value, the theoretical rubber bearing stiffness is determined by Eq. (35). The comparison of theoretical and experimental rubber bearing stiffness is shown in Fig. 25. And the maximum deviation is 8%. It is seen from Figs. 24 and 25 that the theoretical dynamic stiffness matches the experimental dynamic stiffness very well for both the 50.8 mm (2-inch) and 101.6 mm (4-inch) rubber bearings. As a summary of the new WLRB model developed in this work, a flowchart of determination of rubber bearing dynamic coefficients is given in Fig. 26.

Fig. 23. Mixed lubrication of long WLRBs.

Fig. 24. Comparison of theoretical and experimental values of α.

Please cite this article as: S. Liu, & B. Yang, A new model of water-lubricated rubber bearings for vibration analysis of flexible multistage rotor systems, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.03.052i

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21

Fig. 25. Comparison of theoretical and experimental direct stiffness.

Fig. 26. Procedure for determination of dynamic coefficients of WLRBs.

Please cite this article as: S. Liu, & B. Yang, A new model of water-lubricated rubber bearings for vibration analysis of flexible multistage rotor systems, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.03.052i

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4. Parameter study on dynamic coefficients of WLRBs The new model of water-lubricated rubber bearings obtained from this effort is useful in bearing design for WLRBsupported rotor systems. In this section, the effects of eccentricity (ε), land region angle (θL ), and number of grooves on the dynamic coefficients of WLRBs are examined. Unless otherwise specified, the bearing parameters used in numerical studies are given in Table 3, which are the same as those of the 101.6 mm (4 in.) rubber bearing.

Table 3 Bearing geometry and operation parameters in numerical studies Parameter

Value

Parameter

Value

D/L C/R Stave thickness (Ts) Stave width (Ws)

0.5 0.0047 9.52 mm 41.28 mm

Stave length (Ls) Land region angle (θL) Rubber Young's modulus (Er) Poisson's ratio (υ)

203.2 mm 191 7,840,000 N m  2 0.47

Fig. 27. Elastohydrodynamic coefficients with respect to Λ for various ε (a) direct stiffness; (b) cross-coupled stiffness; (c) direct damping coefficients; and (d) cross-coupled damping coefficients.

Please cite this article as: S. Liu, & B. Yang, A new model of water-lubricated rubber bearings for vibration analysis of flexible multistage rotor systems, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.03.052i

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23

4.1. Effect of eccentricity Because the eccentricity does not affect the rubber property, the stiffness of rubber staves and the compliance operator are same as shown in Sections 3.1 and 3.2.1, which are given by K r ¼ 1:21   er η ¼ 0:142 1 e0:002 lnð0:4507Þ

(38)

The elastohydrodynamic coefficients are determined by the infinitesimal perturbation method (IPM). Fig. 27 shows the elastohydrodynamic coefficients with respect to Λ for different values of ε. Here Λ ¼ ð6μω=½ps ðC=RÞ2 Þ is a non-dimensional parameter, which is given in the dimensionless Reynolds equation Eq. (14). It is seen from the figure that the elastohydrodynamic coefficients generally increase as the eccentricity increases. However, with high eccentricity (say ε ¼ 0:6), the elastohydrodynamic direct stiffness (K xx ; K yy ) and cross-coupled damping coefficients (C xy ; C yx ) increase at low bearing number (Λ) and decrease at high bearing number (Λ). Also, by the α-method, the overall dynamic stiffness coefficients of the WLRB are plotted in Fig. 28. It is easy to see from Fig. 28 that the stiffness of rubber staves is much larger than the elastohydrodynamic direct stiffness. 4.2. Effect of land region angle In this study, the eccentricity is chosen as ε0 ¼ 0:5. By the shape factor method and the software Abaqus, the stiffness of rubber staves for WLRBs with different land region angles (see Fig. 15) is plotted in Fig. 29. Also, the compliance operator (η) for various land region angles (predicted by Abaqus) is plotted in Fig. 30. It is observed that the stiffness of rubber staves increases with increasing land region angle; but the compliance operator decreases. The multi-region exponential approximation method, which is described in Section 3.2.1, is used to determine the compliance operator for each case. By using the infinitesimal perturbation method, the elastohydrodynamic coefficients with respect to parameter Λ for various land region angles are obtained; see Fig. 31. It is seen that the elastohydrodynamic

Fig. 28. Rubber bearing dynamic stiffness with respect to Λ for various ε.

Fig. 29. Stiffness of rubber staves with respect to land region angle (θL ).

Please cite this article as: S. Liu, & B. Yang, A new model of water-lubricated rubber bearings for vibration analysis of flexible multistage rotor systems, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.03.052i

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Fig. 30. Compliance operator for various land region angle.

Fig. 31. Elastohydrodynamic coefficients with respect to Λ for various θL (a) direct stiffness; (b) cross-coupled stiffness; (c) direct damping coefficients; and (d) cross-coupled damping coefficients

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coefficients increases as the land region angle increase. Fig. 32 shows the overall rubber bearing stiffness, which increases as the land region angle increases. 4.3. Effect of number of grooves In this study, the eccentricity is chosen as ε0 ¼ 0:5 and the land region angle (see Fig. 15) is θL ¼ 19o for bearings with different number of grooves. In addition, assume the properties of the rubber staves are the same for all the cases, e.g. same compliance operator (η) and same stiffness of the rubber staves (K r ). Fig. 33 shows the elastohydrodynamic coefficients with

Fig. 32. Rubber bearing dynamic stiffness with respect to Λ for various θL .

Fig. 33. Elastohydrodynamic coefficients with respect to Λ for different number of grooves (a) direct stiffness; (b) cross-coupled stiffness; (c) direct damping coefficients; and (d) cross-coupled damping coefficients.

Please cite this article as: S. Liu, & B. Yang, A new model of water-lubricated rubber bearings for vibration analysis of flexible multistage rotor systems, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.03.052i

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Fig. 34. Rubber bearing dynamic stiffness with respect to Λ for different number of grooves.

respect to Λ for WLRBs with different number of grooves. It is observed that the elastohydrodynamic coefficients of WLRBs increase as the number of grooves increases. However, since the stiffness of the rubber staves and the value of α do not change, the overall rubber bearing dynamic stiffness does not change too much, as shown in Fig. 34. Therefore, the stiffness of rubber staves and the value of α dominate the overall rubber bearing stiffness. In summary of the parameter study, in the design of flexible rotating systems with water-lubricated rubber bearings (WLRBs), higher operation eccentricity and larger land region angle of the WLRB can increase the dynamic coefficients of WLRBs; but it could be a cost of energy, which is induced by higher friction due to possible more contacts. In addition, the stiffness of rubber staves and the value of α play an important role in the determination of dynamic coefficients of WLRBs. 5. Conclusions This paper presents a new model of water-lubricated rubber bearings (WLRBs) for a class of flexible rotor–bearing systems. The main results obtained as summarized as follows. (I) Unlike many previous investigations that adopt pointwise bearing models, this effort treats a WLRB of large length-todiameter ratio as a distributed viscoelastic foundation. (II) A theoretical model of WLRBs is developed, which for the first time, describes mixed lubrication in the vibration of a WLRB-supported rotor system. The proposed bearing model combines the stiffness of rubber staves caused by direct contact between the rotating shaft and the rubber bearing and the stiffness due to elastohydrodynamic forces of the fluid film in WLRBs. The stiffness of rubber staves is given in an analytical formula that is derived from a rectangular rubber block; the fluid film stiffness is determined through solution of Reynolds equation, with either two-point approximation or multi-region approximation of the compliant operator of the rubber material. The WLRB model developed is applicable to both compressible and incompressible materials. (III) With mixed lubrication taken into account, the α-method is proposed to compute the overall dynamic stiffness of WLRBs, in which the non-dimensional parameter α indicates the percentage contribution of the stiffness of rubber staves to the overall dynamic stiffness of WLRBs. Furthermore, experimental measurement and curve fitting yields a non-dimensional formula for evaluation of α at various shaft rotation speeds. (IV) The proposed bearing model is validated in experiments. The dynamic stiffness coefficients of the two bearings used in experiments that are predicted by the proposed bearing model are in good agreement with those that are experimentally determined as shown see Fig. 25. (V) The new bearing model and the distributed transfer function formulation presented in this paper lay a platform for vibration analysis and optimal design of a class of flexible multistage rotating systems in engineering applications.

Acknowledgments This research was partially supported by the ITT Corporation. Appendix A. Transfer function formulation of WLRB-supported rotor systems The schematic of a flexible rotating shaft–bearing system is shown in Fig. A.1. In system modeling via the distributed transfer function method (DTFM), divide the shaft into a number of segments by nodes z0 ; z1 ; z2 …, which are the locations of Please cite this article as: S. Liu, & B. Yang, A new model of water-lubricated rubber bearings for vibration analysis of flexible multistage rotor systems, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.03.052i

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Fig. A1. Schematic of a flexible rotating shaft–bearing system .

Fig. A2. Basic elements of the rotor–bearing system .

mounted rigid disks (e.g. impellers of vertical pumps), boundaries of long bearings, and the ends of the shaft. Unlike in FEM modeling, the segment length Li does not have to be small. The rotor–bearing system thus has three basic elements as shown in Fig. A.2: shaft segment with or without long bearings, and mounted rigid disks. As a demonstrative example, consider the Rayleigh beam theory for a rotating shaft segment. The governing equations of motion of the shaft segment are ∂4 U ∂3 V ∂4 U ∂2 U  2ρI Ω 2  ρI 2 2 þ ρA 2 ¼ f x þqx 4 ∂z ∂z ∂t ∂z ∂t ∂t ∂4 V ∂3 U ∂4 V ∂2 V EI 4 þ 2ρI Ω 2  ρI þ ρA 2 ¼ f y þqy 2 2 ∂z ∂z ∂t ∂z ∂t ∂t

EI

(A1)

where E; I; ρ; Ω and A is Young0 s modulus, moment of inertial, density, rotation speed of the shaft and cross-section area respectively; uðz; t Þ and vðz; tÞ are the transverse displacements of the shaft; f x and f y are the external force applied on the shaft; and qx and qy are the bearing forces of the form ∂U ∂V C xy  K xx U  K xy V ∂t ∂t ∂U ∂V qy ¼  C yx  C yy  K yx U K yy V ∂t ∂t

qx ¼  C xx

(A2)

with C αβ and K αβ being the dynamic coefficients of the bearing, which in general are functions of z. Unlike conventional pointwise bearing models, Eqs. (A1) and (A2) are for bearings with a large length-to-diameter ratio. By taking Laplace transform of Eqs. (A1) and (A2) and cast into the spatial state form 

 ∂

η^ ðz; sÞ ¼ ½F ðz; sÞ η^ ðz; sÞ þ fpðz; sÞg ∂z

(A3)

where s is the complex Laplace transform parameter, ½FðsÞ is an eight-by-eight state matrix oT

 n 0 0 0 0 η^ ðz; sÞ ¼ U; U ; U″; U ″; V; V ; V″; V ″

 pðz; sÞ ¼ f0; 0; 0; f x ðz; sÞ=EI; 0; 0; 0; f y ðz; sÞ=EIg 0

with the over-bar standing for Laplace transformation with respect to t and U ¼ ∂U=∂z. By the DTFM [26–27], the s-domain solution of Eq. (A3) takes the form Z Le

    

 η^ ðz; sÞ ¼ G z; ξ; s p ξ; s dξ þ½H ðz; sÞ γ ðsÞ (A4) 

0

 where Le is the length of the shaft segment, ½G z; ξ; s  and ½H ðz; sÞ are the distributed transfer functions that can be obtained in exact and closed form, and fγ ðsÞg contains the modal displacements at the two ends of the shaft segment. Finally, apply Please cite this article as: S. Liu, & B. Yang, A new model of water-lubricated rubber bearings for vibration analysis of flexible multistage rotor systems, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.03.052i

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Eq. (A4) to obtain the unbalanced mass response of a WLRB-supported flexible rotor–bearing system as follows: #

 " A1 B1 U cos Ωt ¼ A2 B2 V sin Ωt

(A5)

Appendix B. Geometry and operation parameters of the water-lubricated rubber bearings

Parameters Sleeve diameter (Ds ) Bearing inside diameter (D) Bearing length (L) Radial clearance (C) Fluid viscosity (μ) Supply pressure (ps ) Stave thickness (T s ) Stave width (W s ) Land region angle (θL ) Rubber Young0 s modulus (Er ) Poisson0 s ratio (ν)

101.6 mm rubber Bearing 101.56 mm 102.04 mm 203.2 mm 0.24 mm 0.001002 Pa s 93079.2 Pa 9.52 mm 41.28 mm 191

50.8 mm rubber Bearing 49.28 mm 49.50 mm 190.5 mm 0.11 mm 0.001002 Pa s 93079.2 Pa 4.76 mm 20.64 mm 191

7:84  106 N m  2 0.47

7:84  106 N m  2 0.47

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31]

T. Someya, Journal-Bearing Databook, Springer-Verlag, Berlin, Germany, 1989. D. Childs, Turbomachinery Rotordynamics: Phenomena, Modeling, and Analysis, John Wiley & Sons, Inc, New York, 1993. B.J. Hamrock, Fundamentals of Fluid Film Lubrication, Marcel Dekker, New York, 2004. W.J. Chen, E.J. Gunter, Introduction to Dynamics of Rotor–Bearing Systems, Trafford Publishing, Victoria, Canada, 2007. S.M. Zakharow, Hydrodynamic lubrication research: current situation and future prospects, Journal of Friction and Wear 31 (2010) 56–67. L. San Andrés, Modern Lubrication Theory, Texas A & M University Digital Libraries, 2012 〈http://repository.tamu.edu/handle/1969.1/93197〉 accessed 20.08.2012. A.Z. Szeri, Fluid Film Lubrication, Cambridge University Press, New York, 2011. R. Tiwari, A.W. Lees, M.I. Friswell, Identification of dynamic bearing parameters: a review, The Shock and Vibration Digest 36 (2004) 99–124. T.W. Dimond, P.N. Sheth, P.E. Allaire, M. He, Identification methods and test results for tilting pad and fixed geometry journal bearing dynamic coefficients — a review, Shock and Vibration 16 (2009) 13–43. Y.-Qi Wang, M.-S. Wang, Experimental study on the tribological properties of water-lubricated rubber bearings, Advanced Material Research 219–294 (2011) 1454–1457. Y. Cheng, Z. Liu, J. Wang, Y. Dai, E. Peng, Experimental study on temperature characteristics of water-lubricated rubber bearing, Advanced Material Research 479–481 (2012) 1097–1101. N. Wang, Q. Meng, P. Wang, T. Geng, X. Yuan, Experimental research on film pressure distribution of water-lubricated rubber bearing with multiaxial grooves, Journal of Fluid Engineering (ASME) 135 (2013). D.L. Cabrera, N.H. Woolley, D.R. Allanson, Y.D. Tridimas, Film pressure distribution in water-lubricated rubber journal bearings, Proceedings of the Institution of Mechanical Engineers, Part J: Journal of Engineering Tribology 219 (2005) 125–132. B.C. Majumdar, R. Pai, D.J. Hargreaves, Analysis of water-lubricated journal bearings with multiple axial grooves, Proceedings of the Institution of Mechanical Engineers, Part J: Journal of Engineering Tribology 218 (2004) 135–146. R. Pai, D.J. Hargreaves, Water-lubricated bearings, Green Tribology, Springer-Verlag, Berlin Heidelberg, 2012, 347–391. R.S. Pai, R. Pai, Nonlinear transient analysis of multiple axial groove water-lubricated journal bearings, Proceedings of the IMechE Part J: Journal of Engineering Tribology 222 (2008) 549–557. R.S. Pai, R. Pai, Stability of four-axial and six-axial grooved water-lubricated journal bearings under dynamic load, Proceedings of the IMechE Part J: Journal of Engineering Tribology 222 (2008) 683–691. M. Lahmar, A. Haddad, D. Nicolas, Elastohydrodynamic analysis of one-layered journal bearing, Proceedings of the IMechE Part J: Journal of Engineering Tribology 212 (1988) 193–205. M. Lahmar, S. Ellagoune, B. Bou-Said, Elastohydrodynamic lubrication analysis of a compliant journal bearing considering static and dynamic deformations of the bearing liner, Tribology Transactions 53 (2010) 349–368. M. Zhang, G. Zhang, Z. Liu, Study on dynamic characteristics of large ship rotor–bearing system based on an advanced water-lubricated rubber bearing model, Proceedings of the ASME Turbo Expo: Marine (2013). C.J. Hooke, D.K. Brighton, J.P. O'Donoghue, Effect of elastic distortions on the performance of thin shell bearings, Proceedings of the Institution of Mechanical Engineers, Conference Proceedings 181 (1966) 63–69. H.D. Conway, H.C. Lee, The analysis of the lubrication of a flexible journal bearing, Journal of Lubrication Technology 97 (1975) 599–604. S. Liu, B. Yang, P.W. Behnke, L. Ding, Vertically-suspended pumps with water-lubricated rubber bearings – experimental identification of dynamic stiffness coefficients, Proceedings of the Twenty-Ninth International Pump Users Symposium, 2013. S. Liu, B. Yang, Modeling and analysis of flexible multistage rotor systems with water-lubricated rubber bearings, Proceedings of the ASME International Mechanical Engineering Congress & Exposition, 2014. A. Muszynska, Rotordynamics, CRC Press, Taylor & Francis Group, Boca Raton, FL, 2005. B. Yang, C.A. Tan, Transfer functions of one-dimensional distributed parameter systems, ASME Journal of Applied Mechanics 59 (1992) 1009–1014. H. Fang, B. Yang, Modeling, synthesis and dynamic analysis of complex flexible rotor system, Journal of Sound and Vibration 211 (1998) 571–592. G. Genta, Vibration Dynamic and Control, Springer, New York, 2009. M.I. Friswell, J.E.T. Penny, S.D. Garvey, A.W. Lees, Dynamic of Rotating Machine, Cambridge University Press, New York, 2010. J. Vance, F. Zeidan, B. Murphy, Machinery Vibration and Rotordynamics, John Wiley & Sons, Inc., Hoboken, NJ, 2010. A.N. Gent, E.A. Meinecke, Compression, bending, and shear of bonded rubber blocks, Polymer Engineering & Science 10 (1970) 48–53.

Please cite this article as: S. Liu, & B. Yang, A new model of water-lubricated rubber bearings for vibration analysis of flexible multistage rotor systems, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.03.052i

S. Liu, B. Yang / Journal of Sound and Vibration ] (]]]]) ]]]–]]] [32] [33] [34] [35] [36] [37]

29

P.B. Lindley, Small-strain compression and rotating moduli of bonded rubber blocks, Plastics and Rubber: Processing & Applications 1 (1981) 331–337. J.M. Hill, A.I. Lee, Large elastic compression of finite rectangular block of rubber, Journal of Mechanics and Applied Mathematics 42 (1989) 267–287. C.M. Harris, A.G. Piersol, Harris' Shock and Vibration Handbook, McGraw-Hill, New York, 2002. Z.L. Qiu, A theoretical and experimental study on dynamic characteristics of journal bearings PhD Thesis, University of Wollongong, 1995. F. Martelli, G. Manfrida, A new approach to the theoretical calculation of the dynamic coefficients of tilting-pad bearings, Wear 70 (1981) 249–258. F.K. Choy, M.J. Braun, Y. Hu, Nonlinear effects in a plain journal bearing: Part I-analytical study, Journal of Tribology 113 (1991) 555–561.

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