Dynamical analysis of an elastic ring squeeze film damper-rotor system

Mechanism and Machine Theory 131 (2019) 406–419

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Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmachtheory

Research paper

Dynamical analysis of an elastic ring squeeze film damper-rotor system Zhifei Han a,b, Qian Ding a,b,∗, Wei Zhang a,b a b

Department of Mechanics, Tianjin University, Tianjin 300350, China Tianjin Key Laboratory of Nonlinear Dynamics and Control, Tianjin 300350, China

a r t i c l e

i n f o

Article history: Received 20 April 2018 Revised 10 September 2018 Accepted 12 October 2018

Keywords: Elastic ring squeeze film damper Rotor system Dynamical characteristic

a b s t r a c t In this paper, a calculation procedure is proposed to investigate the dynamic characteristics and response of an elastic ring squeeze film damper supported rotor system (ERSFD-rotor). First, the oil film pressure and force of ERSFD are analyzed by solving the Reynolds equations. Next, deformation of elastic ring is calculated by using the finite element method (FEM) based on Kirchhoff assumption. Then the dynamical equations of rotor system are numerically simulated with cooperation of the former two steps. Using the procedure, both the response of rotor and the oil film coefficients of the ERSFD can be determined simultaneously. Calculations show that the nonlinearity of oil coefficients of ERSFD are smaller than that of classic squeeze film damper (SFD). Thus, the ERSFD behaves better than the SFD in preventing bi-stable vibration of rotor by suppressing nonlinear effects of the oil film. Influences of the ERSFD parameters, namely the oil film thickness, ring thickness and number of elastic ring bosses, on the dynamical characteristics and response of the rotor are also discussed. The study reveals that the elastic ring enable better dynamical performance of the ERSFD than the SFD in three aspects. © 2018 Elsevier Ltd. All rights reserved.

1. Introduction Squeeze film dampers are widely used in aircraft engines to control the vibration and stability of rotor systems [1–5]. However, the SFD has highly nonlinear characteristics, which creates high-amplitude bi-stable and nonsynchronous responses for the rotors [6–9]. The large-amplitude response can result in rotor/state rubs and fatigue [10]. In Ref. [11], nonlinear responses such as quasi-periodic vibrations, sub-harmonic and jump phenomena were predicted for a range of mass unbalance and bearing parameters by Zhao [12]. Bonello [13] argued that the SFD-rotor system is prone to a periodic motion with sub-synchronous frequency components with high eccentricity around the first critical speed. The nonlinear response of a flexible rotor supported on a damper with centralizing springs was studied in Ref. [14] using the synchronous circular centered-orbit motion solution, slow acceleration method and numerical integration method. The influence of the mass ratio and bearing parameters on response bifurcation of a flexible SFD-rotor were analyzed by Inayat–Hussain [15,16]. Motion bifurcation of a rigid SFD-rotor system considering the effect of fluid inertia was analyzed in [17], to provide a fundamental theory for designing an effective SFD. The work of San Andrés [18] and R. Tiwari [19] clearly demonstrated methods to identify the oil film coefficients of both SFDs and bearings. Various suggestions have been proposed to improve the performance

∗

Corresponding author at: Department of Mechanics, Tianjin University, 92 Weijin Road, Tianjin 300350, China. E-mail address: [email protected] (Q. Ding).

https://doi.org/10.1016/j.mechmachtheory.2018.10.011 0094-114X/© 2018 Elsevier Ltd. All rights reserved.

Z. Han et al. / Mechanism and Machine Theory 131 (2019) 406–419

Housing

Outer Oil Film

407

Housing Bearing

Inner Oil Film

Rotor Mouse Cage Boss

Elastic Ring Outer Race of Bearing

(a) ERSFD

(b) ERSFD-rotor

Fig. 1. Schematic model of ERSFD.

of SFDs, as detailed in Ref. [20]. Zhou [21,22] investigated the response of rotor-floating ring squeeze film dampers. Numerical results show that the floating ring squeeze film damper behaves better than the classic SFD in preventing bi-stable response. By placing an elastic ring inside the oil chamber one creates an elastic ring SFD or ERSFD, as shown in Fig. 1(a). The ERSFD not only retains the advantages of SFDs, but also has better performance in reducing nonlinear characteristics [23]. However, there is still no numerical investigation on how ERSFD could be better than SFD in restraining nonlinear response. Generally, a number of bosses are included, uniformly spaced on the outside and inside surfaces of the elastic ring, which contact the housing and the outer bearing race, respectively. The lubricating oil film inside the chamber between the outer bearing race and ring is defined as the inner oil film, and that between the housing and ring is defined as the outer oil film. The outer and inner oil chambers are connected with each other through oil holes on the elastic ring. The ERSFD, successfully used in aero-engines technology, integrates the functions of damping and frequency modulation (FM), as revealed in Ref. [24] which analyzed the Reynolds equations of the outer/inner oil films. Hong [25] found that an ERSFD can automatically adjust the oil film thickness by making use of the transverse deformation of the elastic ring, which suppresses the nonlinear characteristics of the oil film markedly. In Ref. [26], the dynamical response of an ERSFD-rotor subject to varying axial force, structural parameters, unbalance mass and oil supply were experimentally investigated. On the basis of the Moody friction factor equation and Hirs bulk flow theory, Kang analyzed the oil film characteristics of ERSFDs by considering the shearing effect of the outer squeeze film [27]. But there is still no a useful results about influences of different structure parameters on the nonlinear performance and suppressing vibration, thus an optimizing strategy has not be provided for engineering designing. Zhang [20] analyzed the bearing capacity and the oil film characteristics with different pedestal contact status during a whirl period. Furthermore, in the past studies, oil film coefficients were always defined as linear functions, which do not describe the oil film characteristics adequately. Deformation of the elastic ring is important in determining the dynamic characteristic of the ERSFD. Finite element software, such as ANSYS, can analyze the elastic ring deformation, by modeling the elastic ring using solid elements [20,25,26]. However, the computation is time-consuming, especially for a large number of elements. To simplify the deformation estimation, the elastic ring can be modeled by several thin plates with simple supports [28]. Although computation speed is faster, the accuracy of the deformation calculation is not high enough because the boundary conditions adopted are rather rough. On top of that, the deformation analyses by both computation procedures are not easy to combine with the dynamic analysis of the ERSFD-rotor and the hydrodynamic lubrication analysis of the ERSFD oil film, based on Reynolds equations. Up to now, nearly all works focus on oil characteristics analyses, deformation of elastic ring and experimental response. Owing to lack of efficient numerical procedure for analyzing the response of ERSFD-rotor, in-depth knowledge on the dynamical characteristics of the ERSFD-rotor is still an open problem. In this paper, based on the Kirchhoff assumption of the FEM, the flat shell element (FSE) is employed to establish the model of the elastic ring to investigate its deformation. The coupled Reynolds equations of the outer and inner oil films are established to calculate oil coefficients and pressure. Eccentricity and offset angle are investigated by the rotor equations. Then the equivalent restoring force of the ERSFD is estimated by the Reynolds equations. The Runge–Kutta and finite difference method (FDM) are used to solve the above equations, obtaining oil film dynamic coefficients of the ERSFD and the response of the rotor system. 2. Basic principle and method 2.1. Elastic ring model The elastic ring inside the ERSFD can be modeled as a thin annular shell structure, due to its thickness is 0.05 times less than the radius and its deflection is very small during whole process. Following the theory of plate and shell [29], four-node rectangular flat shell elements (FSE), composed of plane stress elements and Kirchhoff plane bending elements, are used to analyze the deformation of the elastic ring fast accurately with dense enough grids, as shown in Fig. 2. A FSE has six

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Fig. 2. Flat shell elements.

degrees of freedom (6 DOFs) per node in the local coordinate system, consisting of u and v of the plane stress element, w, θ x and θ y of the Kirchhoff plane bending element, and an additional rotational degree of freedom per node, which is set to be 0 for convenient calculation. Thus the local DOF per node is denoted as



ai = ui ,

T vi , wi , θxi , θyi , 0 , i = 1, . . . , 4.

(1)

The sub-stiffness matrix of a flat shell element in local coordinate system is expressed as

⎡

Kij(p)

0 0

⎢ ⎢ ⎢ 0 Kij =⎢ ⎢ 0 ⎣ 0

0 0 0 0

0 (p )

0 0

0 0

Kij(b) 0

0

0

⎤

0 0⎥ ⎥ 0⎥ ⎥, (i, j = 1, . . . , 4) 0⎥ ⎦ 0 0

(2)

(b )

where Kij and Ki j are stiffness sub-matrices of the plane stress element and the Kirchhoff plane bending element, respectively. The flat shell stiffness matrix is

⎡

K11 ⎢K21 K =⎣ K31 K41

K12 K22 K32 K42

⎤

K13 K23 K33 K43

K14 K24 ⎥ K34 ⎦ K44

(3)

To calculate the deformation of elastic ring, one must convert actions and displacements from the local coordinate system (unprimed) to a global coordinate system (primed). According to Fig. 2, the transformation relations between the local and global coordinates are

⎡ γ ⎢0  ai = H ai , H =⎣ 0 0

0

⎤

0 0

γ

0 0⎥ 0⎦

γ

0 0

(4)

γ

0

where γ is the matrix of cosines of the local axes with global axes



γ

1 = s14



s14 0 0 

0 y4  − y1  −z4  + z1   2





0 z4  − z1  , y4  − y1   2

(5)

where s14 = (y4 −y1 ) +(z4 −z1 ) . In order to increase the calculation accuracy and speed of the elastic ring’s deformation, dividing the elastic ring into 48 elements circumferentially and 9 elements axially, the elastic ring model consists of 432 flat shell elements. Suppose there are 8 outer bosses and 8 inner bosses on the ring outside and internal surface respectively. Suppose the elastic ring always touches the housing and the outer bearing race during the whole period of operation, as shown in Fig. 3(b). The boundary condition of the model is that the tangential, radial and axial displacements of all bosses are constrained. Our assumption is acceptable while the journal eccentricity ratio ε < 1 and amplitude of vibration Dmax /c1 < 1. Both conditions are satisfied for almost all practical use of dampers.

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Fig. 3. Movements of the ERSFD.

2.2. Oil film pressure Based on the short π -film assumption, the transient Reynolds equations of the inner and outer oil films of the ERSFD are given by [24,30]

1 R2 and

1 R2



  ∂ ∂ ∂ 2h ∂ h1 ∂ r ∂ h1 3 ∂ p1 3 ∂ p1 h1 + h1 = −12μ − +12μ +ρ h2 2 ∂θ ∂θ ∂x ∂x ∂θ ∂θ ∂t ∂t



 ∂ ∂ ∂r 3 ∂ p2 3 ∂ p2 h2 + h2 = 12η , ∂θ ∂θ ∂x ∂x ∂t

(6)

(7)

respectively. Here μ is the fluid viscosity, R the radius and  the whirling velocity, ρ the lubricating oil density. As shown in Fig. 3(a) and Fig. 2(a), θ and x are the circumferential and the axial coordinates. The variables p1 and p2 represent the inner and outer oil film pressures to be solved, while h1 and h2 are the inner and outer oil film thicknesses, and c1 and c2 are the initial inner and outer oil chamber clearances, respectively, as shown in Fig. 3(b). The following relations hold:

⎧ ⎧ h1 = c1 + r (θ ) + ecosθ h2 = c2 − r ( θ ) ⎪ ⎪ ⎪ ⎪ ⎨ ∂ h1 ∂ r ⎨ ∂ h2 ∂r = − esinθ and =− , ∂θ ∂θ ∂θ ∂θ ⎪ ⎪ ∂h ⎪ ⎪ 2 ⎩ ∂ h1 = r˙ + e˙ cosθ ⎩ = −r˙ ∂t ∂t

(8)

where r and e are the deformation of the elastic ring and the whirling eccentricity of the journal, respectively, and r˙ and e˙ are the first derivatives of r and e. Define ε = e/c1 as the eccentricity ratio. To obtain the inner and outer oil film pressures, the finite difference method (FDM) is used to solve the Reynolds Eqs. (6) and (7) [20].

2.3. Oil film force Integrating the oil film pressure from θ 1 to θ 2 , the lower and upper boundary angles of the squeezing oil film, gives the oil film forces in the radial and circumferential directions as

  Fr Fθ

=−

 θ2  θ1

L 2

− 2L



P dx



− cos θ Rdθ , sin θ

(9)

which can be transformed to the x and y directions

 

Fz F · cos ϕ − Fθ · sin ϕ = r , Fy Fr · sin ϕ + Fθ · cos ϕ

(10)

where θ is the circumferential coordinate, and ϕ is offset angle of the journal and the z axis, φ is the circumferential coordinate of the displacement perturbation, as shown in Fig. 3(a).

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Fig. 4. Model of a rigid rotor supported on two ERSFDs.

e



Fig. 5. Procedure of calculation.

2.4. Equation of motion The dynamical equations of motion of a rigid Jeffcott rotor supported on ERSFDs, shown in Fig. 4, are [31]



mz¨ + (k + ke )z + cz˙ = mω2 eu cos (ωt ) + Fz my¨ + (k + ke )y + cy˙ = mω2 eu sin (ωt ) + Fy − G

(11)

where k is the stiffness of the bearing mouse cage (see Fig. 1(b)), ke the equivalent stiffness of the elastic ring, c the damping of the system, and m the half mass of the disk with the unbalance distance eu . The fourth-order Runge-Kutta method is used to solve Eqs. (11). The speed ratio is defined by

λ=

ω pn

where pn =

,

k+ke m

(12) is the natural frequency of the shaft/disk system.

2.5. Solution procedure The solution procedure is shown in Fig. 5. First, dynamic responses are obtained by solving Eq. (11), and the oil film thickness is calculated under the current eccentric whirling state of the rotor and deformation of the elastic ring. Then the oil film pressure and the resulted force are determined by the FDM under the current oil film thickness distribution, and

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Table 1 Deformation of three models.

δ under single load/10 mm δ under distributed load/10−5 mm −2

FSE

Shell 181

Solid 187

1.72 5.63

1.48 5.54

1.82 5.6

Table 2 Stiffness of the elastic ring.

Angle

δ /10−2 (mm)

0° 7.5° 15° 22.5° 30°

0.968 1.28 1.72 1.28 0.968

FSE (N/m)

Numerical stiffness (N/m)

Difference

Experimental stiffness (N/m)

Difference

8.06 × 106

9.4 × 106

14.2%

8.4 × 106

4%

the deformation of the elastic ring is determined by FEM under the obtained inner and outer oil film pressures. After that, the inner oil film force is adopted to calculate the new dynamic response. The simulation procedure will be carried out until a steady-state response is obtained. 3. Deformation and stiffness of elastic ring Based on the model established above, the deformation of the elastic ring induced by both a single/concentrated load and a distributed load will be calculated, and compared with deformations simulated by the FSE with Shell 181 elements and Solid 187 elements in ANSYS modeling. The stiffness of the elastic ring will also be analyzed in the following. 3.1. Deformation of elastic ring In order to compare the deformation with that obtained in Refs. [25,32], the same parameters of elastic ring are used in the following analysis. Deformation curvatures and maximum deformation δ of the ring, under a single load of 100 N and a distributed load, are shown in Fig. 6 and Table 1. One finds that the results acquired by the three models, FSE, Shell 181 and Solid 187 in ANSYS, are almost same, with minor differences resulting from different models themselves. Because the width and thickness of bosses is quite smaller than perimeter of the elastic ring, the boundary conditions of bosses are set at the axial line in shell elements, FSE and shell 181, and the bosses are not displayed in shell element’s models, as shown in Fig. 6(a)–(d). According to the result, bosses mainly influence the boundary condition rather than the deformation of the elastic ring directly. Thus, the FSE model can be adopted to analyze the deformation of the elastic ring with high enough accuracy. While the accuracies of three models almost same, the computation speed of FSE is faster than the other methods. 3.2. Stiffness of elastic ring Because the elastic ring is an axisymmetric structure, the single force is loaded on different angles between two adjacent bosses of elastic ring. The stiffness is defined as

ke =

1 n F , i δi n

(13)

where F = 100 N and δ i is the maximum deformation. The stiffness of the elastic ring with eight bosses is calculated using (13 ) for different journal eccentricities, with the results listed in Table 2. The differences between stiffness calculated by FSE and the numerical simulation stiffness in Ref. [25], and differences between it and the experimental stiffness in Ref. [32], are 14.2% and 4%, respectively. The results suggest that the proposed method is quite accurate. For the current investigation, the parameter values of the ERSFD are listed in Table 4. The stiffness of the elastic ring is calculated by the proposed method and listed in Table 3 for different boss numbers and ring thicknesses. One finds that the elastic ring stiffness increases with increasing boss number and elastic ring thickness. 4. Pressure and thickness of the inner oil film The oil film thickness and pressure of the SFD and ERSFD with oil film thickness cs = 0.2 mm are presented in Fig. 7 for comparison. Because the oil chamber is separated into several small chambers, the peak pressure value of ERSFD is less than that of SFD, and the pressure distribution of SFD is circumferentially asymmetrical due to the journal whirl. In order to reveal the influence of elastic ring deformation, calculation result of rigid ring squeeze film damper (RRSFD), the ring has

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Fig. 6. Deformation of the elastic ring. (a), (c) and (e) are results of the single load case, and (b), (d) and (f) the distributed load case.

almost no deformation under oil film pressure, is also presented. Fig. 7(c) presents the influence of deformation/elasticity of the ring on the inner oil film pressure. The red/black lines represent the film thickness of the rigid/elastic ring. The corresponding inner pressures of the ERSFD are Pr and Pe . One finds that the oil film thickness increases with the deformation of the elastic ring at the squeezing area. Correspondingly, the oil film pressures are reduced. Contrasting Pr and Pe , the asymmetrical distribution of pressure is lessened by the elastic ring deformation. Thus the influence of the bosses separation and deformation of elastic ring could effectively reduce the asymmetrical distribution of oil film. That is the internal reason why ERSFD could performance better than SFD in preventing nonlinear phenomenon.

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Table 3 Stiffness of elastic ring. Boss number 4

6

8

Elastic ring thickness/mm

ke

0.5 0.7 0.95 0.5 0.7 0.95 0.5 0.7 0.95

1.72 × 105 4.65 × 105 1.14 × 106 4.10 × 105 1.13 × 106 2.75 × 106 9.80 × 105 2.68 × 106 6.7 × 106

Table 4 Parameters of the ERSFD. Parameters

Value

Material Density(kg/m3 ) Width(mm) Radius(mm) Poisson ration Elasticity modulus(MPa) Fluid viscosity/cp Boss width/mm

1Cr11Ni2W2MoA 785 10 32.5 0.3 1.96 × 105 18.82 4

Fig. 7. Inner pressure and oil film thickness.

5. Dynamical coefficients The oil film damping and inertia force coefficients are defined by Center Circular Orbit (CCO) method [33,34]. Noted that there is no direct interaction between the journal and the outer oil film, thus the outer oil film dynamic coefficients are quite small [35,36]. When the static eccentricity ratio ε < 0.51 and the amplitude of vibration Dmax / c1 < 0.51, oil film

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Fig. 8. Oil film characteristics of ERSFD and SFD.

Fig. 9. Influence of boss number on dynamic characteristics of the ERSFD with 0.95 mm elastic ring thickness.

stiffness has been ignored. Thus the equations of motion of ERSFD are



Mzz Mzy

Myz Myy

 



C z¨ + zz Czy y¨

Cyz Cyy

 

 

z˙ F = z y˙ Fy

(14)

The direct flourier transform (DFT) is applied as

 

 

z z¯ iωt = e , y y¯

 

Fz = Fy

 

F¯z iωt e Fy

(15)

After some algebraic operation, following equations are obtained

−ω2 Mzz + ωCzy + i



−ω2 Myy + ωCyz + i

 F¯ ω2 Mzy + ωCzz = z





ωCyy − ω2 Myz =

z¯ F¯z z¯

(16)

Z. Han et al. / Mechanism and Machine Theory 131 (2019) 406–419

Fig. 10. Influence of oil film thickness on dynamic characteristics of the ERSFD with 8-boss elastic ring.

Fig. 11. Influence of elastic ring thickness on dynamic characteristics of the ERSFD with 8-boss elastic ring.

415

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Fig. 12. Dynamical responses of the SFD-rotor systems, with/without damper and with ERSFD (with 4 boss and elastic ring thickness 0.5 mm).

Fig. 13. Dynamical responses of ERSFD-rotor system. (a) inner oil film thickness is 0.3, 0.4 and 0.5 mm (8 bosses and 0.5 mm thickness of elastic ring); (b) elastic ring thickness is 0.5 mm, 0.7 mm and 0.95 mm (4 bosses, c1 = 0.4 mm and c2 = 0.15 mm); (c) elastic ring with 4, 6 and 8 bosses (0.5 mm thickness, c1 = 0.4 mm and c2 = 0.15 mm).

The cross coefficients are negligible as detailed in Ref. [37], the direct oil film coefficients are obtained

Mzz = − Myy

1

ω

Re 2

 F¯z z¯

Czz =



F¯y = − 2 Re y ω 1

Cyy =

1

ω

 Im

1

ω

F¯z z¯

 Im

F¯y y

(17)

5.1. Basic oil film characteristics The original clearances of the inner and outer chambers of ERSFD are chosen as c1 = 0.3 mm and c2 = 0.15 mm, and the boss number is 4.

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Fig. 14. Bi-stable response of the system for k = 1 × 106 N/m, c = 20N · m/s, μ = 1cp, m = 3kg, c1 = cs = 0.2mm, c2 = 0.15mm.

From Fig. 8, we find that the nonlinear degree of the coefficients of SFD is higher than that of ERSFD as the eccentricity increases further. The results mean that the nonlinearity can be suppressed by using of the elastic ring inside SFD. For example, oil film damping and inertia of SFD are almost twice larger than that of ERSFD when ɛ = 0.38.

5.2. Influences of structure parameters on oil film coefficients Influences of the boss amount, film thickness and ring thickness on oil film inertia and damping coefficients of ERSFD are analyzed in this section. From Fig. 9, one finds that oil film coefficients of ERSFD, with elastic ring thickness 0.95 mm and boss amount 4, 6 and 8 respectively, increase nonlinearly with the eccentricity ratio. Besides, increase of the elastic ring boss amount can also reduce the level of nonlinearity of ERSFD. The reason is that the un-symmetrical distribution of pressure is decreased by the divided oil chamber/film and deformation of the elastic ring. Oil film dynamic coefficients of ERSFD with different oil film thickness and different elastic ring thickness are showed in Figs. 10 and 11 respectively. We find that both the increase of oil film thickness and decrease of elastic ring thickness can result in increase of the ability to restrain the nonlinear characteristics for ERSFD.

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6. Dynamical response of the ERSFD-rotor system Let the system parameter values be

k = 8 × 106 N/m, c = 80N m/s, μ=18.82cp, m = 10kg eu = 1 × 10−6 m, c1 = cs = 0.4mm, c2 = 0.15mm. The displacement of the disk from its static equilibrium position is determined by



D=



x2 + y −

mg k + ke

2

.

(18)

Denoting the maximum of D by Dmax , which is also known as amplitude of vibration of rotor, the amplitude-frequency responses of the rotor systems with different dampers, operating through the critical speeds, are numerical simulated and presented in Fig. 12. In the small picture, the resonant amplitudes of the three systems are also illustrated near the speed ratio λ = 1. One finds that both the ERSFD and SFD can suppress Dmax of the rotor response. In addition, the effect of frequency modulation of the ERSFD-rotor system is more obvious than that of the SFD-rotor system. Correspondingly, in Figs. 8–12, the ERSFD has better performance than the SFD in decreasing the nonlinearity of the oil film, while the damping capacity of ERSFD are as good as that of SFD. 6.1. Influences of ERSFD parameters on the system dynamics Fig. 13(a) shows the dynamic response of the ERSFD-rotor systems with 8 bosses, 0.5 mm elastic ring thickness and oil film thickness 0.3, 0.4 and 0.6 mm. The value Dmax decreases with the decrease of oil film thickness, while the critical speed is hardly affected by the change of oil film thickness. By contrasting Fig. 13(a) and Fig. 10, the influence of oil film thickness on reducing vibration and nonlinearity is different; thus, oil film thickness should be chosen according to the actual demand of vibration reduction and nonlinearity suppression. Fig. 13(b) presents the response of ERSFD-rotor systems with 4 bosses, c1 = 0.4 mm and c2 = 0.15 mm and different values of elastic ring thickness, namely 0.5 mm, 0.7 mm and 0.95 mm. We find that Dmax for the system with elastic ring thickness 0.5 mm and 0.7 mms are almost same, but smaller than the case where the elastic ring thickness is 0.95 mm. Fig. 13(c) shows the amplitude-frequency responses of the ERSFD-rotor systems with elastic ring thickness 0.5 mm and 4, 6 and 8 bosses. One finds that Dmax increases with the increase of number of elastic ring bosses. Moreover, the increase of support stiffness with the increase of boss number results in an increase of critical speed. By comparing with Fig. 9, the boss number should be chosen according to the actual demand of the vibration reduction and the nonlinearity suppression. 6.2. Suppressing bi-stable response Fig. 14 shows the amplitude-frequency responses of the SFD-system and ERSFD-system under different unbalance distance eu , with 4 bosses and 0.5 mm elastic ring thickness. From Fig. 14(e) and (f), there is no nonlinear response in SFDrotor and ERSFD-rotor when the eu is 0.83 × 10−6 m. While the eu is 2.48 × 10−6 m, as shown in Fig. 14(c) and (d), there is a jump phenomenon in SFD-rotor but not in ERSFD-rotor. Fig. 14(a) shows that there is an obvious jump phenomenon for the SFD-system as the frequency sweeps up and down across the natural frequency of the system, but response of ERSFDrotor is seak nonlinear. The jump is a typical vibration behavior for nonlinear systems, which behaves as softening stiffness characteristic in SFD-rotor, and can have a harmful impact on the system. Admittedly, the level of nonlinear response of SFD-rotor raise with increase of amplitude Dmax caused by the increase of the eu . However, this jump does not appear in the ERSFD-system. That is, the elastic ring can decrease the nonlinearity of the SFD. 7. Conclusion In this paper, the flat shell element, based on the Kirchhoff assumption, was used to establish the model of the elastic ring. The deformation of the elastic ring, oil film force and rotor motion were determined simultaneously to analyze the oil film coefficients of the ERSFD and the response of the rotor system. By contrasting the dynamical characteristics of ERSFDrotor with that of the SFD-rotor, RRSFD-rotor and no-damper-rotor. The conclusions are summarized as follows. (1) For ERSFD, direct oil film damping and inertia are larger than the cross damping and inertia ones, and the nonlinear level of oil film coefficients is less than the that of the SFD. So ERSFD performance better in preventing bi-stable or jump vibration of rotor by decreasing the nonlinear effects of the oil film. (2) Elastic ring brings three effects to ERSFD. First, deformation of the elastic ring adjusts oil film thickness. Second, the bosses of the elastic ring divide the oil chamber into several sections. As a result, unbalanced distribution of oil film thickness is lightened and the nonlinear degree of the oil film is decreased. Third, the elastic ring can modulate the natural frequency of the rotor system by changing the support stiffness.

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