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Elastic ring deformation and pedestal contact status analysis of elastic ring squeeze film damper
Journal of Sound and Vibration 346 (2015) 314–327
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Elastic ring deformation and pedestal contact status analysis of elastic ring squeeze film damper Wei Zhang a,b, Qian Ding a,b,n a b
Department of Mechanics, Tianjin University, Tianjin 300072, China Tianjin Key Laboratory of Nonlinear Dynamics and Chaos Control, Tianjin 300072, China
a r t i c l e in f o
abstract
Article history: Received 30 November 2014 Received in revised form 5 February 2015 Accepted 8 February 2015 Handling Editor: L.G. Tham Available online 3 March 2015
This paper investigates the dynamic parametric characteristic of the elastic ring squeeze film damper (ERSFD). Firstly, the coupled oil film Reynolds equations and dynamic equations of an ERSFD supported rotor system are established. The finite differential method and numerical simulation are used to analyze the oil film pressure distribution, bearing capacity of ERSFD, oil film stiffness and damping characteristics during a vibration period. Then, based on the oil film pressure results, the deformation of elastic ring is revealed by the finite element method. Finally, pedestal contact status is analyzed according to the change of oil film thickness during a vibration period. The results reveal that the oil film pressure is sectionally continuous, the deformation of elastic ring is complex under the compression of inner and outer oil film, and different pedestal contacts occur in a vibration period. The level of nonlinearity of the bearing capacity, oil film stiffness and damping can be effectively lightened by application of the elastic ring. & 2015 Elsevier Ltd. All rights reserved.
1. Introduction Squeeze film damper (SFD) is a kind of passive vibration isolator widely used in rotor supporting structures of aeroengines for the stabilization and vibration control [1–4]. However, the stiffness and damping coefficients of the conventional SFDs are highly nonlinear, which can result in complex responses of the SFD supported rotor during operation. Zhao [5] and Zhu [3] investigated the jump phenomenon of SFD supported rotor by using the Levenberg–Marquardt nonlinear least squares method and the synchronous circular centered-orbit motion solution, respectively. Inayat-Hussain [6,7] investigated the influences of various parameters on bifurcations of the response of a rotors supported on SFD with and without retainer springs. An efficient computation method is also adopted to investigate the nonlinear responses of a rotor with SFD by Chu and Holmes [8]. The complicated response can be non-synchronous, which induces cyclic stress of the rotating shaft, and even fatigue problem [9,10]. So various suggestions have been proposed to improve the performance of SFD. A porous squeeze film damper (PSFD) was developed by Zhang and Yan [11], by making holes on the outer shell of oil film chamber. The main shortage is that the holes are easily blocked up by the impure oil for actual application. The duel clearance squeeze film damper (DCSFD) and the spiral foil multi-squeeze film damper (SFMSFD) were proposed by Fleming [12] and Hooshang and Waltton [13] respectively, the former can protect the rotor system, and the latter can improve the highly nonlinearity of oil
n
Corresponding author at: Department of Mechanics, Tianjin University, Tianjin 300072, China. Tel.: þ86 22 27401099. E-mail address: [email protected] (Q. Ding).
http://dx.doi.org/10.1016/j.jsv.2015.02.015 0022-460X/& 2015 Elsevier Ltd. All rights reserved.
W. Zhang, Q. Ding / Journal of Sound and Vibration 346 (2015) 314–327
315
film stiffness. Nevertheless, requirement of high level manufacturing technique limits the wide application. Hybrid squeeze film damper (HSFD) was put forward successively by Zhu and Wang [14]. Whereas its structure is too complex to be applied in aero-engine. Comparatively, the elastic ring squeeze film damper (ERSFD) [15] is effective and simple designed and the ERSFDs were already used in aircraft engine successfully. By locating an elastic ring in the oil chamber to separate the previous large chamber into two small oil chambers, the shortage of high nonlinearity existed in conventional SFD is efficiently overcome. Dai and Wang [16] derived the corrugated ring's stiffness coefficient and proposed a bent correcting coefficient using the FE method. Zhou [17] established the oil film pressure governing equation and investigated the vibration damping mechanism of the ERSFD. By simplifying every segment of elastic ring as a clamped beam and calculating its deformation, Cao and Gao [18–21] investigated the elastic ring stiffness and oil film pressure characteristics of an ERSFD, and the dynamic characteristic of a ERSFD supported rotor system as well. Hong [22] and Xu [23] investigated the influences of the number of pedestals, the initial oil film thickness and the orifice distribution on oil film stiffness and damping characteristics of ERSFD using FE method. All these researches reveal that by adding an elastic ring into SFD, the critical speed and peak response amplitude of rotor system are effectively adjusted through the coupling effects of the squeeze film damping, pedestal contacts and elastic ring deformation. In other words, the equivalent stiffness and damping of ERSFD vary with pedestal contact, which can greatly influence the dynamics of rotor system. Nevertheless, estimating such a fluid–solid coupling process is extremely complex, especially for the fact that the elastic ring is dipped in the lubricating oil and both of them are sealed inside the bearing. It is very hard to observe the oil film distribution, elastic ring's deformation and pedestal contact through experiments. So analyzing the static and dynamic characteristics of ERSFD has to rely on numerical simulations presently. As mentioned above, most works on elastic ring deformation of ERSFD are based on simplifying every single segment as a clamed support beam. Such a simplification brings inaccurate result in most cases. In this paper, dynamic equations of ERSFD are established to obtain the dynamic response of ERSFD. Each precession period of ERSFD is divided into 10 time steps. At each time step, oil film coefficients are obtained by solving Reynolds equation using finite differential method, the deformation of elastic ring is calculated through FE method without any simplifications. A numerical method for observing pedestal contact status is proposed. Numerical results are presented to analyze the static characteristics and pedestal contact status of ERSFD. 2. Mathematical model Schematic model of an ERSFD is illustrated in Fig. 1. An ERSFD is composed of the elastic ring, the inner and outer oil films. Both the governing equations of inner and outer oil film pressure distributions can be deduced based on the generalized Reynolds equation. 2.1. Reynolds equation To simplify the governing equation, several assumptions are set as follows: 1) 2) 3) 4)
Newtonian fluid. Thin-film geometry. Negligible inertial, Laminar. Negligible body forces. The generalized Reynolds equation can be expressed in cylindrical coordinates as [24] 1 ∂ ∂ ∂h ∂h 3 ∂p 3 ∂p h ¼ 6μ ψ_ þ h þ ∂z ∂z ∂θ ∂θ ∂t R2 ∂θ
Fig. 1. Schematic model of ERSFD.
(1)
316
W. Zhang, Q. Ding / Journal of Sound and Vibration 346 (2015) 314–327
where R is radius, p the pressure to be solved, h the oil-film thickness, μ the fluid viscosity, and z and θ are the axial and the circumferential coordinates. e and Ψ are the eccentricity and attitude angle of the journal with the rotating speed Ω. e_ and ψ_ are their first derivatives respectively. Fig. 2 illustrates the position variations of the parts of an ERSFD, in which the solid and dotted lines present the initial and the present positions, respectively. C1 and C2 are the initial clearance of inner and outer oil film, e is the eccentricity of journal with respect to the bearing, h1 and h2 are thicknesses of the inner and outer oil films. Both the radial and circumferential deformation of the elastic ring will be considered. r is the radial deformation. The inner and outer oil film thicknesses can be estimated respectively as 8 h1 ðθÞ ¼ C 1 þrðθÞ þ e cos θ > > > > > ∂h1 ∂r < ¼ e sin θ (2) ∂θ ∂θ > > > ∂h1 ∂r > > _ þ e cos ¼ θ : ∂t ∂t and
8 h ðθÞ ¼ C 2 rðθÞ > > > 2 > > ∂r < ∂h2 ¼ ∂θ ∂θ > > > ∂h2 > > ¼0 : ∂t
(3)
Introducing non-dimensional parameters λ ¼ 2z=l, H ¼ h=C 1 , P ¼ p C 1 2 =2Ωμr 2 , where l is the elastic ring width, Eq. 1 can be rewritten in non-dimensional form as 2 ∂ ∂P 2R ∂ ∂P ∂H 6 ∂H (4) H3 H3 þ þ ¼6 l ∂θ ∂θ ∂λ ∂λ ∂θ Rcψ_ ∂t Eqs. (2) and (3) are rewritten as
(
H 1 ðθÞ ¼ 1 þ wðθÞ þ ε cos θ H 2 ðθÞ ¼ c3 wðθÞ
(5)
where ε ¼e/c, c3 ¼C2/C1 is the eccentricity ratio, w(θ) is the deformation function of the elastic ring. Thus the inner oil film provides damping due to the circumferential shear flow in inner chamber, while the outer oil film provides stiffness mainly [18]. 2.2. Equation of motion The oil film force acting on journal can be equivalent to two concentrating forces along x and y directions respectively, Fx and Fy, as shown in Fig. 2. The equation of journal motion with unbalanced exciting force, gravity and oil film force can be
Fig. 2. Movements of the ERSFD.
W. Zhang, Q. Ding / Journal of Sound and Vibration 346 (2015) 314–327
deduced as
8 <
mx€ þk1 x þ c1 x_ ¼ f Ω cos Ω t þF x
317
2
(6)
: my€ þ k2 y þc2 x_ ¼ f Ω sin Ω t þ F y þ G 2
where m is the mass, k1 and k2 are the equivalent stiffness of ERSFD, c1 and c2 are the equivalent damping of which are estimated under different pedestal contact statues [19], f is the eccentricity of rotor. Fx and Fy are oil film forces along x and y directions, G ¼mg is the gravity of journal. Introducing nondimensional variables τ ¼ Ωt, X ¼ 2x=l, Eq. (6) are rewritten into the following nondimensional form: 8 2f 2 > > X€ þ ω21 X þ n1 X_ ¼ cos τ þ F > < 2 x ml mlΩ (7) 2 2g > 2 > € _ 2f > F þ y : Y þ ω2 Y þ n2 X ¼ ml sin τ þ 2 2 mlΩ lΩ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 where ωi ¼ ki =ðmΩ Þ, i¼1, 2. 3. Solving methods 3.1. Reynolds equation Flattening the internal surface of cylindrical bearing into a rectangle plane along generatrix [25] and partitioning the rectangle plane, totally m grids along circumferential direction and n grids along axial direction are obtained as shown in Fig. 3. By means of the classical five-point finite difference method, the partial derivatives, ∂P=∂θ and ∂P=∂λ of any node can be expressed into differential expressions as 8 P i þ ð1=2Þ;j P i ð1=2Þ;j P i;j ð1=2Þ P i;j þ ð1=2Þ ∂P ∂P > > ¼ ; ¼ > > > ∂θ i;j ∂λ i;j Δθ Δλ > > > > < P P P i;j P i 1;j ∂P ∂P i þ 1;j i;j H3 ¼ H 3i þ ð1=2Þ;j ; H3 ¼ H 3i ð1=2Þ;j (8) θ Δθ θ Δθ ∂ ∂ > i þ ð1=2Þ;j i ð1=2Þ;j > > > > P i;j P i;j þ 1 P i;j 1 P i;j ∂P > 3 ∂P > > ¼ H 3i;j þ ð1=2Þ ; H3 ¼ H 3i;j ð1=2Þ > H ∂λ : Δθ Δθ ∂λ i;j ð1=2Þ i;j þ ð1=2Þ Accordingly, the differential expressions of the second order and some other partial derivatives are 8 > H 3i þ ð1=2Þ;j P i þ 1;j þ H 3i ð1=2Þ;j P i 1;j ðH 3i þ ð1=2Þ;j þ H3i ð1=2Þ;j ÞP i;j ∂ ∂P > > > H3 ¼ > < ∂θ ∂θ ðΔθÞ2 i;j > H 3i;j þ ð1=2Þ P i;j þ 1 þH 3i;j ð1=2Þ P i;j 1 ðH 3i;j þ ð1=2Þ þ H 3i;j ð1=2Þ ÞP i;j > ∂ 3 ∂P > > ¼ > : ∂λ H ∂λ ðΔλÞ2 i;j
∂H ∂θ
¼
H i þ ð1=2Þ;j H i ð1=2Þ;j
(9)
(10)
Δθ
Then the non-dimensional differential equations of Reynolds equation for both inner and outer oil films can be represented by Ai;j ðP i þ 1;j Þ1 þ Bi;j P i 1;j 1 þ C i;j P i;j þ 1 1 þDi;j P i;j 1 1 þ Ei;j P i;j 1 ¼ ðF i;j Þ1 (11) Ai;j ðP i þ 1;j Þ2 þ Bi;j P i 1;j 2 þ C i;j P i;j þ 1 2 þDi;j P i;j 1 2 þ Ei;j P i;j 2 ¼ ðF i;j Þ2
(12)
where A¼
H 3i þ ð1=2Þ;j
Δθ2
; A¼
H 3i ð1=2Þ;j
ðF i;j Þ1 ¼ 6
Δθ2
; C¼
2 3 2 3 2R H i;j þ ð1=2Þ 2R H i;j ð1=2Þ ; D ¼ ; E ¼ A þB þC þ D 2 l l Δλ Δλ2
H i þ ð1=2Þ;j H i ð1=2Þ;j
Δθ
þ
6e_ cos θ ; Rlψ_
ðF i;j Þ2 ¼ 6
H i þ ð1=2Þ;j H i ð1=2Þ;j
Δθ
where Ai,j to Fi,j are coefficients concerning with the oil film thickness and other structural parameters. P ¼Pi,j is dimensionless oil film pressure which can be expressed as Ai;j P i þ 1;j 1 þ Bi;j P i 1;j 1 þ C i;j P i;j þ 1 1 þDi;j P i;j 1 1 ðF i;j Þ1 (13) ðP i;j Þ1 ¼ Ei;j
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Fig. 3. Half-step difference sketch.
P i;j
¼ 2
Ai;j ðP i þ 1;j Þ2 þBi;j P i 1;j 2 þC i;j P i;j þ 1 2 þ Di;j P i;j 1 2 ðF i;j Þ2 Ei;j
(14)
Eqs. (13) and (14) can be solved by the interactive method. 3.2. Oil film forces, stiffness and damping The non-dimensional expressions of carrying capacity are displayed in Eq. (15) " #
Fr R θ R 1 ¼ θ12 1 P dλ dθ Fθ " # !" # Z θ2 Z 1 Fx cos θ P dλ ¼ dθ Fy sin θ θ1 1
(15)
where Fr is the radial force, Fθ is the circumferential force. Simpson integral method is used to estimate the inner integration along axial direction: Gi
1 ðP þ 4P i;2 þ2P i;3 þ⋯ þ 2P i;n 1 þ4P i;n þ P i;n þ 1 ÞΔλ 3 i;1
The outer integrations along the circumferential direction " # " # " # " # Fx sin θ1 sin θ2 sin θ3 1 G1 þ4G2 þ 2G3 þ⋯ Fy cos θ1 cos θ2 cos θ3 3 " # " # " #! sin θn þ 1 sin θn 1 sin θn þ 4Gn þ Gn þ 1 Δθ þ 2Gn 1 cos θn þ 1 cos θn 1 cos θn
(16)
(17)
Accordingly, oil film stiffness and damping can be obtained by the following equations [26]: k¼
Fr ; e
F c¼ θ eΩ
(18)
3.3. Deformation of the elastic ring The deformation of elastic ring is calculated by the finite element method. The three-dimensional model of elastic ring with 8 pedestals uniformly distributed on both inner and outer sides is adopted, as shown in Fig. 4(a). The sweep method is applied to mesh the three-dimension model. The number of nodes and elements along the circumferential and axial directions are the same with the number of the difference points in calculating the oil film pressure. So the oil film forces can be loaded into the deformation calculation of the elastic ring accurately. Actually 8200 nodes and 2500 elements are adopted, as shown in Fig. 4(b). For the boundary conditions on the FEM model, it is defined that the axial displacement of all the nodes are restrained and the radial constraints imposed on the end of shaft.
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319
Fig. 4. The elastic ring: (a) elastic ring model and (b) FEM model. Table 1 Structure parameters of ERSFD. Title
Value
Pedestal width (mm) Pedestal number Elastic ring width/thickness (mm) Elastic ring inner/outer diameter (mm) Inner/outer oil film inertial thickness (mm) Rotor speed (rev/min) Fluid viscosity(Pa s)
5 8 15.7/0.95 65/68 0.4/0.15 50,000 1.884 10 2
Fig. 5. Calculation schedule.
3.4. Solution procedure Structure parameters of the ERSFD are listed in Table 1. All the characteristic parameters, oil film thickness and elastic ring deformation are calculated one after another as the schedule shown in Fig. 5. Dynamic response of journal is
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numerically solved from Eq. (7). 10 simulated points are picked out from a stable-state vibration period to reveal the changes of oil film distribution, oil film thickness and elastic deformation condition during continuous procedure. Considering Somerfield semi-membrane hypothesis, SOR iterative method is used to solve Eqs. (13) and (14) to get the oil film force distribution. The iterative initial values are usually zeros or supply pressure. The oil film pressure results are loaded to finite element model to calculate the deformation of the elastic ring. Then the new oil film thickness can be obtained by Eqs. (9) and (10). 4. Numerical results Eq. (7) is numerically solved using the fourth-order Runge–kutta method. Omitting the first 800 periods of transient state data, the eccentricities and offset angles in one period of stable state are computed. Based on the calculations using the finite differential method and FE method, the inner and outer oil film pressure distribution can be obtained by Eqs. (13) and (14) respectively, the deformation of elastic ring is calculated by FE method, oil film force, stiffness and damping is calculated by Eqs. (17) and (18). The vibration data during a period is extracted, the discrete dots is summarized by least square method into characteristic curves. 4.1. Oil film pressure distributions Figs. 6 and 7 present selected circumferential pressure distributions of the inner and outer oil films during one period of the journal's whirl. Both the inner and outer oil films are sectionalized due to the existence of pedestals, which exhibit circumferential sealing oil, as a result the oil film thickness is very thin at the pedestal position and the oil film pressure is nearly 0. The oil film pressure value is varied under different eccentricities and offset angles. Oil film pressure gradient is
Fig. 6. Oil film pressure distributions of inner oil film: (a) e¼ 0.2701, ψ¼ 0.4125; (b) e ¼0.2136, ψ¼ 0.1113; (c) e¼ 0.0859, ψ¼ 1.8493 and (d) e¼ 0.2136, ψ ¼1.0311.
W. Zhang, Q. Ding / Journal of Sound and Vibration 346 (2015) 314–327
321
Fig. 7. Oil film pressure distributions of outer oil film: (a) e¼0.2701, ψ¼ 0.4125; (b) e¼0.2136, ψ¼ 0.1113; (c) e¼ 0.0859, ψ¼ 1.8493 and (d) e¼ 0.2136, ψ¼ 1.0311.
-3
3.5
x 10
0.95 0.85
3
0.75 0.65
2.5
0.55 0.45 hmin
Pmax
2 1.5
0.35 0.25
1
0.15
0.5
-0.05
0.05 -0.15
0 -0.5
-0.25 1
2
3
4
5
6 t
7
8
9
10
-0.35
1
2
3
4
5
6
7
8
9
10
t
Fig. 8. Maximum oil film pressure and minimum oil film thickness with varying time: (a) maximum oil film pressure and (b) minimum oil film thickness.
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lower when the eccentricity is small. There are pressure peaks in both inner and outer oil film, the pressure peaks distribute alternately, which corresponds to the alternately distribution of the inner and outer pedestals. In addition, the pressure peak positions do not match with the opposite position of offset angle, they become closer with the increase of eccentricity. The maximum oil film pressure and minimum oil film thickness curves during one period are presented in Fig. 8. Oil film thickness varies during a vibration period. Comparatively the variation of inner oil film thickness is more obvious due to the actions of eccentricity of bearing and the deformation of elastic ring. On the contrary, the outer oil film thickness changes slightly, because it is only influenced by the deformation of elastic ring. The time point where the oil film thickness is negative indicates pedestal–journal or pedestal–housing contact. At the same time, oil film pressure varies during a vibration period, inversely proportional to that of the oil film thickness. Fig. 9 illustrates the relationship between the journal eccentricity and maximum oil film pressure and bearing capacity of ERSFD. One finds that both the maximum oil film pressure and bearing capacity of the inner film are closed to that of the outer ones, which increase linearly under small eccentricities. That is the main difference between conventional SFDs and ERSFDs. Usually, the oil film pressure and bearing capacity are highly nonlinear for a conventional SFD as the rotor eccentricity increases [26]. For the investigated ERSFD, the bearing capacity exhibits nonlinear only when the eccentricity is large. Altogether, the elastic ring can lighten the nonlinearity of the increase of oil film pressure and capacity force. In other words, the linear range of oil film pressure and capacity force are widen through introducing the elastic ring into SFD. The relationship between the eccentricity and equivalent oil film stiffness and damping are shown in Fig. 10. Both the oil film stiffness and damping vary linearly with eccentricity within an eccentricity range. But they increase nonlinearly when
-3
3
x 10
0.1 inner oil force computing result inner oil force fitting result outer oil force computing result outer oil force fitting result
2.5
inner oil force computing result inner oil force fitting result outer oil force computing result outer oil force fitting result
0.05
1.5
F
Pmax
2
0
1 0.5 0
-0.05 0.08
0.1
0.12 0.14 0.16 0.18
0.2
0.22
0.24 0.26
0.08
0.1
0.12 0.14
e
0.16 0.18
0.2
0.22 0.24 0.26
e
Fig. 9. Maximum oil film pressure and bearing capacity with increase of eccentricity: (a) maximum oil film pressure and (b) bearing capacity.
0.18
0.16 inner oil stiffness computing result inner oil stiffness fitting result outer oil stiffness computing result outer oil stiffness fitting result
0.16 0.14
0.12 0.1 c
k
0.12 0.1
0.08 0.06
0.08
0.04
0.06 0.04
inner oil force computing result inner oil force fitting result outer oil force computing result outer oil force fitting result
0.14
0.02
0.08
0.1
0.12 0.14 0.16 0.18 e
0.2
0.22 0.24 0.26
0
0.08
0.1
0.12 0.14
0.16 0.18
0.2
0.22 0.24 0.26
e
Fig. 10. Equivalent oil film stiffness and damping with increase of eccentricity: (a) oil film stiffness and (b) oil film damping.
W. Zhang, Q. Ding / Journal of Sound and Vibration 346 (2015) 314–327
323
Fig. 11. Deformation of elastic ring: (a) e¼ 0.2701, ψ¼ 0.4125; (b) e¼0.2136, ψ¼ 0.1113; (c) e ¼0.0859, ψ¼ 1.8493 and (d) e¼0.2136, ψ¼ 1.0311.
the eccentricity is larger. Both the inner oil film stiffness and damping is larger than the outer one. Especially the outer oil film damping is much smaller than the inner oil film, which corresponds to the conclusion gained from the outer oil film Reynolds equations in Section 2.1.
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4.2. Deformation of elastic ring Fig. 11 presents the deformation of elastic ring. As the journal whirls, the deformation section is changing circumferentially along the ring with the displacement angle. Deformations can occur simultaneously in several sections on the elastic ring, inward and outward alternately, under compression of inner and outer oil films. When the eccentricity of journal is small, inner and outer oil film pressures are nearly equal, the shape of the deformed elastic ring tends to be elliptic. One notes that the inner oil film is influenced by both the eccentricity and elastic ring deformation, whereas the outer oil film is influenced only by the latter. When the eccentricity becomes larger and larger, the inner oil film pressure increases sharply, yet the outer oil film pressure keeps constant. The difference between the outer and inner pressures leads to the occurrence of big deformation of the elastic ring. When inner and outer oil film pressures are nearly equal, the shape of the deformed elastic ring tends to be elliptic. Otherwise it is complex. There is obvious deformation on the pedestal due to the cross distribution of the pedestals. Under this condition, pedestals contact happens more easily. Thus, simplifying single ring segment into a clamped supported beam or simply supported beam to investigate the deformation of the elastic ring may bring too large inaccuracy for the analysis.
4.3. Pedestal contacting status From Figs. 12 and 13, one finds that the inner oil film thickness alters obviously during a vibration period, while outer oil film thickness changes tiny. The points where the oil film thickness is negative implies pedestal contact happens, there are three pedestal contacting statuses: inner contact, outer contact, and both contacts. The inner pedestal contact happens more frequently, because inner oil film thickness is influenced by eccentricity more than that of the outer oil film thickness.
Fig. 12. Inner oil film thickness. (a) e¼ 0.2701, ψ¼ 0.4125; (b) e ¼0.2136, ψ¼ 0.1113; (c) e¼ 0.0859, ψ¼ 1.8493 and (d) e ¼0.2136, ψ¼ 1.0311.
W. Zhang, Q. Ding / Journal of Sound and Vibration 346 (2015) 314–327
Fig. 13. Outer oil film thickness: (a) e¼ 0.2701, ψ ¼0.4125; (b) e¼ 0.2136, ψ¼ 0.1113; (c) e¼ 0.0859, ψ¼ 1.8493 and (d) e¼ 0.2136, ψ¼ 1.0311.
inner oil force computing result inner oil force fitting result outer oil force computing result outer oil force fitting result
0.8 0.7 0.6
0.67
0.5 hmin
0.65
0.4
0.63 0.61
0.3
0.1 0.15 0.2 0.25
0.2 0.1 0 -0.1
0.08
0.1
0.12
0.14
0.16
0.18
0.2
e
Fig. 14. Minimum oil film thickness.
0.22
0.24
0.26
325
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Fig. 14 presents the relationship between eccentricity and oil film thickness. Minimum thickness of inner oil film is inversely proportional to eccentricity. Pedestal contact happens sometimes during a vibration period as the positive minimum oil film thickness, which corresponds to the phenomenon found in the experiment [17]. 5. Conclusion Oil film parametric characteristic and deformation of elastic ring of elastic ring squeeze film damper (ERSFD) are investigated during a vibration period by adopting the finite differential method and finite element method in this paper. Conclusions are summarized as follows: (1) Deformations can occur simultaneously in several sections on the elastic ring, inward and outward alternately, under compression of the inner and outer oil films. When inner and outer oil film pressures are nearly equal, the shape of the deformed elastic ring tends to be elliptic. Otherwise it is complex. Thus, simplifying single ring segment into a clamped supported beam or simply supported beam to investigate the deformation of the elastic ring may bring too large inaccuracy for the analysis. (2) Oil film thickness alters obviously during a vibration period. So the inner oil film thickness is influenced by eccentricity of bearing and the elastic ring deformation, while the outer one is influenced only by the elastic ring deformation. Consequently the pedestal contact status varies during a continuous running process, and the inner pedestal contact occurs more frequently. (3) Both the inner and outer oil films are sectionalized by the pedestals and the pressure peaks of the inner and outer oil films distribute alternately. Oil film stiffness is provided by both the inner and outer oil films, while oil film damping is provided mainly by inner oil film. The pressure peak positions do not match the opposite position of offset angle, which guarantees sufficient damping. (4) The maximum oil film pressure, bearing capacity, oil film stiffness and damping increase linearly within a certain eccentricity range. The fact implies that the deformation of elastic ring and different pedestal contacts can moderate nonlinear characteristics of ERSFD.
Acknowledgments This research is supported by the National Natural Science of China under Grant no.11272228 and 11332008, and the NSF of Tianjin under Grant no. 13JCZDJC34900. References [1] Y.S. Chen, H.B. Zhang, Review and prospect on the research of dynamic of complete aero-engine systems, Acta Aeronautica et Astronautica Sinica 32 (8) (2011) 1371–1391. (in Chinese). [2] R.A. Ibrahim, Recent advances in nonlinear passive vibration isolators, Journal of Sound and Vibration 314 (2008) 371–452. [3] C.S. Zhu, D.A. Robb, D.J. Ewins, Analysis of the multiple-solution response of a flexible rotor supported on non-linear squeeze film dampers, Journal of Sound and Vibration 252 (2002) 389–408. [4] C.C. Siew, M. Hill, R. Holmes, Evaluation of various fluid-film models for use in the analysis of squeeze film dampers with a central groove, Tribology International 35 (2002) 533–547. [5] J.Y. Zhao, I.W. Linnett, L.J. Mclean, Stability and bifurcation of unbalanced response of a squeeze film damped flexible rotor, ASME Journal Tribology 116 (1994) 361–368. [6] J.I. Inayat-Hussain, Bifurcations of a flexible rotor response in squeeze-film dampers without centering springs, Chaos, Solitons and Fractals 24 (2005) 583–596. [7] J.I. Inayat-Hussain, Bifurcations in the response of a flexible rotor in squeeze-film dampers with retainer springs, Chaos, Solitons and Fractals 39 (2009) 519–532. [8] F.L. Chu, R. Holmes, Efficient computation on nonlinear responses of a rotating assembly incorporating the squeeze-film damper, Computer Methods in Applied Mechanics and Engineering 164 (1988) 363–373. [9] J.Y. Zhao, E.J. Hahn, Eccentric operation and blade-loss simulation of a rigid rotor supported by an improved squeeze film damper, ASME Journal of Tribology 117 (1995) 490–497. [10] M.A. Rezvani, E.J. Hahn, Floating ring film damper: theoretical analysis, Tribology International 33 (2000) 249–258. [11] S.P. Zhang, L.T. Yan, Development of an efficient squeeze film damper for improving the control of rotor vibtation, Journal of Engineering for Gas Turbines and Power 113 (1991) 557–562. [12] P. David, D. Fleming, Clearance squeeze film damper for high load conditions, ASME Journal of Tribology 107 (1985) 275–279. [13] H. Hooshang, J.F. Walton, Advanced multi-squeeze film damper for rotor vibration control, Tribology Transaction 34 (1991) 489–496. [14] C.S. Zhu, X.X. Wang, Vibration control of flexible rotor by hybrid squeeze film damper, Journal of Mechanical Engineering 32 (3) (1996) 76–83. (in Chinese). [15] M. Leontiev, Damper supports, Samsung aerospace, Engine Research & Development Center, 1996. [16] X.J. Dai, G.M. Wang, Analysis on stiffness of corrugated ring in a rotor-bearing support, Journal of Harbin Institute of Technology 26 (5) (1994) 40–43. (in Chinese). [17] M. Zhou, Experimental investigation and application of an elastic ring squeeze film damper, Journal of Aerospace Power 13 (4) (1998) 403–407. [18] L. Cao, D.P. Gao, Investigation on critical speed characteristics of elastic ring SFD-rotor system, Journal of Propulsion Technology 29 (2) (2008) 235–239. [19] L. Cao, D.P. Gao, Damping machanism of elastic ring squeeze film damper, Journal of Vibration Engineering 20 (6) (2007) 584–588. [20] L. Cao, D.P. Gao, Investigation on stiffness characteristics of elastic ring, Gas Turbine Experiment and Research 20 (1) (2007) 36–38. [21] L. Cao, D.P. Gao, Investigation on design factor of elastic ring squeeze film damper, Gas Turbine Experiment and Research 19 (4) (2006) 30–34.
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[22] J. Hong, Y. Deng, Dynamic design method of elastic ring squeeze film damper, Journal of Beijing University of Aeronautics and Astronautics 32 (6) (2006) 649–653. [23] Y.X. Xu, X. Chen, Influence of orifice distribution on the characteristics of elastic ring squeeze film dampers for flywheel energy storage system, IEEE Transactions on Plasma Science 41 (5) (2013) 1272–1279. [24] F. Chorlton, Textbook of Fluid Dynamics, Van Nostrand, London, UK, 1967, 310–353. [25] H.T. Zhu, Q. Ding, Numerical analysis of static characteristics of herringbone grooved hydrodynamic journal bearing, Applied Mechanics and Materials 105–107 (2012) 2259–2262. [26] G. Adiletta, L.P. Della, The squeeze film damper over four decades of investigations. Part I: characteristics and operating features, Shock and Vibration Digest 34 (2002) 3–26.
Contents lists available at ScienceDirect
Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi
Elastic ring deformation and pedestal contact status analysis of elastic ring squeeze film damper Wei Zhang a,b, Qian Ding a,b,n a b
Department of Mechanics, Tianjin University, Tianjin 300072, China Tianjin Key Laboratory of Nonlinear Dynamics and Chaos Control, Tianjin 300072, China
a r t i c l e in f o
abstract
Article history: Received 30 November 2014 Received in revised form 5 February 2015 Accepted 8 February 2015 Handling Editor: L.G. Tham Available online 3 March 2015
This paper investigates the dynamic parametric characteristic of the elastic ring squeeze film damper (ERSFD). Firstly, the coupled oil film Reynolds equations and dynamic equations of an ERSFD supported rotor system are established. The finite differential method and numerical simulation are used to analyze the oil film pressure distribution, bearing capacity of ERSFD, oil film stiffness and damping characteristics during a vibration period. Then, based on the oil film pressure results, the deformation of elastic ring is revealed by the finite element method. Finally, pedestal contact status is analyzed according to the change of oil film thickness during a vibration period. The results reveal that the oil film pressure is sectionally continuous, the deformation of elastic ring is complex under the compression of inner and outer oil film, and different pedestal contacts occur in a vibration period. The level of nonlinearity of the bearing capacity, oil film stiffness and damping can be effectively lightened by application of the elastic ring. & 2015 Elsevier Ltd. All rights reserved.
1. Introduction Squeeze film damper (SFD) is a kind of passive vibration isolator widely used in rotor supporting structures of aeroengines for the stabilization and vibration control [1–4]. However, the stiffness and damping coefficients of the conventional SFDs are highly nonlinear, which can result in complex responses of the SFD supported rotor during operation. Zhao [5] and Zhu [3] investigated the jump phenomenon of SFD supported rotor by using the Levenberg–Marquardt nonlinear least squares method and the synchronous circular centered-orbit motion solution, respectively. Inayat-Hussain [6,7] investigated the influences of various parameters on bifurcations of the response of a rotors supported on SFD with and without retainer springs. An efficient computation method is also adopted to investigate the nonlinear responses of a rotor with SFD by Chu and Holmes [8]. The complicated response can be non-synchronous, which induces cyclic stress of the rotating shaft, and even fatigue problem [9,10]. So various suggestions have been proposed to improve the performance of SFD. A porous squeeze film damper (PSFD) was developed by Zhang and Yan [11], by making holes on the outer shell of oil film chamber. The main shortage is that the holes are easily blocked up by the impure oil for actual application. The duel clearance squeeze film damper (DCSFD) and the spiral foil multi-squeeze film damper (SFMSFD) were proposed by Fleming [12] and Hooshang and Waltton [13] respectively, the former can protect the rotor system, and the latter can improve the highly nonlinearity of oil
n
Corresponding author at: Department of Mechanics, Tianjin University, Tianjin 300072, China. Tel.: þ86 22 27401099. E-mail address: [email protected] (Q. Ding).
http://dx.doi.org/10.1016/j.jsv.2015.02.015 0022-460X/& 2015 Elsevier Ltd. All rights reserved.
W. Zhang, Q. Ding / Journal of Sound and Vibration 346 (2015) 314–327
315
film stiffness. Nevertheless, requirement of high level manufacturing technique limits the wide application. Hybrid squeeze film damper (HSFD) was put forward successively by Zhu and Wang [14]. Whereas its structure is too complex to be applied in aero-engine. Comparatively, the elastic ring squeeze film damper (ERSFD) [15] is effective and simple designed and the ERSFDs were already used in aircraft engine successfully. By locating an elastic ring in the oil chamber to separate the previous large chamber into two small oil chambers, the shortage of high nonlinearity existed in conventional SFD is efficiently overcome. Dai and Wang [16] derived the corrugated ring's stiffness coefficient and proposed a bent correcting coefficient using the FE method. Zhou [17] established the oil film pressure governing equation and investigated the vibration damping mechanism of the ERSFD. By simplifying every segment of elastic ring as a clamped beam and calculating its deformation, Cao and Gao [18–21] investigated the elastic ring stiffness and oil film pressure characteristics of an ERSFD, and the dynamic characteristic of a ERSFD supported rotor system as well. Hong [22] and Xu [23] investigated the influences of the number of pedestals, the initial oil film thickness and the orifice distribution on oil film stiffness and damping characteristics of ERSFD using FE method. All these researches reveal that by adding an elastic ring into SFD, the critical speed and peak response amplitude of rotor system are effectively adjusted through the coupling effects of the squeeze film damping, pedestal contacts and elastic ring deformation. In other words, the equivalent stiffness and damping of ERSFD vary with pedestal contact, which can greatly influence the dynamics of rotor system. Nevertheless, estimating such a fluid–solid coupling process is extremely complex, especially for the fact that the elastic ring is dipped in the lubricating oil and both of them are sealed inside the bearing. It is very hard to observe the oil film distribution, elastic ring's deformation and pedestal contact through experiments. So analyzing the static and dynamic characteristics of ERSFD has to rely on numerical simulations presently. As mentioned above, most works on elastic ring deformation of ERSFD are based on simplifying every single segment as a clamed support beam. Such a simplification brings inaccurate result in most cases. In this paper, dynamic equations of ERSFD are established to obtain the dynamic response of ERSFD. Each precession period of ERSFD is divided into 10 time steps. At each time step, oil film coefficients are obtained by solving Reynolds equation using finite differential method, the deformation of elastic ring is calculated through FE method without any simplifications. A numerical method for observing pedestal contact status is proposed. Numerical results are presented to analyze the static characteristics and pedestal contact status of ERSFD. 2. Mathematical model Schematic model of an ERSFD is illustrated in Fig. 1. An ERSFD is composed of the elastic ring, the inner and outer oil films. Both the governing equations of inner and outer oil film pressure distributions can be deduced based on the generalized Reynolds equation. 2.1. Reynolds equation To simplify the governing equation, several assumptions are set as follows: 1) 2) 3) 4)
Newtonian fluid. Thin-film geometry. Negligible inertial, Laminar. Negligible body forces. The generalized Reynolds equation can be expressed in cylindrical coordinates as [24] 1 ∂ ∂ ∂h ∂h 3 ∂p 3 ∂p h ¼ 6μ ψ_ þ h þ ∂z ∂z ∂θ ∂θ ∂t R2 ∂θ
Fig. 1. Schematic model of ERSFD.
(1)
316
W. Zhang, Q. Ding / Journal of Sound and Vibration 346 (2015) 314–327
where R is radius, p the pressure to be solved, h the oil-film thickness, μ the fluid viscosity, and z and θ are the axial and the circumferential coordinates. e and Ψ are the eccentricity and attitude angle of the journal with the rotating speed Ω. e_ and ψ_ are their first derivatives respectively. Fig. 2 illustrates the position variations of the parts of an ERSFD, in which the solid and dotted lines present the initial and the present positions, respectively. C1 and C2 are the initial clearance of inner and outer oil film, e is the eccentricity of journal with respect to the bearing, h1 and h2 are thicknesses of the inner and outer oil films. Both the radial and circumferential deformation of the elastic ring will be considered. r is the radial deformation. The inner and outer oil film thicknesses can be estimated respectively as 8 h1 ðθÞ ¼ C 1 þrðθÞ þ e cos θ > > > > > ∂h1 ∂r < ¼ e sin θ (2) ∂θ ∂θ > > > ∂h1 ∂r > > _ þ e cos ¼ θ : ∂t ∂t and
8 h ðθÞ ¼ C 2 rðθÞ > > > 2 > > ∂r < ∂h2 ¼ ∂θ ∂θ > > > ∂h2 > > ¼0 : ∂t
(3)
Introducing non-dimensional parameters λ ¼ 2z=l, H ¼ h=C 1 , P ¼ p C 1 2 =2Ωμr 2 , where l is the elastic ring width, Eq. 1 can be rewritten in non-dimensional form as 2 ∂ ∂P 2R ∂ ∂P ∂H 6 ∂H (4) H3 H3 þ þ ¼6 l ∂θ ∂θ ∂λ ∂λ ∂θ Rcψ_ ∂t Eqs. (2) and (3) are rewritten as
(
H 1 ðθÞ ¼ 1 þ wðθÞ þ ε cos θ H 2 ðθÞ ¼ c3 wðθÞ
(5)
where ε ¼e/c, c3 ¼C2/C1 is the eccentricity ratio, w(θ) is the deformation function of the elastic ring. Thus the inner oil film provides damping due to the circumferential shear flow in inner chamber, while the outer oil film provides stiffness mainly [18]. 2.2. Equation of motion The oil film force acting on journal can be equivalent to two concentrating forces along x and y directions respectively, Fx and Fy, as shown in Fig. 2. The equation of journal motion with unbalanced exciting force, gravity and oil film force can be
Fig. 2. Movements of the ERSFD.
W. Zhang, Q. Ding / Journal of Sound and Vibration 346 (2015) 314–327
deduced as
8 <
mx€ þk1 x þ c1 x_ ¼ f Ω cos Ω t þF x
317
2
(6)
: my€ þ k2 y þc2 x_ ¼ f Ω sin Ω t þ F y þ G 2
where m is the mass, k1 and k2 are the equivalent stiffness of ERSFD, c1 and c2 are the equivalent damping of which are estimated under different pedestal contact statues [19], f is the eccentricity of rotor. Fx and Fy are oil film forces along x and y directions, G ¼mg is the gravity of journal. Introducing nondimensional variables τ ¼ Ωt, X ¼ 2x=l, Eq. (6) are rewritten into the following nondimensional form: 8 2f 2 > > X€ þ ω21 X þ n1 X_ ¼ cos τ þ F > < 2 x ml mlΩ (7) 2 2g > 2 > € _ 2f > F þ y : Y þ ω2 Y þ n2 X ¼ ml sin τ þ 2 2 mlΩ lΩ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 where ωi ¼ ki =ðmΩ Þ, i¼1, 2. 3. Solving methods 3.1. Reynolds equation Flattening the internal surface of cylindrical bearing into a rectangle plane along generatrix [25] and partitioning the rectangle plane, totally m grids along circumferential direction and n grids along axial direction are obtained as shown in Fig. 3. By means of the classical five-point finite difference method, the partial derivatives, ∂P=∂θ and ∂P=∂λ of any node can be expressed into differential expressions as 8 P i þ ð1=2Þ;j P i ð1=2Þ;j P i;j ð1=2Þ P i;j þ ð1=2Þ ∂P ∂P > > ¼ ; ¼ > > > ∂θ i;j ∂λ i;j Δθ Δλ > > > > < P P P i;j P i 1;j ∂P ∂P i þ 1;j i;j H3 ¼ H 3i þ ð1=2Þ;j ; H3 ¼ H 3i ð1=2Þ;j (8) θ Δθ θ Δθ ∂ ∂ > i þ ð1=2Þ;j i ð1=2Þ;j > > > > P i;j P i;j þ 1 P i;j 1 P i;j ∂P > 3 ∂P > > ¼ H 3i;j þ ð1=2Þ ; H3 ¼ H 3i;j ð1=2Þ > H ∂λ : Δθ Δθ ∂λ i;j ð1=2Þ i;j þ ð1=2Þ Accordingly, the differential expressions of the second order and some other partial derivatives are 8 > H 3i þ ð1=2Þ;j P i þ 1;j þ H 3i ð1=2Þ;j P i 1;j ðH 3i þ ð1=2Þ;j þ H3i ð1=2Þ;j ÞP i;j ∂ ∂P > > > H3 ¼ > < ∂θ ∂θ ðΔθÞ2 i;j > H 3i;j þ ð1=2Þ P i;j þ 1 þH 3i;j ð1=2Þ P i;j 1 ðH 3i;j þ ð1=2Þ þ H 3i;j ð1=2Þ ÞP i;j > ∂ 3 ∂P > > ¼ > : ∂λ H ∂λ ðΔλÞ2 i;j
∂H ∂θ
¼
H i þ ð1=2Þ;j H i ð1=2Þ;j
(9)
(10)
Δθ
Then the non-dimensional differential equations of Reynolds equation for both inner and outer oil films can be represented by Ai;j ðP i þ 1;j Þ1 þ Bi;j P i 1;j 1 þ C i;j P i;j þ 1 1 þDi;j P i;j 1 1 þ Ei;j P i;j 1 ¼ ðF i;j Þ1 (11) Ai;j ðP i þ 1;j Þ2 þ Bi;j P i 1;j 2 þ C i;j P i;j þ 1 2 þDi;j P i;j 1 2 þ Ei;j P i;j 2 ¼ ðF i;j Þ2
(12)
where A¼
H 3i þ ð1=2Þ;j
Δθ2
; A¼
H 3i ð1=2Þ;j
ðF i;j Þ1 ¼ 6
Δθ2
; C¼
2 3 2 3 2R H i;j þ ð1=2Þ 2R H i;j ð1=2Þ ; D ¼ ; E ¼ A þB þC þ D 2 l l Δλ Δλ2
H i þ ð1=2Þ;j H i ð1=2Þ;j
Δθ
þ
6e_ cos θ ; Rlψ_
ðF i;j Þ2 ¼ 6
H i þ ð1=2Þ;j H i ð1=2Þ;j
Δθ
where Ai,j to Fi,j are coefficients concerning with the oil film thickness and other structural parameters. P ¼Pi,j is dimensionless oil film pressure which can be expressed as Ai;j P i þ 1;j 1 þ Bi;j P i 1;j 1 þ C i;j P i;j þ 1 1 þDi;j P i;j 1 1 ðF i;j Þ1 (13) ðP i;j Þ1 ¼ Ei;j
318
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Fig. 3. Half-step difference sketch.
P i;j
¼ 2
Ai;j ðP i þ 1;j Þ2 þBi;j P i 1;j 2 þC i;j P i;j þ 1 2 þ Di;j P i;j 1 2 ðF i;j Þ2 Ei;j
(14)
Eqs. (13) and (14) can be solved by the interactive method. 3.2. Oil film forces, stiffness and damping The non-dimensional expressions of carrying capacity are displayed in Eq. (15) " #
Fr R θ R 1 ¼ θ12 1 P dλ dθ Fθ " # !" # Z θ2 Z 1 Fx cos θ P dλ ¼ dθ Fy sin θ θ1 1
(15)
where Fr is the radial force, Fθ is the circumferential force. Simpson integral method is used to estimate the inner integration along axial direction: Gi
1 ðP þ 4P i;2 þ2P i;3 þ⋯ þ 2P i;n 1 þ4P i;n þ P i;n þ 1 ÞΔλ 3 i;1
The outer integrations along the circumferential direction " # " # " # " # Fx sin θ1 sin θ2 sin θ3 1 G1 þ4G2 þ 2G3 þ⋯ Fy cos θ1 cos θ2 cos θ3 3 " # " # " #! sin θn þ 1 sin θn 1 sin θn þ 4Gn þ Gn þ 1 Δθ þ 2Gn 1 cos θn þ 1 cos θn 1 cos θn
(16)
(17)
Accordingly, oil film stiffness and damping can be obtained by the following equations [26]: k¼
Fr ; e
F c¼ θ eΩ
(18)
3.3. Deformation of the elastic ring The deformation of elastic ring is calculated by the finite element method. The three-dimensional model of elastic ring with 8 pedestals uniformly distributed on both inner and outer sides is adopted, as shown in Fig. 4(a). The sweep method is applied to mesh the three-dimension model. The number of nodes and elements along the circumferential and axial directions are the same with the number of the difference points in calculating the oil film pressure. So the oil film forces can be loaded into the deformation calculation of the elastic ring accurately. Actually 8200 nodes and 2500 elements are adopted, as shown in Fig. 4(b). For the boundary conditions on the FEM model, it is defined that the axial displacement of all the nodes are restrained and the radial constraints imposed on the end of shaft.
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319
Fig. 4. The elastic ring: (a) elastic ring model and (b) FEM model. Table 1 Structure parameters of ERSFD. Title
Value
Pedestal width (mm) Pedestal number Elastic ring width/thickness (mm) Elastic ring inner/outer diameter (mm) Inner/outer oil film inertial thickness (mm) Rotor speed (rev/min) Fluid viscosity(Pa s)
5 8 15.7/0.95 65/68 0.4/0.15 50,000 1.884 10 2
Fig. 5. Calculation schedule.
3.4. Solution procedure Structure parameters of the ERSFD are listed in Table 1. All the characteristic parameters, oil film thickness and elastic ring deformation are calculated one after another as the schedule shown in Fig. 5. Dynamic response of journal is
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numerically solved from Eq. (7). 10 simulated points are picked out from a stable-state vibration period to reveal the changes of oil film distribution, oil film thickness and elastic deformation condition during continuous procedure. Considering Somerfield semi-membrane hypothesis, SOR iterative method is used to solve Eqs. (13) and (14) to get the oil film force distribution. The iterative initial values are usually zeros or supply pressure. The oil film pressure results are loaded to finite element model to calculate the deformation of the elastic ring. Then the new oil film thickness can be obtained by Eqs. (9) and (10). 4. Numerical results Eq. (7) is numerically solved using the fourth-order Runge–kutta method. Omitting the first 800 periods of transient state data, the eccentricities and offset angles in one period of stable state are computed. Based on the calculations using the finite differential method and FE method, the inner and outer oil film pressure distribution can be obtained by Eqs. (13) and (14) respectively, the deformation of elastic ring is calculated by FE method, oil film force, stiffness and damping is calculated by Eqs. (17) and (18). The vibration data during a period is extracted, the discrete dots is summarized by least square method into characteristic curves. 4.1. Oil film pressure distributions Figs. 6 and 7 present selected circumferential pressure distributions of the inner and outer oil films during one period of the journal's whirl. Both the inner and outer oil films are sectionalized due to the existence of pedestals, which exhibit circumferential sealing oil, as a result the oil film thickness is very thin at the pedestal position and the oil film pressure is nearly 0. The oil film pressure value is varied under different eccentricities and offset angles. Oil film pressure gradient is
Fig. 6. Oil film pressure distributions of inner oil film: (a) e¼ 0.2701, ψ¼ 0.4125; (b) e ¼0.2136, ψ¼ 0.1113; (c) e¼ 0.0859, ψ¼ 1.8493 and (d) e¼ 0.2136, ψ ¼1.0311.
W. Zhang, Q. Ding / Journal of Sound and Vibration 346 (2015) 314–327
321
Fig. 7. Oil film pressure distributions of outer oil film: (a) e¼0.2701, ψ¼ 0.4125; (b) e¼0.2136, ψ¼ 0.1113; (c) e¼ 0.0859, ψ¼ 1.8493 and (d) e¼ 0.2136, ψ¼ 1.0311.
-3
3.5
x 10
0.95 0.85
3
0.75 0.65
2.5
0.55 0.45 hmin
Pmax
2 1.5
0.35 0.25
1
0.15
0.5
-0.05
0.05 -0.15
0 -0.5
-0.25 1
2
3
4
5
6 t
7
8
9
10
-0.35
1
2
3
4
5
6
7
8
9
10
t
Fig. 8. Maximum oil film pressure and minimum oil film thickness with varying time: (a) maximum oil film pressure and (b) minimum oil film thickness.
322
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lower when the eccentricity is small. There are pressure peaks in both inner and outer oil film, the pressure peaks distribute alternately, which corresponds to the alternately distribution of the inner and outer pedestals. In addition, the pressure peak positions do not match with the opposite position of offset angle, they become closer with the increase of eccentricity. The maximum oil film pressure and minimum oil film thickness curves during one period are presented in Fig. 8. Oil film thickness varies during a vibration period. Comparatively the variation of inner oil film thickness is more obvious due to the actions of eccentricity of bearing and the deformation of elastic ring. On the contrary, the outer oil film thickness changes slightly, because it is only influenced by the deformation of elastic ring. The time point where the oil film thickness is negative indicates pedestal–journal or pedestal–housing contact. At the same time, oil film pressure varies during a vibration period, inversely proportional to that of the oil film thickness. Fig. 9 illustrates the relationship between the journal eccentricity and maximum oil film pressure and bearing capacity of ERSFD. One finds that both the maximum oil film pressure and bearing capacity of the inner film are closed to that of the outer ones, which increase linearly under small eccentricities. That is the main difference between conventional SFDs and ERSFDs. Usually, the oil film pressure and bearing capacity are highly nonlinear for a conventional SFD as the rotor eccentricity increases [26]. For the investigated ERSFD, the bearing capacity exhibits nonlinear only when the eccentricity is large. Altogether, the elastic ring can lighten the nonlinearity of the increase of oil film pressure and capacity force. In other words, the linear range of oil film pressure and capacity force are widen through introducing the elastic ring into SFD. The relationship between the eccentricity and equivalent oil film stiffness and damping are shown in Fig. 10. Both the oil film stiffness and damping vary linearly with eccentricity within an eccentricity range. But they increase nonlinearly when
-3
3
x 10
0.1 inner oil force computing result inner oil force fitting result outer oil force computing result outer oil force fitting result
2.5
inner oil force computing result inner oil force fitting result outer oil force computing result outer oil force fitting result
0.05
1.5
F
Pmax
2
0
1 0.5 0
-0.05 0.08
0.1
0.12 0.14 0.16 0.18
0.2
0.22
0.24 0.26
0.08
0.1
0.12 0.14
e
0.16 0.18
0.2
0.22 0.24 0.26
e
Fig. 9. Maximum oil film pressure and bearing capacity with increase of eccentricity: (a) maximum oil film pressure and (b) bearing capacity.
0.18
0.16 inner oil stiffness computing result inner oil stiffness fitting result outer oil stiffness computing result outer oil stiffness fitting result
0.16 0.14
0.12 0.1 c
k
0.12 0.1
0.08 0.06
0.08
0.04
0.06 0.04
inner oil force computing result inner oil force fitting result outer oil force computing result outer oil force fitting result
0.14
0.02
0.08
0.1
0.12 0.14 0.16 0.18 e
0.2
0.22 0.24 0.26
0
0.08
0.1
0.12 0.14
0.16 0.18
0.2
0.22 0.24 0.26
e
Fig. 10. Equivalent oil film stiffness and damping with increase of eccentricity: (a) oil film stiffness and (b) oil film damping.
W. Zhang, Q. Ding / Journal of Sound and Vibration 346 (2015) 314–327
323
Fig. 11. Deformation of elastic ring: (a) e¼ 0.2701, ψ¼ 0.4125; (b) e¼0.2136, ψ¼ 0.1113; (c) e ¼0.0859, ψ¼ 1.8493 and (d) e¼0.2136, ψ¼ 1.0311.
the eccentricity is larger. Both the inner oil film stiffness and damping is larger than the outer one. Especially the outer oil film damping is much smaller than the inner oil film, which corresponds to the conclusion gained from the outer oil film Reynolds equations in Section 2.1.
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4.2. Deformation of elastic ring Fig. 11 presents the deformation of elastic ring. As the journal whirls, the deformation section is changing circumferentially along the ring with the displacement angle. Deformations can occur simultaneously in several sections on the elastic ring, inward and outward alternately, under compression of inner and outer oil films. When the eccentricity of journal is small, inner and outer oil film pressures are nearly equal, the shape of the deformed elastic ring tends to be elliptic. One notes that the inner oil film is influenced by both the eccentricity and elastic ring deformation, whereas the outer oil film is influenced only by the latter. When the eccentricity becomes larger and larger, the inner oil film pressure increases sharply, yet the outer oil film pressure keeps constant. The difference between the outer and inner pressures leads to the occurrence of big deformation of the elastic ring. When inner and outer oil film pressures are nearly equal, the shape of the deformed elastic ring tends to be elliptic. Otherwise it is complex. There is obvious deformation on the pedestal due to the cross distribution of the pedestals. Under this condition, pedestals contact happens more easily. Thus, simplifying single ring segment into a clamped supported beam or simply supported beam to investigate the deformation of the elastic ring may bring too large inaccuracy for the analysis.
4.3. Pedestal contacting status From Figs. 12 and 13, one finds that the inner oil film thickness alters obviously during a vibration period, while outer oil film thickness changes tiny. The points where the oil film thickness is negative implies pedestal contact happens, there are three pedestal contacting statuses: inner contact, outer contact, and both contacts. The inner pedestal contact happens more frequently, because inner oil film thickness is influenced by eccentricity more than that of the outer oil film thickness.
Fig. 12. Inner oil film thickness. (a) e¼ 0.2701, ψ¼ 0.4125; (b) e ¼0.2136, ψ¼ 0.1113; (c) e¼ 0.0859, ψ¼ 1.8493 and (d) e ¼0.2136, ψ¼ 1.0311.
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Fig. 13. Outer oil film thickness: (a) e¼ 0.2701, ψ ¼0.4125; (b) e¼ 0.2136, ψ¼ 0.1113; (c) e¼ 0.0859, ψ¼ 1.8493 and (d) e¼ 0.2136, ψ¼ 1.0311.
inner oil force computing result inner oil force fitting result outer oil force computing result outer oil force fitting result
0.8 0.7 0.6
0.67
0.5 hmin
0.65
0.4
0.63 0.61
0.3
0.1 0.15 0.2 0.25
0.2 0.1 0 -0.1
0.08
0.1
0.12
0.14
0.16
0.18
0.2
e
Fig. 14. Minimum oil film thickness.
0.22
0.24
0.26
325
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Fig. 14 presents the relationship between eccentricity and oil film thickness. Minimum thickness of inner oil film is inversely proportional to eccentricity. Pedestal contact happens sometimes during a vibration period as the positive minimum oil film thickness, which corresponds to the phenomenon found in the experiment [17]. 5. Conclusion Oil film parametric characteristic and deformation of elastic ring of elastic ring squeeze film damper (ERSFD) are investigated during a vibration period by adopting the finite differential method and finite element method in this paper. Conclusions are summarized as follows: (1) Deformations can occur simultaneously in several sections on the elastic ring, inward and outward alternately, under compression of the inner and outer oil films. When inner and outer oil film pressures are nearly equal, the shape of the deformed elastic ring tends to be elliptic. Otherwise it is complex. Thus, simplifying single ring segment into a clamped supported beam or simply supported beam to investigate the deformation of the elastic ring may bring too large inaccuracy for the analysis. (2) Oil film thickness alters obviously during a vibration period. So the inner oil film thickness is influenced by eccentricity of bearing and the elastic ring deformation, while the outer one is influenced only by the elastic ring deformation. Consequently the pedestal contact status varies during a continuous running process, and the inner pedestal contact occurs more frequently. (3) Both the inner and outer oil films are sectionalized by the pedestals and the pressure peaks of the inner and outer oil films distribute alternately. Oil film stiffness is provided by both the inner and outer oil films, while oil film damping is provided mainly by inner oil film. The pressure peak positions do not match the opposite position of offset angle, which guarantees sufficient damping. (4) The maximum oil film pressure, bearing capacity, oil film stiffness and damping increase linearly within a certain eccentricity range. The fact implies that the deformation of elastic ring and different pedestal contacts can moderate nonlinear characteristics of ERSFD.
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