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Analytical study on effect of a circumferential feeding groove on unbalance response of a flexible rotor in squeeze film damper
Tribology International 32 (1999) 559–566 www.elsevier.com/locate/triboint
Analytical study on effect of a circumferential feeding groove on unbalance response of a flexible rotor in squeeze film damper Qingchang Tan *, Xiaohua Li Department of Mechanical Engineering, Jilin University of Technology, Changchun 130025, People’s Republic of China Received 20 April 1999; received in revised form 6 August 1999; accepted 20 October 1999
Abstract Of recent years, a series of researches have shown that a circumferential feeding groove of squeeze film damper (SFD) has evident effect on fluid film forces in SFD. Therefore, the feeding groove also affects dynamic responses of a rotor in SFD. Present work studies the effect of the feeding groove on unbalance response of a flexible rotor in SFD based on new film force models that include effects of the feeding groove and fluid inertia on dynamic characteristics of the fluid film in SFD. Compared with the published work, unbalance responses predicated under considering effect of the feeding groove on the dynamic characteristics in SFD are small, and rotor speed region for unbalance responses with multiple solutions is different, affecting the stability of a rotor system. And the effect of the feeding groove on the unbalance response is related to action of fluid inertia. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Unbalance response; Squeeze film damper; Film force; Rotor
1. Introduction Squeeze Film Dampers (SFDs) are widely used in high speed turbo-machinery due to their capability to isolate vibrations, to reduce forces transmitted to structural components, and to improve the stability characteristics of rotor-bearing systems. The behavior of a rotor system in service is dependent upon values of SFD dynamic coefficients in the governing differential equations. Therefore, that theoretical models for SFD dynamic coefficients are correct and accurate is very important for predicting rotor-system behavior. Controlled tests have been performed to verify SFD performance and to determine the validity of theoretical models when applied to actual hardware. However, experience has shown that the theoretical models based on traditional lubrication theory are limited in their applicability and fail to explain the performance of SFDs over a wide margin of operating conditions. Besides other reasons, investigators [1,2] have shown that main reason for such deviation is that in traditional lubrication
* Corresponding author. Tel. 431-5680278. E-mail address: [email protected] (Q. Tan).
theory, the feeding groove in SFD only provides a constant pressure around the journal surface and effectively splits SFD width into two film lands, resulting in the two film lands operating independently. Much experimental work [3–9] involving SFDs with a circumferential feeding groove has reported significant levels of dynamic pressure at the feeding groove. Recently, experimental work [9] has shown that levels of dynamic pressures measured at the feeding groove are as large as those generated at film lands. Therefore, the feeding groove affects dynamic characteristic of SFD and unbalance responses of a rotor system in SFDs. Since the theoretical models for dynamic coefficients of SFD have been developed all based on traditional lubrication theory, for authors’ knowledge there is not a literature that considers the effect of the feeding groove on SFD dynamic coefficients in studying rotor-system responses. Of recent years, San Andres [1] and authors [10] have advanced new models that include the effect of the feeding groove on SFD coefficients and yield good predications of the coefficients. This article investigates the effects of the feeding groove on unbalance response of a flexible rotor in SFDs based on authors’ previous work [10]. Fluid force models developed in previous work [10] include the effect
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Q. Tan, X. Li / Tribology International 32 (1999) 559–566
Nomenclature B wb c cg 2c1 C1 e Fd, Ft, r1 2m1 m2 U M1 M2 k1 k2 K1 K2 L l lg R Re t e w wc wr h j g
dimensional coefficient of fluid film force, B=RL3hw/c2 non-dimensional SFD parameter; wb=RL3h/c3(m1+m2) radial clearance between the journal and bearing groove depth viscous damping of disc in Fig. 1 =c1/[w(m1+m2)] eccentricity of journal with respective to bearing Fr resultant of fluid film forces, and its components in the t and r directions, respectively, |Fd|=F unbalance eccentricity lumped mass of disc in Fig. 1 lumped mass at bearing stations in Fig. 1 unbalance parameter U=r1m1/c(m1+m2) =m1/(m1+m2) =m2/(m1+m2) rotor stiffness in Fig. 1 radial stiffness of retainer spring in Fig. 1 =k1/[(w2(m1+m2)] =k2/[(w2(m1+m2)] whole length of a damper one-side length of film land one-side length of feeding groove journal radius gap Reynolds number, Re=c2rw/h time eccentricity ratio with respect to bearing, e=e/c rotor speed =√k1/m1 =√k2/(m1+m2) dynamic viscosity of oil =lg/l =c/cg
of fluid inertia on dynamic characteristics of SFD film. Therefore, present work is carried out under considering the effect of fluid inertia on unbalance response. The traditional Jeffcott rotor is carried on SFDs with linear centering springs and circumferential-feeding grooves, as shown in Fig. 1. The centering spring is assumed to be adjusted to offset any gravitational load. Alternatively, the axis of rotation is vertical, resulting in no gravitational loading.
2. Analysis model In the analysis of rotor-system performance, the following assumptions are made: (a) a rotor may be subjected to external synchronous rotating loads; (b) the rolling element bearings are radically rigid; (c) the damping is viscous; (d) torsional and axial vibrations of a rotor is negligible; (e) the rotor is axially symmetric. Variables for describing the rotor system are defined, as shown in
Fig. 1. Fig. 1 is a diagram of the rotor system with node 1 being taken at the central disc of mass 2m1, and nodes 2 and 3 at ends of the rotor which is supported by identical SFDs. Lumped mass at bearing ends is m2, the retainer spring for central preloading has stiffness k2 and the rotor stiffness between central and either end node is k1. All unbalance is assumed to be at the disc, resulting in a disc mass eccentricity r1. Viscous damping at the disc is 2C1. Since the rotor is symmetric about the disc, it will suffice to consider one half of the system only. The equations of motion, appropriately ordered, for this system at a rotor speed w are given by: m1x¨1⫹c1x˙1⫹k1(x1⫺x3)⫽r1m1w2 cos(wt)
(1)
m1x¨2⫹c1x˙2⫹k1(x2⫺x4)⫽r1m1w2 sin(wt)
(2)
m2x¨3⫹k1(x3⫺x1)⫹k2x3⫽F cos(wt⫹j)
(3)
m2x¨4⫹k1(x4⫺x2)⫹k2x4⫽F sin(wt⫹j)
(4)
Eq. (2) is multiplied by imaginary j, then the Eq. (2) plus Eq. (1) yields:
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Fig. 1. Details of physical system.
m1z¨1⫹c1z˙1⫹k1(z1⫺z2)⫽r1m1w2ejwt
(5)
where z1=x1+jx2 is vector of system displacements at the disc, and z2=x3+jx4 is vector of relative system displacements at SFD station. Real and imaginary parts of z1 stand for, respectively, x1 and x2. Similarly, from Eq. (3) and Eq. (4), one may get: m2z¨2⫹k1(z2⫺z1)⫹k2z2⫽Fej(wt+j) As shown in the Appendix A, Fe expressed as: Fej(wt+j)⫽z2(Fr⫹jFt)/e
(6) j(wt+j)
may be (7)
where Fr and Ft are fluid film forces in SFD. From [10], Fr and Ft for π fluid film can be expressed as: Fr=−Bfr
(8)
Ft=−Bft
Ilm n in the above expressions are defined by the famous Booker [16] journal bearing integral form (see Appendix B). fr and ft in Eq. (8) are dimensionless radial and tangential forces of SFD fluid film. Gt and Gr in expressions of fr and ft show respectively effects of the feeding groove on viscosity damping part and inertia part of fluid film forces, as shown in Fig. 2. Also, Reynolds number Re reflects effect of fluid inertia on film forces. Therefore, compared with the published work, for example [11–13], present work investigates effects of the feeding groove and fluid inertia on unbalance response of a flexible rotor system in SFDs. For circular synchronous motion, solutions of Eq. (5) and Eq. (6) can be written as: z1⫽z¯1ejwt and z2⫽z¯2ejwt The above expressions are substituted into Eq. (5) and Eq. (6), then the two equations are transformed into nondimension forms using dividing each equation by (m1+m2)cw2 and letting t=wt:
where fr⫽(GtI11 3 ⫺ReGrIr)e; ft(GtI20 3 ⫺ReGrIt)e; Ir⫽I /12⫹I e/5; 02 1
21 2
30 It⫽I11 1 /12⫹I2 e/5.
Gt⫽
2+3j+3A1j(1+2j) 2(1+j)3
Gr⫽
2+3j 2(1+j)3
g2(3−2g) A1⫽ 1+(Re/10g 2)2
冋
K1−M1+jC1 −K1 −K1
K1+K2−M2
册冋 册 冤
U Z¯1 ⫽ wb fr+jft Z¯2 · Z¯ w e 2
冥
(9)
From Eq. (9), one may get (K1⫺M1⫹jC1)Z¯1⫺K1Z¯2⫽U ⫺K1Z¯1⫹(K1⫹K2⫺M2)Z¯2⫽
wb Z¯2(fr+jft) · w e
(10) (11)
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Fig. 2.
Flexible, symmetric unbalance rotor supported on identical squeeze film dampers and retainer springs.
From Eq. (10) and Eq. (11), Z¯2 can be expressed as:
冋冉 冊
冉
2 1
冊册
K wb fr+jft ⫺ K1⫹K2⫺M2⫺ · Z¯2⫽ w e K1−M1+jC1 ⫺
(12)
K1U K1−M1+jC1
Now, to let ⫺
K1U =V +jV K1−M1+jC1 R I
(K1⫹K2⫺M2⫺
K 21 )⫽ER⫹jEI K1−M1+jC1
where VR, VI, ER and EI stand for respectively real and imaginary parts of the above expressions. Substituting the above expressions into Eq. (12) gives Z¯2⫽
VR+jVI wb fr w bf t −E +j −E we R we I
冉
冊冉
冊
(13)
From Eq. (6), magnitude of Z¯2 is eccentricity ratio ⑀ of SFD. Therefore, equating magnitudes on both sides of Eq. (13) and squaring, one obtains:
冉冊
冉冊
wb 2 2 2 wb (f r ⫹f t )⫺2e (f E ⫹f E )⫹e2(E2R⫹E2I ) w w r R t t
(14)
⫺(V2R⫹V2I )⫽0
assumed e but unspecified wb, the equation is a quadratic in wb. Hence, the value of wb for which Eq. (14) is satisfied for some assumed values of e can be determined from Eq. (14). A repetition of this for other values of w would give relations of eccentricity ratio e with wb and w. The models for radial and tangential film forces, fr and ft, in the Eq. (14) are different from those in the published work. From the Eq. (8) and Fig. 2, one may note that Gt and Gr express the effects of the feeding groove on the film forces. As Fig. 1 shows, with j=lg/l→0, that is without the feeding groove, Gt and Gr will approach 1, neglecting the effect of the feeding groove. And at the same time if Reynolds number Re is taken as 0 (that is neglecting the effect of fluid inertia), the models for fr and ft in the Eq. (14) will simplify to the same with those in the published work. Therefore, under the conditions of Gt=Gr=1 and Re=0, the relations of e with wb and w predicted by the Eq. (14) should be the same with those from the published work. The data in terms of non-dimensional quantities are presented by using a non-dimensional bearing parameter wb/wc and the non-dimensional rotor speed w/wc, where wc is some characteristic natural frequency of the rotor system. Using the following values of non-dimensional system parameters [14]: M2⫽0.25 M1⫽1⫺M2⫽0.75 K1⫽(1⫺M2)/(w/wc)2⫽0.75/(w/wc)2
Eq. (14) determines the relation between eccentricity ratio ⑀ and dynamic parameters of SFD and the rotor system.
K2⫽(wr/wc)2⫽0.25/(w/wc)2
3. Calculations and discussion
Fig. 3 shows that under the conditions of Gt=Gr=1 and Re=0, the relation curves of e with w/wc and wb/wc from Eq. (14) are compared with those from the published work [14]. The curves in Fig. 3 show that the Eq. (14) is correct in indicating the unbalance response of the flexible rotor in SFD.
Firstly, correctness of the Eq. (14) is checked by using the published work [11–14] . Eq. (14) is a nonlinear equation in e, and could be solved iteratively by some appropriate technique. But, one may note that for some
C⫽0.0075/(w/wc) U⫽0.3,
Q. Tan, X. Li / Tribology International 32 (1999) 559–566
Fig. 3.
Comparison of response e in [14] with that from Eq. (14).
Fig. 4 shows effects of the feeding groove on unbalance response to compare with the response (the response curves are expressed by the symbol - - - - without considering the groove effect) predicated by tradition lubrication theory. From the figure, the responses ⑀ predicated by Eq. (14) under considering the effect of the feeding groove are smaller than those predicated by
Fig. 4.
563
traditional lubrication theory such as the published work [12,13]. The effect of the feeding groove on film force in SFD causes the predicated response to be smaller about 15% than that from traditional lubrication theory. The rotor speed region for multiple solutions predicated by Eq. (14) for considering the effect of the feeding groove is smaller than that from traditional lubrication theory. From Fig. 4b, the response predicated by traditional lubrication theory for wb/wc=0.3 is multiple solutions, but the predication from the Eq. (14) is single solution, improving the stability of the rotor system. In traditional lubrication theory, the feeding groove in SFD only provides a low pressure around the journal surface and effectively splits SFD width into two film lands, resulting in the film lands operating independently. But, experimental work [3–9] showed that significant levels of dynamic pressure at the feeding groove. This shows that dynamic pressure at the feeding groove is not low pressure and that the feeding groove has hydrodynamic effect on the fluid in the feeding groove, not splitting SFD width into two film lands. Therefore, SFD force predicated by traditional lubrication theory will be smaller than that practically generated in SFD. However, the models of film forces, fr and ft, in the Eq. (14) contains the hydrodynamic effect of the feeding groove on SFD forces. So that the unbalance response predicated by traditional lubrication theory is larger than that from the Eq. (14). And previous work [15] also pointed out that the performance of SFDs is better than the predictions from tradition lubrication theory.
Effect of the groove on unbalance response of a flexible rotor system.
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Fig. 5 shows that changing of the groove width affects the unbalance responses for Reynolds number Re=1 and 15. With the groove width increasing, levels of SFD force reduce, causing SFD performance to drop. Therefore, the unbalance response predicated by Eq. (14) becomes large. The calculations from the Eq. (14) show that the effect of the groove depth on the responses is very small. This is because effect of the groove depth on SFD forces is very small, from [10]. From Fig. 5a and b, one may note that the groove effect on the response for Reynolds number Re=15 is more evident than that for Re=1. For the model of 2p film, radial and tangential forces of fluid film in SFD can be expressed based on [10] as: Fr=−Bfr
(15)
Ft=−Bft
by the tangential fluid force Ft, whereas the radial fluid force Fr provides a dynamic stiffness. The nonlinear dependence of the radial fluid force on the eccentricity ratio complicates the dynamics of SFD system. Therefore, the large discrepancy between radial fluid force Fr in Eq. (8) from the work [10] and the experimental result will affect predicting the unbalance response of the flexible rotor. Since practical radial fluid force is larger than one predicted by Eq. (8), the magnitudes of practical unbalance response of the flexible rotor may be smaller than those predicted, and the unstability of practical operation of the flexible rotor may be more complicated than the predicted.
4. Conclusions Based on the work [10] that studied effect of the feeding groove on film forces and dynamic characteristics in SFD, present work investigates effect of the feeding groove on unbalance responses of a flexible rotor in SFD and obtains the following results:
where fr⫽⫺ReGrIre ft⫽GtI20 3 e 21 Ir⫽I02 1 /12⫹I2 e/5
fr and ft of the Eq. (14) are replaced by corresponding ones in the Eq. (15), indicating the unbalance response of a flexible rotor in SFD with 2p film. Fig. 6 shows response curves predicated by the model of 2p film. From the figure, one may note that the effect of increasing the groove width is also evident. The damping effect of the squeeze film is provided
Fig. 5.
1. Compared with the predications from traditional lubrication theory, it is small that the response and the rotor speed region for multiple solutions are predicated by considering the effect of the feeding groove on SFD forces, and for wb/wc=0.3, the response predications from traditional lubrication theory is multiple solutions, but present work yields the response with single solution, improving the stability design of a rotor system.
Effect of groove width on unbalance response of a flexible rotor system for fluid inertia.
Q. Tan, X. Li / Tribology International 32 (1999) 559–566
Fig. 6.
565
Effect of the groove on unbalance response of a flexible rotor system for 2p film.
2. With the groove width increasing, the response predicated by Eq. (14) becomes large, showing that the capability of SFD isolating vibration decreases, and for large Reynolds number, the groove effect is evident. 3. The effect of the groove depth on the unbalance response is very small.
where j is the phase difference between the force and displacement. As may be seen from Fig. A1 and Nomenclature: Fr⫽F cos j
(A3)
Ft⫽F sin j
(A4)
cos j⫽x/e; sin j⫽y/e
(A5)
Hence: Fx⫽F cos(wt⫹j)⫽(xFr⫹yFt)/e
Appendix A
Fy⫽F sin(wt⫹j)⫽(xFt⫹yFr)/e
From Fig. A1, vector of SFD displacement and resultant vector of SFD forces can be expressed, respectively, as: z2⫽z¯2ejwt⫽x⫹jy Fd⫽|Fd|e
⫽Fe
j(wt+j)
Substituting Fx and Fy into Eq. (A2) yields: Fd⫽Fej(wt+j)⫽z2(Fr⫹jFt)/e
(A1) ⫽Fx⫹jFy
j(wt+j)
(A2) Appendix B From [16], Ilm n can be expressed as:
冕
J1
Ilm n ⫽
sinl J cosm J dJ (1+e cos J)n
J2
For p film:
冋
册
p(1−r) 1 1 Ir⫽ ⫺ (1⫺r) ; re2 12 5 It⫽ Fig. A1.
冉 | |冊
1+e 19 2e⫺ln ; 2 60e 1−e
(A6)
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2e I11 3 ⫽ 4; r
[6]
p I20 3 ⫽ 3; 2r
[7]
2 1/2
r⫽(1⫺e ) . For 2p film: Ir⫽
冋
[8]
册
2p(1−r) 1 1 ⫺ (1⫺r) ; re2 12 5
p 2 1/2 I20 3 ⫽ 3; r⫽(1⫺e ) . r
[9]
[10]
[11]
[12]
References [13] [1] San Andres LA. Analysis of squeeze film dampers with a central groove. ASME J Tribol 1992;114(4):659–65. [2] Arauz GL, San Andres LA. Effect of a circumferential feeding groove on the dynamic force response of a short squeeze film damper. ASME J Tribol 1994;116(2):369–77. [3] Walton JF II, Walowwit JA, Zorzi ES, Schtrand J. Experimental observation of cavitating squeeze-film dampers. ASME J Tribol 1987;109:290–5. [4] Roberts JB, Holmes R, Mason PJ. Estimation of squeeze film damping and inertia coefficient from experimental free-decay data. Proc Inst Mechan Engs 1986;200(2c):123–33. [5] Ramil MD, Roberts JB, Ellis J. Determination of squeeze-film
[14]
[15]
[16]
dynamic coefficients from experimental transient data. ASME J Tribol 1987;109(1):155–63. San Andres LJ, Vance JM. Experimental measurement of the dynamic pressure distribution in a squeeze-damper executing circular-centered orbit. ASLE Trans 1987;30(3):373–83. Zeidan FY, Vance JM. Cavitation and air entrainment effects on the response of squeeze film supported rotors. ASME J Tribol 1990;112:347–53. Rouch KE. Experimental evaluation of squeeze film damper coefficients with frequency domain techniques. ASLE Trans 1990;33(1):67–75. Arauz GL, San Andres L. Experimental study on the effect of a circumferential feeding groove on the dynamic force response of a sealed squeeze film damper. ASME J Tribol 1996;118:900–5. Tan Q, Chang Y, Wang L. Effect of a cicumferential feeding groove on fluid force in short squeeze film dampers. Tribol Int 1997;30(6):409–16. Taylor DL, Kumar BRK. Closed-formed, steady-state solution for the unbalance response of a flexible rotor in squeeze film damper. ASME J Engng Power 1983;105:551–9. Mclean LJ, Hanhn EJ. Stability of squeeze film damped multimass flexible rotor bearing systems. ASME J Tribol 1985;107:402–10. Hahn EJ, Chen PYP. Harmonic balance analysis of general squeeze film damped multidegree-of-freedom rotor bearing systems. ASME J Tribol 1994;116:499–507. Mclean LJ, Hahn EJ. Unbalance behavior of squeeze film damped multi-mass flexible rotor bearing systems. ASME J Lubric Technol 1983;105:22–8. Holmes R, Sykes J. The effects of manufactiring tolerances on the vibration of are-engine rotor-damper assemblies. In: 6th Workshop on Rotor-Dynamic Instability Problems in High-Performance Turbo-Machinery, TEXAS A and M University, College Station, TX, May 21–23, 1990. Booker JF. A table of the journal-bearing integral. ASME J Basic Engng 1965;87:533–5.
Analytical study on effect of a circumferential feeding groove on unbalance response of a flexible rotor in squeeze film damper Qingchang Tan *, Xiaohua Li Department of Mechanical Engineering, Jilin University of Technology, Changchun 130025, People’s Republic of China Received 20 April 1999; received in revised form 6 August 1999; accepted 20 October 1999
Abstract Of recent years, a series of researches have shown that a circumferential feeding groove of squeeze film damper (SFD) has evident effect on fluid film forces in SFD. Therefore, the feeding groove also affects dynamic responses of a rotor in SFD. Present work studies the effect of the feeding groove on unbalance response of a flexible rotor in SFD based on new film force models that include effects of the feeding groove and fluid inertia on dynamic characteristics of the fluid film in SFD. Compared with the published work, unbalance responses predicated under considering effect of the feeding groove on the dynamic characteristics in SFD are small, and rotor speed region for unbalance responses with multiple solutions is different, affecting the stability of a rotor system. And the effect of the feeding groove on the unbalance response is related to action of fluid inertia. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Unbalance response; Squeeze film damper; Film force; Rotor
1. Introduction Squeeze Film Dampers (SFDs) are widely used in high speed turbo-machinery due to their capability to isolate vibrations, to reduce forces transmitted to structural components, and to improve the stability characteristics of rotor-bearing systems. The behavior of a rotor system in service is dependent upon values of SFD dynamic coefficients in the governing differential equations. Therefore, that theoretical models for SFD dynamic coefficients are correct and accurate is very important for predicting rotor-system behavior. Controlled tests have been performed to verify SFD performance and to determine the validity of theoretical models when applied to actual hardware. However, experience has shown that the theoretical models based on traditional lubrication theory are limited in their applicability and fail to explain the performance of SFDs over a wide margin of operating conditions. Besides other reasons, investigators [1,2] have shown that main reason for such deviation is that in traditional lubrication
* Corresponding author. Tel. 431-5680278. E-mail address: [email protected] (Q. Tan).
theory, the feeding groove in SFD only provides a constant pressure around the journal surface and effectively splits SFD width into two film lands, resulting in the two film lands operating independently. Much experimental work [3–9] involving SFDs with a circumferential feeding groove has reported significant levels of dynamic pressure at the feeding groove. Recently, experimental work [9] has shown that levels of dynamic pressures measured at the feeding groove are as large as those generated at film lands. Therefore, the feeding groove affects dynamic characteristic of SFD and unbalance responses of a rotor system in SFDs. Since the theoretical models for dynamic coefficients of SFD have been developed all based on traditional lubrication theory, for authors’ knowledge there is not a literature that considers the effect of the feeding groove on SFD dynamic coefficients in studying rotor-system responses. Of recent years, San Andres [1] and authors [10] have advanced new models that include the effect of the feeding groove on SFD coefficients and yield good predications of the coefficients. This article investigates the effects of the feeding groove on unbalance response of a flexible rotor in SFDs based on authors’ previous work [10]. Fluid force models developed in previous work [10] include the effect
0301-679X/99/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 3 0 1 - 6 7 9 X ( 9 9 ) 0 0 0 8 6 - 9
560
Q. Tan, X. Li / Tribology International 32 (1999) 559–566
Nomenclature B wb c cg 2c1 C1 e Fd, Ft, r1 2m1 m2 U M1 M2 k1 k2 K1 K2 L l lg R Re t e w wc wr h j g
dimensional coefficient of fluid film force, B=RL3hw/c2 non-dimensional SFD parameter; wb=RL3h/c3(m1+m2) radial clearance between the journal and bearing groove depth viscous damping of disc in Fig. 1 =c1/[w(m1+m2)] eccentricity of journal with respective to bearing Fr resultant of fluid film forces, and its components in the t and r directions, respectively, |Fd|=F unbalance eccentricity lumped mass of disc in Fig. 1 lumped mass at bearing stations in Fig. 1 unbalance parameter U=r1m1/c(m1+m2) =m1/(m1+m2) =m2/(m1+m2) rotor stiffness in Fig. 1 radial stiffness of retainer spring in Fig. 1 =k1/[(w2(m1+m2)] =k2/[(w2(m1+m2)] whole length of a damper one-side length of film land one-side length of feeding groove journal radius gap Reynolds number, Re=c2rw/h time eccentricity ratio with respect to bearing, e=e/c rotor speed =√k1/m1 =√k2/(m1+m2) dynamic viscosity of oil =lg/l =c/cg
of fluid inertia on dynamic characteristics of SFD film. Therefore, present work is carried out under considering the effect of fluid inertia on unbalance response. The traditional Jeffcott rotor is carried on SFDs with linear centering springs and circumferential-feeding grooves, as shown in Fig. 1. The centering spring is assumed to be adjusted to offset any gravitational load. Alternatively, the axis of rotation is vertical, resulting in no gravitational loading.
2. Analysis model In the analysis of rotor-system performance, the following assumptions are made: (a) a rotor may be subjected to external synchronous rotating loads; (b) the rolling element bearings are radically rigid; (c) the damping is viscous; (d) torsional and axial vibrations of a rotor is negligible; (e) the rotor is axially symmetric. Variables for describing the rotor system are defined, as shown in
Fig. 1. Fig. 1 is a diagram of the rotor system with node 1 being taken at the central disc of mass 2m1, and nodes 2 and 3 at ends of the rotor which is supported by identical SFDs. Lumped mass at bearing ends is m2, the retainer spring for central preloading has stiffness k2 and the rotor stiffness between central and either end node is k1. All unbalance is assumed to be at the disc, resulting in a disc mass eccentricity r1. Viscous damping at the disc is 2C1. Since the rotor is symmetric about the disc, it will suffice to consider one half of the system only. The equations of motion, appropriately ordered, for this system at a rotor speed w are given by: m1x¨1⫹c1x˙1⫹k1(x1⫺x3)⫽r1m1w2 cos(wt)
(1)
m1x¨2⫹c1x˙2⫹k1(x2⫺x4)⫽r1m1w2 sin(wt)
(2)
m2x¨3⫹k1(x3⫺x1)⫹k2x3⫽F cos(wt⫹j)
(3)
m2x¨4⫹k1(x4⫺x2)⫹k2x4⫽F sin(wt⫹j)
(4)
Eq. (2) is multiplied by imaginary j, then the Eq. (2) plus Eq. (1) yields:
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Fig. 1. Details of physical system.
m1z¨1⫹c1z˙1⫹k1(z1⫺z2)⫽r1m1w2ejwt
(5)
where z1=x1+jx2 is vector of system displacements at the disc, and z2=x3+jx4 is vector of relative system displacements at SFD station. Real and imaginary parts of z1 stand for, respectively, x1 and x2. Similarly, from Eq. (3) and Eq. (4), one may get: m2z¨2⫹k1(z2⫺z1)⫹k2z2⫽Fej(wt+j) As shown in the Appendix A, Fe expressed as: Fej(wt+j)⫽z2(Fr⫹jFt)/e
(6) j(wt+j)
may be (7)
where Fr and Ft are fluid film forces in SFD. From [10], Fr and Ft for π fluid film can be expressed as: Fr=−Bfr
(8)
Ft=−Bft
Ilm n in the above expressions are defined by the famous Booker [16] journal bearing integral form (see Appendix B). fr and ft in Eq. (8) are dimensionless radial and tangential forces of SFD fluid film. Gt and Gr in expressions of fr and ft show respectively effects of the feeding groove on viscosity damping part and inertia part of fluid film forces, as shown in Fig. 2. Also, Reynolds number Re reflects effect of fluid inertia on film forces. Therefore, compared with the published work, for example [11–13], present work investigates effects of the feeding groove and fluid inertia on unbalance response of a flexible rotor system in SFDs. For circular synchronous motion, solutions of Eq. (5) and Eq. (6) can be written as: z1⫽z¯1ejwt and z2⫽z¯2ejwt The above expressions are substituted into Eq. (5) and Eq. (6), then the two equations are transformed into nondimension forms using dividing each equation by (m1+m2)cw2 and letting t=wt:
where fr⫽(GtI11 3 ⫺ReGrIr)e; ft(GtI20 3 ⫺ReGrIt)e; Ir⫽I /12⫹I e/5; 02 1
21 2
30 It⫽I11 1 /12⫹I2 e/5.
Gt⫽
2+3j+3A1j(1+2j) 2(1+j)3
Gr⫽
2+3j 2(1+j)3
g2(3−2g) A1⫽ 1+(Re/10g 2)2
冋
K1−M1+jC1 −K1 −K1
K1+K2−M2
册冋 册 冤
U Z¯1 ⫽ wb fr+jft Z¯2 · Z¯ w e 2
冥
(9)
From Eq. (9), one may get (K1⫺M1⫹jC1)Z¯1⫺K1Z¯2⫽U ⫺K1Z¯1⫹(K1⫹K2⫺M2)Z¯2⫽
wb Z¯2(fr+jft) · w e
(10) (11)
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Fig. 2.
Flexible, symmetric unbalance rotor supported on identical squeeze film dampers and retainer springs.
From Eq. (10) and Eq. (11), Z¯2 can be expressed as:
冋冉 冊
冉
2 1
冊册
K wb fr+jft ⫺ K1⫹K2⫺M2⫺ · Z¯2⫽ w e K1−M1+jC1 ⫺
(12)
K1U K1−M1+jC1
Now, to let ⫺
K1U =V +jV K1−M1+jC1 R I
(K1⫹K2⫺M2⫺
K 21 )⫽ER⫹jEI K1−M1+jC1
where VR, VI, ER and EI stand for respectively real and imaginary parts of the above expressions. Substituting the above expressions into Eq. (12) gives Z¯2⫽
VR+jVI wb fr w bf t −E +j −E we R we I
冉
冊冉
冊
(13)
From Eq. (6), magnitude of Z¯2 is eccentricity ratio ⑀ of SFD. Therefore, equating magnitudes on both sides of Eq. (13) and squaring, one obtains:
冉冊
冉冊
wb 2 2 2 wb (f r ⫹f t )⫺2e (f E ⫹f E )⫹e2(E2R⫹E2I ) w w r R t t
(14)
⫺(V2R⫹V2I )⫽0
assumed e but unspecified wb, the equation is a quadratic in wb. Hence, the value of wb for which Eq. (14) is satisfied for some assumed values of e can be determined from Eq. (14). A repetition of this for other values of w would give relations of eccentricity ratio e with wb and w. The models for radial and tangential film forces, fr and ft, in the Eq. (14) are different from those in the published work. From the Eq. (8) and Fig. 2, one may note that Gt and Gr express the effects of the feeding groove on the film forces. As Fig. 1 shows, with j=lg/l→0, that is without the feeding groove, Gt and Gr will approach 1, neglecting the effect of the feeding groove. And at the same time if Reynolds number Re is taken as 0 (that is neglecting the effect of fluid inertia), the models for fr and ft in the Eq. (14) will simplify to the same with those in the published work. Therefore, under the conditions of Gt=Gr=1 and Re=0, the relations of e with wb and w predicted by the Eq. (14) should be the same with those from the published work. The data in terms of non-dimensional quantities are presented by using a non-dimensional bearing parameter wb/wc and the non-dimensional rotor speed w/wc, where wc is some characteristic natural frequency of the rotor system. Using the following values of non-dimensional system parameters [14]: M2⫽0.25 M1⫽1⫺M2⫽0.75 K1⫽(1⫺M2)/(w/wc)2⫽0.75/(w/wc)2
Eq. (14) determines the relation between eccentricity ratio ⑀ and dynamic parameters of SFD and the rotor system.
K2⫽(wr/wc)2⫽0.25/(w/wc)2
3. Calculations and discussion
Fig. 3 shows that under the conditions of Gt=Gr=1 and Re=0, the relation curves of e with w/wc and wb/wc from Eq. (14) are compared with those from the published work [14]. The curves in Fig. 3 show that the Eq. (14) is correct in indicating the unbalance response of the flexible rotor in SFD.
Firstly, correctness of the Eq. (14) is checked by using the published work [11–14] . Eq. (14) is a nonlinear equation in e, and could be solved iteratively by some appropriate technique. But, one may note that for some
C⫽0.0075/(w/wc) U⫽0.3,
Q. Tan, X. Li / Tribology International 32 (1999) 559–566
Fig. 3.
Comparison of response e in [14] with that from Eq. (14).
Fig. 4 shows effects of the feeding groove on unbalance response to compare with the response (the response curves are expressed by the symbol - - - - without considering the groove effect) predicated by tradition lubrication theory. From the figure, the responses ⑀ predicated by Eq. (14) under considering the effect of the feeding groove are smaller than those predicated by
Fig. 4.
563
traditional lubrication theory such as the published work [12,13]. The effect of the feeding groove on film force in SFD causes the predicated response to be smaller about 15% than that from traditional lubrication theory. The rotor speed region for multiple solutions predicated by Eq. (14) for considering the effect of the feeding groove is smaller than that from traditional lubrication theory. From Fig. 4b, the response predicated by traditional lubrication theory for wb/wc=0.3 is multiple solutions, but the predication from the Eq. (14) is single solution, improving the stability of the rotor system. In traditional lubrication theory, the feeding groove in SFD only provides a low pressure around the journal surface and effectively splits SFD width into two film lands, resulting in the film lands operating independently. But, experimental work [3–9] showed that significant levels of dynamic pressure at the feeding groove. This shows that dynamic pressure at the feeding groove is not low pressure and that the feeding groove has hydrodynamic effect on the fluid in the feeding groove, not splitting SFD width into two film lands. Therefore, SFD force predicated by traditional lubrication theory will be smaller than that practically generated in SFD. However, the models of film forces, fr and ft, in the Eq. (14) contains the hydrodynamic effect of the feeding groove on SFD forces. So that the unbalance response predicated by traditional lubrication theory is larger than that from the Eq. (14). And previous work [15] also pointed out that the performance of SFDs is better than the predictions from tradition lubrication theory.
Effect of the groove on unbalance response of a flexible rotor system.
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Fig. 5 shows that changing of the groove width affects the unbalance responses for Reynolds number Re=1 and 15. With the groove width increasing, levels of SFD force reduce, causing SFD performance to drop. Therefore, the unbalance response predicated by Eq. (14) becomes large. The calculations from the Eq. (14) show that the effect of the groove depth on the responses is very small. This is because effect of the groove depth on SFD forces is very small, from [10]. From Fig. 5a and b, one may note that the groove effect on the response for Reynolds number Re=15 is more evident than that for Re=1. For the model of 2p film, radial and tangential forces of fluid film in SFD can be expressed based on [10] as: Fr=−Bfr
(15)
Ft=−Bft
by the tangential fluid force Ft, whereas the radial fluid force Fr provides a dynamic stiffness. The nonlinear dependence of the radial fluid force on the eccentricity ratio complicates the dynamics of SFD system. Therefore, the large discrepancy between radial fluid force Fr in Eq. (8) from the work [10] and the experimental result will affect predicting the unbalance response of the flexible rotor. Since practical radial fluid force is larger than one predicted by Eq. (8), the magnitudes of practical unbalance response of the flexible rotor may be smaller than those predicted, and the unstability of practical operation of the flexible rotor may be more complicated than the predicted.
4. Conclusions Based on the work [10] that studied effect of the feeding groove on film forces and dynamic characteristics in SFD, present work investigates effect of the feeding groove on unbalance responses of a flexible rotor in SFD and obtains the following results:
where fr⫽⫺ReGrIre ft⫽GtI20 3 e 21 Ir⫽I02 1 /12⫹I2 e/5
fr and ft of the Eq. (14) are replaced by corresponding ones in the Eq. (15), indicating the unbalance response of a flexible rotor in SFD with 2p film. Fig. 6 shows response curves predicated by the model of 2p film. From the figure, one may note that the effect of increasing the groove width is also evident. The damping effect of the squeeze film is provided
Fig. 5.
1. Compared with the predications from traditional lubrication theory, it is small that the response and the rotor speed region for multiple solutions are predicated by considering the effect of the feeding groove on SFD forces, and for wb/wc=0.3, the response predications from traditional lubrication theory is multiple solutions, but present work yields the response with single solution, improving the stability design of a rotor system.
Effect of groove width on unbalance response of a flexible rotor system for fluid inertia.
Q. Tan, X. Li / Tribology International 32 (1999) 559–566
Fig. 6.
565
Effect of the groove on unbalance response of a flexible rotor system for 2p film.
2. With the groove width increasing, the response predicated by Eq. (14) becomes large, showing that the capability of SFD isolating vibration decreases, and for large Reynolds number, the groove effect is evident. 3. The effect of the groove depth on the unbalance response is very small.
where j is the phase difference between the force and displacement. As may be seen from Fig. A1 and Nomenclature: Fr⫽F cos j
(A3)
Ft⫽F sin j
(A4)
cos j⫽x/e; sin j⫽y/e
(A5)
Hence: Fx⫽F cos(wt⫹j)⫽(xFr⫹yFt)/e
Appendix A
Fy⫽F sin(wt⫹j)⫽(xFt⫹yFr)/e
From Fig. A1, vector of SFD displacement and resultant vector of SFD forces can be expressed, respectively, as: z2⫽z¯2ejwt⫽x⫹jy Fd⫽|Fd|e
⫽Fe
j(wt+j)
Substituting Fx and Fy into Eq. (A2) yields: Fd⫽Fej(wt+j)⫽z2(Fr⫹jFt)/e
(A1) ⫽Fx⫹jFy
j(wt+j)
(A2) Appendix B From [16], Ilm n can be expressed as:
冕
J1
Ilm n ⫽
sinl J cosm J dJ (1+e cos J)n
J2
For p film:
冋
册
p(1−r) 1 1 Ir⫽ ⫺ (1⫺r) ; re2 12 5 It⫽ Fig. A1.
冉 | |冊
1+e 19 2e⫺ln ; 2 60e 1−e
(A6)
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2e I11 3 ⫽ 4; r
[6]
p I20 3 ⫽ 3; 2r
[7]
2 1/2
r⫽(1⫺e ) . For 2p film: Ir⫽
冋
[8]
册
2p(1−r) 1 1 ⫺ (1⫺r) ; re2 12 5
p 2 1/2 I20 3 ⫽ 3; r⫽(1⫺e ) . r
[9]
[10]
[11]
[12]
References [13] [1] San Andres LA. Analysis of squeeze film dampers with a central groove. ASME J Tribol 1992;114(4):659–65. [2] Arauz GL, San Andres LA. Effect of a circumferential feeding groove on the dynamic force response of a short squeeze film damper. ASME J Tribol 1994;116(2):369–77. [3] Walton JF II, Walowwit JA, Zorzi ES, Schtrand J. Experimental observation of cavitating squeeze-film dampers. ASME J Tribol 1987;109:290–5. [4] Roberts JB, Holmes R, Mason PJ. Estimation of squeeze film damping and inertia coefficient from experimental free-decay data. Proc Inst Mechan Engs 1986;200(2c):123–33. [5] Ramil MD, Roberts JB, Ellis J. Determination of squeeze-film
[14]
[15]
[16]
dynamic coefficients from experimental transient data. ASME J Tribol 1987;109(1):155–63. San Andres LJ, Vance JM. Experimental measurement of the dynamic pressure distribution in a squeeze-damper executing circular-centered orbit. ASLE Trans 1987;30(3):373–83. Zeidan FY, Vance JM. Cavitation and air entrainment effects on the response of squeeze film supported rotors. ASME J Tribol 1990;112:347–53. Rouch KE. Experimental evaluation of squeeze film damper coefficients with frequency domain techniques. ASLE Trans 1990;33(1):67–75. Arauz GL, San Andres L. Experimental study on the effect of a circumferential feeding groove on the dynamic force response of a sealed squeeze film damper. ASME J Tribol 1996;118:900–5. Tan Q, Chang Y, Wang L. Effect of a cicumferential feeding groove on fluid force in short squeeze film dampers. Tribol Int 1997;30(6):409–16. Taylor DL, Kumar BRK. Closed-formed, steady-state solution for the unbalance response of a flexible rotor in squeeze film damper. ASME J Engng Power 1983;105:551–9. Mclean LJ, Hanhn EJ. Stability of squeeze film damped multimass flexible rotor bearing systems. ASME J Tribol 1985;107:402–10. Hahn EJ, Chen PYP. Harmonic balance analysis of general squeeze film damped multidegree-of-freedom rotor bearing systems. ASME J Tribol 1994;116:499–507. Mclean LJ, Hahn EJ. Unbalance behavior of squeeze film damped multi-mass flexible rotor bearing systems. ASME J Lubric Technol 1983;105:22–8. Holmes R, Sykes J. The effects of manufactiring tolerances on the vibration of are-engine rotor-damper assemblies. In: 6th Workshop on Rotor-Dynamic Instability Problems in High-Performance Turbo-Machinery, TEXAS A and M University, College Station, TX, May 21–23, 1990. Booker JF. A table of the journal-bearing integral. ASME J Basic Engng 1965;87:533–5.