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A new derivation for journal bearing stiffness and damping coefficients in polar coordinates
ARTICLE IN PRESS
JOURNAL OF SOUND AND VIBRATION Journal of Sound and Vibration 290 (2006) 500–507 www.elsevier.com/locate/jsvi
Short Communication
A new derivation for journal bearing stiffness and damping coefficients in polar coordinates J.K. Wang, M.M. Khonsari Department of Mechanical Engineering, Louisiana State University, Baton Rouge, LA 70803, USA Received 23 March 2005; received in revised form 5 April 2005; accepted 12 April 2005 Available online 1 August 2005
Abstract A general derivation for obtaining bearing dynamic coefficients in polar coordinates is presented. It is shown that this derivation is simple and consistent with the commonly used formulae for the linearized stiffness and damping coefficients. What is more important is that the physical meaning of this derivation is clearer and simpler. r 2005 Elsevier Ltd. All rights reserved.
1. Introduction The existing definitions of the linearized stiffness and damping coefficients in polar coordinates are derived from the Taylor series expansions of the radial component fe and tangential component ff of the fluid force f (Fig. 1) and neglecting higher than first order terms. There are two commonly used approaches for arriving at definitions of these coefficients (see Section 2). However, they lead to inconsistencies in the final expressions for the stiffness and damping coefficients. Especially, when applied to infinitely short bearings, the inconsistencies become so obvious. This paper presents a more general and straightforward derivation of bearing stiffness and damping coefficients in polar coordinates. The final expressions for the stiffness and damping coefficients are consistent with those in both Refs. [1,2]. Corresponding author. Tel.: +1 225 578 9192; fax: +1 225 578 5924.
E-mail address: [email protected] (M.M. Khonsari). 0022-460X/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jsv.2005.04.019
ARTICLE IN PRESS J.K. Wang, M.M. Khonsari / Journal of Sound and Vibration 290 (2006) 500–507
Nomenclature bij
damping coefficients of fluid film bearing, N s/m, ði; j ¼ ; fÞ (i is the direction of the force, j is the direction of the speed)
b¯ ij
ðC=RÞ3 mL
C fe ff
radial clearance, m radial component of the fluid force, N tangential component of the fluid force, N stiffness coefficients of fluid film bearing, N/m, ði; j ¼ ; fÞ (i is the direction of the force, j is the direction of the displacement)
kij
bij ði; j ¼ ; fÞ
501
k¯ ij
ðC=RÞ3 moL
L Ob Oj R t x
bearing length, m center of the journal bearing center of the journal journal radius, m time, s the coordinate in the horizontal direction the coordinate in the vertical direction lubricant viscosity, Pa s running speed of the rotor, rad/s eccentricity ratio attitude angle
y m o e f
kij ði; j ¼ ; fÞ
2. Existing inconsistencies between two kinds of existing definitions Fig. 1 shows the sketch of the decomposition of the fluid force in journal bearing. In Fig. 1, Ob is the center of the bearing; Ojs is the steady-state equilibrium position of the journal center; Oj is the dynamic position of the journal center; e and f are the eccentricity ratio and attitude angle of the journal center, respectively; subscript s represents the steady-state equilibrium position; C is the radial clearance; fe and ff are the radial and tangential components of the fluid forces f in journal bearing; x and y are the coordinates in the horizontal direction and vertical direction, respectively; ee and ef are the unit vectors in radial and tangential directions, respectively. Referring to Fig. 1, the fluid force f is " # h i f . (1) f ¼ f e þ f f ef ¼ e ef ff y x
Ob
fφ
C(εs+ε∆ ) φs
f φ'
∆φ
(f )
Oj
Cεs (fφ)s
fε'
eφ
fε eε
O js (f)s (fε) s
Fig. 1. Sketch of the decomposition of the fluid force in journal bearing.
ARTICLE IN PRESS J.K. Wang, M.M. Khonsari / Journal of Sound and Vibration 290 (2006) 500–507
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The two commonly used definitions mentioned in Section 1 are both based on the same Taylor series expansions of the radial component and tangential component of the fluid force f separately while neglecting higher than first-order terms as follows [1–3]. 3 2 D 7 6 Df 7 qf qf qf qf 6 7, 6 f ¼ ðf Þs þ 7 _ 6 D_ q qf q_ qf 5 4 _ Df 2 f f ¼ ðf f Þs þ
D
3
7 6 @f f @f f @f f @f f 6 Df 7 7. 6 7 _ 6 @ @f @_ @f 4 D_ 5 _ Df
The above equations can be written in the following form: 2 3 2 qf qf qf ! ! 6 Cq Cqf 7 CD 6 Cq_ f ðf Þs 6 7 6 þ ¼ 6 qf 7 6 qf qf f 5 CDf f f ðf f Þs 4 f 4 f Cq Cqf Cq_
3 qf ! _ 7 CD_ Cqf 7 _ , qf f 7 5 CDf _ Cqf
(2)
where all the first-order derivatives are evaluated at the steady-state equilibrium position (es, fs).
2.1. Direct deduction of the stiffness and damping coefficients One of the approaches in the existing literature arrives at the definition of the stiffness coefficients (kij) and damping coefficients (bij) from the Taylor series expansions—Eq. (2)— directly [3]: ! " # ! " # ! f ðf Þs k kf b bf CD CD_ ¼ (3) _ , f f ðf f Þs kf kff bf bff CDf CDf where "
k
kf
kf
kff
#
2
qf 6 Cq 6 ¼ 6 qf 4 f Cq
3 qf " b Cqf 7 7 and 7 qf f 5 bf Cqf
bf bff
#
2
qf 6 Cq_ 6 ¼ 6 qf 4 f Cq_
3 qf _7 Cqf 7 . qf f 7 5 _ Cqf
(4)
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2.2. Transformation approach for deriving the stiffness and damping coefficients From Fig. 1, the following transformation is used [1,2]: ! ! ! f f 0 cos Df sin Df ¼ , ff f 0f sin Df cos Df
(5)
where f 0 is the component of the fluid force f in the same direction as the fluid force component 0 ðf Þs at the steady-state equilibrium position, and f f is the component of the fluid force f in the same direction as the fluid force component f f s at the steady-state equilibrium position. For a small perturbation, Df 1, so that cos Df 1, sin Df ¼ Df. Using these two approximations, Eq. (5) can be rewriten as ! ! ! f f 0 f f ¼ þ Df . ff f 0f f Then, the following definitions of the stiffness and damping coefficients are developed [1,2]. ! " # ! " # ! f 0 ðf Þs k kf b bf CD CD_ ¼ (6) _ , f 0f ðf f Þs kf kff bf bff CDf CDf where 3 3 2 2 qf qf ff qf qf " # " # _7 6 Cq Cqf þ C 7 6 C@_ b bf k kf C@f 7 7 6 6 and ¼6 ¼ 6 qf 7 7. (7) qf qf qf kf kff b b f 5 4 4 f f 5 f ff f f _ Cq_ Cq Cqf C Cqf 2.3. Inconsistencies between two kinds of existing definitions From Eqs. (4) and (7), it is very clear that some inconsistencies between the definitions of kf and kff exist. The second definitions of kf and kff each has one more coupled term than does the first set of equations presented in Section 2.1. 3. A new derivation of the definitions of stiffness and damping coefficients Taking the derivative of both sides of Eq. (1) with respect to time t yields: df f def df df de ¼ e þ ef þ f þff , dt dt dt dt dt
(8)
_ df qf d qf df qf d_ qf df þ þ þ ; ¼ dt q dt qf dt q_ dt qf_ dt df f qf f d qf f df qf f d_ qf f df_ þ þ þ : ¼ _ dt dt q dt qf dt q_ dt qf
(9)
where
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The derivatives of unit vectors e and ef are [4] de df ef ; ¼ dt dt def df e : ¼ dt dt
(10)
Substituting Eqs. (9) and (10) into Eq. (8) yields df ¼ dt
! _ df qf d qf df qf d_ qf df þ þ þ f e _ dt dt f q dt qf dt q_ dt qf
! _ df qf f d qf f df qf f d_ qf f df þ þ þ þ f ef . þ _ dt dt q dt qf dt q_ dt qf Simplifying further, we obtain 3 qf d qf df qf d_ qf df_ þ þ þ ff i6 _ dt 7 dt qf q_ dt qf 7 6 q dt 7 ef 6 6 qf d _ 7. 4 f þ qf f þ f df þ qf f d_ þ qf f df 5 _ dt dt q dt qf q_ dt qf 2
df h ¼ e dt
(11)
For a very short-time interval Dt when a small perturbation is applied to the journal that was in steady-state equilibrium position s ; fs , Eq. (11) can be approximated as follows through multiplying the two sides by Dt. 3 2 qf qf qf qf _ D þ f f Df þ D_ þ Df 7 _ h i6 q qf q_ qf 7 6 e e f (12) f fs ¼ 7. 6 qf qf f qf f qf f 4 f _5 Df D þ þ f Df þ D_ þ q qf q_ qf_ The fluid force at the steady-state equilibrium position s ; fs is 2 3 f s h i 5. 4 e e fs ¼ f ff s
Substituting Eqs. (1) and (13) into Eq. (12), yields 2 3 qf qf qf qf _ 3 2 D f Df þ D þ f D_ þ f 7 h i f f s h i6 _ qf q_ qf 7 5 ¼ e ef 6 q e ef 4 6 qf 7, qf f qf f qf f ff ff 4 f 5 _ D þ þ f Df þ D_ þ Df s _ q qf q_ qf
(13)
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i.e., 3 2 qf qf qf qf _ 3 D þ f D_ þ D f Df þ f f f s 7 6 _ qf q_ qf 7 5 ¼ 6 q 4 7. 6 qf qf f qf f qf f ff ff 4 f _5 Df D þ þ f Df þ D_ þ s _ q qf q_ qf 2
Rewriting Eq. (14), one arrives at the following equation: 3 2 2 qf ff qf qf 3 2 " # f f s 7 6 Cq_ 6 5 ¼ 6 Cq Cqf C 7 CD þ 6 4 7 6 qf 6 qf qf f f 5 CDf ff ff 4 f 4 f s Cq_ Cq Cqf Cq
3 qf " # Cqf_ 7 7 CD_ _ . qf f 7 5 CDf Cqf_
Now, the stiffness and damping coefficients can be defined as follows: 3 2 " #" # " #" # f f s k kf b bf CD CD_ 4 5¼ _ , kf kff bf bff CDf CDf ff ff
(14)
(15)
(16)
s
where "
k
kf
kf
kff
#
2
qf 6 Cq 6 ¼6 4 qf f Cq
3 ff qf þ Cqf C 7 7 7 qf f f 5 Cqf C
" and
b
bf
bf
bff
#
2
qf 6 Cq_ 6 ¼ 6 qf 4 f Cq_
3 qf _7 Cqf 7 qf f 7 5 _ Cqf
(17)
The final expressions for the stiffness and damping coefficients described by Eq. (17) are consistent with the commonly used expressions described by Eq. (7) in Section 2.2. However, the physical meaning in this derivation is clearer and simpler. In Eq. (16), the change of the fluid force component fe is exactly the relative force change in the direction of ee due to the small perturbation, and the change of the fluid force component ff is exactly the relative force change in the direction of ef due to the small perturbation. 4. Verification and discussion In this section, as an example, the definitions of stiffness and damping coefficients described by Eqs. (16) and (17) are applied to infinitely short bearing. Assuming that the short bearing theory with half-Sommerfeld boundary conditions applies [5], the fluid forces in the journal bearing are [6] " # _ p 1 þ 22 _ RL3 m 22 o 2f , (18) þ f ¼ 5=2 2C 2 ð1 2 Þ2 ð1 2 Þ
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" # RL3 m p o 2f_ 4_ . ff ¼ þ 2C 2 2ð1 2 Þ3=2 ð1 2 Þ2
(19)
Substituting Eqs. (18) and (19) into Eqs. (17) and simplifying the resulting expressions, it can be shown that elements of the linearized stiffness matrix kij ði; j ¼ ; fÞ are k ¼
2omRL3 ð1 þ 2 Þ
kf ¼ kf ¼
C 3 ð1 2 Þ3 pomRL3
;
; 3=2 4C 3 ð1 2 Þ pomRL3 ð1 þ 22 Þ
kff ¼
5=2
4C 3 ð1 2 Þ omRL3
2
C 3 ð1 2 Þ
(20) ;
:
The elements of the damping matrix bij ði; j ¼ ; fÞ are b ¼
pmRL3 ð1 þ 22 Þ
; 2C 3 ð1 2 Þ5=2 2mRL3 ; bf ¼ 3 C ð1 2 Þ2 (21) 2mRL3 ; bf ¼ 3 2 C ð1 2 Þ pmRL3 bff ¼ : 2C 3 ð1 2 Þ3=2 3 3 Using k¯ ij ¼ C=R =moL kij ; ði; j ¼ ; fÞ and b¯ ij ¼ C=R =mL bij ; ði; j ¼ ; fÞ, the linearized stiffness coefficients and damping coefficients can be normalized. It has been shown that the normalized expressions for the stiffness and damping coefficients are consistent with those in Refs. [1,2]. However, if the definitions described by Eqs. (3) and (4) are applied to the same infinitely short bearing theory, kf 0 and kff 0 since both of the fluid force components f and f f are not an explicit function of f.
5. Conclusion The derivations presented in this paper offer a simple procedure for arriving at the definitions of the stiffness and damping coefficients in polar coordinates. The method also offers a clear physical meaning of the dynamic coefficient and yields consistent results when applied to infinitely short bearings.
ARTICLE IN PRESS J.K. Wang, M.M. Khonsari / Journal of Sound and Vibration 290 (2006) 500–507
507
References [1] R. Holmes, The vibration of a rigid rotor on short journal bearings, Journal of Mechanical Engineering Science 2 (1960) 337–341. [2] A. Szeri, Linearized force coefficients of a 1101 partial journal bearing, Proceedings of the Institution of Mechanical Engineers 181 (Part 3B) (1966–67) 130–133. [3] D. Childs, H. Moes, H.V. Leeuwen, Journal bearing impedance description for rotor dynamic applications, ASME Journal of Lubrication Technology 99 (1977) 198–214. [4] D.T. Greenwood, Intermediate Dynamics, second ed., Prentice-Hall, Englewood Cliffs, NJ, 1988. [5] M.M. Khonsari, E.R. Booser, Applied Tribology: Bearing Design and Lubrication, Wiley, New York, 2001. [6] M.M. Khonsari, Y.J. Chang, Stability boundary of non-linear orbits within clearance circle of journal bearings, ASME Journal of Vibration and Acoustics 115 (1993) 303–307.
JOURNAL OF SOUND AND VIBRATION Journal of Sound and Vibration 290 (2006) 500–507 www.elsevier.com/locate/jsvi
Short Communication
A new derivation for journal bearing stiffness and damping coefficients in polar coordinates J.K. Wang, M.M. Khonsari Department of Mechanical Engineering, Louisiana State University, Baton Rouge, LA 70803, USA Received 23 March 2005; received in revised form 5 April 2005; accepted 12 April 2005 Available online 1 August 2005
Abstract A general derivation for obtaining bearing dynamic coefficients in polar coordinates is presented. It is shown that this derivation is simple and consistent with the commonly used formulae for the linearized stiffness and damping coefficients. What is more important is that the physical meaning of this derivation is clearer and simpler. r 2005 Elsevier Ltd. All rights reserved.
1. Introduction The existing definitions of the linearized stiffness and damping coefficients in polar coordinates are derived from the Taylor series expansions of the radial component fe and tangential component ff of the fluid force f (Fig. 1) and neglecting higher than first order terms. There are two commonly used approaches for arriving at definitions of these coefficients (see Section 2). However, they lead to inconsistencies in the final expressions for the stiffness and damping coefficients. Especially, when applied to infinitely short bearings, the inconsistencies become so obvious. This paper presents a more general and straightforward derivation of bearing stiffness and damping coefficients in polar coordinates. The final expressions for the stiffness and damping coefficients are consistent with those in both Refs. [1,2]. Corresponding author. Tel.: +1 225 578 9192; fax: +1 225 578 5924.
E-mail address: [email protected] (M.M. Khonsari). 0022-460X/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jsv.2005.04.019
ARTICLE IN PRESS J.K. Wang, M.M. Khonsari / Journal of Sound and Vibration 290 (2006) 500–507
Nomenclature bij
damping coefficients of fluid film bearing, N s/m, ði; j ¼ ; fÞ (i is the direction of the force, j is the direction of the speed)
b¯ ij
ðC=RÞ3 mL
C fe ff
radial clearance, m radial component of the fluid force, N tangential component of the fluid force, N stiffness coefficients of fluid film bearing, N/m, ði; j ¼ ; fÞ (i is the direction of the force, j is the direction of the displacement)
kij
bij ði; j ¼ ; fÞ
501
k¯ ij
ðC=RÞ3 moL
L Ob Oj R t x
bearing length, m center of the journal bearing center of the journal journal radius, m time, s the coordinate in the horizontal direction the coordinate in the vertical direction lubricant viscosity, Pa s running speed of the rotor, rad/s eccentricity ratio attitude angle
y m o e f
kij ði; j ¼ ; fÞ
2. Existing inconsistencies between two kinds of existing definitions Fig. 1 shows the sketch of the decomposition of the fluid force in journal bearing. In Fig. 1, Ob is the center of the bearing; Ojs is the steady-state equilibrium position of the journal center; Oj is the dynamic position of the journal center; e and f are the eccentricity ratio and attitude angle of the journal center, respectively; subscript s represents the steady-state equilibrium position; C is the radial clearance; fe and ff are the radial and tangential components of the fluid forces f in journal bearing; x and y are the coordinates in the horizontal direction and vertical direction, respectively; ee and ef are the unit vectors in radial and tangential directions, respectively. Referring to Fig. 1, the fluid force f is " # h i f . (1) f ¼ f e þ f f ef ¼ e ef ff y x
Ob
fφ
C(εs+ε∆ ) φs
f φ'
∆φ
(f )
Oj
Cεs (fφ)s
fε'
eφ
fε eε
O js (f)s (fε) s
Fig. 1. Sketch of the decomposition of the fluid force in journal bearing.
ARTICLE IN PRESS J.K. Wang, M.M. Khonsari / Journal of Sound and Vibration 290 (2006) 500–507
502
The two commonly used definitions mentioned in Section 1 are both based on the same Taylor series expansions of the radial component and tangential component of the fluid force f separately while neglecting higher than first-order terms as follows [1–3]. 3 2 D 7 6 Df 7 qf qf qf qf 6 7, 6 f ¼ ðf Þs þ 7 _ 6 D_ q qf q_ qf 5 4 _ Df 2 f f ¼ ðf f Þs þ
D
3
7 6 @f f @f f @f f @f f 6 Df 7 7. 6 7 _ 6 @ @f @_ @f 4 D_ 5 _ Df
The above equations can be written in the following form: 2 3 2 qf qf qf ! ! 6 Cq Cqf 7 CD 6 Cq_ f ðf Þs 6 7 6 þ ¼ 6 qf 7 6 qf qf f 5 CDf f f ðf f Þs 4 f 4 f Cq Cqf Cq_
3 qf ! _ 7 CD_ Cqf 7 _ , qf f 7 5 CDf _ Cqf
(2)
where all the first-order derivatives are evaluated at the steady-state equilibrium position (es, fs).
2.1. Direct deduction of the stiffness and damping coefficients One of the approaches in the existing literature arrives at the definition of the stiffness coefficients (kij) and damping coefficients (bij) from the Taylor series expansions—Eq. (2)— directly [3]: ! " # ! " # ! f ðf Þs k kf b bf CD CD_ ¼ (3) _ , f f ðf f Þs kf kff bf bff CDf CDf where "
k
kf
kf
kff
#
2
qf 6 Cq 6 ¼ 6 qf 4 f Cq
3 qf " b Cqf 7 7 and 7 qf f 5 bf Cqf
bf bff
#
2
qf 6 Cq_ 6 ¼ 6 qf 4 f Cq_
3 qf _7 Cqf 7 . qf f 7 5 _ Cqf
(4)
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2.2. Transformation approach for deriving the stiffness and damping coefficients From Fig. 1, the following transformation is used [1,2]: ! ! ! f f 0 cos Df sin Df ¼ , ff f 0f sin Df cos Df
(5)
where f 0 is the component of the fluid force f in the same direction as the fluid force component 0 ðf Þs at the steady-state equilibrium position, and f f is the component of the fluid force f in the same direction as the fluid force component f f s at the steady-state equilibrium position. For a small perturbation, Df 1, so that cos Df 1, sin Df ¼ Df. Using these two approximations, Eq. (5) can be rewriten as ! ! ! f f 0 f f ¼ þ Df . ff f 0f f Then, the following definitions of the stiffness and damping coefficients are developed [1,2]. ! " # ! " # ! f 0 ðf Þs k kf b bf CD CD_ ¼ (6) _ , f 0f ðf f Þs kf kff bf bff CDf CDf where 3 3 2 2 qf qf ff qf qf " # " # _7 6 Cq Cqf þ C 7 6 C@_ b bf k kf C@f 7 7 6 6 and ¼6 ¼ 6 qf 7 7. (7) qf qf qf kf kff b b f 5 4 4 f f 5 f ff f f _ Cq_ Cq Cqf C Cqf 2.3. Inconsistencies between two kinds of existing definitions From Eqs. (4) and (7), it is very clear that some inconsistencies between the definitions of kf and kff exist. The second definitions of kf and kff each has one more coupled term than does the first set of equations presented in Section 2.1. 3. A new derivation of the definitions of stiffness and damping coefficients Taking the derivative of both sides of Eq. (1) with respect to time t yields: df f def df df de ¼ e þ ef þ f þff , dt dt dt dt dt
(8)
_ df qf d qf df qf d_ qf df þ þ þ ; ¼ dt q dt qf dt q_ dt qf_ dt df f qf f d qf f df qf f d_ qf f df_ þ þ þ : ¼ _ dt dt q dt qf dt q_ dt qf
(9)
where
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The derivatives of unit vectors e and ef are [4] de df ef ; ¼ dt dt def df e : ¼ dt dt
(10)
Substituting Eqs. (9) and (10) into Eq. (8) yields df ¼ dt
! _ df qf d qf df qf d_ qf df þ þ þ f e _ dt dt f q dt qf dt q_ dt qf
! _ df qf f d qf f df qf f d_ qf f df þ þ þ þ f ef . þ _ dt dt q dt qf dt q_ dt qf Simplifying further, we obtain 3 qf d qf df qf d_ qf df_ þ þ þ ff i6 _ dt 7 dt qf q_ dt qf 7 6 q dt 7 ef 6 6 qf d _ 7. 4 f þ qf f þ f df þ qf f d_ þ qf f df 5 _ dt dt q dt qf q_ dt qf 2
df h ¼ e dt
(11)
For a very short-time interval Dt when a small perturbation is applied to the journal that was in steady-state equilibrium position s ; fs , Eq. (11) can be approximated as follows through multiplying the two sides by Dt. 3 2 qf qf qf qf _ D þ f f Df þ D_ þ Df 7 _ h i6 q qf q_ qf 7 6 e e f (12) f fs ¼ 7. 6 qf qf f qf f qf f 4 f _5 Df D þ þ f Df þ D_ þ q qf q_ qf_ The fluid force at the steady-state equilibrium position s ; fs is 2 3 f s h i 5. 4 e e fs ¼ f ff s
Substituting Eqs. (1) and (13) into Eq. (12), yields 2 3 qf qf qf qf _ 3 2 D f Df þ D þ f D_ þ f 7 h i f f s h i6 _ qf q_ qf 7 5 ¼ e ef 6 q e ef 4 6 qf 7, qf f qf f qf f ff ff 4 f 5 _ D þ þ f Df þ D_ þ Df s _ q qf q_ qf
(13)
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i.e., 3 2 qf qf qf qf _ 3 D þ f D_ þ D f Df þ f f f s 7 6 _ qf q_ qf 7 5 ¼ 6 q 4 7. 6 qf qf f qf f qf f ff ff 4 f _5 Df D þ þ f Df þ D_ þ s _ q qf q_ qf 2
Rewriting Eq. (14), one arrives at the following equation: 3 2 2 qf ff qf qf 3 2 " # f f s 7 6 Cq_ 6 5 ¼ 6 Cq Cqf C 7 CD þ 6 4 7 6 qf 6 qf qf f f 5 CDf ff ff 4 f 4 f s Cq_ Cq Cqf Cq
3 qf " # Cqf_ 7 7 CD_ _ . qf f 7 5 CDf Cqf_
Now, the stiffness and damping coefficients can be defined as follows: 3 2 " #" # " #" # f f s k kf b bf CD CD_ 4 5¼ _ , kf kff bf bff CDf CDf ff ff
(14)
(15)
(16)
s
where "
k
kf
kf
kff
#
2
qf 6 Cq 6 ¼6 4 qf f Cq
3 ff qf þ Cqf C 7 7 7 qf f f 5 Cqf C
" and
b
bf
bf
bff
#
2
qf 6 Cq_ 6 ¼ 6 qf 4 f Cq_
3 qf _7 Cqf 7 qf f 7 5 _ Cqf
(17)
The final expressions for the stiffness and damping coefficients described by Eq. (17) are consistent with the commonly used expressions described by Eq. (7) in Section 2.2. However, the physical meaning in this derivation is clearer and simpler. In Eq. (16), the change of the fluid force component fe is exactly the relative force change in the direction of ee due to the small perturbation, and the change of the fluid force component ff is exactly the relative force change in the direction of ef due to the small perturbation. 4. Verification and discussion In this section, as an example, the definitions of stiffness and damping coefficients described by Eqs. (16) and (17) are applied to infinitely short bearing. Assuming that the short bearing theory with half-Sommerfeld boundary conditions applies [5], the fluid forces in the journal bearing are [6] " # _ p 1 þ 22 _ RL3 m 22 o 2f , (18) þ f ¼ 5=2 2C 2 ð1 2 Þ2 ð1 2 Þ
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" # RL3 m p o 2f_ 4_ . ff ¼ þ 2C 2 2ð1 2 Þ3=2 ð1 2 Þ2
(19)
Substituting Eqs. (18) and (19) into Eqs. (17) and simplifying the resulting expressions, it can be shown that elements of the linearized stiffness matrix kij ði; j ¼ ; fÞ are k ¼
2omRL3 ð1 þ 2 Þ
kf ¼ kf ¼
C 3 ð1 2 Þ3 pomRL3
;
; 3=2 4C 3 ð1 2 Þ pomRL3 ð1 þ 22 Þ
kff ¼
5=2
4C 3 ð1 2 Þ omRL3
2
C 3 ð1 2 Þ
(20) ;
:
The elements of the damping matrix bij ði; j ¼ ; fÞ are b ¼
pmRL3 ð1 þ 22 Þ
; 2C 3 ð1 2 Þ5=2 2mRL3 ; bf ¼ 3 C ð1 2 Þ2 (21) 2mRL3 ; bf ¼ 3 2 C ð1 2 Þ pmRL3 bff ¼ : 2C 3 ð1 2 Þ3=2 3 3 Using k¯ ij ¼ C=R =moL kij ; ði; j ¼ ; fÞ and b¯ ij ¼ C=R =mL bij ; ði; j ¼ ; fÞ, the linearized stiffness coefficients and damping coefficients can be normalized. It has been shown that the normalized expressions for the stiffness and damping coefficients are consistent with those in Refs. [1,2]. However, if the definitions described by Eqs. (3) and (4) are applied to the same infinitely short bearing theory, kf 0 and kff 0 since both of the fluid force components f and f f are not an explicit function of f.
5. Conclusion The derivations presented in this paper offer a simple procedure for arriving at the definitions of the stiffness and damping coefficients in polar coordinates. The method also offers a clear physical meaning of the dynamic coefficient and yields consistent results when applied to infinitely short bearings.
ARTICLE IN PRESS J.K. Wang, M.M. Khonsari / Journal of Sound and Vibration 290 (2006) 500–507
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References [1] R. Holmes, The vibration of a rigid rotor on short journal bearings, Journal of Mechanical Engineering Science 2 (1960) 337–341. [2] A. Szeri, Linearized force coefficients of a 1101 partial journal bearing, Proceedings of the Institution of Mechanical Engineers 181 (Part 3B) (1966–67) 130–133. [3] D. Childs, H. Moes, H.V. Leeuwen, Journal bearing impedance description for rotor dynamic applications, ASME Journal of Lubrication Technology 99 (1977) 198–214. [4] D.T. Greenwood, Intermediate Dynamics, second ed., Prentice-Hall, Englewood Cliffs, NJ, 1988. [5] M.M. Khonsari, E.R. Booser, Applied Tribology: Bearing Design and Lubrication, Wiley, New York, 2001. [6] M.M. Khonsari, Y.J. Chang, Stability boundary of non-linear orbits within clearance circle of journal bearings, ASME Journal of Vibration and Acoustics 115 (1993) 303–307.