5.2.3 Linearization of the Oil Film Force To discuss the linear stability of a shaft, the oil film force is linearized beforehand in the neighborhood of the equilibrium point of the journ
Trang 172 5 Stability of a Rotating Shaft — Oil Whip
The oil film force of a finite length bearing can also be written in the following form, similar to those of an infinitely long bearing and a short bearing [20]:
Pκ= − 6µ
R j
c
2
L R j
(ω − 2˙θ)P(1)
κ + ˙κP(2)
Pθ= 6µ
R j
c
2
L R j
(ω − 2˙θ)P(1)
θ + ˙κP(2)
In this case, P(1)κ , P(2)κ , P(1)θ , and P(2)θ are functions ofκ with the bearing dimensions
as parameters These are usually calculated numerically
5.2.3 Linearization of the Oil Film Force
To discuss the linear stability of a shaft, the oil film force is linearized beforehand in the neighborhood of the equilibrium point of the journal center Oj0(κ0, θ0)
Fig 5.4 Rectangular coordinates (radial direction, circumferential direction)
Equations 5.23 and 5.24 are used to express the oil film force Then, variable substitutionsκ ⇒ κ0+ κ, θ ⇒ θ0+ θ are made in these expressions, and assumptions that new variablesκ, θ, and their time derivatives ˙κ and ˙θ are so small that their second power, third power, and products can be disregarded give the linearized oil film
force Pκand Pθas follows:
Pκ= − 6µ
R j
c
2
L R j×
0 ω (2+ κ )(1− κ )
1+ κ
2
2κ0
2+ κ +
2κ0
1− κ
Trang 2
− 4κ0 θ˙ (2+ κ0 )(1− κ0 )+ 2˙κ
(1− κ0 )3/2
π
2 −π(2 + κ8
Pθ= + 6µ
R j
c
2
L R j×
⎡
⎢⎢⎢⎢⎢
(2+ κ0 )
1− κ0
1+ κ
1
κ0
− 2κ0
2+ κ0
+ κ0
1− κ0
− 2πκ0θ˙ (2+ κ0 )
1− κ0
+ 4κ0˙κ (2+ κ0 )(1− κ0 )
⎤
⎥⎥⎥⎥⎥
Now, to consider the journal motion in the rectangular coordinate system (x , y)
shown in Fig 5.4, let us transform the above components of the oil film force to the
rectangular components P x and P yusing the following expressions:
P x = Pκ− Pθ0 y j
cκ0, P y = Pθ+ Pκ0 y j
cκ0
(5.31)
where Pθ0 and Pκ0are the stationary values of the oil film force at the equilibrium point, i.e., the constant terms of Eqs 5.29 and 5.30 Now, let us perform the variable transformationκ ⇒ x j /c, θ ⇒ y j /cκ0, ˙κj ⇒ ˙x j /c, and ˙θ j ⇒ ˙y j /cκ0, and assume that
x j , y j, ˙x j, and ˙y jare sufficiently small, considering small vibrations
Then the oil film forces P x and P ycan be written as follows:
P x = P x0 + K xx
P0
c x j + K xy
P0
c y j + C xx
P0
ωc ˙x j + C xy
P0
ωc ˙y j (5.32)
P y = P y0 + K yx
P0
c x j + K yy
P0
c y j + C yx
P0
ωc ˙x j + C yy
P0
ωc ˙y j (5.33)
where P x0 and P y0 in the above equations are the stationary values of the oil film force at the equilibrium point, and are given as follows:
P x0= − 6µ
R j
c
2
R j L 2κ0 ω
P y0= + 6µ
R j
c
2
R j L πκ0ω (2+ κ0 )
1− κ0
(5.35)
P0is their resultant P0= P2
x0 + P2
y0and is given as follows:
P0= 6µ
R j
c
2
R j Lω κ0
π2− (π2− 4)κ0
where c is the radial clearance of the bearing andω is the angular velocity of the rotating shaft The coefficients Kxx , · · ·, C xx, · · · are nondimensional coefficients, and,
as shown below, are given as functions ofκ only This is important:
Trang 374 5 Stability of a Rotating Shaft — Oil Whip
K xx= − 4κ0
π2− (π2− 4)κ0
1
κ0 − κ0
2+ κ0 + κ0
1− κ0
(5.37)
1− κ0
κ0
π2− (π2− 4)κ0
(5.38)
C xx= − 2(2+ κ0 )
κ0
1− κ0
π2− (π2− 4)κ0
π
2−π(2 + κ8
0 )
(5.39)
π2− (π2− 4)κ0
(5.40)
K yx= + π
1− κ0
π2− (π2− 4)κ0
1
κ0
− 2κ0
2+ κ0
+ κ0
1− κ0
(5.41)
K yy = − 1
Fig 5.5 Nondimensional spring constants and nondimensional damping constants of an oil
film
Trang 4Equations 5.32 and 5.33 can be written in an easier form to grasp as follows:
P x = P x0 + k xx x j + k xy y j + c xx x˙j + c xy y˙j (5.43)
P y = P y0 + k yx x j + k yy y j + c yx x˙j + c yy y˙j (5.44)
Or, in a matrix representation:
P x
P y =
P x0
P y0 +
k xx k xy
k yx k yy
x j
y j +
c xx c xy
c yx c yy
˙
x j
˙
where
k i j = K i j
P0
c , c i j = C i j
P0
In the above equations, k xx , k xy , k yx , and k yyare called oil film spring constants and
c xx , c xy , c yx , and c yy are called oil film damping constants These eight constants
are collectively called oil film constants Of these eight oil film constants, k xx , k yy,
c xx , and c yy are called diagonal terms and k xy , k yx , c xy , and c yxare called coupling
terms The existence of a coupling term means that the direction of the force is
different from that of the displacement and hence causes circumferential whirling of
a shaft
K xx , · · ·, C xx, · · · given by Eqs 5.37 – 5.42 are called the nondimensional spring
constants and the nondimensional damping constants of the oil film, respectively.
Fig 5.5 shows the nondimensional spring constants and the nondimensional damping constants graphically The horizontal axes indicate the eccentricity ratio κ0 of the equilibrium point
In the case of the short bearing approximation and for a finite length bearing also, the oil film coefficients can be written in the form of Eq 5.32 and Eq 5.33, and it is known that the nondimensional coefficients K xx , · · ·, C xx, · · · are functions of κ0only This is important
It is recognized that the relation C xy = C yxin Eq 5.42 holds quite generally in various cases, in finite length bearings or under Reynolds’ boundary condition, for example [54] [56]
5.2.4 Equations of Motion
By using the oil film force obtained in the preceding section, the equations of mo-tion of a rotating shaft supported by journal bearings can be derived and dynamic characteristics of the rotating shaft can thereby be analyzed
For simplicity, let us consider a system with one rotor and two bearings as shown
in Fig 5.6 (a system composed of a rotor supported by two journal bearings) and assume the following:
1 The shaft of the rotor is a thin bar of a circular section and its mass can be neglected
2 A disk with mass is attached to the shaft at the center
Trang 576 5 Stability of a Rotating Shaft — Oil Whip
Fig 5.6 A system with one rotor and two bearings
3 Both ends of the shaft are supported by journal bearings of the same specifica-tion
4 The whole system is completely symmetrical with respect to the disk and has no imbalance
The equations of motion of the system can be written as follows in the coordinate system of Fig 5.7:
k(x − x j)+ P x + P1 cosθ0= 0 (5.49)
k(y − y j) + P y − P1 sinθ0= 0 (5.50)
where m is the mass of the disk, k is the spring constant of the shaft, (x , y) and (x j , y j) are the coordinates of the disk center and the journal center, respectively;
P x and P y are the x and y components of the oil film force P acting on the journal, respectively; and P1= mg is the bearing load (g is the acceleration of gravity) From the balance of the bearing load and the oil film force at the equilibrium point, P1and
P0of Eq 5.36 must be equal, i.e., P1= P0 Here, P and P1denote the sum of the oil film forces of the two bearings and the sum of the two bearing loads, respectively
If the center of mass of the disk has a deviationδ, the right-hand side of Eqs 5.47
and 5.48 should read mδω2cosωt and −mδω2sinωt, respectively ω is the angular
velocity of the shaft
5.2.5 Stability Limit
It is not easy to discuss the stability of the shaft in terms of the equation of motion
of the previous section because the oil film force P x and P yare complicated nonlin-ear functions Therefore, we divide the vibrations into two categories for which the equation of motion can be simplified, namely into sufficiently small vibrations and sufficiently large vibrations, and then discuss the stability of the two cases separately Small vibrations mean such vibrations of the journal center around the equilib-rium point that the amplitude is sufficiently small compared with its eccentricity from
Trang 6Fig 5.7 Coordinates of the system (radial and circumferential direction)
the bearing center The situation is shown in Fig 5.8a,ba In this case, the oil film force Eqs 5.23 and 5.24 can be approximated by the linear expressions of Eqs 5.32 and 5.33 in the neighborhood of the equilibrium point of the journal, as stated before
Fig 5.8a,b Small vibrations (a) and large vibrations (b) [14]
Large vibrations mean such vibrations (whirling) of the shaft that it bends con-siderably as shown in Fig 5.8a,bb In this case, the journal may tilt in the bearing, and the journal center inevitably circles around the bearing center for the majority of the bearing length (conical motion) In this case, the oil film force Eqs 5.23 and 5.24 can be simplified by approximating the journal motion by a steady revolution The stability limits (diverging criteria) of small vibrations and that of large vibrations are different
By combining the stability limits of small vibrations and that of large vibrations,
it is possible to explain the complicated process of the occurrence of oil whip
Trang 778 5 Stability of a Rotating Shaft — Oil Whip
a Stability of Small Vibrations
Linearization of Equation of Motion The following relations are obtained from Eqs 5.47 and 5.48:
x j= m
k ¨x + x, y j= m
k ¨y + y, ˙x j=m
k
x + ˙x, ˙y j=m
k
y + ˙y (5.51)
By means of these relations, it is possible to eliminate the coordinates of the
jour-nal center (x j , y j) from the equation of motion Eqs 5.47 and 5.48 Thus, substitution
of Eqs 5.49 and 5.50 into Eqs 5.47 and 5.48, respectively, use of the oil film force for small vibrations Eqs 5.32 and 5.33, and additional use of Eq 5.51 give the fol-lowing linearized form of Eqs 5.47 and 5.48 on the coordinates of the disk center
(x, y) only:
C xx
P1
k ωc
x+K xx
P1
kc − 1¨x + C xx
P1
m ωc ˙x + K xx
P1
mc x +C xy
P1
kωc
y +K xy
P1
kc ¨y + C xy
P1
mωc ˙y + K xy
P1
C yx
P1
kωc
x +K yx
P1
kc ¨x + C yx
P1
mωc ˙x + K yx
P1
mc x +C yy
P1 kωc
y+K yy
P1
kc − 1¨y + C yy
P1 mωc ˙y + K yy
P1
The stability of the rotor can be investigated by solving these equations simultane-ously
Stability Criterion The above simultaneous equations can be written in the following general form:
A x + B ¨x + C ˙x + Dx + E y + F ¨y + G˙y + Hy = 0 (5.54)
a x + b ¨x + c ˙x + d x + e y + f ¨y + g ˙y + h y = 0 (5.55)
When solutions of the form x = αe st
and y = βe st
are assumed, the following
equa-tion must hold for the existence of soluequa-tions other than x ≡ 0 and y ≡ 0:
As3+ Bs2+ Cs + D, Es3+ Fs2+ Gs + H
a s3+ b s2+ c s + d, e s3+ f s2+ g s + h
This is called the characteristic equation, and it will become the sixth-order equation below, if the determinant is developed:
A0s6+ A1s5+ A2s4+ A3s3+ A4s2+ A5s + A6= 0 (5.57)
where it is assumed that A0> 0 If A0 < 0, then the sign of the whole equation will
be changed so that A0> 0
For the solutions x and y to be stable (i.e., they do not diverge), it is necessary
and sufficient if the real part of all roots of the characteristic equation Eq 5.57 are
Trang 8negative, and the Routh–Hurwitz criterion is known as a criterion for this In the
case of Eq 5.57, it can be written as follows:
A0, A1, A2, A3, A4, A5, A6> 0 (5.58)
A1 A3 A5 0 0
A0 A2 A4 A6 0
0 A1 A3 A5 0
0 A0 A2 A4 A6
0 0 A1 A3 A5
A1 A3 A5
A0 A2 A4
0 A1 A3
In other words, all the coefficients A0, A1, · · · must be positive and the two above determinants of these coefficients must also be positive
In the case of Eqs 5.52 and 5.53, the coefficients of the characteristic equation
Eq 5.57, A0, A1, · · · will be as follows:
A0= B0
1
ω2ω1
P 1
mc
2
A1= B1
1
ωω1
P 1
mc
2
− B2
1
ωω1
P 1
mc
A2= 2B0
1
ω2ω1
P 1
mc
2
+ B3
1
ω1
P 1
mc
2
− B4
1
ω1
P 1
mc
+ 1
A3= 2B1
1
ωω1
P 1
mc
2
− B2
1 ω
P 1
mc
A4= B0
1
ω2
P 1
mc
2
+ 2B3
1
ω1
P 1
mc
2
− B4
P 1
mc
A5= B1
1 ω
P 1
mc
2 , A6= B3
P 1
mc
2
where B0, B1, · · · are as follows:
B0= C xx C yy − C xy C yx, ω1= k/m
B1= K xx C yy + K yy C xx − K xy C yx − K yx C xy
B2= C xx + C yy, B3= K xx K yy − K xy K yx
B4= K xx + K yy
When K xx , · · · C xx, · · · are given by Eqs 5.37 – 5.42, the actual calculations show that Eq 5.58 of the Routh–Hurwitz criterion always holds, and if Eq 5.59 holds, Eq 5.60 always holds Therefore, only Eq 5.59 need be considered as a sta-bility condition If the determinant of Eq 5.59 is developed, and the coefficients
A0, A1, · · · are substituted into it, a comparatively simple result is obtained as fol-lows This is the stability criterion:
Trang 980 5 Stability of a Rotating Shaft — Oil Whip
1
ω2
P 1
mc
> K1(κ0)
K2(κ0)+ω1
1
P 1
where K1(κ0) and K2(κ0) are given as follows:
K1(κ0)= B1 − B1B2B4+ B2 B3
B0B2
, K2(κ0)= B2
Thus, finally, the shaft will be stable if Eq 5.61 is satisfied
Fig 5.9 Stability chart for an infinitely long bearing [14] [15]
Since K1and K2are functions ofκ0only, Eq 5.61 can be expressed in a chart as
shown in Fig 5.9 This is called a stability chart The eccentricity ratioκ0is taken downward along a vertical axis, and on the other vertical axis to the left, a scale for
the relation between nondimensional bearing load P1/ 6µ(R/c)2RLω!and the ec-centricity ratioκ0(cf Eq 5.36 where P0= P1) is shown The horizontal axis shows the nondimensional quantity (1/ω2)(P1/mc) which is made up of the rotational
an-gular velocity, the bearing load, mass of the disk, and the bearing clearance Three curves in the chart are the stability limit curves for three different values, 0, 5 and
10 of nondimensional parameter (1/ω1 )(P1/mc), which is formed from the critical
speed of the shaft, the bearing load, mass of the disk, and the bearing clearance The leftmost curve corresponds to a rigid shaft
To the lower right of each stability curve is the stable region and to the upper left
is the unstable region If the dimensions of the bearing and the shaft and the rotating speed, etc are given, the position of the point corresponding to the operational con-dition is determined on the chart, and the stability can be judged by the side of the curve on which the point falls
Although infinitely long bearings under G¨umbel’s boundary condition have been considered so far, short bearings or finite length bearings under other boundary
Trang 10con-Fig 5.10 Locus of the journal center for finite length bearings [18]
Fig 5.11 Eccentricity of the journal for finite length bearings [18]
ditions can be discussed in a similar way For example, when G¨umbel’s boundary condition is used for a finite length bearing, the locus of the journal center will be
as shown in Fig 5.10; the eccentricity ratio can be obtained from Fig 5.11 The stability chart in this case is shown in Fig 5.12 (Gotoda [18] [20]) As in the case
of Fig 5.9, the point corresponding to the dimensions of the bearings and rotating shafts, the rotating speed, and so forth is first determined on the stability chart, and then stability of the shaft is determined by the side of the stability curve on which the point falls
Trang 1182 5 Stability of a Rotating Shaft — Oil Whip
Fig 5.12a-c Stability charts for finite length bearings for three different values of the
param-eter 1
ω 2
P1
mc
[18]
Trang 12From the stability chart, the following can be said:
1 The larger the eccentricity ratio, the more stable the shaft is If the eccentricity ratio is larger than 0.8, in paticular, the shaft is always stable
2 The higher the critical speed, the higher the stability limit is
3 The smaller the (length L /diameter D) ratio of the bearings, the more stable the
shaft is
4 There is no such simple rule that the stability limit is equal to twice the critical speed
It can be seen from the above analysis that a low oil viscosity, high bearing
pres-sure, large bearing clearance, high critical shaft speed, and a small L/D ratio are
recommended for high shaft stability
b Stability of Large Vibrations
When a shaft bends and whirls with a large amplitude, the journal center performs steady revolution around the bearing center for the major part of the bearing length,
as shown in Fig 5.8a,bb Therefore, by setting the time derivative ˙κ of the eccentricity ratio to be 0 in Eq 5.24 of the oil film force, the circumferential component of the
oil film force Pθcan be written in the following simple form:
whereω is the rotating speed of the shaft and Ω = ˙θ is the whirling speed of the shaft
In the case of large vibrations (or whirling), stability means whether the whirling radius of the journal diverges or converges under the oil film force mentioned above, and in this case, twice the critical speed has an important meaning as seen from the above equation
Fig 5.13 Modeling of large vibrations [15]
For large vibrations, the shaft system can be modeled by a cylinder of mass m tied to the bearing center with a spring of spring constant k as shown in Fig 5.13.
... Eqs 5. 47 and 5. 48 Thus, substitutionof Eqs 5. 49 and 5. 50 into Eqs 5. 47 and 5. 48, respectively, use of the oil film force for small vibrations Eqs 5. 32 and 5. 33, and additional use of Eq 5. 51... given by Eqs 5. 37 – 5. 42, the actual calculations show that Eq 5. 58 of the Routh–Hurwitz criterion always holds, and if Eq 5. 59 holds, Eq 5. 60 always holds Therefore, only Eq 5. 59 need be considered... coefficients must also be positive
In the case of Eqs 5. 52 and 5. 53, the coefficients of the characteristic equation
Eq 5. 57, A0, A1, · · ·