A coupling vibration model of multi-stage pump rotor system based on FEM.
Wang, Leqin ; Zhou, Wenjie ; Wei, Xuesong 等
1. Introduction
The vibration issue of multi-stage pump rotor system has been one
of the main problems in multi-stage pump units. Compared with the
single-stage pumps or turbines, the effect of annular seals on the
vibration behavior of multi-stage pump rotor system needs additional
consideration [1]. The multi-stage pump rotor system can be simplified
into a rotor-bearing-seal system. As an important factor influencing the
vibration status of the rotor system, the dynamic characteristics of
bearings and the dynamic vibration of bearing-rotor system have been the
target of many researchers [2, 3].
Fluid flows in the annular seals can generate reaction force, which
increases the stiffness of the rotating shaft system [4]. In fact, the
seal fluid forces are usually expressed in the form of dynamic
coefficients. Childs [5, 6] proposed a finite-length method to calculate
the dynamic coefficients of annular seals based on the Hirs'
turbulent lubrication equations and perturbation method, and this method
has been widely applied to the analysis of the stable motion of
rotor-bearing-seal system. Kanemori and Iwatsubo [7, 8] conducted groups
of experiments to research the dynamic coefficients of long annular
seals at various rotating speeds, whirling speeds, and pressure drops,
they found the experimental results coincided fairly well with
Childs' theory. However, the Childs' linear model will be no
longer applicable in the case of large disturbance. Muszynska and Bently
[9, 10] proposed a classic nonlinear seal fluid dynamic force model on
the basis of experiments, which could get satisfactory results even if
the whirl motion was in large disturbance. The Muszynska nonlinear seal
force model was commonly used to study the nonlinear vibration of
rotor-bearing-seal system [11, 12].
Although some FSI models of the rotor system were proposed in the
previous researches, most of them were used to study the transient
vibration. In this paper, a novel coupling vibration model of
multi-stage pump rotor-bearing-seal system for stable motion was
proposed. The dynamic coefficients of short annular seals and bearings
were calculated by Childs' finite-length method, Hertz contact
theory and Elasto-Hydrodynamic Lubrication (EHL), respectively. The
calculated results were consistent with the experimental results
conducted by a model rotor system. The influence of rotating speed on
the dynamic coefficients of short annular seals and bearings was also
analyzed.
2. Coupling vibration model of the system
2.1. Ball bearing model
The rollers and raceways of ball bearing will deform when the ball
bearing suffers from radial or axial loads. Considering that the
double-row self-aligning ball bearings in this paper are not under
conditions of high rotating speed, gyroscopic moments and centrifugal
forces on the rolling elements are ignored. Similarly, the axial
deformations are also ignored due to the small axial force and only the
radial load is taken into consideration. Fig. 1 shows the contact
deformations between the raceways and rollers under radial force
[F.sub.r].
[FIGURE 1 OMITTED]
The values of the contact deformations can be calculated by Hertz
contact theory, which can be calculated by the following form [13]:
[delta] = 2K/[pi][m.sub.a] [cube root of (1/8[(3/E').sup.2]
[Q.sup.2][SIGMA][rho]], (1)
where [m.sub.a] = [cube root of (2Lk/[pi])], E' = 2/((1 -
[[mu].sup.2.sub.1])/[E.sub.1] + (1 - [[mu].sup.2.sub.2])/E;
[SIGMA][rho] is curvature sum; Q is contact load; [E.sub.1] and
[E.sub.2] are elastic moduli of rolling balls and races, respectively;
[[mu].sub.1] and [[mu].sub.2] are Poisson's ratio for rolling balls
and races, respectively, k is ellipticity; K and L are first and second
kind of complete elliptic integrals, respectively.
The curvature sum [SIGMA][rho] is determined by raceway groove
curvature radius factor and geometric structure of the ball bearing and
it can be got by Zhou et al [14], thus the sum contact deformations can
be calculated as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2)
where [k.sub.p] = [(3/E').sup.2/3]
1/[pi][[K.sub.i]/[m.sub.a,i][([SIGMA][[rho].sub.i].sup.1/3] +
[K.sub.o]/[m.sub.a,o][([SIGMA] [[rho].sub.o]).sup.1/3]];
[sigma] is contact deformation and the subscript 'i' and
'o' denote inner race and outer race, respectively.
From Fig. 2, the load equilibrium equation of the ball bearing can
be described as:
[F.sub.r] = [Q.sub.0] + 2 [n.summation over (j=1)] [Q.sub.j] cos
j[psi], (3)
where [Q.sub.0], [Q.sub.1], [Q.sub.2], ..., [Q.sub.j] are loads on
the roller, respectively, [[sigma].sub.r] is contact deformation in
maximum load direction, [psi] is the angle of the adjacent rolling
balls.
[FIGURE 2 OMITTED]
Furthermore, due to the symmetrical structure of double-row
self-aligning ball bearing and the radial clearance, the largest ball
bearing load can be summarized based on massive computation:
[Q.sub.max] = 2.5[F.sub.r]/z cos [alpha], (4)
where z is the number of rolling balls, [alpha] is contact angle.
According to the definition of contact stiffness, the Hertz contact
stiffness of ball radial bearing can be obtained as:
[k.sup.b.sub.c] = d[F.sub.r]/d[[delta].sub.r =
3/3.684[k.sub.p][(cos [alpha]).sup.-5/3] [z.sup.-2/3][F.sup.-1/3.sub.r]
=-^-T-7, (5)
where superscript 'b' denotes bearing.
The rolling balls and the raceways are separated by a layer of oil
film. Based on EHL, the systemic numerical simulations were carried out
by Hamrock and Dowson to research the isothermal elliptical contact
issue and propose the film thickness formula [15]. The minimum oil film
thickness can be expressed as:
[h.sub.min] = 3.63[R.sub.x][U.sup.0.68]
[G.sup.0.49][W.sup.-0.073](1 - [e.sup.-0.68k]), (6)
where U = [[eta].sup.[bar.u].sub.0]/E'[R.sub.x] ; W =
Q/E'[R.sup.2.sub.x] ; G = [lambda]E' ; [R.sub.x] is equivalent
radius of curvature in ball rolling direction, [[eta].sub.0] is dynamic
viscosity, [bar.u] is surface velocity, [lambda] is pressure viscosity
coefficient.
The total minimum film thickness can be calculated as:
h = [h.sub.min,i] + [h.sub.min,o] = [k.sub.q] [Q.sup.-0.073], (7)
where
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
The calculation of lubrication oil film stiffness is similar to
Hertz contact stiffness and the lubrication oil film stiffness of
double-row self-aligning ball bearing can be expressed according to the
definition of oil film stiffness:
[k.sup.b.sub.f] = 1/0.0683[k.sub.q][(cos
[alpha]).sup.1.073][z.sup.0.073] [F.sup.-1.073.sub.r], (8)
Finally, an overall radial stiffness of double-row self-aligning
ball bearing can be calculated as:
[k.sup.b.sub.t] = [k.sup.b.sub.xx] = [k.sup.b.sub.yy] =
[(1/[k.sup.b.sub.c] + 1/[k.sup.b.sub.f]).sup.-1]. (9)
2.2. Annular seal model
The finite-length theory was propose by Childs [5, 6], which has
been widely applied to calculate the dynamic coefficients of annular
seals and can be expressed as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (10)
where superscript 's' denotes short annular seal.
The values of dynamic coefficients are determined by geometry size
and working condition of the short annular seals.
2.3. System motion equations
For a rotor-bearing-seal system with n degrees of freedom, the
motion state can be described by n independent generalized coordinates
[u.sub.j] (j = 1, 2, ..., n), which can be expressed by non-conservative
Lagrange equation:
d/dt ([partial derivative]T/[partial derivative][[??].sub.j] -
[partial derivative](T - V)/[partial derivative][u.sub.j] = [q.sub.j],
(11)
where T is kinectic energy, V is strain energy, [q.sub.j] are
generalized forces.
The motion equations of rigid disc can be obtained by Eq. (11) and
they have been described in [3]:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (12)
where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; m is
lumped
mass, [OMEGA] is rotating speed, [J.sub.d], [J.sub.p] are diametric
moment of inertia and polar moment of inertia, respectively, [u.sub.1]
and [u.sub.2] are the generalized coordinates vectors, [q.sub.1] and
[q.sub.2] are the generalized forces vectors, superscript 'd'
denotes rigid disc.
The motion equations of elastic shaft can be derived from FEM and
each element can be expressed as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (13)
where superscript 'e' denotes elastic shaft element,
[M.sup.e], [OMEGA][J.sup.e] and [K.sup.e] are 4 x 4 matrices.
Combining the motion equations of the rigid discs and elastic shaft
elements, the motion equations of rotor system can be expressed as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (14)
where [[M.sub.1]], [[J.sub.1]] and [[K.sub.1]] have the similar
structure, and dimensions of these three coefficient matrixes are 2n x
2n; n is the total numbers of nodes.
Eq. (14) can be described in the following form:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (15)
In order to introduce the dynamic coefficients of bearings and
short annular seals, a FSI method [1] is used to establish the final
vibration model of rotor-bearing-seal system. Assuming the seal is on
node j, the displacement of node j can be expressed as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (16)
where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Substituting Eqs. (16) and (10) into Eq. (15) and the motion
equation of the rotor system including seal force can be expressed as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (17)
For the general case of multi-stage annular seals and considering
the effect of ball bearings, the final motion equation of
rotor-bearing-seal system can be written as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (18)
where
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII];
and [n.sub.s], [n.sub.b] are numbers of annular seals and bearings,
respectively.
The solving process of coupling vibration mode for multi-stage pump
is shown in Fig. 3.
[FIGURE 3 OMITTED]
3. Experimental test facilities
A model rotor system was established to verify the coupling
vibration model described above. Fig. 4 shows the experimental model
rotor system test facilities. The model rotor system is driven by an AC
motor and its rotating speed is controlled by a variable-frequency
electric cabinet. The maximum rotating speed of the drive motor and the
model rotor are 3,000 r/min and 6,000 r/min, respectively, and the
torque is transmitted by a synchronous belt.
In order to observe and record the real-time vibration data for
further data processing, a data acquisition system including data
acquisition and analysis instrument with multi-channel, notebook
computer, dis-placement sensors and power supply is applied. The data
acquisition and analysis instrument combines the multi-DSP parallel
processing technology, low-noise design technology and high-speed data
transmission, which ensures the real-time and accuracy of collected
vibration data. Furthermore, the amplitude and phase of the fundamental
frequency can be calculated by correlation analysis from the collected
data.
[FIGURE 4 OMITTED]
[FIGURE 5 OMITTED]
The model rotor system is the key component in the experiment
facilities. As shown in Fig. 5, the model rotor system is designed in
segmental structure. There are nine chambers in the system, including
five low-pressure chambers and four high-pressure chambers. There are
five disks instead of impellers in the corresponding five low-pressure
chambers. The only fluid flow passage between high-pressure chamber and
low-pressure chamber is the seal channel. The model shaft is supported
by two double-row self-aligning ball bearings. In order to produce the
pressure drop between the two ends of each annular seal channel, a
booster pump is used to supply the high-pressure fluid, which is
connected with the high-pressure chambers. The displacement sensor is
used to measure the vibration amplitude of the model shaft.
Fig. 6 shows the sealing ring and its inner seal channels. The
inlets of the sealing ring are placed in high-pressure chamber and the
outlets are connected with low-pressure chamber. Each sealing ring has
two seal channels between the inlets and outlets. For each sealing ring,
the high-pressure fluid from the booster pump is first pumped into the
high-pressure chamber and then flows into the low-pressure chambers
through the two seal channels. There are totally four sealing rings in
the rotor system, which means that there are eight sealing channels,
i.e., eight sealing forces acting on the model shaft. The radius
clearance of each sealing channel [R.sub.c] ranges from 0.3 to 0.5 mm
and the pressure drop ranges from 0.1 MPa to 0.5 MPa.
[FIGURE 6 OMITTED]
4. Results and discussion
In order to verify the accuracy of the coupling vibration model
mentioned above, the experimental critical speed of model rotor system
was compared with the simulation results calculated by the coupling
vibration model. The damping coefficients and cross stiffness of ball
bearings used in the paper were set to zero. The calculation parameters
of bearings and each sealing channel used in the paper are listed in
Table 1 and Table 2, respectively (the number of each sealing channel
increases successively from the drive-side).
Fig. 7 shows the dynamic relationships of Q with [k.sup.b.sub.f]
and [k.sup.b.sub.t], it can be seen that the oil film stiffness of right
bearing is larger than those of left bearing and both of them decrease
rapidly when rotating speed is less than 500 r/min. As rotating speed
increases, the oil film stiffness difference of the two bearing
decreases and tends to reach the same constant. However, the overall
radial stiffness of two bearings approximately decreases linearly as
rotating speed increases, this is because the contact stiffness is the
main factor influencing the overall radial stiffness. The results imply
that the stiffness of the bearings could not be regarded as a constant
under different rotating speed and it relates with the rotating speed.
[FIGURE 7 OMITTED]
The short annular sealing channels are divided into three forms
according to the length-diameter ratio. The first form includes channel
1, channel 2 and channel 6 to channel 8. The second form includes
channel 3. Channel 4 and channel 5 belong to the third form. In Fig. 8,
we can see that the sealing channel with larger length-diameter ratio
has bigger dynamic characteristics. The principal stiffness decreases
gradually as the rotating speed increases, but the dynamic
characteristics of cross stiffness increases linearly when rotating
speed increases. Similarly, The principal damping and cross damping have
the same varying tendency. The results also show that the rotating speed
is an important factor influencing the sealing dynamic characteristics.
[FIGURE 8 OMITTED]
Fig. 9 shows the Campbell chart in which the first critical speeds
under different radius clearances at pressure drop [??]p = 0.3 MPa were
calculated by the coupling vibration model. The critical speeds increase
from 2054.6 r/min to 2315.5 r/min as the radius clearances change from
0.5 mm to 0.3 mm, which means that under the same condition of pressure
drop, the smaller radius clearance will cause greater sealing force
acting on the rotor system and Lomakin effect will be more remarkable.
[FIGURE 9 OMITTED]
[FIGURE 10 OMITTED]
According to Fig. 10, It can be obviously seen that the first
critical speeds under different pressure drops when [R.sub.c] = 0.3 mm
were calculated. Unlike the calculated results of radius clearances, the
first critical speeds increase from 1969.4 r/min to 2605.3 r/min as the
pressure drop increases from 0.1 MPa to 0.5 MPa. The calculated results
illustrate that greater pressure drop will cause higher critical speed,
which is consistent with the calculated results of Jiang et al [1].
Table 3 shows the experimental and calculated first critical speeds
under different drop pressures and radius clearances. Comparing the
experimental results and the calculated results, it is evident through
the Table 3 that the radius clearance and pressure drop play a
significant role in the dynamic vibration characteristics of the
rotor-bearing-seal system.
Based on the coupling vibration model developed in the paper, the
maximum error percentage and minimum error percentage are 5.5% and 0.2%,
respectively and most error percentages are 1%~3%. The reasons of
calculated error are mainly due to the ignorance of the eccentricity of
shaft and the gyroscopic moments and centrifugal forces of rolling
elements, which cause the increase in calculated error of dynamic
coefficients of annular seals and ball bearing, respectively. Even so,
the error percentages are smaller than those calculated by Jiang et al
[1], in which they ignored the effect of rotating speed on the bearings
and regarded the dynamic coefficients of the bearings as constants. Even
if some other effects on the shaft are ignored in the paper, such as the
rotating damping of the fluids and mechanical seals, the calculated
results under different operating conditions are accurate enough to meet
the engineering needs, which means the coupling vibration model proposed
in the paper is feasible.
5. Conclusions
In order to predict the dynamic vibration characteristics of
multi-stage pumps, a coupling vibration model of multi-stage pump rotor
system was proposed in this paper. The multi-stage pump rotor system was
simplified to a rotor-bearing-seal system and the dynamic coefficients
of bearings and short annular seals were considered changing with the
rotating speed. Based on FEM and Lagrange equation, the motion equations
of the coupling vibration model were established. The dynamic
relationships of [OMEGA] with the stiffness of bearings and annular
seals were calculated and the results shown that the rotating speed was
an important factor influencing the dynamic coefficients of bearing and
annular seals. In order to verify the numerical results of the coupling
vibration model, the first critical speed of a model rotor system was
measured. The calculated results implied that smaller radius clearance
and greater pressure drop would lead to remarkable Lomakin effect and
the coefficients of bearings and annular seals had an important impact
on the system. The experimental results of the model rotor showed good
agreement with the simulated results. Most error percentages were
between 1%~3%. In fact, the stable vibration of rotor system supported
by journal bearings and other structure form bearings can also be
calculated by the coupling vibration model.
Received April 10, 2015
Accepted January 06, 2016
Acknowledgements
This work was supported by the National Natural Science Foundation
of China (Grant No. 51479167) and Science and Technology Key Project of
Zhejiang Province (Grant No. 2014C01066). The support is gratefully
acknowledged. The authors are also thankful to Zhejiang Keer Pump Stock
Co.,Ltd for technical support.
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Leqin Wang *, Wenjie Zhou **, Xuesong Wei ***, Lulu Zhai ****,
Guangkuan Wu *****
* Institute of Process Equipment, Zhejiang University, Hangzhou
310027, China, E-mail: hj_wlq2@zju.edu.cn
** Institute of Process Equipment, Zhejiang University, Hangzhou
310027, China, E-mail: zhouwenjiezwj@zju.edu.cn
*** Institute of Process Equipment, Zhejiang University, Hangzhou
310027, China, Faculty of Engineering, Kyushu Institute of Technology,
Kitakyushu 804-8550, Japan, E-mail: wxs0773@126.com
**** Institute of Process Equipment, Zhejiang University, Hangzhou
310027, China, E-mail: zju_zhailulu@foxmail.com
***** Institute of Process Equipment, Zhejiang University, Hangzhou
310027, China, Institute of Water Resources and Hydro-electric
Engineering, Xi'an University of Technology, Xi'an 710048,
China, E-mail: wuguangkuan@163.com
cross http://dx.doi.org/10.5755/j01.mech.22.1.11420
Table 1
Ball bearing parameters
Ball bearing D, m d, m [alpha], rad
Left 6.2 x [10.sup.-2] 5.2 x [10.sup.-2] 0.2217
Right 2.5 x [10.sup.-2] 2 x [10.sup.-2] 0.2217
Ball bearing [f.sub.i] [f.sub.o] z [D.sub.b], )
Left 0.52 2.965 11 1.03 x 10-2
Right 0.52 2.965 11 8.65 x 10-3
Table 2
Sealing parameters of each sealing channel
No. of seal channel 1 2
Length of seal channel, m 1.8 x [10.sup.-2] 1.8 x [10.sup.-2]
Diameter of seal channel, 2.5 x [10.sup.-2] 2.5 x [10.sup.-2]
m (not including clearance)
No. of seal channel 3 4
Length of seal channel, m 1.3 x [10.sup.-2] 1.8 x [10.sup.-2]
Diameter of seal channel, 2.5 x [10.sup.-2] 2.8 x [10.sup.-2]
m (not including clearance)
No. of seal channel 5 6
Length of seal channel, m 1.8 x [10.sup.-2] 1.8 x [10.sup.-2]
Diameter of seal channel, 2.8 x [10.sup.-2] 2.5 x [10.sup.-2]
m (not including clearance)
No. of seal channel 7 8
Length of seal channel, m 1.8 x [10.sup.-2] 1.8 x [10.sup.-2]
Diameter of seal channel, 2.5 x [10.sup.-2] 2.5 x [10.sup.-2]
m (not including clearance)
Table 3
Experimental and calculated results of first critical speed
Rc [DELTA]p Calculated results Experimental results
0.3 mm 0.1 MPa 1969.4 2022.5
0.2 MPa 2150 2213.1
0.3 MPa 2313.5 2306.7
0.4 MPa 2464.5 2565.9
0.5 MPa 2605.3 2616.3
0.4 mm 0.1 MPa 1912.5 1989.6
0.2 MPa 2040.8 2082.4
0.3 MPa 2158 2129
0.4 MPa 2267 2230.5
0.5 mm 0.1 MPa 1874.4 1982.7
0.2 MPa 1968.3 2024.4
0.3 MPa 2054.6 2051.3
Rc [DELTA]p Error percentage
0.3 mm 0.1 MPa 2.6%
0.2 MPa 2.9%
0.3 MPa 0.3%
0.4 MPa 4.0%
0.5 MPa 0.4%
0.4 mm 0.1 MPa 3.9%
0.2 MPa 2%
0.3 MPa 1.4%
0.4 MPa 1.6%
0.5 mm 0.1 MPa 5.5%
0.2 MPa 2.8%
0.3 MPa 0.2%
