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Suppression of low-frequency vibration for rotor-bearing system of flywheel energy storage system
Mechanical Systems and Signal Processing 121 (2019) 496–508
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Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp
Suppression of low-frequency vibration for rotor-bearing system of flywheel energy storage system Yujiang Qiu, Shuyun Jiang ⇑ School of Mechanical Engineering, Southeast University, Nanjing 211189, Jiangsu, PR China
a r t i c l e
i n f o
Article history: Received 30 May 2017 Received in revised form 19 October 2018 Accepted 20 November 2018
Keywords: Flywheel energy storage system Rotor dynamics Low-frequency vibration suppression Magnetic pendulum tuned mass damper
a b s t r a c t A nonsynchronous low-frequency whirling occurs frequently for flywheel energy storage system (FESS) with permanent magnetic bearing (PMB) and spiral groove bearing. To suppress low-frequency vibration of a large-mass FESS, a radial magnetic pendulum tuned mass damper (TMD) and an axial magnetic pendulum TMD have been developed; and identification method of dynamic parameter for the magnetic pendulum TMD has been proposed. Dynamic model for the FESS has been established using the Lagrange’s equation; and effectiveness of the two magnetic pendulum TMD’s has been evaluated by analyzing mode damping ratios of the FESS. The results show that the two TMD’s can be effective for the low-frequency vibration of the high-capacity FESS with PMB and spiral groove bearing; and radial magnetic pendulum TMD is superior to the axial magnetic pendulum TMD due to its lower resonant frequency and larger mode damping ratio for the FESS. Ó 2018 Elsevier Ltd. All rights reserved.
1. Introduction Flywheel energy storage system (FESS) supported by permanent magnetic bearing (PMB) and spiral groove bearing has many merits, such as low frictional power loss, simple structure and easy maintenance [1]. Fig. 1 shows a schematic of the FESS with PMB and spiral groove bearing. The flywheel is supported on the spiral groove bearing by an elastic pivot; the PMB is on top of the rotor, which offers a suspend force being equal to 90 percent of rotor weight, meanwhile, only 10 percent of the rotor weight acts on the spiral groove bearing, so that friction power loss of the spiral groove bearing is at low level. From rotor dynamic theory, the FESS supported by PMB and spiral groove bearing is super-critical due to low radial stiffness of the PMB. As a consequent, the FESS must pass its 1st critical speed to reach the rated speed. Moreover, previous study [2] showed that a nonsynchronous low-frequency whirling occurs frequently when the rotor runs at high speeds. To solve this issue, an upper damper must be used to help the rotor to pass 1st critical speed and to suppress the nonsynchronous low frequency whirling, as shown in Fig. 1. Presently, several kinds of dampers have been widely applied to the high speed rotor systems to suppress the vibration, which include active electromagnetic damper (AMD) [3–6], passive electromagnetic damper (PMD) [7–10], squeeze oil-film damper (SFD) [1,11–14], tuned mass damper (TMD) [15–18], etc. However, the active electromagnetic dampers need an elaborate active control system. Compared to the oil film damper, both the AMD and the PMD offer a smaller damping with a same space volume. Squeeze oil-film damper is not suitable for the suppression of the low-frequency vibration.
⇑ Corresponding author. E-mail address: [email protected] (S. Jiang). https://doi.org/10.1016/j.ymssp.2018.11.033 0888-3270/Ó 2018 Elsevier Ltd. All rights reserved.
Y. Qiu, S. Jiang / Mechanical Systems and Signal Processing 121 (2019) 496–508
497
Nomenclature k1 k2 ku k3 k4 M m1 m4 Jp Jd c1 c4 L B l’ lu X
x
f Y A ðxÞ YsR YsI mp kp
xd x y
Stiffness of the pendulum TMD Stiffness of the axial PMB Stiffness of the radial magnetic rings Stiffness of the pivot Stiffness of the O-rings Mass of the flywheel rotor Vibration mass of the pendulum TMD Vibration mass of the lower damper Diameter moment inertia of the flywheel rotor Polar moment inertia of the flywheel rotor Damping coefficient of the pendulum TMD Damping coefficient of the lower damper Total length of the lower shaft and flywheel rotor Length of the flywheel Length of the lower shaft Length of the upper shaft Rotating speed of the rotor Modal frequency or exciting frequency Modal damping ratio Acceleration mobility Real part of residual mobility Image part of residual mobility Additional mass of the exciting system Additional stiffness of the exciting system Natural frequency of the pendulum TMD Vibration in the direction of x-axis Vibration in the direction of y-axis
Fig. 1. The schematic of FESS supported by PMB and spiral groove bearing.
As a type of classical damper, tuned mass damper has many merits, such as simple structure, good damping performance etc. For the FESS with PMB and spiral groove bearing, a type of passive pendulum oil TMD, named the upper damper, as shown in Fig. 1, has been developed to suppress the low-frequency vibration [1,2,17,18]. The upper damper consists of a permanent magnet suspended by three elastic rod through slip spherical joint, and the housing of the upper damper is filled with the damper oil; the test studies indicated that the upper damper is effective for a 1 kg-class FESS [17]. However, when the upper damper is introduced in a 100 kg-class FESS, it does not work well, because the slip joint bears too large frictional resistance to oscillate normally, meanwhile, the resonant frequency of the damper is obviously greater than the frequency of nonsynchronous low-frequency whirling of the FESS, i.e., the upper damper configuration isn’t efficient for the 100 kg-class FESS. Thus, it is an important issue to develop a new damper for the large-sized FESS.
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In this paper, a new type of magnetic pendulum TMD is proposed to suppress the low-frequency vibration of the largemass FESS with PMB and spiral groove bearing. The identification method of dynamic parameters for the magnetic pendulum TMD is studied. Dynamic model of the FESS is established using the Lagrange’s equation, and damping effect of the magnetic pendulum TMD is evaluated by analyzing mode damping ratio of the FESS. 2. Physical description of magnetic pendulum TMD The proposed TMD is called radial magnetic pendulum TMD, as shown in Fig. 2. It can be seen that the TMD consists of axial PMB and TMD. In details, the axial PMB includes two axial magnetic rings and flywheel top, which is used to levitate the flywheel’s rotor. The TMD includes lots of elements, such as rolling spherical joint, suspension rod, radial magnetic rings, damper housing, damping oil, journal, etc. It should be mentioned that its core part is the journal, which is suspended by three suspension rods with the rolling spherical joints. The damper housing is filled with the damping oil, and the journal is submerged in the damping oil. The inner radial magnetic ring is installed at the upper shaft connected to the flywheel top, and the outer magnetic ring is installed at the inside surface of the journal; the radial magnetic rings, i.e. radial magnetic bearing, is served as an exciter for the TMD. Based on this consideration, the new TMD is defined as the radial magnetic pendulum TMD. In addition, the upper end of the flywheel rotor is supported by the radial magnetic pendulum TMD and the axial PMB in parallel. Specially, the flywheel’s weight is entirely lifted by the axial PMB, meaning that the radial magnetic pendulum TMD is free of carrying the flywheel’s weight, which ensures the rolling balls run freely and smoothly, consequently, the nature frequency of the TMD is able to be within 10 Hz. For the flywheel rotor system, once the nonsynchronous low-frequency whirling of appears, the TMD will resonate in the damping oil under the exciting force from the radial magnetic rings to suppress the vibration of the FESS. For the comparison, an axial magnetic pendulum TMD is developed according to the principle of the upper damper for the 1 kg-class FESS, as shown in Fig. 3. Just like the radial magnetic pendulum TMD in Fig. 2, the axial PMB is also used to levitate the flywheel’s rotor. However, being different from the radial magnetic pendulum TMD, the axial PMB of the axial magnetic pendulum TMD is directly connected to the pendulum TMD in series, which means that the journal of the damper is excited by the axial PMB. Thus, the flywheel with 100 kg has to be carried by the hooked suspension rods and the rolling spherical joints, in results, an excessive contact deformation will form due to the point contact under the large load. The axial magnetic pendulum TMD tends to be failing in smooth running due to the excessive contact deformation or rolling frictional resistance. In the following sections, a comparative study will be conducted using the theory analysis and experiment manner, the aim is to illustrate the effectiveness of the proposed TMD for the suppression of the low-frequency vibration of 100 kg-class FESS. 3. Design theory of the pendulum TMD for FESS From the vibration theory, the effect of the TMD can be evaluated using mode damping ratio of the FESS with the TMD employed, so free vibration of the FESS is analyzed in this section. Fig. 4 shows the schematics of the FESS with two type of TMD. Both systems seem to be similar in configuration, such as the flywheel, the upper shaft, the lower support (including the spiral groove bearing and the lower damper), and a brushless DC motor/generator, but an obviously different configuration can be observed between the radial magnetic pendulum TMD and the axial magnetic pendulum TMD, thus, this difference should be considered to establish the dynamic models for the FESS. From the applied mechanics, the FESS is both a rotor system and a multibody system. The dynamic model of the FESS is built under following assumptions: (1) only lateral vibration is considered; (2) the flywheel and upper shaft are rigid; (3) the axial PMB and radial magnetic rings are simplified as spring elements; (4) the pendulum TMD or the lower damper is simplified as a mass-spring-damping system; (5) the pivot acts as an elastic element with infinitesimal mass; (6) lower end of the pivot is fixed to vibrating body of the lower damper, because the oil film stiffness of the spiral groove bearing is high enough. The dynamic model for the FESS is shown in Fig. 5. The kinetic energy of the flywheel rotor system is
T¼
" " 2 2 # 2 2 # 1 1 x_ 2 þ x_ 3 y_ þ y_ 3 x_ 2 x_ 3 y2 y3 x_ 2 x_ 3 y_ y_ 3 þ Jp X X þ 2 þ 2 þ 2 þ Jd 2 2 2 2 B B B B 1 1 2 2 2 2 þ m1 x_ 1 þ y_ 1 þ m4 x_ 4 þ y_ 4 2 2
1 M 2
For the FESS with axial magnetic pendulum TMD, the potiential energy of the flywheel rotor system is
i 1 1 h 1 U ¼ k1 x21 þ y21 þ k2 ðx2 x1 Þ2 þ ðy2 y1 Þ2 þ k3 2 2 2 1 þ k4 x24 þ y24 2
(
0
l x3 x4 ðx2 x3 Þ B
2
2 ) 0 l þ ðy3 y4 ðy2 y3 Þ B
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Fig. 2. Axial magnetic pendulum TMD.
Fig. 3. Radial magnetic pendulum TMD.
Fig. 4. The schematic of the FESS with: (a) the radial magnetic pendulum TMD, (b) the axial magnetic pendulum TMD.
For the FESS with radial magnetic pendulum TMD, the potiential energy of the flywheel rotor system is
U¼
( 2 2 ) 1 1 2 lu lu 1 k1 x1 þ y21 þ ku ðx2 x1 Þ þ ðx2 x3 Þ þ ðy2 y1 Þ þ ðy2 y3 Þ þ k2 x22 þ y22 2 2 2 B B ( 2 2 ) 0 0 1 l l 1 þ k4 x24 þ y24 þ k3 x3 x4 ðx2 x3 Þ þ ðy3 y4 ðy2 y3 Þ 2 2 B B
499
500
Y. Qiu, S. Jiang / Mechanical Systems and Signal Processing 121 (2019) 496–508
Fig. 5. Dynamic model of the FESS with: (a) the radial magnetic pendulum TMD, (b) the axial magnetic pendulum TMD.
The consumed energy of the flywheel rotor system is
Z¼
1 1 2 2 2 2 c1 x_ 1 þ y_ 1 þ c4 x_ 4 þ y_ 4 2 2
Free vibration equation for the FESS can be obtained using the Lagrange’s equation
d dt
@T @ q_ i
@T @U @q þ @q þ @@Zq_ i ¼ 0 and writi i
ten as
M €z þ ðiXH þ CÞz_ þ Kz ¼ 0
ð1Þ
where,
2
0
m1
6 0 6 M¼6 4 0
0
m2
m3
m3
m2
0
0
0
2
3
0 0 6 0 Jp 7 6 0 7 B2 7; H ¼ 6 6 J 0 5 4 0 Bp2 m4 0 0
0
0 Jp B2 Jp
B2 0
3
2 c1 7 60 07 6 7; C ¼ 6 7 40 05 0 0
0
0 0 0 0 0 0
0
3
07 7 7; 05
0 0 c4
8 9 > > > > z1 > > > < > = z z ¼ 2 > z3 > > > > > > : > ; z4
m2 ¼ M=4 þ J d =B2 ; m3 ¼ M=4 J d =B2 zj
¼ xj þ i yj
ðj ¼ 1; 2; 3; 4Þ
For the FESS with axial magnetic pendulum TMD,
2
k1 þ k2 6 6 k2 6 K ¼6 6 0 4
k2 0 2 k2 þ lB k3
0
0 0
lB Bl k3
lB Bl k3 l 2 k3 B
l0 k B 3
Bl k3
0
0
3
7 7 7 7 Bl k3 7 5 0
l k B 3
k3 þ k4
For the FESS with radial magnetic pendulum TMD,
2
k1 þ ku 6 6 1 þ lu k 6 u B 6 K ¼6 6 lu k 6 B u 6 4 0
3 lu 1 þ lBu ku k 0 B u h i 7 2 0 2 0 0 7 l0 ½k2 þ 1 þ lBu ku þ lB k3 lBu 1 þ lBu ku þ 1 þ lB lB k3 k 7 B 3 7 7 h 0 i 2 0 0 0 7 lu lu lu 2 l l l l 1 þ B 1 þ B ku þ 1 þ B B k3 k þ 1 þ k k 7 u 3 3 B B B 7 5 l0 l0 k 1 þ k þ k k 3 3 3 4 B B
Because the Eq. (1) cannot be decoupled by modal synthesis method for gyroscopic moment, the state variable method is applied to solve it,. The state equation is
Y. Qiu, S. Jiang / Mechanical Systems and Signal Processing 121 (2019) 496–508
AZ_ þ BZ ¼ 0
501
ð2Þ
where,
_ ;A¼ Z ¼ fz; zg T
iXH þ C
K ;B¼ 0 0
M
M
0
M
By solving Eq. (2), the complex frequencies k and the complex vibration amplitude z are
( kr ¼ nr þ jxr zr
¼ X r þ jY r
; r ¼ 1; 2; 3; 4
ð3Þ
where, n is mode decaying exponent, x is mode angular frequency. If x > 0, the mode is a forward mode, or it is a backward mode. Modal damping ratio is
f¼
n
ð4Þ
x
In order to efficiently suppress the low-frequency vibration of the FESS and ensure the stability of low-frequency mode, the damping ratio of low-frequency mode should have a large value. Therefore, one design strategy for the pendulum TMD is to increase the damping ratio of low-frequency mode by selecting the damper proper parameters, including k1, k2, m1 and c1. Since the backward mode is always stable, only the damping ratios of the forward modes with low frequencies need to be discussed. From the vibration theory, the resonant frequencies of the upper dampers are as follows, qffiffiffiffiffiffiffiffiffiffi For the FESS with axial magnetic pendulum TMD, xd ¼ k1mþk1 2 ; qffiffiffiffiffiffiffiffiffiffi For the FESS with radial magnetic pendulum TMD, xd ¼ k1mþk1 u It can be seen that the damper resonant frequency depends on its parameters, such as k1, k2, ku and m1. Therefore, another design strategy for the TMD is to ensure its resonant frequency to be close to the nonsynchronous low-frequency vibration by selecting the damper parameters.
4. Case study Fig. 6 shows the 100 kg-class FESS developed in our lab, and the structure and performance parameters of FESS except the pendulum TMD are listed in Table 1. Here, the parameters of flywheel rotor were obtained with measuring instrument for geometrical quantity, and the parameters of lower damper were detected using test rigs specially designed in our lab. The parameters of pendulum TMD will be given in the next section. 4.1. Dynamic parameter identification of the pendulum TMD The developed magnetic pendulum TMD’s are shown in Fig. 7. Fig. 7(a) shows the axial magnetic pendulum TMD, and Fig. 7(b) and (c) show the radial magnetic pendulum TMD. The dynamic parameters of the TMD’s should be obtained by an effective identification method.
Fig. 6. The 100 kg-class FESS developed in our lab.
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Table 1 The structure and performance parameters of the FESS. Lower damper
Flywheel rotor
Item
Value
Item
Value
Item
Value
k3 (N/m) k4 (N/m) m4 (kg) c4 (Ns/m)
0.50 106 1.60 106 0.75 400
M (kg) Jp (kg.m2) Jd (kg.m2) B (mm)
110 1.24 0.99 200
l (mm) l’ (mm) lu (mm)
390 190 70
4.1.1. Parameter identification system The test system is established using the developed 100 kg-class FESS, as shown in Fig. 8. The flywheel is fixed on the platform by a pair of magnetic chucks, so the TMD can be considered as a single degree-of-freedom vibration system. The TMD is excited by a electromagnetic vibration exciter (JZK-2, produced by China Jiangsu Sinocera Piezotronics, Inc.) with sine chirp signal (generated by YE1311B signal generator produced by Jiangsu Sinocera Piezotronics, Inc.). An impendence head (CLYD-331A, produced by Jiangsu Sinocera Piezotronics, Inc.) is mounted on the rod to measure exciting force and acceleration of vibrator of the TMD. Measured signals are sent to the computer by an signal conditioner (AZ804B, produced by the China Nanjing Anzheng Software Co., Ltd) and an data collector (AZ308R, produced by Nanjing Anzheng Software Co., Ltd), then the acceleration mobility of test system is calculated by the CRAS V7.0 Software on the vibration signal. The dynamic parameters of the damper can be obtained through identifying the test acceleration mobility. 4.1.2. Parameter identification method Dynamic model for the parameters identification system is illustrated in Fig. 9. An additional mass (mp) and stiffness (kp) of the exciting system are introduced to improve the accuracy of parameter identification. By vibration theory, the acceleration mobility of the test system is,
Y A ð xÞ ¼
m x2 2 m1 mx þ jc1 x þ k
ð5Þ
where,
m ¼ m1 þ mp ; k ¼ k1 þ ku þ kp After obtaining the mobility of test system, a reasonable method must be selected to identify the system’s parameters. Particle swarm optimization (PSO), as a biologically inspired evolutionary computation algorithm, has many merits, such as easy programming, high precision, rapid convergence and so on, the PSO has been applied widely [19–22]. Therefore, iterative method based on the PSO is used to identify the dynamic parameters of the system under the over-damped and underdamped conditions from the test mobility curves of system. Firstly, a fitting equation should be built, since the test system isn’t a strict single freedom degree system, there is a noise interference, so an acceleration mobility fitting equation containing the residual mobility (YsR and YsI) is built as follows:
Y A ðx; pÞ ¼
m x2 þ Y sR þ jY sI 2 m1 mx þ jc1 x þ k
ð6Þ
where, p is an vector of unknown parameters, p = {m, k, c1, YsR, YsI}.
Then, an error between the fitted value of mobility (Y Ai ) and the test value of mobility (Y Ai ) at the frequency xi can be obtained, ið
e xi Þ ¼ Y Ai ðxi Þ Y Ai ðxi Þ
ð7Þ
Therefore, the total variance between the fitted value and the test value is used to estimate the optimal parameter, which is defined as
mind ¼ min
N X
ð8Þ
e i ei
i¼1
where, ei is conjugate number of ei . The optimal parameters {m, k, c1} can be achieved by the PSO [19]; here, specific iterating process of PSO is not stated. Finally, the parameters of the TMD can be obtained, which are
kd ¼ k kp ; m1 ¼ m mp
ð9Þ
where, kd represents the total stiffness of the TMD, which is equal to the sum of k1 and ku (or k2 for radial magnetic pendulum TMD).
Y. Qiu, S. Jiang / Mechanical Systems and Signal Processing 121 (2019) 496–508
503
Fig. 7. The developed two types of TMD’s: (a) the axial magnetic pendulum TMD, (b) and (c) the radial magnetic pendulum TMD.
Fig. 8. Parameter identification system for the pendulum TMD: (a) schematic; (b) and (c) photos.
4.1.3. Identification results The test values of dynamic parameters for two TMD’s are listed in Table 2. The stiffness of the axial PMB and radial magnetic rings (namely, k2 and ku) are list in Table 2. Being limited with the space, the detail of the test method is not given in the paper. The results show that the natural frequency of radial magnetic pendulum TMD is less than that of the axial magnetic pendulum TMD. This can be attributed to the fact that the stiffness of the radial magnetic pendulum TMD is obviously less than that of axial magnetic pendulum TMD. As expected, the axial magnetic pendulum TMD cannot be excited smoothly by the axial PMB due to large friction resistance on the ball joint. On the contrary, the radial magnetic pendulum TMD can run
504
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Fig. 9. Dynamic model of the parameters identification system.
Table 2 The experimental results of the dynamic parameters of two TMD’s. Types
m1/kg
k1/104N/m
ku/104N/m
k2/104N/m
xn/Hz
Radial magnetic pendulum TMD Axial magnetic pendulum TMD
4.65 4.20
0.15 2.55
2.17 –
1.75 1.75
11.24 16.10
flexibly due to smaller axial load on the joint compared to the former, which benefits from the parallel connection of the axial PMB and the radial TMD. The experimental damping coefficients of the two TMD’s are shown in Fig. 10. It can be seen that three curves are almost parallel under the log-log coordinates. The damping coefficients increase with the viscosity of damping oil and decrease with damper’s radial gap (i.e. the radial clearance between the journal and damping housing).
4.2. Damping effect of the pendulum TMD for low-frequency vibration Fig. 11 shows modal test system for the 100 kg-class FESS at static state. Calibrated eddy current displacement probes (CWY-DO-502, produced by Jiangsu Sinocera Piezotronics, Inc.) are employed to detect the system vibration. The response points are located at the upper damper (number 1), the flywheel top (number 2), flywheel bottom (number 3) and lower damper (number 4). A hammer (CL-YD-305, produced by Jiangsu Sinocera Piezotronics, Inc.) is used to impact the system at the back of response 4 of the lower damper. All the signals are sent to the computer by a (AZ804B) signal conditioner and a data collector (AZ308R). The mobilities (namely, frequency response function) of four response points are calculated and recorded by the CRAS V7.0 Software on the vibration signals. Modal frequencies, shapes and damping ratios of the rotor system are estimated from the test mobility curves by the modal iterating identification method [23]. Fig. 12 gives the mobility amplitude-frequency curves for four detection points of the FESS with the radial magnetic pendulum TMD. The No.32 machine oil (produced by oil refinery, China) is used for the radial magnetic pendulum TMD, and the damping coefficient c1 is identified to be 120Ns/m. In the figure, Y14, Y24, Y34 and Y44 represent the mobility of the response point 1, 2, 3 and 4 due to a force at point 4, separately. Obviously, there are four peaks within 400 Hz corresponding to the four modes. For the FESS with the axial magnetic pendulum TMD, the mobility amplitude-frequency curves are similar with the radial magnetic pendulum TMD, so we do not repeat here. Tables 3 and 4 give the theoretical and test values of the previous four orders modal frequencies and the corresponding mode shapes of the FESS, which are also shown in the Figs. 13 and 14. It can be seen that the test results of the model frequencies and the mode shapes nearly agree with the theoretical ones. However, small differences still exist in two results due to some uncertainties in the measurements besides the several assumptions made for dynamic modeling, for example, the test mobility curves of the pendulum TMD, and the test frequency response data of the FESS are unavoidably interfered by the noise, nonlinearity and accuracy of test system. As shown in Figs. 13 and 14, the 1st-order and 2nd-order modes are related to the low-frequency vibration of the FESS. For the FESS with the radial magnetic pendulum TMD, the maximal displacement of the 1st-order mode locates at the TMD, as shown in Fig. 13, but for the FESS with the axial magnetic pendulum TMD, the maximal displacement locates at the flywheel top, as shown in Fig. 14. Therefore, the radial magnetic pendulum TMD should be easier to be excited than the axial magnetic pendulum TMD, which means that the radial magnetic pendulum TMD is more efficient to help the rotor to pass the 1st critical speed smoothly. However, the 1st-order mode frequency (2.58 Hz) of the FESS with the radial magnetic pendulum TMD is slightly greater than that (1.96 Hz) of the axial magnetic pendulum TMD; this can be attributed to the fact that the radial stiffness of the radial magnetic pendulum TMD is greater than that of the axial magnetic pendulum TMD. Fig. 15 shows the 1st-order mode damping ratios of the flywheel rotor system at the different damping coefficients. During the test process, series of damping coefficients of the pendulum TMD are obtained by a combination of several damper radial gaps and different types of damping oils. It can be seen from Fig. 15 that the test values are in generally agreeable with the theoretical ones. Similarly, the differences between two results are also seen, it may be caused by the noise, nonlinearity
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(b)
Gap= 0.75mm Gap= 1.00mm Gap= 1.50mm
1000
Damping coefficient (Ns/m)
Damping coefficient (Ns/m)
(a)
100
10 0.01
Gap = 1.0 mm Gap = 1.5 mm Gap = 2.0 mm
1,000
100
10
1 0.01
0.1 Viscostiy (Pa.s)
0.1 Viscosity (Pa.s)
Fig. 10. Damping coefficients under different damping oil viscosities and the damper radial gaps: (a) the radial magnetic pendulum TMD, (b) the axial magnetic pendulum TMD.
Fig. 11. The schematic (a) and photo (b) of modal identification system of the FESS.
7
1st
|Y|(x 10-5m/N)
6 5
2nd
4
3rd
|Y14| |Y24| |Y34| |Y44|
3 2 4th
1 0
1
10 100 Frequency ¹ (Hz)
Fig. 12. Amplitude-frequency curves of mobility of FESS with radial magnetic pendulum TMD.
and accuracy of test system besides precision of the dynamic modeling. In addition, compared with the radial TMD, the axial TMD has a larger coulomb damping due to larger axial load, which is not considered in the theory model, so it is reasonable that the difference of the predicted result and the test one with the axial TMD is larger than that with the radial TMD.
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Table 3 The modal parameters of FESS with radial magnetic pendulum TMD. Type
Order
Frequency (Hz)
Mode
R1
R2
R3
R4
Experimental values
1 2 3 4
2.58 11.23 31.49 270.0
1.000\0.0° 1.000\0.0° 0.161\3.2° 0.000\0.0°
0.811\5.5° 0.069\161.6° 0.492\173.6° 0.000\0.0°
0.417\5.6° 0.061\175.2° 1.000\0° 0.000\0.0°
0.001\8.1° 0.0006\164.6° 0.528\8.0° 1.000\0°
Theoretical values
1 2 3 4
2.43 11.44 30.41 263.44
1.000\0° 1.000\0° 0.152\2.1° 0.1E4\143.5°
0.864\1.5° 0.075\172.3° 0.548\180.0° 0.004\18.6°
0.425\1.6° 0.052\174.6° 1.000\0° 0.007\161.5°
0.002\5.8° 0.007\179.2° 0.596\2.1° 1.000\0.0
Table 4 The modal parameters of FESS with axial magnetic pendulum TMD. Type
Order
Frequency (Hz)
Mode
R1
R2
R3
R4
Experimental values
1 2 3 4
1.96 15.84 31.76 267.52
0.322\7.5° 1.000\0.0° 0.032\1.7° 0.000\0.0°
1.000\0.0° 0.012\153.1° 0.49\176.6° 0.000\0.0°
0.542\5.2° 0.009\156.1° 1.000\0.0° 0.000\0.0°
0.0003\18.5° 0.001\151.1° 0.532\5.9° 1.000\0.0°
Theoretical values
1 2 3 4
1.89 15.66 30.17 263.44
0.379\1.3° 1.000\0° 0.073\6.9° 0.5E5\143.8°
1.000\0.0° 0.021\167.0° 0.532\180° 0.004\18.5°
0.490\0.0° 0.016\173.1° 1.000\0.0° 0.007\161.5°
0.0013\0.65° 0.003\177.7° 0.592\2.1° 1.000\0.0°
1st 2.58 HZ
3rd 31.49 HZ
4th 270 HZ
0.6
0.6
0.4
0.4
0.4
0.4
0.2
x
0
0.2 0 -1
1st 2.44 HZ
(b)
0.2 0 -1
0 1 x 1 -1 0 y
1 -1 0 y 1
z(m)
0.6 z(m)
0.6
0 -1
x
2nd 11.28 HZ
0
0.2 0 -1
1 -1 0 y 1
x
3rd 30.35 HZ 0.6
0.4
0.4
0.4
0.4
0 -1
0
1 x1 -1 0 y
0.2 0 -1
0
1 x1 -1 0 y
z(m)
0.6 z(m)
0.6
0.2
0.2 0 -1
0
1 x1 -1 0 y
0
1 -1 0 y 1
4th 263.44 HZ
0.6 z(m)
z(m)
2nd 11.23 HZ
z(m)
z(m)
(a)
0.2 0 -1
0
1 x1 -1 0 y
Fig. 13. Modal frequencies and shapes of flywheel rotor system with radial magnetic pendulum TMD: a) test results, b) theoretical results.
Obviously, the 1st-order mode damping ratio of flywheel rotor system increases at first and then decreases with the damping coefficient, so there is an optimal damping coefficient corresponding to the maximal mode damping ratio. The maximal 1st-order mode damping ratios of two flywheel rotor systems with pendulum TMD are more than 0.1. Moreover, the maximal 1st-order mode damping ratio of flywheel rotor system the with the radial magnetic pendulum TMD is larger than
Y. Qiu, S. Jiang / Mechanical Systems and Signal Processing 121 (2019) 496–508
1st 1.96 HZ
2nd 15.84 HZ
4th 267.52 HZ 0.6
0.4
0.4
0.4
0.4
0.2
0.2 0 -1
1 x0 1 -1 0 y
0.2 0 -1
1 x0 1 -1 0 y
1st 1.89 HZ
(b)
z(m)
0.6 z(m)
0.6
0 -1
0.2 0 -1
x 0 1 -1 0 1 y
2nd 15.66 HZ
1 x0 1 -1 0 y
3rd 30.17 HZ
4th 263.44 HZ
0.6
0.6
0.4
0.4
0.4
0.4
0.2 0 -1
0 x
1 -1 0 y 1
0.2 0 -1
0
1 x 1 -1 0 y
z(m)
0.6 z(m)
0.6 z(m)
z(m)
3rd 31.76 HZ
0.6 z(m)
z(m)
(a)
0.2 0 -1
0
1 x 1 -1 0 y
507
0.2 0 -1
x0 1 -1 0 y 1
Fig. 14. Modal frequencies and shapes of flywheel rotor system with axial magnetic pendulum TMD: a) test results, b) theoretical results.
0.5
Expt., Ther., Expt., Ther.,
§1
0.4 0.3
axial TMD axial TMD radial TMD radial TMD
0.2 0.1 0.0
0
2000
4000 6000 c1 (Ns/m)
8000
Fig. 15. The 1st-order mode damping ratios of the flywheel rotor system with different damping coefficients.
that of flywheel rotor system with the axial magnetic pendulum TMD, so the radial magnetic pendulum TMD is superior to the axial magnetic pendulum TMD in the damping effect on the low-frequency vibration. 5. Conclusion In this paper, two types of magnetic pendulum TMD’s have been developed to suppress the low frequency vibration of the large-mass FESS with PMB and spiral groove bearing. The identification method of the dynamic parameters for the TMD’s has been established, and evaluating method has been proposed to compare the effectiveness for two TMD’s. The test study has been conducted using the 100 kg-class FESS in our lab to verified the theoretical results, the main conclusion can be given as follows, 1) Compared with the axial magnetic pendulum TMD, the radial magnetic pendulum TMD has a lower resonant frequency and contributes a greater 1st-order mode damping ratio for the FESS. Thus, the radial magnetic pendulum TMD is more efficient for the suppression of low-frequency vibration of the large-mass FESS. 2) Considering the FESS is both a rotor system and a rigid multi-body system, the dynamic model of the FESS has been established using the Lagrange’s equation; the effectiveness of the pendulum TMD can be evaluated by analyzing the 1st mode damping ratio of the FESS with the pendulum TMD employed.
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Y. Qiu, S. Jiang / Mechanical Systems and Signal Processing 121 (2019) 496–508
3) The pendulum TMD is simplified as a single degree-of-freedom system, and its dynamic parameters can be identified from the mobility curves by the iterative identification method based on the PSO.
Acknowledgment The authors gratefully acknowledge the support provided by the National Natural Science Foundation of China through grant Nos. 51635004, 11472078, 11172065. Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.ymssp.2018.11.033. References [1] X. Dai, Z. Shen, H. Wei, On the vibration of rotor-bearing system with squeeze film damper in an energy storage flywheel, Int. J. Mech. Sci. 43 (11) (2001) 2525–2540. [2] X. Dai, H. Wei, Z. Shen, Dynamics design and experiment study of the rotor-bearing system of a flywheel energy storage system, Chin. J. Mech. Eng. (China) 39 (4) (2003) 97–101. [3] J.M. Vance, D. Ying, J.L. Nikolajsen, Actively controlled bearing dampers for aircraft engine applications, J. Eng. Gas Turbines Power Trans. ASME 122 (3) (2000) 466–472. [4] L. Chen, C. Zhu, M. Wang, et al, Vibration control for active magnetic bearing high-speed flywheel rotor system with modal separation and velocity estimation strategy, J. Vibroengineering 17 (2) (2015) 757–775. [5] A. Looser, A. Tüysüz, C. Zwyssig, et al, Active magnetic damper for ultrahigh-speed permanent-magnet machines with gas bearings, IEEE Trans. Ind. Electron. 64 (4) (2017) 2982–2991. [6] R.S. Srinivas, R. Tiwari, C. Kannababu, Application of active magnetic bearings in flexible rotordynamic systems–A state-of-the-art review, Mech. Syst. Sig. Process. 106 (2018) 537–572. [7] M. Silvagni, A. Tonoli, A. Bonfitto, Self-powered eddy current damper for rotor dynamic applications, J. Vibr. Acoust. 137 (1) (2015) 011015. [8] F.N. Werfel, U. Floegel-Delor, T. Riedel, et al, A compact HTS 5 kWh/250 kW flywheel energy storage system, IEEE Trans. Appl. Supercond. 17 (2) (2007) 2138–2141. [9] S.K. Cheah, H.A. Sodano, Novel eddy current damping mechanism for passive magnetic bearings, J. Vib. Control 14 (11) (2008) 1749–1766. [10] X. Liu, X. Feng, Y. Shi, et al, Development of a semi-active electromagnetic vibration absorber and its experimental study, J. Vib. Acoust. 135 (5) (2013) 051015. [11] D. Luo, Y. Liu, X. Sun, D. Hung, The design and analysis of supercritical carbon dioxide centrifugal turbine, Appl. Therm. Eng. 127 (12) (2017) 527–535. [12] S. Nagesh, A.M. Junaid Basha, T.D. Singh, Dynamic performance analysis of high speed flexible coupling of gas turbine engine transmission system, J. Mech. Sci. Technol. 29 (1) (2015) 173–179. [13] Chang-Liang Tang, Xing-Jian Dai, Xiao-Zhang Zhang, ect. Rotor dynamics analysis and experiment study of the flywheel spin test system, J. Mech. Sci. Technol. 26 (9) (2012) 2669–2677. [14] S. Jiang, H. Wang, S. Wen, Flywheel energy storage system with a permanent magnet bearing and a pair of hybrid ceramic ball bearings,, J. Mech. Sci. Technol. 28 (12) (2014) 5043–5053. [15] C. Shi, R.G. Parker, S.W. Shaw, Tuning of centrifugal pendulum vibration absorbers for translational and rotational vibration reduction, Mech. Mach. Theory 66 (2013) 56–65. [16] H. Xiujin, H. Lidong, H. Wenchao, Study on semi-active TMD controlling vibration of two-span rotors over critical speeds, J. Mech. Electr. Eng. 31 (10) (2014) 1244–1248. [17] X. Dai, Z. Shen, H. Wei, Study on whirl modal damping of energy storage flywheel-bearing system, J. Vib. Eng. (In Chinese) 15 (1) (2002) 98–101. [18] H. Wang, S. Jiang, Z. Shen, The dynamic analysis of an energy storage flywheel system with hybrid bearing support, J. Vib. Acoust. 131 (5) (2009) 051006. [19] Y. Del Valle, G.K. Venayagamoorthy, S. Mohagheghi, et al, Particle swarm optimization: basic concepts, variants and applications in power systems, IEEE Trans. Evol. Comput. 12 (2) (2008) 171–195. [20] Y. Zheng, Y. Liao, Parameter identification of nonlinear dynamic systems using an improved particle swarm optimization, Optik-Int. J. Light Electron. Opt. 127 (19) (2016) 7865–7874. [21] V.S. Özsoy, H.H. Örkcü, H. Bal, Particle Swarm Optimization Applied to Parameter Estimation of the Four-Parameter Burr III Distribution, Iran. J. Sci. Technol. Trans. A Sci. 42 (1) (2017) 1–15. [22] M.A. Galewski, Modal parameters identification with particle swarm optimization, Key Eng. Mater. 597 (2016) 119–124. [23] D. Li, Q. Lu, Analysis of Experiments in Engineering Vibration, Tsinghua University Press, Beijing, China, 2004.
Contents lists available at ScienceDirect
Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp
Suppression of low-frequency vibration for rotor-bearing system of flywheel energy storage system Yujiang Qiu, Shuyun Jiang ⇑ School of Mechanical Engineering, Southeast University, Nanjing 211189, Jiangsu, PR China
a r t i c l e
i n f o
Article history: Received 30 May 2017 Received in revised form 19 October 2018 Accepted 20 November 2018
Keywords: Flywheel energy storage system Rotor dynamics Low-frequency vibration suppression Magnetic pendulum tuned mass damper
a b s t r a c t A nonsynchronous low-frequency whirling occurs frequently for flywheel energy storage system (FESS) with permanent magnetic bearing (PMB) and spiral groove bearing. To suppress low-frequency vibration of a large-mass FESS, a radial magnetic pendulum tuned mass damper (TMD) and an axial magnetic pendulum TMD have been developed; and identification method of dynamic parameter for the magnetic pendulum TMD has been proposed. Dynamic model for the FESS has been established using the Lagrange’s equation; and effectiveness of the two magnetic pendulum TMD’s has been evaluated by analyzing mode damping ratios of the FESS. The results show that the two TMD’s can be effective for the low-frequency vibration of the high-capacity FESS with PMB and spiral groove bearing; and radial magnetic pendulum TMD is superior to the axial magnetic pendulum TMD due to its lower resonant frequency and larger mode damping ratio for the FESS. Ó 2018 Elsevier Ltd. All rights reserved.
1. Introduction Flywheel energy storage system (FESS) supported by permanent magnetic bearing (PMB) and spiral groove bearing has many merits, such as low frictional power loss, simple structure and easy maintenance [1]. Fig. 1 shows a schematic of the FESS with PMB and spiral groove bearing. The flywheel is supported on the spiral groove bearing by an elastic pivot; the PMB is on top of the rotor, which offers a suspend force being equal to 90 percent of rotor weight, meanwhile, only 10 percent of the rotor weight acts on the spiral groove bearing, so that friction power loss of the spiral groove bearing is at low level. From rotor dynamic theory, the FESS supported by PMB and spiral groove bearing is super-critical due to low radial stiffness of the PMB. As a consequent, the FESS must pass its 1st critical speed to reach the rated speed. Moreover, previous study [2] showed that a nonsynchronous low-frequency whirling occurs frequently when the rotor runs at high speeds. To solve this issue, an upper damper must be used to help the rotor to pass 1st critical speed and to suppress the nonsynchronous low frequency whirling, as shown in Fig. 1. Presently, several kinds of dampers have been widely applied to the high speed rotor systems to suppress the vibration, which include active electromagnetic damper (AMD) [3–6], passive electromagnetic damper (PMD) [7–10], squeeze oil-film damper (SFD) [1,11–14], tuned mass damper (TMD) [15–18], etc. However, the active electromagnetic dampers need an elaborate active control system. Compared to the oil film damper, both the AMD and the PMD offer a smaller damping with a same space volume. Squeeze oil-film damper is not suitable for the suppression of the low-frequency vibration.
⇑ Corresponding author. E-mail address: [email protected] (S. Jiang). https://doi.org/10.1016/j.ymssp.2018.11.033 0888-3270/Ó 2018 Elsevier Ltd. All rights reserved.
Y. Qiu, S. Jiang / Mechanical Systems and Signal Processing 121 (2019) 496–508
497
Nomenclature k1 k2 ku k3 k4 M m1 m4 Jp Jd c1 c4 L B l’ lu X
x
f Y A ðxÞ YsR YsI mp kp
xd x y
Stiffness of the pendulum TMD Stiffness of the axial PMB Stiffness of the radial magnetic rings Stiffness of the pivot Stiffness of the O-rings Mass of the flywheel rotor Vibration mass of the pendulum TMD Vibration mass of the lower damper Diameter moment inertia of the flywheel rotor Polar moment inertia of the flywheel rotor Damping coefficient of the pendulum TMD Damping coefficient of the lower damper Total length of the lower shaft and flywheel rotor Length of the flywheel Length of the lower shaft Length of the upper shaft Rotating speed of the rotor Modal frequency or exciting frequency Modal damping ratio Acceleration mobility Real part of residual mobility Image part of residual mobility Additional mass of the exciting system Additional stiffness of the exciting system Natural frequency of the pendulum TMD Vibration in the direction of x-axis Vibration in the direction of y-axis
Fig. 1. The schematic of FESS supported by PMB and spiral groove bearing.
As a type of classical damper, tuned mass damper has many merits, such as simple structure, good damping performance etc. For the FESS with PMB and spiral groove bearing, a type of passive pendulum oil TMD, named the upper damper, as shown in Fig. 1, has been developed to suppress the low-frequency vibration [1,2,17,18]. The upper damper consists of a permanent magnet suspended by three elastic rod through slip spherical joint, and the housing of the upper damper is filled with the damper oil; the test studies indicated that the upper damper is effective for a 1 kg-class FESS [17]. However, when the upper damper is introduced in a 100 kg-class FESS, it does not work well, because the slip joint bears too large frictional resistance to oscillate normally, meanwhile, the resonant frequency of the damper is obviously greater than the frequency of nonsynchronous low-frequency whirling of the FESS, i.e., the upper damper configuration isn’t efficient for the 100 kg-class FESS. Thus, it is an important issue to develop a new damper for the large-sized FESS.
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In this paper, a new type of magnetic pendulum TMD is proposed to suppress the low-frequency vibration of the largemass FESS with PMB and spiral groove bearing. The identification method of dynamic parameters for the magnetic pendulum TMD is studied. Dynamic model of the FESS is established using the Lagrange’s equation, and damping effect of the magnetic pendulum TMD is evaluated by analyzing mode damping ratio of the FESS. 2. Physical description of magnetic pendulum TMD The proposed TMD is called radial magnetic pendulum TMD, as shown in Fig. 2. It can be seen that the TMD consists of axial PMB and TMD. In details, the axial PMB includes two axial magnetic rings and flywheel top, which is used to levitate the flywheel’s rotor. The TMD includes lots of elements, such as rolling spherical joint, suspension rod, radial magnetic rings, damper housing, damping oil, journal, etc. It should be mentioned that its core part is the journal, which is suspended by three suspension rods with the rolling spherical joints. The damper housing is filled with the damping oil, and the journal is submerged in the damping oil. The inner radial magnetic ring is installed at the upper shaft connected to the flywheel top, and the outer magnetic ring is installed at the inside surface of the journal; the radial magnetic rings, i.e. radial magnetic bearing, is served as an exciter for the TMD. Based on this consideration, the new TMD is defined as the radial magnetic pendulum TMD. In addition, the upper end of the flywheel rotor is supported by the radial magnetic pendulum TMD and the axial PMB in parallel. Specially, the flywheel’s weight is entirely lifted by the axial PMB, meaning that the radial magnetic pendulum TMD is free of carrying the flywheel’s weight, which ensures the rolling balls run freely and smoothly, consequently, the nature frequency of the TMD is able to be within 10 Hz. For the flywheel rotor system, once the nonsynchronous low-frequency whirling of appears, the TMD will resonate in the damping oil under the exciting force from the radial magnetic rings to suppress the vibration of the FESS. For the comparison, an axial magnetic pendulum TMD is developed according to the principle of the upper damper for the 1 kg-class FESS, as shown in Fig. 3. Just like the radial magnetic pendulum TMD in Fig. 2, the axial PMB is also used to levitate the flywheel’s rotor. However, being different from the radial magnetic pendulum TMD, the axial PMB of the axial magnetic pendulum TMD is directly connected to the pendulum TMD in series, which means that the journal of the damper is excited by the axial PMB. Thus, the flywheel with 100 kg has to be carried by the hooked suspension rods and the rolling spherical joints, in results, an excessive contact deformation will form due to the point contact under the large load. The axial magnetic pendulum TMD tends to be failing in smooth running due to the excessive contact deformation or rolling frictional resistance. In the following sections, a comparative study will be conducted using the theory analysis and experiment manner, the aim is to illustrate the effectiveness of the proposed TMD for the suppression of the low-frequency vibration of 100 kg-class FESS. 3. Design theory of the pendulum TMD for FESS From the vibration theory, the effect of the TMD can be evaluated using mode damping ratio of the FESS with the TMD employed, so free vibration of the FESS is analyzed in this section. Fig. 4 shows the schematics of the FESS with two type of TMD. Both systems seem to be similar in configuration, such as the flywheel, the upper shaft, the lower support (including the spiral groove bearing and the lower damper), and a brushless DC motor/generator, but an obviously different configuration can be observed between the radial magnetic pendulum TMD and the axial magnetic pendulum TMD, thus, this difference should be considered to establish the dynamic models for the FESS. From the applied mechanics, the FESS is both a rotor system and a multibody system. The dynamic model of the FESS is built under following assumptions: (1) only lateral vibration is considered; (2) the flywheel and upper shaft are rigid; (3) the axial PMB and radial magnetic rings are simplified as spring elements; (4) the pendulum TMD or the lower damper is simplified as a mass-spring-damping system; (5) the pivot acts as an elastic element with infinitesimal mass; (6) lower end of the pivot is fixed to vibrating body of the lower damper, because the oil film stiffness of the spiral groove bearing is high enough. The dynamic model for the FESS is shown in Fig. 5. The kinetic energy of the flywheel rotor system is
T¼
" " 2 2 # 2 2 # 1 1 x_ 2 þ x_ 3 y_ þ y_ 3 x_ 2 x_ 3 y2 y3 x_ 2 x_ 3 y_ y_ 3 þ Jp X X þ 2 þ 2 þ 2 þ Jd 2 2 2 2 B B B B 1 1 2 2 2 2 þ m1 x_ 1 þ y_ 1 þ m4 x_ 4 þ y_ 4 2 2
1 M 2
For the FESS with axial magnetic pendulum TMD, the potiential energy of the flywheel rotor system is
i 1 1 h 1 U ¼ k1 x21 þ y21 þ k2 ðx2 x1 Þ2 þ ðy2 y1 Þ2 þ k3 2 2 2 1 þ k4 x24 þ y24 2
(
0
l x3 x4 ðx2 x3 Þ B
2
2 ) 0 l þ ðy3 y4 ðy2 y3 Þ B
Y. Qiu, S. Jiang / Mechanical Systems and Signal Processing 121 (2019) 496–508
Fig. 2. Axial magnetic pendulum TMD.
Fig. 3. Radial magnetic pendulum TMD.
Fig. 4. The schematic of the FESS with: (a) the radial magnetic pendulum TMD, (b) the axial magnetic pendulum TMD.
For the FESS with radial magnetic pendulum TMD, the potiential energy of the flywheel rotor system is
U¼
( 2 2 ) 1 1 2 lu lu 1 k1 x1 þ y21 þ ku ðx2 x1 Þ þ ðx2 x3 Þ þ ðy2 y1 Þ þ ðy2 y3 Þ þ k2 x22 þ y22 2 2 2 B B ( 2 2 ) 0 0 1 l l 1 þ k4 x24 þ y24 þ k3 x3 x4 ðx2 x3 Þ þ ðy3 y4 ðy2 y3 Þ 2 2 B B
499
500
Y. Qiu, S. Jiang / Mechanical Systems and Signal Processing 121 (2019) 496–508
Fig. 5. Dynamic model of the FESS with: (a) the radial magnetic pendulum TMD, (b) the axial magnetic pendulum TMD.
The consumed energy of the flywheel rotor system is
Z¼
1 1 2 2 2 2 c1 x_ 1 þ y_ 1 þ c4 x_ 4 þ y_ 4 2 2
Free vibration equation for the FESS can be obtained using the Lagrange’s equation
d dt
@T @ q_ i
@T @U @q þ @q þ @@Zq_ i ¼ 0 and writi i
ten as
M €z þ ðiXH þ CÞz_ þ Kz ¼ 0
ð1Þ
where,
2
0
m1
6 0 6 M¼6 4 0
0
m2
m3
m3
m2
0
0
0
2
3
0 0 6 0 Jp 7 6 0 7 B2 7; H ¼ 6 6 J 0 5 4 0 Bp2 m4 0 0
0
0 Jp B2 Jp
B2 0
3
2 c1 7 60 07 6 7; C ¼ 6 7 40 05 0 0
0
0 0 0 0 0 0
0
3
07 7 7; 05
0 0 c4
8 9 > > > > z1 > > > < > = z z ¼ 2 > z3 > > > > > > : > ; z4
m2 ¼ M=4 þ J d =B2 ; m3 ¼ M=4 J d =B2 zj
¼ xj þ i yj
ðj ¼ 1; 2; 3; 4Þ
For the FESS with axial magnetic pendulum TMD,
2
k1 þ k2 6 6 k2 6 K ¼6 6 0 4
k2 0 2 k2 þ lB k3
0
0 0
lB Bl k3
lB Bl k3 l 2 k3 B
l0 k B 3
Bl k3
0
0
3
7 7 7 7 Bl k3 7 5 0
l k B 3
k3 þ k4
For the FESS with radial magnetic pendulum TMD,
2
k1 þ ku 6 6 1 þ lu k 6 u B 6 K ¼6 6 lu k 6 B u 6 4 0
3 lu 1 þ lBu ku k 0 B u h i 7 2 0 2 0 0 7 l0 ½k2 þ 1 þ lBu ku þ lB k3 lBu 1 þ lBu ku þ 1 þ lB lB k3 k 7 B 3 7 7 h 0 i 2 0 0 0 7 lu lu lu 2 l l l l 1 þ B 1 þ B ku þ 1 þ B B k3 k þ 1 þ k k 7 u 3 3 B B B 7 5 l0 l0 k 1 þ k þ k k 3 3 3 4 B B
Because the Eq. (1) cannot be decoupled by modal synthesis method for gyroscopic moment, the state variable method is applied to solve it,. The state equation is
Y. Qiu, S. Jiang / Mechanical Systems and Signal Processing 121 (2019) 496–508
AZ_ þ BZ ¼ 0
501
ð2Þ
where,
_ ;A¼ Z ¼ fz; zg T
iXH þ C
K ;B¼ 0 0
M
M
0
M
By solving Eq. (2), the complex frequencies k and the complex vibration amplitude z are
( kr ¼ nr þ jxr zr
¼ X r þ jY r
; r ¼ 1; 2; 3; 4
ð3Þ
where, n is mode decaying exponent, x is mode angular frequency. If x > 0, the mode is a forward mode, or it is a backward mode. Modal damping ratio is
f¼
n
ð4Þ
x
In order to efficiently suppress the low-frequency vibration of the FESS and ensure the stability of low-frequency mode, the damping ratio of low-frequency mode should have a large value. Therefore, one design strategy for the pendulum TMD is to increase the damping ratio of low-frequency mode by selecting the damper proper parameters, including k1, k2, m1 and c1. Since the backward mode is always stable, only the damping ratios of the forward modes with low frequencies need to be discussed. From the vibration theory, the resonant frequencies of the upper dampers are as follows, qffiffiffiffiffiffiffiffiffiffi For the FESS with axial magnetic pendulum TMD, xd ¼ k1mþk1 2 ; qffiffiffiffiffiffiffiffiffiffi For the FESS with radial magnetic pendulum TMD, xd ¼ k1mþk1 u It can be seen that the damper resonant frequency depends on its parameters, such as k1, k2, ku and m1. Therefore, another design strategy for the TMD is to ensure its resonant frequency to be close to the nonsynchronous low-frequency vibration by selecting the damper parameters.
4. Case study Fig. 6 shows the 100 kg-class FESS developed in our lab, and the structure and performance parameters of FESS except the pendulum TMD are listed in Table 1. Here, the parameters of flywheel rotor were obtained with measuring instrument for geometrical quantity, and the parameters of lower damper were detected using test rigs specially designed in our lab. The parameters of pendulum TMD will be given in the next section. 4.1. Dynamic parameter identification of the pendulum TMD The developed magnetic pendulum TMD’s are shown in Fig. 7. Fig. 7(a) shows the axial magnetic pendulum TMD, and Fig. 7(b) and (c) show the radial magnetic pendulum TMD. The dynamic parameters of the TMD’s should be obtained by an effective identification method.
Fig. 6. The 100 kg-class FESS developed in our lab.
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Table 1 The structure and performance parameters of the FESS. Lower damper
Flywheel rotor
Item
Value
Item
Value
Item
Value
k3 (N/m) k4 (N/m) m4 (kg) c4 (Ns/m)
0.50 106 1.60 106 0.75 400
M (kg) Jp (kg.m2) Jd (kg.m2) B (mm)
110 1.24 0.99 200
l (mm) l’ (mm) lu (mm)
390 190 70
4.1.1. Parameter identification system The test system is established using the developed 100 kg-class FESS, as shown in Fig. 8. The flywheel is fixed on the platform by a pair of magnetic chucks, so the TMD can be considered as a single degree-of-freedom vibration system. The TMD is excited by a electromagnetic vibration exciter (JZK-2, produced by China Jiangsu Sinocera Piezotronics, Inc.) with sine chirp signal (generated by YE1311B signal generator produced by Jiangsu Sinocera Piezotronics, Inc.). An impendence head (CLYD-331A, produced by Jiangsu Sinocera Piezotronics, Inc.) is mounted on the rod to measure exciting force and acceleration of vibrator of the TMD. Measured signals are sent to the computer by an signal conditioner (AZ804B, produced by the China Nanjing Anzheng Software Co., Ltd) and an data collector (AZ308R, produced by Nanjing Anzheng Software Co., Ltd), then the acceleration mobility of test system is calculated by the CRAS V7.0 Software on the vibration signal. The dynamic parameters of the damper can be obtained through identifying the test acceleration mobility. 4.1.2. Parameter identification method Dynamic model for the parameters identification system is illustrated in Fig. 9. An additional mass (mp) and stiffness (kp) of the exciting system are introduced to improve the accuracy of parameter identification. By vibration theory, the acceleration mobility of the test system is,
Y A ð xÞ ¼
m x2 2 m1 mx þ jc1 x þ k
ð5Þ
where,
m ¼ m1 þ mp ; k ¼ k1 þ ku þ kp After obtaining the mobility of test system, a reasonable method must be selected to identify the system’s parameters. Particle swarm optimization (PSO), as a biologically inspired evolutionary computation algorithm, has many merits, such as easy programming, high precision, rapid convergence and so on, the PSO has been applied widely [19–22]. Therefore, iterative method based on the PSO is used to identify the dynamic parameters of the system under the over-damped and underdamped conditions from the test mobility curves of system. Firstly, a fitting equation should be built, since the test system isn’t a strict single freedom degree system, there is a noise interference, so an acceleration mobility fitting equation containing the residual mobility (YsR and YsI) is built as follows:
Y A ðx; pÞ ¼
m x2 þ Y sR þ jY sI 2 m1 mx þ jc1 x þ k
ð6Þ
where, p is an vector of unknown parameters, p = {m, k, c1, YsR, YsI}.
Then, an error between the fitted value of mobility (Y Ai ) and the test value of mobility (Y Ai ) at the frequency xi can be obtained, ið
e xi Þ ¼ Y Ai ðxi Þ Y Ai ðxi Þ
ð7Þ
Therefore, the total variance between the fitted value and the test value is used to estimate the optimal parameter, which is defined as
mind ¼ min
N X
ð8Þ
e i ei
i¼1
where, ei is conjugate number of ei . The optimal parameters {m, k, c1} can be achieved by the PSO [19]; here, specific iterating process of PSO is not stated. Finally, the parameters of the TMD can be obtained, which are
kd ¼ k kp ; m1 ¼ m mp
ð9Þ
where, kd represents the total stiffness of the TMD, which is equal to the sum of k1 and ku (or k2 for radial magnetic pendulum TMD).
Y. Qiu, S. Jiang / Mechanical Systems and Signal Processing 121 (2019) 496–508
503
Fig. 7. The developed two types of TMD’s: (a) the axial magnetic pendulum TMD, (b) and (c) the radial magnetic pendulum TMD.
Fig. 8. Parameter identification system for the pendulum TMD: (a) schematic; (b) and (c) photos.
4.1.3. Identification results The test values of dynamic parameters for two TMD’s are listed in Table 2. The stiffness of the axial PMB and radial magnetic rings (namely, k2 and ku) are list in Table 2. Being limited with the space, the detail of the test method is not given in the paper. The results show that the natural frequency of radial magnetic pendulum TMD is less than that of the axial magnetic pendulum TMD. This can be attributed to the fact that the stiffness of the radial magnetic pendulum TMD is obviously less than that of axial magnetic pendulum TMD. As expected, the axial magnetic pendulum TMD cannot be excited smoothly by the axial PMB due to large friction resistance on the ball joint. On the contrary, the radial magnetic pendulum TMD can run
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Fig. 9. Dynamic model of the parameters identification system.
Table 2 The experimental results of the dynamic parameters of two TMD’s. Types
m1/kg
k1/104N/m
ku/104N/m
k2/104N/m
xn/Hz
Radial magnetic pendulum TMD Axial magnetic pendulum TMD
4.65 4.20
0.15 2.55
2.17 –
1.75 1.75
11.24 16.10
flexibly due to smaller axial load on the joint compared to the former, which benefits from the parallel connection of the axial PMB and the radial TMD. The experimental damping coefficients of the two TMD’s are shown in Fig. 10. It can be seen that three curves are almost parallel under the log-log coordinates. The damping coefficients increase with the viscosity of damping oil and decrease with damper’s radial gap (i.e. the radial clearance between the journal and damping housing).
4.2. Damping effect of the pendulum TMD for low-frequency vibration Fig. 11 shows modal test system for the 100 kg-class FESS at static state. Calibrated eddy current displacement probes (CWY-DO-502, produced by Jiangsu Sinocera Piezotronics, Inc.) are employed to detect the system vibration. The response points are located at the upper damper (number 1), the flywheel top (number 2), flywheel bottom (number 3) and lower damper (number 4). A hammer (CL-YD-305, produced by Jiangsu Sinocera Piezotronics, Inc.) is used to impact the system at the back of response 4 of the lower damper. All the signals are sent to the computer by a (AZ804B) signal conditioner and a data collector (AZ308R). The mobilities (namely, frequency response function) of four response points are calculated and recorded by the CRAS V7.0 Software on the vibration signals. Modal frequencies, shapes and damping ratios of the rotor system are estimated from the test mobility curves by the modal iterating identification method [23]. Fig. 12 gives the mobility amplitude-frequency curves for four detection points of the FESS with the radial magnetic pendulum TMD. The No.32 machine oil (produced by oil refinery, China) is used for the radial magnetic pendulum TMD, and the damping coefficient c1 is identified to be 120Ns/m. In the figure, Y14, Y24, Y34 and Y44 represent the mobility of the response point 1, 2, 3 and 4 due to a force at point 4, separately. Obviously, there are four peaks within 400 Hz corresponding to the four modes. For the FESS with the axial magnetic pendulum TMD, the mobility amplitude-frequency curves are similar with the radial magnetic pendulum TMD, so we do not repeat here. Tables 3 and 4 give the theoretical and test values of the previous four orders modal frequencies and the corresponding mode shapes of the FESS, which are also shown in the Figs. 13 and 14. It can be seen that the test results of the model frequencies and the mode shapes nearly agree with the theoretical ones. However, small differences still exist in two results due to some uncertainties in the measurements besides the several assumptions made for dynamic modeling, for example, the test mobility curves of the pendulum TMD, and the test frequency response data of the FESS are unavoidably interfered by the noise, nonlinearity and accuracy of test system. As shown in Figs. 13 and 14, the 1st-order and 2nd-order modes are related to the low-frequency vibration of the FESS. For the FESS with the radial magnetic pendulum TMD, the maximal displacement of the 1st-order mode locates at the TMD, as shown in Fig. 13, but for the FESS with the axial magnetic pendulum TMD, the maximal displacement locates at the flywheel top, as shown in Fig. 14. Therefore, the radial magnetic pendulum TMD should be easier to be excited than the axial magnetic pendulum TMD, which means that the radial magnetic pendulum TMD is more efficient to help the rotor to pass the 1st critical speed smoothly. However, the 1st-order mode frequency (2.58 Hz) of the FESS with the radial magnetic pendulum TMD is slightly greater than that (1.96 Hz) of the axial magnetic pendulum TMD; this can be attributed to the fact that the radial stiffness of the radial magnetic pendulum TMD is greater than that of the axial magnetic pendulum TMD. Fig. 15 shows the 1st-order mode damping ratios of the flywheel rotor system at the different damping coefficients. During the test process, series of damping coefficients of the pendulum TMD are obtained by a combination of several damper radial gaps and different types of damping oils. It can be seen from Fig. 15 that the test values are in generally agreeable with the theoretical ones. Similarly, the differences between two results are also seen, it may be caused by the noise, nonlinearity
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(b)
Gap= 0.75mm Gap= 1.00mm Gap= 1.50mm
1000
Damping coefficient (Ns/m)
Damping coefficient (Ns/m)
(a)
100
10 0.01
Gap = 1.0 mm Gap = 1.5 mm Gap = 2.0 mm
1,000
100
10
1 0.01
0.1 Viscostiy (Pa.s)
0.1 Viscosity (Pa.s)
Fig. 10. Damping coefficients under different damping oil viscosities and the damper radial gaps: (a) the radial magnetic pendulum TMD, (b) the axial magnetic pendulum TMD.
Fig. 11. The schematic (a) and photo (b) of modal identification system of the FESS.
7
1st
|Y|(x 10-5m/N)
6 5
2nd
4
3rd
|Y14| |Y24| |Y34| |Y44|
3 2 4th
1 0
1
10 100 Frequency ¹ (Hz)
Fig. 12. Amplitude-frequency curves of mobility of FESS with radial magnetic pendulum TMD.
and accuracy of test system besides precision of the dynamic modeling. In addition, compared with the radial TMD, the axial TMD has a larger coulomb damping due to larger axial load, which is not considered in the theory model, so it is reasonable that the difference of the predicted result and the test one with the axial TMD is larger than that with the radial TMD.
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Table 3 The modal parameters of FESS with radial magnetic pendulum TMD. Type
Order
Frequency (Hz)
Mode
R1
R2
R3
R4
Experimental values
1 2 3 4
2.58 11.23 31.49 270.0
1.000\0.0° 1.000\0.0° 0.161\3.2° 0.000\0.0°
0.811\5.5° 0.069\161.6° 0.492\173.6° 0.000\0.0°
0.417\5.6° 0.061\175.2° 1.000\0° 0.000\0.0°
0.001\8.1° 0.0006\164.6° 0.528\8.0° 1.000\0°
Theoretical values
1 2 3 4
2.43 11.44 30.41 263.44
1.000\0° 1.000\0° 0.152\2.1° 0.1E4\143.5°
0.864\1.5° 0.075\172.3° 0.548\180.0° 0.004\18.6°
0.425\1.6° 0.052\174.6° 1.000\0° 0.007\161.5°
0.002\5.8° 0.007\179.2° 0.596\2.1° 1.000\0.0
Table 4 The modal parameters of FESS with axial magnetic pendulum TMD. Type
Order
Frequency (Hz)
Mode
R1
R2
R3
R4
Experimental values
1 2 3 4
1.96 15.84 31.76 267.52
0.322\7.5° 1.000\0.0° 0.032\1.7° 0.000\0.0°
1.000\0.0° 0.012\153.1° 0.49\176.6° 0.000\0.0°
0.542\5.2° 0.009\156.1° 1.000\0.0° 0.000\0.0°
0.0003\18.5° 0.001\151.1° 0.532\5.9° 1.000\0.0°
Theoretical values
1 2 3 4
1.89 15.66 30.17 263.44
0.379\1.3° 1.000\0° 0.073\6.9° 0.5E5\143.8°
1.000\0.0° 0.021\167.0° 0.532\180° 0.004\18.5°
0.490\0.0° 0.016\173.1° 1.000\0.0° 0.007\161.5°
0.0013\0.65° 0.003\177.7° 0.592\2.1° 1.000\0.0°
1st 2.58 HZ
3rd 31.49 HZ
4th 270 HZ
0.6
0.6
0.4
0.4
0.4
0.4
0.2
x
0
0.2 0 -1
1st 2.44 HZ
(b)
0.2 0 -1
0 1 x 1 -1 0 y
1 -1 0 y 1
z(m)
0.6 z(m)
0.6
0 -1
x
2nd 11.28 HZ
0
0.2 0 -1
1 -1 0 y 1
x
3rd 30.35 HZ 0.6
0.4
0.4
0.4
0.4
0 -1
0
1 x1 -1 0 y
0.2 0 -1
0
1 x1 -1 0 y
z(m)
0.6 z(m)
0.6
0.2
0.2 0 -1
0
1 x1 -1 0 y
0
1 -1 0 y 1
4th 263.44 HZ
0.6 z(m)
z(m)
2nd 11.23 HZ
z(m)
z(m)
(a)
0.2 0 -1
0
1 x1 -1 0 y
Fig. 13. Modal frequencies and shapes of flywheel rotor system with radial magnetic pendulum TMD: a) test results, b) theoretical results.
Obviously, the 1st-order mode damping ratio of flywheel rotor system increases at first and then decreases with the damping coefficient, so there is an optimal damping coefficient corresponding to the maximal mode damping ratio. The maximal 1st-order mode damping ratios of two flywheel rotor systems with pendulum TMD are more than 0.1. Moreover, the maximal 1st-order mode damping ratio of flywheel rotor system the with the radial magnetic pendulum TMD is larger than
Y. Qiu, S. Jiang / Mechanical Systems and Signal Processing 121 (2019) 496–508
1st 1.96 HZ
2nd 15.84 HZ
4th 267.52 HZ 0.6
0.4
0.4
0.4
0.4
0.2
0.2 0 -1
1 x0 1 -1 0 y
0.2 0 -1
1 x0 1 -1 0 y
1st 1.89 HZ
(b)
z(m)
0.6 z(m)
0.6
0 -1
0.2 0 -1
x 0 1 -1 0 1 y
2nd 15.66 HZ
1 x0 1 -1 0 y
3rd 30.17 HZ
4th 263.44 HZ
0.6
0.6
0.4
0.4
0.4
0.4
0.2 0 -1
0 x
1 -1 0 y 1
0.2 0 -1
0
1 x 1 -1 0 y
z(m)
0.6 z(m)
0.6 z(m)
z(m)
3rd 31.76 HZ
0.6 z(m)
z(m)
(a)
0.2 0 -1
0
1 x 1 -1 0 y
507
0.2 0 -1
x0 1 -1 0 y 1
Fig. 14. Modal frequencies and shapes of flywheel rotor system with axial magnetic pendulum TMD: a) test results, b) theoretical results.
0.5
Expt., Ther., Expt., Ther.,
§1
0.4 0.3
axial TMD axial TMD radial TMD radial TMD
0.2 0.1 0.0
0
2000
4000 6000 c1 (Ns/m)
8000
Fig. 15. The 1st-order mode damping ratios of the flywheel rotor system with different damping coefficients.
that of flywheel rotor system with the axial magnetic pendulum TMD, so the radial magnetic pendulum TMD is superior to the axial magnetic pendulum TMD in the damping effect on the low-frequency vibration. 5. Conclusion In this paper, two types of magnetic pendulum TMD’s have been developed to suppress the low frequency vibration of the large-mass FESS with PMB and spiral groove bearing. The identification method of the dynamic parameters for the TMD’s has been established, and evaluating method has been proposed to compare the effectiveness for two TMD’s. The test study has been conducted using the 100 kg-class FESS in our lab to verified the theoretical results, the main conclusion can be given as follows, 1) Compared with the axial magnetic pendulum TMD, the radial magnetic pendulum TMD has a lower resonant frequency and contributes a greater 1st-order mode damping ratio for the FESS. Thus, the radial magnetic pendulum TMD is more efficient for the suppression of low-frequency vibration of the large-mass FESS. 2) Considering the FESS is both a rotor system and a rigid multi-body system, the dynamic model of the FESS has been established using the Lagrange’s equation; the effectiveness of the pendulum TMD can be evaluated by analyzing the 1st mode damping ratio of the FESS with the pendulum TMD employed.
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3) The pendulum TMD is simplified as a single degree-of-freedom system, and its dynamic parameters can be identified from the mobility curves by the iterative identification method based on the PSO.
Acknowledgment The authors gratefully acknowledge the support provided by the National Natural Science Foundation of China through grant Nos. 51635004, 11472078, 11172065. Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.ymssp.2018.11.033. References [1] X. Dai, Z. Shen, H. Wei, On the vibration of rotor-bearing system with squeeze film damper in an energy storage flywheel, Int. J. Mech. Sci. 43 (11) (2001) 2525–2540. [2] X. Dai, H. Wei, Z. Shen, Dynamics design and experiment study of the rotor-bearing system of a flywheel energy storage system, Chin. J. Mech. Eng. (China) 39 (4) (2003) 97–101. [3] J.M. Vance, D. Ying, J.L. Nikolajsen, Actively controlled bearing dampers for aircraft engine applications, J. Eng. Gas Turbines Power Trans. ASME 122 (3) (2000) 466–472. [4] L. Chen, C. Zhu, M. Wang, et al, Vibration control for active magnetic bearing high-speed flywheel rotor system with modal separation and velocity estimation strategy, J. Vibroengineering 17 (2) (2015) 757–775. [5] A. Looser, A. Tüysüz, C. Zwyssig, et al, Active magnetic damper for ultrahigh-speed permanent-magnet machines with gas bearings, IEEE Trans. Ind. Electron. 64 (4) (2017) 2982–2991. [6] R.S. Srinivas, R. Tiwari, C. Kannababu, Application of active magnetic bearings in flexible rotordynamic systems–A state-of-the-art review, Mech. Syst. Sig. Process. 106 (2018) 537–572. [7] M. Silvagni, A. Tonoli, A. Bonfitto, Self-powered eddy current damper for rotor dynamic applications, J. Vibr. Acoust. 137 (1) (2015) 011015. [8] F.N. Werfel, U. Floegel-Delor, T. Riedel, et al, A compact HTS 5 kWh/250 kW flywheel energy storage system, IEEE Trans. Appl. Supercond. 17 (2) (2007) 2138–2141. [9] S.K. Cheah, H.A. Sodano, Novel eddy current damping mechanism for passive magnetic bearings, J. Vib. Control 14 (11) (2008) 1749–1766. [10] X. Liu, X. Feng, Y. Shi, et al, Development of a semi-active electromagnetic vibration absorber and its experimental study, J. Vib. Acoust. 135 (5) (2013) 051015. [11] D. Luo, Y. Liu, X. Sun, D. Hung, The design and analysis of supercritical carbon dioxide centrifugal turbine, Appl. Therm. Eng. 127 (12) (2017) 527–535. [12] S. Nagesh, A.M. Junaid Basha, T.D. Singh, Dynamic performance analysis of high speed flexible coupling of gas turbine engine transmission system, J. Mech. Sci. Technol. 29 (1) (2015) 173–179. [13] Chang-Liang Tang, Xing-Jian Dai, Xiao-Zhang Zhang, ect. Rotor dynamics analysis and experiment study of the flywheel spin test system, J. Mech. Sci. Technol. 26 (9) (2012) 2669–2677. [14] S. Jiang, H. Wang, S. Wen, Flywheel energy storage system with a permanent magnet bearing and a pair of hybrid ceramic ball bearings,, J. Mech. Sci. Technol. 28 (12) (2014) 5043–5053. [15] C. Shi, R.G. Parker, S.W. Shaw, Tuning of centrifugal pendulum vibration absorbers for translational and rotational vibration reduction, Mech. Mach. Theory 66 (2013) 56–65. [16] H. Xiujin, H. Lidong, H. Wenchao, Study on semi-active TMD controlling vibration of two-span rotors over critical speeds, J. Mech. Electr. Eng. 31 (10) (2014) 1244–1248. [17] X. Dai, Z. Shen, H. Wei, Study on whirl modal damping of energy storage flywheel-bearing system, J. Vib. Eng. (In Chinese) 15 (1) (2002) 98–101. [18] H. Wang, S. Jiang, Z. Shen, The dynamic analysis of an energy storage flywheel system with hybrid bearing support, J. Vib. Acoust. 131 (5) (2009) 051006. [19] Y. Del Valle, G.K. Venayagamoorthy, S. Mohagheghi, et al, Particle swarm optimization: basic concepts, variants and applications in power systems, IEEE Trans. Evol. Comput. 12 (2) (2008) 171–195. [20] Y. Zheng, Y. Liao, Parameter identification of nonlinear dynamic systems using an improved particle swarm optimization, Optik-Int. J. Light Electron. Opt. 127 (19) (2016) 7865–7874. [21] V.S. Özsoy, H.H. Örkcü, H. Bal, Particle Swarm Optimization Applied to Parameter Estimation of the Four-Parameter Burr III Distribution, Iran. J. Sci. Technol. Trans. A Sci. 42 (1) (2017) 1–15. [22] M.A. Galewski, Modal parameters identification with particle swarm optimization, Key Eng. Mater. 597 (2016) 119–124. [23] D. Li, Q. Lu, Analysis of Experiments in Engineering Vibration, Tsinghua University Press, Beijing, China, 2004.