Modeling, analysis, and control of shaft transverse vibration from rotating systems through active bearing concept Junhee Kwon 1 Yeongnam University 280 Daehak-Ro, Gyeongsan, Gyeongbuk 38541, Republic of Korea Dongwoo Hong 2 Yeongnam University 280 Daehak-Ro, Gyeongsan, Gyeongbuk 38541, Republic of Korea Byeongil Kim 3 Yeongnam University 280 Daehak-Ro, Gyeongsan, Gyeongbuk 38541, Republic of Korea

ABSTRACT Rotating parts are widely applied to mechanical systems such as pumped-storage hydroelectricity, nuclear power plant, machining tools and so on. While operating, they can be easily damaged or destroyed by unbalanced mass, bending, torsion and misalignment. In order to solve this problem, rotor vibration control can be conducted through active bearing concept. In this work, active bearing system which consists of piezo actuators and rubber grommets is proposed and applied to a rotating system motivated from a pumped-storage hydroelectricity, for performing active vibration control. The main point of this paper is to prevent damage or failure caused by harsh transverse vibration through active bearings. First, the rotating system is modeled by transfer matrix method (TMM) based on Euler-Bernoulli beam theory and in order to check accuracy of this model, the responses of TMM are compared with the responses from the finite element method (FEM). For implementing active control in real time, normalized least mean square (NLMS) algorithm is utilized. The results show that the proposed active bearing concept shows great performance on the attenuation of shaft transverse vibration.

1. INTRODUCTION

Rotor unbalance can have a critical effect on damage to mechanical systems such as bearings and gears and can significantly shorten the life span of rotating equipment. Since it is impossible to manufacture a perfectly balanced rotor, methods are being sought to minimize the effect of unbalance. To solve this problem, Cao et al [1] proposed Active Magnetic Bearing (AMB) performance comparison of two analytical methods. Kim et al [2] proposed a new type of compact, high-

1 rnjswns02@ynu.ac.kr

2 dongwoo229@naver.com

3 bikim@yu.ac.kr

performance five-axes AMB with solid cores and rotor. Dede et al [3] proposed analytical method for rotor system using FEM and Lumped Parameter Method.

In this paper, Rotor modeling proceeds with FEM and TMM and analyzes the difference between the two analysis methods to verify that the TMM model is correct. Also, a novel active bearing composed piezo actuators and rubbers are proposed. After coupling the active bearing to the verified TMM-based model, the signal is traced through NLMS adaptive algorithms and simulation is performed by controlling the actuator through the traced signal. The simulation results show that the displacement in each direction is greatly reduced.

2. Design of the rotor system

In this section, rotor system motivated from a pumped-storage hydroelectricity (See Figure 1) The blue and green mean shaft, yellow means disk including unbalance mass.

. Figure 1: Rotor system motivated from a pumped-storage hydroelectricity.

2.1. Finite Element Method The FEM-based modeling using Euler-Bernoulli beam theory consists of 26 elements and 27 nodes. Mass matrix, stiffness matrix and state vector used for modeling for one element are as follows.

156 + 𝑚 𝑖 22𝐿 54 −13𝐿 22𝐿 4𝐿 2 + 𝐼 𝑑 13𝐿 −3𝐿 2

𝜌𝐴𝐿 420 [

M 1 =

] (1)

54 13𝐿 156 −22𝐿 −13𝐿 −3𝐿 2 −22𝐿 4𝐿 2

6 + 𝑘 𝑖𝑗 3𝐿 −6 3𝐿 3𝐿 2𝐿 2 −3𝐿 𝐿 2

𝐾 1 = 2𝐸𝐼

]

𝐿 3 [

(2)

−6 −3𝐿 6 −3𝐿 3𝐿 𝐿 2 −3𝐿 2𝐿 2

q = [𝑥 1 𝜓 1 𝑥 2 𝜓 2 ] 𝑇 (3)

The m i , 𝑖 𝑑 and k ij mean additional mass, moment of inertia and stiffness, respectively, such as disks and bearings. Damping matrix is constructed taking in to account the gyroscope effect of the disk element (See Equation 5).

I d 𝜓 ̈ 𝐷 + 𝐼 𝑝 Ω𝜃 ̇ 𝐷 = 𝑀 𝑦𝐷

𝐿 + 𝑀 𝑦𝐷

𝑅 I d 𝜃 ̈ 𝐷 −𝐼 𝑃 Ω𝜓 ̈ 𝐷 = 𝑀 𝑥𝐷

(4)

𝐿 + 𝑀 𝑥𝐷

𝑅

0 0 0 0 0 0 0 Ω𝐼 𝑝 0 0 0 0 0 −Ω𝐼 𝑝 0 0

]

G D = [

(5)

The equation of motion is derived by assembling the matrices mentioned above. The equation of motion is abbreviated as Equation (7), and the value of state at each point can be known through matrix calculation.

𝑞 𝑦 ̇ ] + [𝐾 𝑥 0 0 𝐾 𝑦 ] [𝑞 𝑥

[ 𝑀 𝑥 0 0 𝑀 𝑦 ] [𝑞 𝑥

𝑞 𝑦 ̈ ] + [ 0 𝐺 𝑥 𝐺 𝑦 0 ] [𝑞 𝑥

𝑞 𝑦 ] = [𝐹 𝑥

𝐹 𝑦 ]

(6)

𝐹 𝑦 ] = [𝐾 𝑥 −Ω 2 𝑀 𝑥 Ω 𝐺 𝑥 Ω 𝐺 𝑦 𝐾 𝑦 −Ω 2 𝑀 𝑦

] [ 𝑞 𝑥

[ 𝐹 𝑥

𝑞 𝑦 ]

(7)

−1

𝑞 𝑥 𝑞 𝑦 ] = [𝐾 𝑥 −Ω 2 𝑀 𝑥 Ω 𝐺 𝑥 Ω 𝐺 𝑦 𝐾 𝑦 −Ω 2 𝑀 𝑦

[ 𝐹 𝑥

𝐹 𝑦 ]

[

]

(8)

2.2. Transfer Matrix Method

The TMM-based modeling using Euler-Bernoulli beam theory. The shaft, mass, disk, bearing transfer matrix are expressed in Equations (9) to (12).

[ 1 𝐿 𝐿 2

𝐿 3

6𝐸𝐼 0 0 0 0

2𝐸𝐼

𝐿 2

0 1 𝐿 𝐸𝐼

2𝐸𝐼 0 0 0 0

0 −𝐼 𝑑 Ω 2 1 𝐿 0 jΩ 2 𝐼 𝑝 0 0 0 0 0 1 0 0 0 0

T shaft =

(9)

0 0 0 0 1 𝐿 𝐿 2

𝐿 3

2𝐸𝐼

6𝐸𝐼

𝐿 2

0 0 0 0 0 1 𝐿 𝐸𝐼

2𝐸𝐼 0 −𝑗Ω 2 𝐼 𝑝 0 0 0 −𝐼 𝑑 Ω 2 1 𝐿 0 0 0 0 0 0 0 1 ]

[ 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 m d Ω 2 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 𝑚 𝑑 Ω 2 0 0 1]

T mass =

(10)

[ 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 −𝐼 𝑑 Ω 2 1 0 0 𝑗𝐼 𝑝 Ω 0 0 𝑚 𝑑 Ω 2 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 −𝑗𝐼 𝑝 Ω 0 0 0 −𝐼 𝑑 Ω 2 1 0 0 0 0 0 𝑚 𝑑 Ω 2 0 0 1]

T disk =

(11)

[ 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 −(𝑘 𝑥𝑥 + 𝑗𝑐 𝑥𝑥 Ω) 0 0 1 𝑘 𝑥𝑦 + 𝑗𝑐 𝑥𝑦 Ω 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 𝑘 𝑦𝑥 + 𝑗𝑐 𝑦𝑥 Ω 0 0 0 −(𝑘 𝑦𝑦 + 𝑗𝑐 𝑦𝑦 Ω) 0 0 1]

T bearing =

(12)

Where k ij , 𝑐 𝑖𝑗 , Ω mean stiffness and damping of bearing and rotational speed, respectively. By calculating the above transfer matrices, the overall transfer matrix of the system can be obtained. In order to determine the left-end state vector, the boundary condition is set as the free condition at both ends. The left-end state vector is determined through the transfer matrix reduced by the boundary condition. The value of state at each point can be known through matrix calculation of the obtained left-end state vector (See Equation 17).

𝑧 𝑖 = 𝑇 𝑖 𝑇 𝑖−1 … 𝑇 1 𝑇 0 𝑧 0 (13)

Boundary condition 𝑀 𝑦0 = 𝑀 𝑥0 = 𝑉 𝑥0 = 𝑉 𝑦0 = 𝑀 𝑦𝑖 = 𝑀 𝑥𝑖 = 𝑉 𝑥𝑖 = 𝑉 𝑦𝑖 = 0 (14)

−1

𝑡 31 𝑡 32 𝑡 35 𝑡 36 𝑡 41 𝑡 42 𝑡 45 𝑡 46 𝑡 71 𝑡 72 𝑡 75 𝑡 76 𝑡 81 𝑡 82 𝑡 85 𝑡 86

−𝑡 39 −𝑡 49 −𝑡 79 −𝑡 89

𝑥 𝛹 −𝑦

′ = [

𝑧 0

] (15)

]

= [

]

[

𝜃

0

𝑥 𝛹 𝑀 𝑦 −𝑉 𝑥

′ (1) 𝑧 0

[ 𝑧 0

0 0 ]

[

′ (2)

0 0 𝑧 0

𝑧 0 =

(16)

=

−𝑦

′ (3) 𝑧 0

𝜃 𝑀 𝑥

′ (4)

𝑉 𝑦 ]

0

0

𝑧 𝑛 = 𝑇 𝑛 𝑇 𝑛−1 … 𝑇 2 𝑇 1 𝑧 0 (17)

2.3. Active Bearing

The active bearing system which consists of piezo actuators and rubber grommets is proposed and applied to a rotating system (See Figure 2). In this Figure 2, black, red, yellow correspond to rubber, actuators and bearing, respectively.

Figure 2: The active bearing model & Free Body diagram of active bearing. For active bearing, a free body diagram can be drawn as in the right Figure 2, and the transfer matrix is as Equation 21.

𝑥 𝑦 𝐹 𝑥 𝐹 𝑦

𝑥 𝑦 𝐹 𝑥 𝐹 𝑦

1 0 −𝐺 𝑦𝑦 −𝐺 𝑥𝑦 0 1 −𝐺 𝑦𝑥 −𝐺 𝑥𝑥 0 0 1 0 0 0 0 1

[

]

= [

] [

]

(18)

𝑏𝑒𝑎𝑟𝑖𝑛𝑔

𝑎𝑐𝑡𝑢𝑎𝑡𝑜𝑟

𝑅

𝐿

𝑥 𝑦 𝐹 𝑥 𝐹 𝑦

𝑥 𝑦 𝐹 𝑥 𝐹 𝑦

1 0 0 0 0 1 0 0 𝑚 𝑎𝑐𝑡𝑢𝑎𝑡𝑜𝑟 Ω 2 0 1 0 0 𝑚 𝑎𝑐𝑢𝑡𝑎𝑡𝑜𝑟 Ω 2 0 1

[

]

= [

] [

]

(19)

𝑎𝑐𝑡𝑢𝑎𝑡𝑜𝑟

𝑎𝑐𝑢𝑡𝑎𝑡𝑜𝑟

𝐿

𝑅

𝑥 𝑦 𝐹 𝑥 𝐹 𝑦

𝑥 𝑦 𝐹 𝑥 𝐹 𝑦

1 0 −1/𝑍 𝑠𝑥𝑥 0 0 1 0 −1/𝑍 𝑠𝑦𝑦 0 0 1 0 0 0 0 1

[

]

= [

] [

]

(20)

𝑎𝑐𝑢𝑡𝑎𝑡𝑜𝑟

𝑓𝑖𝑥𝑒𝑑

𝑅

𝐿

𝑥 𝜓 𝑀 𝑦 −𝑉 𝑥

𝑥 𝜓 𝑀 𝑦 −𝑉 𝑥

[ 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 𝑆 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 𝑆 2 0 0 1]

[

[

=

(21)

−𝑦

−𝑦

𝜃 𝑀 𝑥

𝜃 𝑀 𝑥

𝑉 𝑦 ]

𝑉 𝑦 ]

Actuator Grommet Bearing

S 1 = 𝑚 𝑎𝑐𝑡𝑢𝑎𝑡𝑜𝑟 Ω 2 + 𝑍 𝑥𝑥 1 −𝑚 𝑎𝑐𝑡𝑢𝑎𝑡𝑜𝑟 Ω 2 𝐺 𝑦𝑦 + 𝑍 𝑥𝑥 ∗𝐺 𝑦𝑦

S 2 = 𝑚 𝑎𝑐𝑡𝑢𝑎𝑡𝑜𝑟 Ω 2 + 𝑍 𝑦𝑦 1 −𝑚 𝑎𝑐𝑡𝑢𝑎𝑡𝑜𝑟 Ω 2 𝐺 𝑥𝑥 + 𝑍 𝑦𝑦 ∗𝐺 𝑥𝑥

3. Simulation of Active control system

3.1. Check accuracy of model

In this paper, the second paragraph performed modeling for FEM and TMM. In order to control through TMM, the natural frequency of FEM and TMM models were compared (See Table 1). The natural frequency in each mode has a difference of up to 3%, and the mode shape was additionally compared. The accuracy of TMM modeling was verified through the fact the natural frequency and mode shape in each mode did not differ significantly between FEM and TMM

Table 1: Natural Frequency in Mode from 1 to 3.

[Hz] Mode 1 Mode Mode 3

FEM 58.70 131.0 274.7

TMM 60.65 131.4 278.0

Error (%) 3.32 ↑ 0.26 ↑ 1.19 ↑

Figure 3: Mode shape in x and y direction of mode 1 to 3

3.2. Simulation using NLMS adaptive algorithms.

Through Equation 17, the state vector of the node to which the active bearing is coupled can be determined. When the displacement of the actuator is set to 𝑥 𝑎𝑐𝑡𝑢𝑎𝑡𝑜𝑟 , the output of the adaptive filter is input to the actuator. By adaptive filter, the displacement is changed every sampling time. Based on the changed state vector of the active bearing node, the state vector of other nodes is updated using Equation 22.

𝑧 𝑘 = 𝑇 𝑘 𝑇 𝑘−1 … 𝑇 𝑛+1 𝑧 𝑛 𝑧 0 = (𝑇 𝑛 𝑇 𝑛−1 … 𝑇 1 ) −1 𝑧 𝑛

(22)

The adaptive algorithm uses the NLMS adaptive algorithm with the advantage of provides the solution to the solve convergence of the Least Mean Square (LMS) algorithm, and the filter length, learning rate, gain was determined by trial and error. In order to judge whether the result is good or not, the following selection criteria were selected.

Figure 4: Criteria for selecting the better results.

The displacement for each node was investigated through the method proposed in paragraph 3.2. To select the value that shows the best performance, it was performed while changing the gain. Figure 3 shows the gain change when the filter length is 4 and the learning rate is 0.4. When Gain x = 2.4, 𝐺𝑎𝑖𝑛 𝑦 = 0.06 , it can be seen that displacement in each direction is reduced by 33.3% and in y direction by 5.18%.

| Percentage of non-controlled and controlled |» Average of Percentages ———[ Bach position |-—

Figure 5: Displacement changing rate according to gain change

Orbit is indicated based on the displacement obtained through the gain above (See Figure 6). Since there are a total of 27 nodes, only orbits from the node at the midpoint in each stage are shown.

X direct Y direction

Figure 6: Orbit at the midpoint in each stage.

4. Conclusion.

In this paper, we have presented two different analytical methods to verify the model by comparing the natural frequency and shape of each mode. Through those results, it was confirmed that there was no significant difference between the two analysis methods, and it was shown that active control can be performed using the TMM-based model. Using NLMS adaptive algorithms, we can see good performance of active bearing composed of actuators and rubbers. In the current work, the active bearing is used for pumped-storage hydroelectricity, but they can be applied to various rotor system such as machine tools, nuclear power generators and so on.

5. ACKNOWLEDGEMENTS

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2021R1A6A1A03039493) 6. REFERENCES

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