Bi-stable buckled beam nonlinear energy sink applied to rotor system

Mechanical Systems and Signal Processing 138 (2020) 106546

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Bi-stable buckled beam nonlinear energy sink applied to rotor system Hongliang Yao ⇑, Yuwei Wang, Linqing Xie, Bangchun Wen School of Mechanical Engineering and Automation, Northeastern University, Shenyang 110819, PR China

a r t i c l e

i n f o

Article history: Received 29 June 2019 Received in revised form 22 October 2019 Accepted 25 November 2019

Keywords: Bi-stable buckled beam nonlinear energy sink Rotor system Vibration suppression

a b s t r a c t In this paper, a bi-stable nonlinear energy sink (BNES) with buckled beam is developed to suppress the vibration of unbalanced rotor system. The structure of the BNES is developed. The specific structures and working principles of the buckled beam are introduced. Based on these, the dynamic equations of the rotor-BNES system are established. Then, the transient and steady state responses of the rotor-BNES system are numerically studied and compared with the linear visco-elastic damper. In addition, the vibration suppression ability and frequency suppression range of the BNES are analyzed. Finally, experiments are carried out and compared with the simulations to confirm the vibration suppression effectiveness of the BNES. The numerical and experimental results show that the designed BNES has a strong vibration suppression effect on the rotor system and can withstand a wide range of energy. Ó 2019 Elsevier Ltd. All rights reserved.

1. Introduction Vibration absorption plays an important role in vibration suppression of rotor systems, which can be largely divided into passive type [1] and semi-active (or active) type [2]. However, both these two types have some shortcomings: passive absorbers can only work within a narrow frequency band near the anti-resonance point, while the semi-active (or active) absorbers are usually complicated in structure. The nonlinear energy sink (NES), unlike traditional absorbers, can absorb vibration in a wide frequency range and has the potential to be the next generation of passive vibration suppression approach for rotor systems. At present, NES can be divided into ungrounded NES [3–8] and grounded NES [9]. The ungrounded NES is nonlinearly coupled with the main system through the small mass, while the grounded NES is linearly coupled to the main system and nonlinearly coupled to the ground. The NES transmits the vibration energy of the main system to the small mass by the targeted energy transfer (TET) [10–14] mechanism. The energy is directly consumed by the NES damping and will not return to the main system, which improves the vibration suppression efficiency. The NES has been applied in aerospace [15], rotating machinery [16], architecture [17] and other fields. In terms of rotor systems, NESs have been studied for a while now. For example, Guo et al. modeled a three-degree-offreedom hollow shaft-NES system and analyzed the influence of critical parameters to the suppression ability [18]. Bab et al., meanwhile, studied a vibration control method in which a NES is used for a rotating beam [19]; Bergeot et al. analyzed the steady-state response of helicopter blades with a NES [20]; Yao et al. designed a NES with permanent magnetic springs and

⇑ Corresponding author. E-mail address: [email protected] (H. Yao). https://doi.org/10.1016/j.ymssp.2019.106546 0888-3270/Ó 2019 Elsevier Ltd. All rights reserved.

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H. Yao et al. / Mechanical Systems and Signal Processing 138 (2020) 106546

coil springs [21]; Ebrahimzade et al. investigated the effectiveness of NES on the aeroelastic behaviors of rotor blades [22]; The above researches have fully illustrated the effectiveness of the NES for rotor system vibration suppression. However, the traditional NESs can only be effective in a certain energy threshold, thereby limiting their wide range of engineering applications. In order to further expand the engineering applications and solve the narrow threshold limitation of the NES, bi-stable nonlinear energy sinks (BNESs) [23–26] which have two stable equilibrium positions and a critical equilibrium position were developed. Many BNESs have been applied to vibration suppression and energy recovery. For example, AL-Shudeifa et al. proved that BNES maintains its high performance of shock mitigation in a broadband fashion of the input initial energies [24]. Romeo et al. used numerical simulation to prove that the BNES has better transient vibration reduction effect for the main system than the ordinary NES [25]; Fang et al. studied a very important way to realize the rapid consumption of vibration energy [26]; Wang et al. designed a two-degree-of-freedom bi-stable piezoelectric energy harvester to verify that it can absorb the two-order resonance energy of the main system [27]; Yang et al. proposed two new coupled lever bi-stable nonlinear energy harvesters to enhance the inter-sink dynamic response for improvement of vibration energy harvesting [28]; Zhou et al. proposed a flexible bi-stable energy harvester (FBEH) and proved that the FBEH owning a variable potential energy function is beneficial for snap-through [29]. According to different design methods, bi-stable structures can be divided into four main categories: slightly flexible bi-stable beam structure [30], buckled pre-compression deformation bi-stable beam structure [31–33], cantilever bi-stable structure with magnets [34,35] and bi-stable tilted beam structure [36]. However, to authors’ knowledge, there is no BNES applied to rotor system yet. Therefore, in this paper, a BNES with low axial space’s structure for rotor system is proposed, and its structural characteristics, stiffness variation, vibration characteristics and vibration suppression effects are studied. 2. Structure and dynamic model of the rotor-BNES 2.1. Structure of the rotor-BNES system The BNES is applied to the single-span-single-disc rotor system in Fig. 1. The rotor-BNES system is composed of a rotor, a BNES, two support bearings and a motor. As shown in Fig. 2, the specific BNES structure includes a bearing, a bearing support, a BNES mass, a supporting part and three buckled beams. For the buckled beams, they can be divided into three parts: the inner left buckled beam, the inner right buckled beam and the outer buckled beam. 2.2. Bi-stable stiffness of the BNES During operation, the BNES does not rotate with the rotor and suppresses the horizontal and vertical vibration of the rotor system in horizontal and vertical directions, respectively. Compared with the total stiffness of the rotor shaft, the stiffness of BNES is very weak and will not change the natural frequency of rotor system. The buckled beam parameter model is shown in Fig. 3, and the specific inner and outer buckled beam parameters are shown in Tables 2.1 and 2.2. The BNES connection stiffness is a kind of bi-stable stiffness, which contains linear stiffness and nonlinear stiffness. The vertical bi-stable stiffness is provided by outer long straight beam and inner short straight beams, and the horizontal bistable stiffness is provided by outer short straight beam and inner long straight beams. (a) Linear stiffness The linear stiffness is provided by the short straight beams (Fig. 3), so the linear stiffness can be calculated as 3

k¼

Edd hd

ð1Þ

3 4ld

BNES Bearing Motor Disc

Shaft

Fig. 1. Structure of rotor-BNES system.

H. Yao et al. / Mechanical Systems and Signal Processing 138 (2020) 106546

ybnes

Outer buckled beam

Bearing

3

Supporting part

Bearing support

xbnes

O

Inner left buckled beam

Inner right buckled beam

Fig. 2. Specific structure of BNES and buckled beams.

lc dc hc

Long straight beam

Short straight beam ld hd

dd

Fig. 3. Buckled beam parameter model.

According to the parameters in Tables 2.1 and 2.2, the linear force generated by outer buckled beam and inner buckled beams can be respectively expressed as

f bx ðxÞ ¼ 2470x

ð2Þ

f ay ðyÞ ¼ 4320y

ð3Þ

where, f bx represents the outer buckled beam force in the horizontal direction. f ay represents the inner buckled beam force in the vertical direction. x represents the horizontal displacement of outer short straight beam. y represents the vertical displacement of inner short straight beams. All the force and displacement units of Eqs. (2) to (9) are N/m and m, respectively. (b) Nonlinear stiffness The nonlinear stiffness is provided by two long straight beams, and the specific ANSYS buckled beam model can be established as shown in Fig. 4. The mechanical characteristic analysis of buckled beams is a kind of buckling analysis. The main purpose is to solve the relationship of the end load F and displacement UY. The specific analysis procedure is shown in Fig. 5. According to the material and dimension parameters of the inner and outer buckled beams shown in Tables 2.1 and 2.2, the nonlinear buckling analysis is carried out, and the relationship between the nonlinear force and the vibration amplitude of the inner and outer long straight beams are shown in Fig. 6(a) and (b), respectively. Polynomial fitting the ANSYS results of Fig. 6(a) and (b), the relationship between the nonlinear force and the vibration amplitude of the inner and outer long straight beams are respectively expressed as

FðUYÞ ¼ 2:066  109 UY 3  1:122  107 UY 2 þ 1:645  104 UY

ð4Þ

FðUYÞ ¼ 3:793  109 UY 3 þ 2:057  107 UY 2 þ 3:015  104 UY

ð5Þ

(c) Bi-stable stiffness All the buckled beams have a pre-compression (UY ¼ 4:2 mm ) before installation, so the inner long straight beams and outer long straight beams will produce horizontal preload force (F 1 ¼ 7:3 N ) and vertical preload force (F 2 ¼ 13:4 N ), respectively. According to the Eq. (2), Eq. (4) and horizontal preload force, the horizontal direction relationship between force and amplitude of inner and outer buckled beams can be respectively expressed as

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H. Yao et al. / Mechanical Systems and Signal Processing 138 (2020) 106546 Table 2.1 Inner buckled beam model parameters. Parameter

Value

Parameter

Value

lc1 hc1 dd1 E1

30 mm 1 mm 3 mm 3.5 Gpa

dc1 ld1 hd1

3 mm 10.7 mm 1 mm 4°

a1

Table 2.2 Outer buckled beam model parameters. Parameter

Value

Parameter

Value

lc2 hc2 dd2 E2

42 mm 1 mm 25 mm 3.5 Gpa

dc2 ld2 hd2

60 mm 31 mm 1.5 mm 3°

a2

UY F

Fig. 4. ANSYS buckled beam model.

Unit Type Dimension Definition

Material Parameters Buckled Beam Model

Mesh Generation

Constraint Addition

Unit load

Open Prestressing Effect Static Analysis Modal Extension

Block Lanczo Method

Eigenvalue Buckling Analysis Obtain First-Order Critical Load Arc Length Method

Introduce Initial Defect Open Large Deformation

Nonlinear Buckling Analysis

Caculate Load-Displacement Relationship Fig. 5. ANSYS analysis procedure.

f ax ðxbnes Þ ¼ 2:066  109 x3bnes þ 5:022  106 x2bnes þ 210xbnes  7:3

ð6Þ

f bx ðxbnes Þ ¼ 2470xbnes þ 7:3

ð7Þ

where, f ax represents the inner buckled beam force in the horizontal direction. xbnes represents the vibration amplitude of the BNES in horizontal direction.

H. Yao et al. / Mechanical Systems and Signal Processing 138 (2020) 106546

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Fig. 6. ANSYS analysis results.

The variation of the BNES horizontal force and vibration amplitude is shown in Fig. 7. The inner buckled beams provide a kind of nonlinear force in Fig. 7(a), and the outer buckled beam provides the linear force in Fig. 7(b). Again, according to the Eq. (3), Eq. (5) and vertical preload force, the vertical direction relationship between force and amplitude of inner and outer buckled beams can be respectively expressed as

f ay ðybnes Þ ¼ 4320ybnes  13:4

ð8Þ

f by ðybnes Þ ¼ 3:793  109 y3bnes  9:191  106 y2bnes þ 389ybnes þ 13:4

ð9Þ

where, f by represents the inner buckled beam force in the vertical direction. ybnes represents the vibration amplitude of the BNES in vertical direction. The variation of the BNES vertical force and vibration amplitude is shown in Fig. 8. The inner buckled beams provide a kind of linear force in Fig. 8(a), and the outer buckled beam provides the nonlinear force in Fig. 8(b). According to the above analysis, combining the nonlinear force and linear force of the compressed BNES, the relationship between the BNES bi-stable force and its vibration amplitude in the horizontal and vertical directions can be respectively expressed as

F x ðxbnes Þ ¼ f ax ðxbnes Þ þ f bx ðxbnes Þ

ð10Þ

F y ðybnes Þ ¼ f ay ðybnes Þ þ f by ðybnes Þ

ð11Þ

The BNES bi-stable force has three points of zero force, that is, BNES has three equilibrium positions. Among them, A, C (Fig. 9(a)) are stable equilibrium positions in the horizontal direction, D, F (Fig. 9(b)) are stable equilibrium positions in the vertical direction, and B, E are critical equilibrium positions.

Fig. 7. Relationship between force and amplitude of inner and outer buckled beams in the horizontal direction.

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Fig. 8. Relationship between force and amplitude of inner and outer buckled beams in the vertical direction.

Fig. 9. Bi-stable force of the BNES in horizontal and vertical directions.

2.3. The rotor-BNES system dynamic model The dynamic model of rotor-BNES system is composed of the BNES lumped mass model and the rotor system finite element model. The BNES model is a lumped mass model. The buckled beam is extremely light in weight and only provides horizontal and vertical stiffness. At the same time, the BNES mass has a large density and stiffness. Therefore, using the lumped mass method, the BNES dynamic models in horizontal and vertical directions are shown in Fig. 10(a) and (b), respectively. According to the above BNES dynamic model, the BNES dynamic differential equations are written as

Fig. 10. Dynamic model of BNES in horizontal and vertical directions.

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H. Yao et al. / Mechanical Systems and Signal Processing 138 (2020) 106546




mbnes €xbnes þ F x ðxbnes Þ þ cax x_ bnes þ cbx x_ bnes  F xx þ F 1 ¼ 0 €bnes þ F y ðybnes Þ þ cay y_ bnes þ cby y_ bnes  F yy þ F 2 ¼ 0 mbnes y

ð12Þ

where, mbnes represents the BNES mass, xbnes and ybnes represent the BNES vibration amplitude in the horizontal and vertical direction, respectively. cax represents the horizontal direction connection damping between the BNES and the main system. cbx represents the horizontal direction connection damping between the BNES and the supporting part. F xx and F yy represent the rotor-shaft-BNES interaction forces in the horizontal and vertical direction, respectively. cay represents the vertical direction connection damping between the BNES and the main system. cby represents the vertical direction connection damping between the BNES and the supporting part. The rotor system model is a finite element model. The rotor system is divided into a series of shaft segments, which are expressed in length, inner diameter, outer diameter and density. The bearings of rotor system are simplified to some certain stiffness. Ignoring the other influences of the rotor system, the rotor system finite element model is established. Combining the BNES lumped mass model and the rotor system finite element model together, the rotor-BNES model is shown in Fig. 11. As shown in Fig. 12, ignoring the axial deformation, each segment has two nodes, and each node has 4 degrees of freedom. These degrees of freedom are the horizontal displacement, the vertical displacement, and the angular displacements around axes perpendicular to the element rotation axis, respectively. According to literature [37] and Bernoulli-Euler bending theory, the segments are connected to each other at the node, and the dynamic differential equation of the rotor-BNES system can be expressed as

€ þ Du_ þ xGv_ þ Ku ¼ F Mu

ð13Þ

where, M, D, K and G are the mass, damping, stiffness and gyroscopic matrixes of the rotor-BNES system, respectively. u and

v are the vibration response vector. F is the external excitation vector.

The matrixes of the rotor-BNES system are composed of rotor segment unit’s matrix and BNES matrix. The specific segment unit’s matrix expression forms of M, D, K and Gare shown in Appendix A, and the results of M, D, K and G are as follows. The mass matrix M (Fig. 13) assembled by mass units is a symmetrical matrix. The gyroscopic matrix G (Fig. 14) assembled by gyroscopic units is a symmetrical matrix. The stiffness matrix K (Fig. 15) assembled by stiffness units is a non-symmetrical matrix. According to the Rayleigh damping calculation formula, the damping matrix of the rotor-BNES system can be obtained

y9 , k yy

9x

kby

mbnes 2

3

4

5

6

7

9 10

8

19

kay 13 14 15 16 17

11

1

12

mbnes

k xx

kax

19

x9 ,

kbx 9y

Fig. 11. Dynamic model of rotor-BNES.

Fig. 12. Finite element model of shaft segment.

k yy 18

k xx

8

H. Yao et al. / Mechanical Systems and Signal Processing 138 (2020) 106546

Fig. 13. Mass matrix M.

Fig. 14. Gyroscopic matrix G.

Fig. 15. Stiffness matrix K.

H. Yao et al. / Mechanical Systems and Signal Processing 138 (2020) 106546

D ¼ aM þ bK

9

ð14Þ

The BNES stiffness is different in horizontal and vertical direction, so it is necessary to solve the vibration response in horizontal and vertical direction, respectively. (a) Horizontal vibration The unbalance force is decomposed to two adjacent nodes, and each node has a half unbalance force. So, the horizontal excitation vector F caused by the eccentric mass is expressed as

2 16 3T 18 zfflfflffl}|fflfflffl{ 1 zfflfflffl}|fflfflffl{ 1 2 2 F ¼ 40    0; mp ex sinðxt Þ; 0; mp ex sinðxt Þ; 0    05 2 2

ð15Þ

where, mp x and e represent the eccentric mass, the speed of the rotor and the eccentricity, respectively. And the horizontal vibration response vectors are respectively expressed as

 T u ¼ x1 ; hy1 ;    ; x18 ; hy18 ; x19

ð16Þ

v ¼ ½y1 ; hx1 ;    ; y18 ; hx18 ; y19 T

ð17Þ

(b) Vertical vibration At the same time, the vertical excitation vector F caused by the eccentric mass is expressed as

2 16 3T 18 zfflfflffl}|fflfflffl{ 1 zfflfflffl}|fflfflffl{ 1 2 2 4 F ¼ 0    0; mp ex cosðxt Þ; 0; mp ex cosðxt Þ; 0    05 2 2

ð18Þ

And the vertical vibration response vectors are respectively expressed as

u ¼ ½y1 ; hx1 ;    ; y18 ; hx18 ; y19 T

v¼



x1 ; hy1 ;    ; x18 ; hy18 ; x19

ð19Þ

T

ð20Þ

3. Numerical simulation and analysis 3.1. Parameter definition The rotor-BNES system parameters are composed of two parts: BNES parameters and rotor system parameters. The BNES buckled beams use the common 3D printed material. The structural parameters of the buckled beam are shown in Tables 2.1 and 2.2, and the other BNES parameters are shown in Table 3.1. The rotor system parameters in this paper are from the Bently rotor test rig. The total rotor length, rotor diameter, elastic modulus, disk diameter and disk thickness are L, D, E0 , D0 and H0 , respectively. The specific rotor system parameters are shown in Table 3.2. Table 3.3 shows the dimensions of each rotor system shaft segment. The rotor system is divided into 17 shaft segments, and l and r are the shaft length and the shaft radius, respectively. 3.2. Transient vibration suppression analysis of the rotor-BNES system In this section, different transient energies will be applied to the rotor-BNES system. At the same time, the responses of the rotor system with and without BNES will be compared to confirm the vibration absorption capacity of the BNES. (1) A 30 N horizontal pulse force is applied at the ninth node Under the action of the 30 N pulse force in the horizontal direction, the initial displacement of the rotor system is about 0.7 mm. Using the numerical calculation method, the horizontal transient response of the rotor system without and with BNES can be obtained.

Table 3.1 Other BNES parameters. Parameter

Value

Parameter

Value

mbnes bbnes cbx cby

0.1 kg 0 1 Ns/m 1 Ns/m

abnes

3 1 Ns/m 1 Ns/m

cax cay

10

H. Yao et al. / Mechanical Systems and Signal Processing 138 (2020) 106546 Table 3.2 Rotor system parameters. Parameter

Value

Parameter

Value

L E3 H0 kxx

420 mm 210 Gpa 20 mm 1  106N/m 3

D D0

10 mm 80 mm 40 mm 1  106N/m 0

a

e

kyy b

Table 3.3 Dimensions of each shaft section of the rotor system. Number

1

2

3

4

5

6

7

8

9

l/mm r/mm

60 5

20 5

20 5

20 5

20 5

20 5

20 5

20 5

20 40

Number

10

11

12

13

14

15

16

17

l/mm r/mm

20 5

20 5

20 5

20 5

20 5

20 5

20 5

60 5

As is shown in Fig. 16, the rotor system without BNES performs a free-attenuation motion, and the initial input energy is dissipated by its own damping. The vibration amplitude attenuation speed is slow, and it takes about 0.91 s to decay to 0.2 mm. Adding the BNES, the vibration amplitude attenuation of the rotor system (Fig. 17(a)) is extremely fast. It takes about 0.52 s to decay to 0.2 mm, and the attenuation speed is 2.06 times that of the rotor system without BNES. As is shown in Fig. 17(a) and (b), during the vibration attenuation process, the BNES mainly exhibits two states: (a) Within 0–0.52 s, the BNES performs sink-to-sink motion and consumes energy between two adjacent stable equilibrium points. When the energy is large, the snap-through motion [38,39] is fast to consume energy; When the energy is small, the vibration is slowly consumed at a single stable equilibrium point; (b) Within 0.52–1.5 s, the BNES performs in-sink motion. At this time, the rotor system has only a little energy, which cannot make the BNES trigger the snap-through motion. The energy is dissipated at a single stable equilibrium point and the BNES eventually stays at the stable equilibrium point C. (2) A 30 N vertical pulse force is applied at the ninth node The Bently rotor system is isotropic, so the rotor system without the BENS has the same vibration attenuation in the vertical direction as the horizontal direction. The vertical transient response of the rotor system without BNES is shown in Fig. 19. Adding the BNES, the amplitude of the rotor system (Fig. 18 (a)) attenuates fast, the rotor system takes about 0.46 s to decay to 0.2 mm, and the attenuation speed is 2.06 times that of the rotor system without BNES. As is shown in Fig. 18(b), the BNES only has sink-to-sink motion (0–0.13 s) and in-sink motion (0.13–1.5 s), and the BNES eventually stay at the stable equilibrium point F. Comparing Figs. 17 and 18, the BNES sink-to-sink motion has more obvious vibration suppression effect than in-sink motion, so the sink-to-sink motion is the main energy attenuation mode under this excitation force. (3) A 45 N horizontal pulse force is applied at the ninth node

1.0

Amplitude /( mm)

rotor system without BNES

0.5 0.91,0.2

0.0

-0.5

-1.0 0.0

0.5

Time /(s )

1.0

1.5

Fig. 16. Transient response curve of the rotor system under a 30 N horizontal pulse force.

H. Yao et al. / Mechanical Systems and Signal Processing 138 (2020) 106546

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Fig. 17. Transient response curves of the rotor-BNES system under a 30 N horizontal pulse force.

Fig. 18. Transient response curves of the rotor-BNES system under a 30 N vertical pulse force.

In order to verify that the BNES can absorb different impact energy, assuming the rotor is given a 45 N horizontal pulse force, the rotor system will produce a middle vibration amplitude about 1.16 mm. As is shown in Fig. 19, the vibration amplitude of the rotor system without BNES continues to be slow, and it takes about 1.17 s to decay to 0.2 mm. Adding the BNES, the rotor system amplitude (Fig. 20(a)) experiences 0.41 s attenuate to about 0.2 mm. Comparing with small pulse force, the attenuation speed has an obvious improvement, and the attenuation speed is 2.85 times that of the rotor system without BNES. As is shown in Fig. 20(b), the BNES amplitude is larger than small horizontal pulse force and continuously traverses multiple equilibrium positions. This indicates that more energy is transferred to the BNES and consumed in the vibration. Under this pulse force, the BNES vibration attenuation process mainly exhibits three states: (a) Within 0–0.16 s, the BNES performs complete snap-through motion. Its specific perform is completely cross two stable equilibrium points, and does not oscillate at any stable equilibrium point; (b) Within 0.16–0.53 s, the BNES performs sink-to-sink motion and consumes energy between two adjacent stable equilibrium points; (c) Within 0.53–1.5 s, the BNES performs in-sink motion and dissipates the remaining energy slowly at a single stable equilibrium point. Under this instantaneous excitation, the BNES eventually stays at the stable equilibrium point A, which is due to the fact that there are two stable equilibrium positions in the rotorBNES system, and the BNES will eventually settle to an equilibrium position based on the input energy. (4) A 45 N vertical pulse force is applied at the ninth node The vertical transient response of the rotor system without BNES is the same as the Fig. 21. As is shown in Fig. 21(a), the BNES still has obvious vibration absorption effect on the rotor system. The rotor system amplitude experiences 0.42 s attenuate to about 0.2 mm, and the attenuation speed is 2.06 times that of the rotor system without BNES. Under this pulse force, the attenuation speed of the rotor-BNES system is better than small pulse force condition’s attenuation. As is shown in Fig. 21(b), the BNES has complete snap-through motion (0–0.13 s), sink-to-sink motion (0.13–0.4 s) and insink motion (0.4–1.5 s), and the BNES eventually stay at the stable equilibrium point F.

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H. Yao et al. / Mechanical Systems and Signal Processing 138 (2020) 106546

1.4

Amplitude /(mm)

rotor system without BNES

0.7 1.17 ,0.2

0.0

-0.7

-1.4 0.0

0.5

1.0

1.5

Time /(s ) Fig. 19. Transient response curve of the rotor system under a 45 N horizontal pulse force.

Fig. 20. Transient response curves of the rotor-BNES system under a 45 N horizontal pulse force.

Fig. 21. Transient response curves of the rotor-BNES system under a 45 N vertical pulse force.

Comparing Fig. 17, Fig. 18, Fig. 20 and Fig. 21, the increase of external transient force triggers the generation of complete snap-through motion. At the same time, the BNES’s complete snap-through motion has more obvious vibration suppression effect than sink-to-sink motion, which is the main reason for the faster decay speed of the rotor system. (5) A 60 N horizontal pulse force is applied at the ninth node

H. Yao et al. / Mechanical Systems and Signal Processing 138 (2020) 106546

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In order to investigate the relationship between the complete snap-through motion attenuation time and external excitation, the transient excitation in the horizontal direction continues to increase. Under the condition that other parameters are constant, a 60 N pluse force is given to the rotor horizontally. As is shown Fig. 22, this pulse force causes the rotor system generate a displacement of about 1.5 mm, and the rotor system amplitude takes about 1.35 s to decay to 0.2 mm. Adding the BNES, the rotor system (Fig. 23(a)) only needs 0.37 s attenuate to about 0.2 mm, and the attenuation speed is 3.64 times that of the rotor system without BNES. Under the action of this excitation energy, the vibration absorption effect of BNES is further enhanced, and BNES is eventually stays at the stable equilibrium point A. As is shown in Fig. 23(b), the BNES still has three motion states. The BNES performs a complete snap-through motion (0– 0.2 s), completes the sink-to-sink motion (0.2–0.42 s), and performs in-sink motion (0.42–1.5 s). (6) A 60 N vertical pulse force is applied at the ninth node The vertical transient response of the rotor system without BNES is the same as the Fig. 22. As is shown in Fig. 24(a), the BNES continues to enhance the vibration absorption effect of the rotor system. The rotor system experiences 0.21 s attenuate to about 0.2 mm, and the attenuation speed is 6.43 times that of the rotor system without BNES. As is shown in Fig. 24(b), the BNES has almost no sink-to-sink motion, and the main energy of the BNES is consumed by complete snap-through motion in 0–0.31 s. Finally, the BNES stays at equilibrium point D. Comparing Fig. 20, Fig. 21, Fig. 23 and Fig. 24, the large pulse force will cause the complete snap-through movement time obviously increased, the sink-to-sink movement time significantly reduced and the overall vibration suppression effect more obvious. Therefore, as the external energy increases, the complete snap-through motion time is longer, and the BNES has a more effective vibration suppression effect. During the attenuation of the rotor system, the more times of the BNES generates the snap-through motion, the better vibration suppression effect of the BNES will be generated. In summary, in a certain horizontal or vertical range of impact energy, the BNES has a strong vibration suppression effect on the rotor system with different initial energies. Adding the BNES, the rotor system has an extremely fast amplitude attenuation, which is more than double times that of the rotor system without BNES; The working BNES mainly presents three

1.8 rotor system without BNES

Amplitude/(mm)

1.2

0.6

1.35,0.2

0.0

-0.6 -1.2

-1.8 0.0

0.5

Time /(s)

1.0

1.5

Fig. 22. Transient response curve of the rotor system under a 60 N horizontal pulse force.

Fig. 23. Transient response curves of the rotor-BNES system under a 60 N horizontal pulse force.

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H. Yao et al. / Mechanical Systems and Signal Processing 138 (2020) 106546

Fig. 24. Transient response curves of the rotor-BNES system under a 60 N vertical pulse force.

kinds of working states: complete snap-through motion (large energy), sink-to-sink motion (middle energy) and in-sink motion (small energy). Among the three BNES working states, the complete snap-through motion has the best vibration suppression effect; Under the same BNES and different initial energies, the BNES has a strong inhibitory effect on the amplitude of the main system regardless of the final BNES equilibrium position. The final stable equilibrium position does not affect the vibration suppression mode and effect of the BNES on the rotor system. 3.3. Steady-state vibration suppression analysis of the rotor-BNES system The common rotor systems need to experience different vibration frequencies during operation. As shown in Fig. 25, the lines (A, B, C and D) represent the relationship of characteristic frequency and rotating speed, and the line E is used to solve the natural frequency of rotor system. As the disk is installed approximately in the middle of the span of the main bearings, the first order frequency is less affected by the rotating speed and whirling direction. So, within the operating range of the rotor (0~150 Hz), the first order forward whirling natural frequency xF1 and backward whirling natural frequency xB1 are both about 46.5 Hz. In this section, different eccentric mass will be applied to the rotor-BNES system, and the responses of the rotor system with and without BNES will be compared in the frequency domain. (1) Eccentric mass is mp ¼ 2  104 kg Under the eccentric action of the rotor disc, the rotor system will generate the same periodic force in the vertical and horizontal directions. As is shown in Fig. 26, the rotor system without BNES has a 46.5 Hz natural frequency and a 0.82 mm maximum amplitude. As is shown in Fig. 27(a), the horizontal rotor-BNES system exhibits strong modulated response (SMR) [40] phenomenon in the 45.0–47.8 Hz frequency range. Adding the BNES, the overall suppression effect of the rotor system is obvious, and the maximum amplitude is 0.523 mm. So, its vibration suppression rate is more than 36.2%. At the same time, the maximum amplitude of the BNES is 2.41 mm, which is much larger than that of the rotor system in the resonance region, indicating that the vibration energy of the rotor system is transmitted to the BNES.

400 A Frequency/(Hz)

F2

300

B

B2

2

200

C D

100

,

F1

E

B1

0

0

100

300 200 Rotating speed/(r/s)

Fig. 25. Campbell diagram of rotor system.

400

H. Yao et al. / Mechanical Systems and Signal Processing 138 (2020) 106546

15

1.0 rotor system without BNES

Amplitude /( mm)

0.8

0.6 0.4 0.2

0.0 40

42

44 46 48 Frequency/(Hz)

50

52

Fig. 26. Frequency response curve of the rotor system under a weak stable periodic exciting force.

Fig. 27. Frequency response curves of the rotor-BNES system under a weak stable periodic exciting force.

As is shown in Fig. 27(b), the vertical rotor-BNES system also exhibit SMR phenomenon in the frequency range 47.7~49.1 Hz. The rotor system maximum amplitude is 0.47 mm, and its vibration suppression rate is more than 42.6%. (2) Eccentric mass is mp ¼ 2:5  104 kg In order to analyze the influence of the exciting force on the vibration suppression effect and the SMR region, the eccentric mass is increased. As shown in Fig. 28, the natural frequency of the rotor system without BNES is still around 46.5 Hz, and the maximum amplitude is about 1.02 mm. Adding the BNES, there are SMR region (Fig. 29(a)) between the horizontal rotor system and the BNES in the range of 44.6~47.9 Hz, and a wider suppression band than that of Fig. 27(a). The maximum amplitude of the horizontal rotor system is 0.58 mm. Compared with the rotor system without BNES, it is reduced by 43.1%, and it has stronger vibration suppression ability than small energy. At the same time, the BNES amplitude maximum is also increased to 2.53 mm, and the amplitude in resonance region is bigger than the BNES amplitude in Fig. 27(a). So, as the excitation force increases, more energy is transferred to the BNES. As is shown in Fig. 29(b), the vertical rotor-BNES system also exhibit SMR phenomenon in the frequency range 47.5~49.5 Hz, which is wider than that of Fig. 27(b). The rotor system maximum amplitude is 0.58 mm, and its vibration suppression rate is more than 43.1%. (3) Eccentric mass is mp ¼ 3  104 kg In order to further analyze the influence of the exciting force on the vibration suppression effect and SMR region, the eccentric mass is continuously increased. As shown in Fig. 30, the maximum amplitude of the rotor system without BNES is 1.22 mm. As is shown in Fig. 31(a), the rotor system and BNES exhibit SMR phenomenon in the frequency range of 44.4~48.4 Hz. Adding the BNES, the maximum amplitude of the horizontal rotor system is 0.67 mm, and the vibration suppression rate is increased to 45.1%. So, the BNES still has obvious vibration suppression effect for large vibration energy, and the vibration suppression rate will increase with the increase of energy.

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H. Yao et al. / Mechanical Systems and Signal Processing 138 (2020) 106546

1.2

Amplitude /( mm)

rotor system without BNES

0.9

0.6

0.3

0.0 40

42

44 46 48 Frequency/(Hz)

50

52

Fig. 28. Frequency response curve of the rotor system under a middle stable periodic exciting force.

Fig. 29. Frequency response curves of the rotor-BNES system under a middle stable periodic exciting force.

1.5 rotor system without BNES

Amplitude /(mm)

1.2

0.9 0.6 0.3 0.0 40

42

44 46 48 Frequency/(Hz)

50

52

Fig. 30. Frequency response curve of the rotor system under a large stable periodic exciting force.

As is shown in Fig. 31(b), the vertical rotor-BNES system also exhibit SMR phenomenon in the frequency range 47.2~49.8 Hz, which is wider than that of Fig. 29(b). The rotor system maximum amplitude is 0.69 mm, and its vibration suppression rate is more than 43.4%. Under the action of large periodic force, the time responses of the rotor-BNES system in horizontal and vertical directions are solved, and the specific results are shown in Figs. 32 and 33, respectively.

H. Yao et al. / Mechanical Systems and Signal Processing 138 (2020) 106546

Fig. 31. Frequency response curves of the rotor-BNES system under a large stable periodic exciting force.

Fig. 32. Horizontal direction time response curves of the rotor-BNES system under a large stable periodic exciting force.

Fig. 33. Vertical direction time response curves of the rotor-BNES system under a large stable periodic exciting force.

17

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H. Yao et al. / Mechanical Systems and Signal Processing 138 (2020) 106546

As is shown in Fig. 32(a), the horizontal rotor system exhibits a strong nonlinear beat phenomenon at 47 Hz. The BNES in Fig. 32(b) has different phenomenon from the ordinary NES. When the energy is small, the BNES will vibrate around an equilibrium position (A or C). When the energy is large, the BNES will vibrate through two equilibrium positions (A and C) to complete the snap-through motion. As shown in Fig. 33(a), the vertical rotor system also has a strong nonlinear beat phenomenon at 49 Hz; As shown in Fig. 33(b), the BNES performs the same sink-to-sink motion in the vertical direction as the horizontal direction. In summary, the BNES has a significant suppression effect on the entire frequency domain, especially in the resonance region; In a certain range of exciting energy, the small energy can also trigger BNES. With the increase of the energy, the nonlinear phenomenon in the frequency domain becomes more and more obvious, the vibration absorption effect of BNES will continue to increase, and the vibration suppression frequency band will also become wider; Under the action of periodic force and BNES, the time responses of the rotor system in the SMR region is mainly nonlinear beat phenomenon, and the energy is mainly consumed by the BNES sink-to-sink motion.

3.4. Comparison between BNES and linear visco-elastic damper In this section, the same transient energies as the rotor with the BNES will be applied to the rotor with linear visco-elastic damper, and the transient vibration suppression effect of the BNES and linear visco-elastic damper will be compared to confirm the vibration absorption advantages of the BNES. In general, the linear visco-elastic damper provides a large stiffness and large damping (Fig. 34). Under suitable stiffness and damping conditions, the linear visco-elastic damper can achieve the same maximum amplitude of vibration suppression as BNES in frequency domain, but the transient attenuation speed is much lower than that of BNES. In order to ensure the comparability of the BNES and linear visco-elastic damper, the rotor system is selected the same parameters. Among them, the eccentric mass is mp ¼ 2:5  104 kg. For the parameter of linear visco-elastic damper, the mass and stiffness are selected as ma ¼ 0:1 kg and kax ¼ kay ¼ 1500 N=m, respectively. According to the stiffness of linear visco-elastic damper, the damping of linear viscoelastic damper can be calculated as

cax ¼ kax 

1 2x

ð21Þ

cay ¼ kay 

1 2x

ð22Þ

After adding the linear visco-elastic damper, the maximum amplitude of the rotor system is 0.58 mm, as shown in Fig. 35. So, under the above parameters of linear visco-elastic damper, the linear visco-elastic damper could achieve the same vibration suppression effect as BNES (Fig. 29) in frequency domain. Assuming the rotor is given a 30 N, 45 N or 60 N pulse force in the horizontal or vertical direction, it will produce a 0.7 mm, 1.16 mm or 1.5 mm initial amplitude in x or y direction at 0 s. After adding the linear visco-elastic damper, the rotor system vibration amplitude in Fig. 36(a), (b) and (c) experience about 0.52 s, 0.64 s and 0.73 s to decrease to 0.2 mm, respectively. Comparing with the linear visco-elastic damper, the BNES has a more obvious suppression effect. The horizontal rotor system with the BNES in Fig. 17(a), Fig. 20(a) and Fig. 23(a) experience about 0.44 s, 0.41 s and 0.37 s to 0.3 mm, respectively. The attenuation speeds are 1.18 times, 1.56 times and 1.97 times as fast as that of the rotor system with the linear viscoelastic damper, respectively. At the same time, the vertical rotor system with the BNES in Fig. 18(a), Fig. 21(a) and Fig. 24 (a) experience about 0.46 s, 0.42 s and 0.21 s to 0.3 mm, respectively. The attenuation speeds are 1.13 times, 1.52 times and 3.48 times as fast as that of the rotor system with the linear visco-elastic damper, respectively. As the impact energy increases, the BNES vibration suppression effect becomes stronger, and the linear visco-elastic damper vibration suppression effect becomes weaker. So, under the condition that the maximum amplitude of vibration in frequency domain is the same, the transient attenuation speed of linear visco-elastic damper is much lower than that of BNES.

Fig. 34. Schematic model of the rotor with the linear visco-elastic damper.

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H. Yao et al. / Mechanical Systems and Signal Processing 138 (2020) 106546

0.8

Amplitude /(mm)

rotor with linear visco damper

0.6

0.4

0.2

0.0 40

42

44 46 48 Frequency/(Hz)

50

52

Fig. 35. Frequency response curves of the rotor with linear visco-elastic damper.

1.0

Amplitude /(mm)

rotor with linear visco damper

0.5 (0.52,0.2)

0.0

-0.5

-1.0 0.0

0.5

1.0

1.5

Time /(s)

(a) A 30 N pulse force 1.4

1.8 rotor with linear visco damper

rotor with linear visco damper

0.7

Amplitude /(mm)

Amplitude /(mm)

1.2

(0.64,0.2) 0.0

-0.7

0.6

(0.73,0.2)

0.0 -0.6 -1.2

-1.4 0.0

0.5

1.0 Time /(s)

( b ) A 4 5 N p ul s e f o rc e

1.5

-1.8 0.0

0.5

1.0

1.5

Time /(s)

( c ) A 6 0 N p ul se fo rc e

Fig. 36. Transient responses of the rotor system with the linear visco-elastic damper under different pulse forces.

4. Experiments 4.1. Experimental setup The Bently rotor test rig is used to verify the vibration suppression effect of the BNES on the response vibration. In this experiment, the BNES is mounted on the single-disc rotor system, as shown in Fig. 37(a). The experimental devices are consisted of a single disc rotor system, a BNES, two eddy current sensors, a photoelectric encoder, and a set of national instrument (NI) acquisition device.

20

H. Yao et al. / Mechanical Systems and Signal Processing 138 (2020) 106546

Fig. 37. Experimental devices.

According to the parameters in Table 2.1 and Table 2.2, the BNES (Fig. 37(b)) is designed. At the same time, the photoelectric system, the eddy current sensor and NI acquisition card are used to test the vibration frequency, test rotor system vibration displacement and complete the data acquisition, respectively.

4.2. Experimental results In this section, different periodic forces are applied to the rotor-BNES system, and the rotor system experimental results with and without BNES will be compared in the frequency domain. (1) Eccentric mass is mp ¼ 2:5  104 kg According to the simulation parameters, the BNES vibration suppression effects in the vertical and horizontal directions are explored. As shown in Fig. 38, the rotor system without BNES generates a 1.05 mm maximum amplitude at 46.5 Hz. Adding the BNES, the horizontal and vertical experimental results of the rotor system in Fig. 39(a) and (b) have the same trends as the simulation curves, but the experimental suppression rates is a little lower than the simulation results. The maximum vibration amplitudes of the two directions are 0.614 mm and 0.629 mm, respectively, so the vibration suppression rates reach 41.5% and 40.1%, respectively. (2) Eccentric mass is mp ¼ 3:0  104 kg When the eccentric mass increases, the eccentric disk will generate a stronger periodic force. The rotor system vibration suppression effects of the BNES in the vertical and horizontal directions are obtained. The rotor system (Fig. 40) without BNES generates a 1.23 mm maximum amplitude at 46.5 Hz. Adding the BNES, the horizontal and vertical experimental results of the rotor system in Fig. 41(a) and (b) have the same trends as the simulation curves. The maximum vibration amplitudes of the two directions are 0.654 mm and 0.681 mm, respectively, so the vibration suppression rates reach 46.8% and 44.6%, respectively. Under such exciting force, the vibration suppression effect is better than the simulation results.

1.2

Amplitude/(mm)

Experimental results

0.9

0.6

0.3

0.0 35

40

45 50 Frequency/(Hz)

55

60

Fig. 38. Experimental frequency response curve of the rotor system without BNES under a middle stable periodic exciting force.

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H. Yao et al. / Mechanical Systems and Signal Processing 138 (2020) 106546

0.9

0.8

Experimental results Simulation curve

0.6

Amplitude/(mm)

Amplitude/(mm)

Experimental results Simulation curve

0.4

0.2

0.0 42

44

46 48 50 Frequency/(Hz)

52

0.3

0.0 42

54

(a) Horizontal direction frequency response curves

0.6

44

46 48 50 Frequency/(Hz)

52

54

(b) Vertical direction frequency response curves

Fig. 39. Experiment and simulation frequency response curves of the rotor system with BNES under a middle stable periodic exciting force.

1.6

Amplitude /(mm)

Experimental results

1.2

0.8

0.4

0

35

40

45 50 Frequency/(Hz)

55

60

Fig. 40. Experimental frequency response curve of the rotor system without BNES under a large stable periodic exciting force.

0.9

0.9 Experimental results Simulation curve

Amplitude /(mm)

Amplitude /(mm)

Experimental results Simulation curve

0.6

0.3

0.0

0.6

0.3

0.0 42

44

46 48 50 Frequency/(Hz)

52

54

(a) Horizontal direction frequency response curves

42

44

46 48 50 Frequency/(Hz)

52

54

(b) Vertical direction frequency response curves

Fig. 41. Experiment and simulation frequency response curves of the rotor system with BNES under a large stable periodic exciting force.

As is shown in Fig. 41(a), the horizontal rotor system with BNES has SMR phenomenon in 44~50 Hz. At the same time, the vertical rotor system (Fig. 41(b)) with BNES also has very strong SMR regions in 46~48 Hz. Their specific 47 Hz horizontal time domain response and 49 Hz vertical time domain response are shown in Fig. 42(a) and (b), respectively.

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H. Yao et al. / Mechanical Systems and Signal Processing 138 (2020) 106546

Fig. 42. Experimental time response curves of the rotor system with BNES under a large stable periodic exciting force.

So, the steady-state experiments demonstrate that the BNES has a strong vibration suppression effect and realize the rotor system vibration suppression in a wide frequency domain. 5. Conclusions In this paper, a bi-stable buckled beam nonlinear energy sink applied to the rotor system is proposed. The principles and fundamental characteristics of the BNES are studied numerically and experimentally. The mainly results are as follows: (1) The designed buckled beam BNES has a strong vibration suppression effect on the rotor system and can withstand a wide range of energy. (2) With the increase of the external force, the vibration absorption effect of the BNES will be enhanced in a certain energy range. (3) The final stable equilibrium position of the BNES does not affect the vibration suppression effect of the BNES on the rotor system.

Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements The authors would like to gratefully acknowledge the National Natural Science Foundation of China (Grant No. U1708257) and the Fundamental Research Funds for the China Central Universities (Grant No. N180313009) for the financial support for this study. Appendix A For each segment unit, the mass matrix, damping matrix, stiffness matrix and gyroscopic matrix of rotor system can be represented by 2  2 block matrices, respectively. Assembling these segment unit matrixes and the BNES matrix, the total matrix of the rotor-BNES system can be obtained. Each segment unit’s mass matrix can be expressed as

2 Mð i Þ ¼

156

la l 6 6 22l

6 420 4 54

22l

54

4l

2

13l

13l

156

13l 3l

la ¼ pq

 2 d 2

2

22l

13l

3

2

36

3l

2 2 26 4l 3l 7 7 la r 6 3l 7þ 6 22l 5 120l 4 36 3l

4l

2

3l

l

2

36

3l

3

36

7 7 7 3l 5

3l

4l

3l

l

2

ðA1Þ

2

ðA2Þ

H. Yao et al. / Mechanical Systems and Signal Processing 138 (2020) 106546

23

where, la is the rotor shaft segment area density. q is the rotor shaft segment density. d is rotor shaft section diameter. l is the rotor shaft length. r is the rotor shaft section radius. Each segment unit’s stiffness matrix can be expressed as

2 KðiÞ ¼

12

6l

12

2

6l

4l EI 6 6 6l 3 6 l 4 12 6l 6l

2l

6l

2 2l 7 7 7¼ 6l 5

12

2

3

6l

4l

2

"

ð iÞ

K11 ð iÞ

K21

ð iÞ

K12

# ðA3Þ

ð iÞ

K22

where, E is the elastic modulus of the rotor system. I is the segment moment of inertia. The BNES matrix is different in horizontal and vertical direction, so the unit’s stiffness matrix of coupling BNES matrix and rotor segment unit’s stiffness matrix is different in horizontal and vertical direction. The specific coupled matrix in horizontal and vertical direction is as follows. (a) Horizontal direction The unit’s stiffness matrix of coupling BNES matrix and rotor segment unit’s stiffness matrix can be respectively expressed as

 K 1111 ¼ K 1818 ¼

ð11Þ

kxx

0

0

0 

ð12Þ

K 1212 ¼ K 22 þ K 11 þ  K ax2 ¼

kax2

ðA4Þ 0 0

kax1 0

ðA5Þ

ðA6Þ

0

K ax3 ¼ ½ K ax3

0

ðA7Þ

K 19 ¼ K ax4 þ kbx where, kbx ¼

ðA8Þ

df bx ðx19 Þ ; kax1 dx19

¼

@f ax ðx12 x19 Þ ; kax2 @x12

¼

@f ax ðx12 x19 Þ ; kax3 @x19

¼

x12 x19 Þ  @f ax ð@x ; kax4 12

¼

x12 x19 Þ  @f ax ð@x 19

(b) Vertical direction The unit’s stiffness matrix of coupling BNES matrix and rotor segment unit’s stiffness matrix can be respectively expressed as

 K 1919 ¼ K 3535 ¼

ð27Þ

kyy

0

0

0

ð28Þ

K 2929 ¼ K 22 þ K 11  K ay1 ¼

kay



ðA9Þ

kay

0

0

0

ðA10Þ

ðA11Þ

0

 K ay2 ¼ kay

0



ðA12Þ

K ay3 ¼ kay þ kby where, kay ¼

dyay ðy19 Þ ; kby dy19

ðA13Þ ¼

df by ðy19 Þ . dy19

Each segment unit’s gyroscopic matrix can be expressed as

3 36 3l 36 3l " ð iÞ 2 2 7 6 G11 la r 6 3l 4l 3l l 7 ¼ 7¼ 6 ð iÞ 60l 4 36 3l 36 3l 5 G21 2

ð iÞ

G

2

3l

l

2

3l

2

ð iÞ

G12 ð iÞ

G22

# ðA14Þ

4l

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