Theoretical modeling and experimental analysis of nonlinear electromagnetic shunt damping

Journal Pre-proof Theoretical modeling and experimental analysis of nonlinear electromagnetic shunt damping Bo Yan, Hongye Ma, Ning Yu, Lu Zhang, Chuanyu Wu PII:

S0022-460X(20)30015-8

DOI:

https://doi.org/10.1016/j.jsv.2020.115184

Reference:

YJSVI 115184

To appear in:

Journal of Sound and Vibration

Received Date: 4 July 2019 Revised Date:

16 December 2019

Accepted Date: 9 January 2020

Please cite this article as: B. Yan, H. Ma, N. Yu, L. Zhang, C. Wu, Theoretical modeling and experimental analysis of nonlinear electromagnetic shunt damping, Journal of Sound and Vibration (2020), doi: https://doi.org/10.1016/j.jsv.2020.115184. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier Ltd.

Theoretical modeling and experimental analysis of nonlinear electromagnetic shunt damping Bo Yana,b,c*, Hongye Maa,c, Ning Yua,c, Lu Zhangb, Chuanyu Wua,c a.

Faculty of Mechanical Engineering & Automation, Zhejiang Sci-Tech University,

Hangzhou, 310018, China b.

Key Laboratory of Space Utilization, Technology and Engineering Center for

space Utilization, Chinese Academy of Sciences. Beijing, 100094, China. c.

Zhejiang Main Laboratory of Planting Equipment Technology, Hangzhou,

310018, China

*Corresponding author. Faculty of Mechanical Engineering & Automation, Zhejiang Sci-Tech University Emails: [email protected] (B. Yan); [email protected] (H. Ma); [email protected]; [email protected] (L. Zhang); [email protected] (C. Wu)

Abstract Linear electromagnetic shunt damping (L-EMSD) has been investigated deeply for vibration control in previous studies. This paper proposes nonlinear electromagnetic shunt damping (N-EMSD) for vibration isolation enhancement of linear vibration isolators (LVIs), which has not been discussed in existing literature. N-EMSD is composed of a pair of the permanent magnets (PMs) and a pair of the coils, where the two coils are wound in opposite direction and connected in series. The nonlinear electromagnetic coupling coefficient is derived. The coupling governing equations of a LVI with N-EMSD are established and the amplitude-frequency relationship is theoretically derived using the harmonic balance method (HBM). Both the simulations and experiments are carried out to verify the nonlinear damping characteristic of N-EMSD. The results demonstrate that the LVI with N-EMSD can effectively reduce the vibration in the resonance region without affecting the vibration isolation performance in the isolation region compared with the traditional L-EMSD. It is also found in both simulation and experiment for the two

coils configuration that the frequency of the induced voltage is twice the frequency of the displacement. Furthermore, the transmissibility of the LVI with N-EMSD reduces with the increase of the input amplitude in the resonance region, which demonstrates the nonlinearity of N-EMSD. The natural frequency slightly decreases with the decrease of the peak transmissibility. This paper extends the electromagnetic shunt damping (EMSD) technique from linear to nonlinear fields and provides a guideline to design nonlinear damping. Keywords Nonlinear electromagnetic shunt damping; nonlinear damping; shunt damping; vibration isolation; negative resistance 1. Introduction Structural vibration is a common phenomenon existing in nature and our daily life. In many cases, unwanted vibrations can affect the running accuracy of machines. Shunt damping is one of the effective methods to attenuate vibrations [1, 2] and is investigated deeply since Forward [3] experimentally demonstrated the feasibility of utilizing external electronic circuits to control mechanical vibration. Thereafter, Hagood and Flotow [4] discussed the passive electrical circuit connected to piezoelectric material for vibration control. The results implied that the resistive shunt exhibits the viscous damping effect on piezoelectric material and shunting with a resistor and inductor introduces an electrical resonance, whose behavior is like a mechanical vibration absorber. Normally, a single resonance circuit connecting to the terminals of a piezoelectric transducer can suppress single mode vibration [5]. Therefore, an adaptive multi-mode resonant method by minimizing the relative phase difference between a vibration reference signal and the shunt current is proposed to control the multi-mode vibration [6]. If the external circuit is connected to the terminals of an electromagnetic transducer, it will be the electromagnetic shunt damping method [7]. Inoue et al. [8] and Cheng et al. [9] successfully investigated the single mode and the multi-mode vibration suppression. Niederberger et al. [10] proposed two different adaptation strategies to suppress the single mode vibration, one is based on minimizing the root mean square vibration, the other minimizes the

phase difference between the two measured signals. The aforementioned studies mainly focused on the passive shunt, which could be detuned when the mechanical parameters (mass, stiffness and so on) change. The active shunt needs external energy to drive it and can overcome the disadvantage of high dependency on tuning frequencies. Fleming [11-13] investigated the active EMSD with the linear quadratic regulator and H ∞ control strategies. Niu et

al.

[14]

proposed

a

negative

resistance

capacitor-matching-inductance

electromagnetic shunt damper to control vibration of a cantilever beam. The negative resistance can cancel the inherent impedance of the coil, so that the induced current increases, leading to the enhancement of vibration attenuation [15]. Yan et al. [16] proposed a negative resistance EMSD vibration isolator to control the first-order and the third-order vibrations of a beam. Later on, the multi-mode vibration control with the negative inductance negative resistance shunt [17], the broadband vibration absorption to overcome the narrow band of traditional vibration absorbers [18, 19] and self-sensing characteristic of EMSD [20] are studied. McDaid and Mace [21] used an adaptive synthetic shunt impedance to realize the broadband vibration absorption. Zhu et al. [22] numerically studied the similarity and difference between the electromagnetic shunt damper and the tuned mass damper. Moreover, the electromagnetic transducers can also be applied in energy harvesting [23-30] and micro-vibration control of the spacecraft [31, 32]. Apart from these, Li and Zhu [33] investigated the versatile behaviors of electromagnetic shunt damper, Zhou et al. [34] numerically studies the electromagnetic shunt damper according to three strategies of the fixed points theory, H2 optimization criterion and maximum damping criterion. To the best of the authors’ knowledge, these studies mainly focus on the L-EMSD field, in other words, the electromagnetic coupling coefficient is regarded as a constant. However, the existing studies have also suggested that the electromagnetic coupling coefficient is a function of the relative position between the coil and PMs [18] and will be a nonlinear variable when the relative displacement is large. In the field of vibration isolation, traditional linear damping is effective in the resonance

region while harmful in the isolation region. Therefore, the design of nonlinear damping with EMSD for broadband isolation is of great interest. This paper proposes N-EMSD for the vibration isolation enhancement of linear vibration isolators, which has not been investigated deeply in previous studies. N-EMSD consists of a pair of the PMs, a pair of the coils that are wound in opposite direction and connected in series, and a negative resistance shunt circuit. N-EMSD can enhance the vibration isolation performance in the resonance region without affecting the vibration isolation in the isolation region compared with the traditional L-EMSD. This paper differs from the previous studies in following aspects: (1) the electromagnetic coupling coefficient is designed to be nonlinear while others are linear [8, 15, 22, 24]; (2) the frequency of the voltage of the circuit is twice the frequency of the displacement for the novel PMs and coils configuration, however, they are equal to each other for the traditional L-EMSD [23, 25]; (3) the isolation performance of the LVI with N-EMSD in the isolation region is improved compared with the LVI with L-EMSD. The rest of the paper is organized as follows. Section 2 describes the design of N-EMSD, analyzes the nonlinear electromagnetic coupling coefficient and negative resistance shunt circuit, and theoretically solves the response of the LVI with N-EMSD. Section 3 numerically compares the vibration isolation performance of the LVI with L-EMSD and LVI with N-EMSD, discusses the effects of different parameters on the vibration isolation performance of the LVI with N-EMSD. The corresponding experiments are exhibited in section 4. Finally, the conclusions are summarized in section 5. 2. Theoretical modeling 2.1. Design of N-EMSD Two types of the PMs and coils configurations of the LVI are put forward to demonstrate the effectiveness of N-EMSD. Fig. 1 shows the 3D models ((a) and (d)), the distribution of the PMs and coils ((b) and (e)), and prototype of the coil ((c) and (f)) for the LVI with L-EMSD and LVI with N-EMSD. The LVI consists of a pair of

the PMs, three springs and linear bearings that are used to decrease the friction force, the lower base and upper load plates, and a negative resistance shunt circuit. The linear stiffness is provided by the three springs. For the LVI with L-EMSD shown in Fig. 1 (a) and (b), PM 1 and 2 are mounted on the center of the load plate with a nut. The distance D between PM 1 and PM 2 can be adjusted with a spacer. The coil is distributed symmetrically along the center of the spacer. Then the coil is connected to the negative resistance shunt circuit. This configuration can be regarded as L-EMSD. Different from Fig. 1(b), two coils (coilt and coilb) are connected in series but wound in opposite direction as shown in Fig. 1(e) and (f). One can change the distance H between the coilt and coilb with the screws. In addition, the coils are connected to the negative resistance shunt circuit. This configuration can provide N-EMSD. The turns and distribution of the coilt are same with those of the coilb. It is worth noting that the configurations of the LVI with L-EMSD shown in Fig. 1 (a) and with N-EMSD shown in Fig. 1 (d) have same shape apart from the distributions of coils. The dimension and residual flux density of the ring PM are listed in Table 1.

Fig. 1. Two configurations of L-EMSD and N-EMSD. The LVI with L-EMSD for (a) 3D model, (b) distribution of the PMs and coil, and (c) prototype of the coil. The LVI with N-EMSD for (d) 3D model, (e) distribution of the PMs and coils, and (f)

prototype of the coils. Table 1 Dimension and residual flux density of the ring PM. name

Value

Inner radius (mm)

3

Outer radius (mm)

14

Height (mm)

10

Residual flux density (T)

1.45

2.2. Modeling of the nonlinear electromagnetic coupling coefficient In cylindrical coordinate system, the flux density vector B ( r ,φ , x ) generated by a ring PM at any point P ( r ,φ , x ) outside the PM can be written as [19, 35] B ( r , φ , x ) = Br ( r , φ , x ) r + Bφ ( r , φ , x ) φ + Bx ( r , φ , x ) x

(1)

where Br, B φ and Bx denote the radial, circumferential and axial magnetic flux components, respectively.

Br ( r , φ , x ) =

µ0 M 4π

2

2

∑∑ ( −1)

( j+k )

j =1 k =1

Bx ( r , φ , x ) =

µ0 M 4π

∫

2π

0

2

cos (φ − φ ')d ( r , φ , x; Rc ( j ) , φ ', xk ) Rc ( j ) dφ '

(2)

Bφ ( r , φ , x ) = 0

(3)

( j +1)

x2

2π

x1

0

∑ ( −1) ∫ ∫ j =1

 r cos (φ − φ ') − Rc ( j )  (4)

× d ( r , φ , x; Rc ( j ) , φ ', x ')  Rc ( j ) dφ ' dx ' 3

where 2 d ( r , φ , x ; r ', φ ', x ' ) =  r 2 + r '2 − 2 rr ' co s (φ − φ ' ) + ( x − x ' )   

−1 2

r′ and x′ are the radial and axial coordinates of a point at the coil, respectively. j is the number of the surface. Rc(1)=Rin is the inner radius and Rc(2)=Rout is the outer radius of the ring PM. The residual flux density is

Br 0 = µ0M where M is the magnetization vector.

µ 0 is the permeability of vacuum.

(5)

Once there is relative motion between the PMs and the coil, the electromotive force (emf) is induced according to Faraday’s law

dVe = N ( v × B ) dl

(6)

where N is the turns of the coil and v is the relative velocity. The radial magnetic flux density at any point P ( r , φ , x ) contributed by PM 1 and PM 2 can be written as B pr ( r , φ , x ) = B r PM 1 ( r , φ , x ) + B r PM 2 ( r , φ , x )

(7)

Substituting Eqs. (1) and (8) into (6), and integrating Eq. (6), one can obtain Ve = vN ∫ B pr dl φ = vCe φ

(8)

l

2π

Ce = N ⋅ R p ∫ B pr dφ

(9)

0

where Ce is defined as the electro-mechanical coupling coefficient. For the LVI with N-EMSD shown in Fig. 1(d) and (f), the winding directions of the coilt and coilb are opposite. Thus, the induced emf can be obtained according to Eq. (8)

Ve1 = vCe11φ + vCe12 ( −φ ) = vCe1φ

(10)

where Ce11 and Ce12 are the electro-mechanical coupling coefficients of the coilt and coilb, respectively. Considering the directions of the two coils and magnetic flux density, we have Ce1= Ce11-Ce12. Ampere force will be generated correspondingly when there is current flowing in the coil according to Ampere's force theorem, which can be expressed as dF = NI ( t ) dI × B pr

(11)

Inserting Eqs. (1)-(4) and (7) into (11) and integrating Eq. (11), one can obtain

Fe = −xCm I (t ) Cm = N ⋅ R p ∫

2π

0

B pr dφ

(12) (13)

where I(t) is the induced current. Cm is defined as the electromagnetic coupling coefficient. Comparing Eq. (13) with (9), it can be obtained that Ce is equal to Cm. Consequently, Ampere force of the LVI with N-EMSD shown in Fig. 1(e) is

expressed as

Fe1 = −xCm1 I (t ) = − x ( Cm11 − Cm12 ) I (t )

(14)

where Cm11 and Cm12 are the electromagnetic coupling coefficients of the coilt and coilb， respectively. Fig. 2(a) shows the simulated Cm1 under different H for N-EMSD. It demonstrates that Cm1 exhibits the nonlinear characteristic and can be zero at the center position of the spacer shown in Fig. 1(e). It can be also found that C m1 decreases with the increase in H. Fig. 2(b) demonstrates that Cm can be regarded as a constant to provide linear damping. For the forthcoming theoretical investigation, the polynomial fit is used to approximate Cm1, which can be expressed as

Ce1 = c1 x + c3 x3

(15)

The coefficients of Ce1 under different H shown in Fig. 2 are listed in Table 2. (a)

(b)

L-EMSD N-EMSD

Fig. 2. (a) Simulated Cm1 with respect to H for N-EMSD, where the configuration is shown in Fig. 1(d). (b) Simulated Cm for L-EMSD, where the configuration is shown in Fig. 1 (b). Table 2 Coefficients of Ce1 with respect to H. H (mm)

c1

c3

13

-2.8987×103

2.6617×107

15

-2.4622×103

8.1991×106

17

-2.0354×103

-3.2949×106

2.3. Negative resistance shunt circuit The coils are connected to an external negative resistance shunt circuit to

counteract the inherent resistance of the coils. Fig. 3(a) shows the electrical schematic diagram of the negative resistance shunt circuit. An operational amplifier (type OPA-541 AP) is used to construct equivalent negative resistance [36]. The negative resistance can be expressed as

R1 R R2

(16)

dI + ( Re + Rs ) I (t ) dt

(17)

Rs = − when R1 is equal to R2, we have Rs = −R . The electric governing equation is Ve = Le

where Ve is the induced emf. Le, I and Re are the equivalent inductance, induced current and resistance of the coils.

Fig. 3. (a) Electrical schematic diagram and (b) prototype of the negative resistance shunt circuit. 2.3. Governing equations The simplified model of the LVI with N-EMSD is shown in Fig. 4. Three springs provide linear stiffness k and linear structural damping c. N-EMSD is used to provide nonlinear damping. When the isolator is subjected to base motion &x&0 , the governing equation is written as

mx&& + cx& + kx + Fe = −mx&&0

(18)

where m is the isolated mass. x is the relative displacement between the load plate and base plate. Fe can be calculated according to Eqs. (12) and (13).

x+x0 m

c

N-EMSD

k

Cm1I

&x&0 Fig. 4. Simplified model of the LVI with N-EMSD. 2.3.1. The LVI with L-EMSD When Ce and Cm are constants, the shunt circuit provides linear damping. The governing equations are expressed as [20]

mx&& + cx& + kx + Ce I ( t ) = F ( t ) Le

dI dt

+ ( Re + Rs ) I (t ) − Ce x& = 0

(19) (20)

where F (t ) = − mx&&0 is the equivalent excitation force. Using the Fourier Transformation and taking the zero initial conditions into consideration, we have   jCe 2ω 2 − + + + m ω jc ω k   x ( jω ) = F ( jω ) jLeω + ( Re + Rs )  

(21)

Thus, the transfer function of the LVI with L-EMSD is

H ( jω ) =

k   2  Le Ce Ce 2 ( Re + Rs )  k −m−  ω + jω  c +  2 2   ( Leω ) + ( Re + Rs )2  ( Leω ) + ( Re +Rs )2    2

(22)

2.3.2 The LVI with N-EMSD The negative resistance shunt circuit provides nonlinear damping due to the nonlinear characteristics of Ce1 and Cm1. Substituting Eqs. (14) and (15) into (18), and Eqs. (10) and (15) into (17), taking equation Ce1=Cm1 into consideration, the governing equations of the LVI with N-EMSD are

mx&& + cx& + kx + ( c1 x + c3 x3 ) I ( t ) = mx0ω 2 cos (ωt + φ ) Le

dI dt

(

)

+ ( Re + Rs ) I (t ) − c1 x + c3 x 3 x& = 0

(23) (24)

In general, when the LVI with L-EMSD is subjected to an excitation of frequency f. The frequencies of the displacement and voltage are equal to f [25]. However, the frequency of the voltage changes if the proposed N-EMSD is applied in the LVI because the interaction between the coils and PMs for L-EMSD shown in Fig. 1(b) and N-EMSD shown in Fig. 1(e) are different. The numerical simulation and two experiments are carried out to find out this difference. The schematic of the experimental setup is shown in Fig. 5(a). A laser displacement sensor (KEYENCE type IL-065) is used to record the displacement x1 of the load plate. The corresponding displacement and emf URe are recorded via a digital oscilloscope (Tektronix TBS 2000). The parameters used for simulations are as: Re=23.5Ω, Rs=-20Ω, Le=5.41mH, c1=-2.4622×103, c3=8.1991×106, and x0=1.75mm. Fig. 5(b) presents the simulated time histories of the displacement and emf of the LVI with N-EMSD when the excitation frequency and amplitude are 5Hz and 1.75mm, it can be found that the frequency of the emf is 5Hz while that of the displacement is 10Hz. Figs. 5(c) and (d) are experimental time histories of the displacement and frequency for different excitation frequencies. Both the numerical simulation and experiments demonstrate that the frequency of the voltage URe of the coils is twice that of the displacement x1. The HBM is an effective method to estimate the response of nonlinear systems [37-39]. In spite of the nonlinear influence in the system, the steady-state solution is still close to a periodic solution under certain conditions. Hence, ignoring the higher order terms, the first order displacement and current have the following forms:

x = a sin(ωt ) + b cos(ωt )

(25)

I (t ) = p sin(2ωt ) + q cos(2ωt )

(26)

The amplitude coefficients a, b, p and q are assumed to vary very slowly.

Fig. 5. Frequency relationship between the output displacement and emf of the LVI: (a) Experiment setup for testing the frequencies of the displacement and emf, (b) simulated time histories of the displacement and emf when the excitation frequency and amplitude are 5Hz and 1.75mm. Experimental time histories of frequencies of the displacement and emf when the excitation frequency and amplitude are (c) 10Hz and 0.5mm and (d) 8Hz and 1mm. H is 15mm for all cases.

Substituting Eqs. (25) and (26) into (24) and neglecting the harmonic terms of

sin ( 4ωt ) and cos ( 4ω t ) , the constant terms of sin ( 2ωt ) and cos ( 2ωt ) are expressed as 1 1 ωc1 ( − a 2 + b 2 ) + ω c3 ( − a 4 + b 4 ) − 2qω Le + p ( Re + Rs ) = 0 2 4 1 − abω c1 − abω c3 ( a 2 + b 2 ) + 2 pω Le + q ( Re + Rs ) = 0 2

(27) (28)

According to Eqs. (27) and (28), p and q are obtained

 1 1    1  p =  2ω Le ab  c1 + c3r 2  −  c1 ( −a 2 + b 2 ) + c3 ( −a 4 + b 4 )  ( Re + Rs )  ωU (29) 2 2   2   

 1 1     q =  ω Le  c1 ( −a 2 + b2 ) + c3 ( −a 4 + b4 )  + ab ( Re + Rs )  c1 + c3r 2   ωU 2 2      where U =

(30)

1

( 2ω Le ) + ( Re + Rs ) 2

2

Substituting Eqs. (25), (26), (29) and (30) into (23), balancing the constant terms multiplied by sin (ω t ) and cos (ωt ) and ignoring the higher harmonic terms, we have  3 1 1  a  k − mω 2 + U ω 2 Le  r 2 c12 + r 4 c1c3 + r 6 c32   4 4 2    1 1 1  −b  cω + U ω ( Re + Rs )  r 2 c12 + r 4 c1c3 + c32 r 6   = − mx0ω 2 sin φ 4 16 4    3 1 1  b  k − mω 2 + U ω 2 Le  r 2 c12 + r 4 c1c3 + r 6 c32   4 4 2    1 1 1  + a  cω + U ω ( Re + Rs )  r 2 c12 + r 4 c1c3 + r 6 c32   = mx0ω 2 cos φ 4 16 4  

(31)

(32)

where r 2 = a 2 + b 2 , r is the amplitude of the relative displacement. Thus the relationship between the relative displacement r and circular frequency

ω is  3 1 1  r  k − mω 2 + U ω 2 Le  r 2 c12 + r 4 c1c3 + r 6 c32   4 4 2  

2

2

2

 1 1 1  + r 2 cω + U ω ( Re + Rs )  r 2 c12 + r 4 c1c3 + r 6 c32   = ( mx0ω 4 16 4  

 3 1 1  r  k − mω 2 + U ω 2 Le  r 2 c12 + r 4c1c3 + r 6 c32   4 4 2  cos θ =  2 mx0ω

(33)

)

2 2

(34)

where θ is the phase of the displacement. The absolute displacement is defined as x1 = x0 cos (ω t + θ ) + r cos (ω t )

(35)

Consequently, the transmissibility of the LVI with N-EMSD is written as [40]

T=

x0 2 + r 2 + 2 x0 r cos θ x0 2

(36)

3. Numerical simulations This section numerically discusses the advantages of N-EMSD compared with L-EMSD, and also comprehensively investigates the effects of the different parameters (H and x0) on vibration isolation performance. The simulated results are presented using the parameters listed in Tables 2 and 3. In addition, the electromagnetic coupling coefficient of the LVI with L-EMSD is approximated as Cm=7.4 NA-1 according to Fig. 2(b).

Table 3 Parameters of the LVI with L-EMSD and LVI with N-EMSD. Parameters

LVI with L-EMSD

LVI with N-EMSD

Mass, m(kg)

0.611

0.611

Damping ratio, ζ

0.035

0.035

Stiffness of springs, k(N/m)

2360

2360

D(mm)

15

15

H(mm)

-

13/15/17

Resistance Re(Ω)

9.4

23.5

Negative resistance Rs(Ω)

-3

-20

Inductance Le(mH)

1.67

5.41

Total number of turns

110

500

3.1. Comparison of the vibration isolation performance between L-EMSD and N-EMSD Fig. 6 shows the transmissibility comparison between the LVI with L-EMSD and the LVI with N-EMSD when H is 15mm and the excitation amplitude x0 is 1.5mm. The natural frequency and corresponding transmissibility of the passive LVI (No control) are 9.87Hz and 14.39, respectively. It demonstrates that the vibration isolation performances of the LVI with L-EMSD and the LVI with N-EMSD are almost same with each other in the resonance region by adjusting Rs. However, the vibration isolation performance of the LVI with N-EMSD is better than that of the

LVI with L-EMSD in the isolation region. It should be noted that the natural frequency of the LVI with N-EMSD slightly decreases.

with L-EMSD with N-EMSD No control

101

100

10-1

0

10

20

30

40

50

Fig. 6. Transmissibility comparison between the LVI with L-EMSD and the LVI with N-EMSD when H=15mm and x0=1.5mm. 3.2. Effect of the excitation amplitude x0 Fig. 7 shows the transmissibility of the LVI with N-EMSD with respect to x0 for H=15mm. It implies that the vibration isolation performance enhances and corresponding natural frequency slightly decreases with the increase in x0 in the resonance region. However, the vibration isolation performance is hard to change with the change of x0 in the isolation region compared with the passive LVI. In other words, N-EMSD is very small near the static equilibrium position but is large away from the static equilibrium position, which demonstrates N-EMSD is a kind of nonlinear damping. Furthermore, the corresponding numerical transmissibility are calculated with the fourth order Runge-Kutta method to validate the theoretical model. In steady-state, the transmissibility is expressed by T = ( x + x0 ) / x0 . It can be found that the theoretical solutions agree well with the numerical solutions, which demonstrates the effectiveness of analyzing of the frequency relationship between the

displacement and emf, and also the theoretical derivation of the transmissibility with the HBM.

5.5 Numerical solutions x0=0.75mm Theoretical solutions x0=0.75mm

5

Numerical solutions x0=1.00mm Theoretical solutions x0=1.00mm

4.5 4

Numerical solutions x0=1.25mm Theoretical solutions x0=1.25mm

3.5

Numerical solutions x0=1.50mm Theoretical solutions x0=1.50mm

3 2.5 20

22

Numerical solutions x0=1.75mm Theoretical solutions x0=1.75mm

24

2

shift

No control

1.5 1 0.5 0

0

5

10

15

20

25

Fig. 7. Transmissibility of the LVI with N-EMSD with respect to x0 for H=15mm. 3.3. Effect of H Fig. 8 shows the transmissibility of the LVI with N-EMSD with respect to H for x0=1.75mm. It demonstrates that the transmissibility decreases with the decrease of H in the resonance region and the vibration isolation performance is hard to change in the isolation region with the change of H. It should be noted that the natural frequency slightly shifts toward a smaller value when H decreases. One can obtain the expected isolation performance by adjusting H.

4 Theoretical solutions H=13mm Theoretical solutions H=15mm Theoretical solutions H=17mm

3.5 3 2.5 2 1.5 1

20

25

0.5 0

0

5

10

15

20

25

Fig. 8. Transmissibility of the LVI with N-EMSD with respect to H for x0=1.75mm. 4. Experimental verification 4.1. Experimental setup The experiment is setup to verify the vibration isolation performance of the LVI with N-EMSD, and the photograph of the experimental setup is shown in Fig. 9. The LVI is mounted on the exciter. One accelerometer is mounted on the load plate to record the output acceleration. The other one is mounted on the load plate to record the excitation signal, which is also feedback to controller to control the accuracy of the excitation. Then one can easily obtain the acceleration transmissibility with the output and input data. The excitation signal is generated by a professional software (from DynaTronic Corporation Ltd.) and amplified by the power amplifier (VENZO 880), which drives the exciter (TIRA 200N) to generate the designed excitation. A chirp excitation with the bandwidth [2, 25] Hz and sweep speed 1Hz/s is used during the experiment.

The operational amplifier (OPA-541 AP) is used to construct the equivalent negative resistance and the schematic is shown in Fig. 3. A DC source is used to power up OPA-541. The other experimental parameters can be found in Tables 1 and 3.

Fig. 9. Photograph of the experimental setup. 4.2. Comparison of the vibration isolation performance between L-EMSD and N-EMSD Some experiments are performed to illustrate the vibration isolation advantage of N-EMSD over L-EMSD in the isolation region. Fig. 10 shows the experimental transmissibility comparison between the LVI with L-EMSD and the LVI with N-EMSD, where x0 is 1.5mm for the two cases. “No control” case means that N-EMSD and L-EMSD are not used and the isolator is passive. When H=13mm (Fig. 10(a)), the LVI with L-EMSD and the LVI with N-EMSD have same vibration isolation performance in the resonance region. However, the isolation performance becomes better for the LVI with N-EMSD compared with that of the LVI with L-EMSD in the isolation region. In addition, the natural frequency of the LVI with N-EMSD decreases to 9.34Hz, which is slightly smaller than that of the LVI with L-EMSD. If H increases to 15mm (Fig. 10(b)), the vibration isolation performance of the LVI also becomes better in the isolation region when N-EMSD is used. The peak frequency also decreases from 9.86Hz to 9.46Hz, which is different from the traditional L-EMSD [15]. The two cases also show that the isolation performance of

the LVI with N-EMSD is similar to the passive isolator in the isolation region. All these demonstrates that N-EMSD can decrease the response in the resonance region and enhance the vibration isolation performance in the isolation region compared with that of the LVI with L-EMSD, which verifies that N-EMSD can provide nonlinear damping.

Fig. 10. Experimental transmissibility comparison between the LVI with L-EMSD and the LVI with N-EMSD for (a) Rs=-2.7Ω and (b) Rs=-3Ω. 4.3. Effect of x0 Fig. 11 shows the experimental transmissibility of the LVI with N-EMSD with respect to x0 when H=15mm. The natural frequency and corresponding transmissibility of the passive LVI are 9.87Hz and 14.39, respectively. It can be found that the vibration isolation performance improves when N-EMSD is used. When the excitation amplitude x0 increases from 0.25mm to 1.75mm, the peak transmissibility decreases from 8.87 to 3.29, which is about 77% vibration isolation improvement. The natural frequency decreases from 9.94Hz to 9.27Hz. Fig. 11 also implies that N-EMSD exhibits the nonlinear damping characteristic that can provide relative large damping to reduce the vibration in the resonance region but small damping to improve the isolation performance in the isolation region.

No control with N-EMSD x0=0.25mm with N-EMSD x0=0.5mm

fp=9.87;T=14.39

14 12 10

fp=9.94;T=8.87

8

f p=9.76;T=6.65

6

fp=9.46;T=3.96

4

fp=9.46;T=3.50

2

with N-EMSD x0=0.75mm with N-EMSD x0=1mm with N-EMSD x0=1.25mm with N-EMSD x0=1.5mm with N-EMSD x0=1.75mm

fp=9.74;T=5.28 fp=9.58;T=4.50

fp=9.27;T=3.29

5

10

15

20

25

Fig. 11. Experimental transmissibility of the LVI with N-EMSD with respect to x0 when H=15mm. 4.4. Effect of H Fig. 12 shows the experimental transmissibility of the LVI with N-EMSD with respect to H. When x0=1.5mm, the experimental results with different H are shown in Fig. 12(a). When H increases from 13mm to 17mm, the peak transmissibility decreases from 3.62 to 3.31 and the natural frequency slightly decreases from 9.46Hz to 9.14Hz. When x0=1.75mm, we can find the similar trend as shown in Fig. 12(b). Fig. 12 also implies that one can get desirable vibration isolation performance via adjusting H. 4.5. Comparison between the theoretical and experimental results It is very important to appreciate how well the results from the theoretical equations match the real experimental data. Fig. 13 shows the comparison between the experimental and theoretical transmissibility of the LVI with N-EMSD when H=15mm. It can be seen that the experimental results agree well with the theoretical results.

Fig. 12. Experimental transmissibility of the LVI with N-EMSD with respect to H for (a) x0=1.5mm and (b) x0=1.75mm. Experimental Theoretical

5

5

Experimental Theoretical x0=0.75mm

x0=0.50mm

0 4

0 4 Experimental Theoretical

2

Experimental Theoretical

2

x0=1.00mm

0 4

x0=1.25mm

0 4

Experimental Theoretical

Experimental Theoretical

2 0

x0=1.50mm

5

10

15

20

2 0 25

x0=1.75mm

5

10

15

20

25

Fig. 13. Comparison between the experimental and theoretical transmissibility of the LVI with N-EMSD for H=15mm.

5. Conclusions This paper proposes novel N-EMSD to achieve a kind of nonlinear damping for vibration isolation enhancement. The nonlinear electromagnetic coupling coefficient

of N-EMSD is theoretically derived. The emf frequency is twice the displacement frequency, which is totally different from that of L-EMSD. The coupling governing equations are established and the HBM is used to derive the amplitude-frequency relationship. The numerical simulations and experiment are performed to verify the nonlinear damping performance of the proposed N-EMSD. The results demonstrate that: (1) The LVI with N-EMSD can effectively enhance the vibration isolation performance in the resonance region without affecting the vibration isolation performance in the isolation region; (2) The transmissibility of the LVI with N-EMSD decreases with the increase of the excitation amplitude in the resonance region, which demonstrates the nonlinearity of N-EMSD; (3) the natural frequency slightly decreases with the decrease of the peak transmissibility of the LVI with N-EMSD; (4) The vibration isolation performance can be changed by adjusting the distance between the coilt and coilb. This paper extends the investigation of EMSD from linear to nonlinear fields and also provides an effective method to design and optimize nonlinear damping.

Acknowledgment This work was supported by the National Natural Science Foundation of China under grant no.11602223 and the Zhejiang Provincial Natural Science Foundation of China under grant no. LY20E050001.

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Nonlinear electromagnetic shunt damping (N-EMSD) is proposed and modeled




The linear and nonlinear electromagnetic coupling coefficients are discussed




The voltage frequency of the coils is two times of the displacement frequency




A linear vibration isolator with N-EMSD is analyzed via the harmonic balance method




N-EMSD can achieve nonlinear damping to improve vibration isolation performance

Bo Yan and Hongye Ma designed research; Bo Yan and Hongye Ma performed research; Hongye Ma, Lu Zhang and Ning Yu performed the experiment; Chuanyu Wu and Ning Yu analyzed data; Bo Yan and Hongye Ma wrote the paper; Lu Zhang and Ning Yu checked the language.

The authors declare that they have no conflict of interest.