Design and experiment of a noncontact electromagnetic vibration isolator with controllable stiffness

Acta Astronautica 168 (2020) 130–137

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Research paper

Design and experiment of a noncontact electromagnetic vibration isolator with controllable stiffness

T

Yu Chen, Hao Wen∗, Dongping Jin State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, 29 Yudao Street, Nanjing, 210016, Jiangsu, China

A R T I C LE I N FO

A B S T R A C T

Keywords: Noncontact electromagnetic isolator Satellite Low-frequency vibration Controllable stiffness

Isolating low-frequency vibration in current satellites that has a negative effect on the performance of precise payloads has increasing demand. In this work, a simplified model of a noncontact electromagnetic isolator for low-frequency vibration, which has a controllable stiffness with a minimum of zero, is proposed. Starting with the designing process of the vibration isolator, a series of ground-based experiments are carried out to verify the controllable stiffness. Two typical operating scenarios, the vibration isolation and rapid maneuver modes, are experimentally demonstrated. The experimental results indicate that the stiffness has a practically proportional relationship with the input current and the proposed design works well for both of the operation modes.

1. Introduction The space missions of earth observation and deep space observation are calling for an unprecedented high stability and high accuracy [1]. The satellite itself contains various types of sources of vibration, such as solar panels, reaction wheels, and so on [2]. These devices can produce vibration that covers a wide frequency band. It's hard for traditional passive vibration isolators to cope with this situation because they'll be effective only when excitation frequencies are greater than 2 ωn , where ωn = k /m is the natural frequency of the system. In order to solve this problem, many new techniques have been proposed. Among the techniques, noncontact vibration isolator or actuator has gained great attention from many researchers [3–15]. In these works, the precise payload is separated from the main body of the satellite and between them are noncontact actuators. With no physical connection to the main body, the payload can have a more desirable working environment. For example, Pedreiro [3] et al. designed a space architecture called disturbance-free payload (DFP). In their works, the satellite is separated into two modules by the noncontact isolator, namely the Support Module (SM) and the Payload Module (PM). SM is the main body of the satellite and it contains almost all the disturbance sources, whereas PM is the precise payload on the satellite which needs a stable working environment. Their works promoted the development of noncontact spacecraft. Xu [6] et al. designed a Stewart spacecraft based on noncontact actuators with mixed sensitivity robust H∞ control and achieved stringent pointing accuracy and pointing stability.

∗

Guelman [7] et al. designed a system that can provide 3-axis stabilization with only one magnetometer as sensor. The optimization of noncontact actuators also has been performed by many researchers [8,10,13], which is beneficial to the design of noncontact actuators. Some other researchers put their efforts into modifying the stiffness of the system [16–21]. Onoda [16,17] et al. focused on the system with hysteretic variable stiffness. Liu [19] et al. designed a variable stiffness system whose stiffness is controlled by changing the damping coefficient. These variable stiffness systems still have physical connection, so it is always hard to design a system that has stiffness down to zero. Wu [21] et al. designed a variable stiffness mechanism by adjusting the symmetric rotation of two parallel connected springs, and a vibration isolation system with relatively small size and large stiffness range was provided. This system successfully achieved zero stiffness with a contact-type vibration isolator but zero stiffness can only be realized in a small range of motion. Noncontact isolator would be necessary if zero stiffness is to be realized within a large motion range. This paper proposes a noncontact vibration isolator with controllable stiffness. The isolator stiffness can be adjusted from zero within the entire range of motion. For sophisticated space mission, the isolator can work in two different modes, namely isolation mode and maneuver mode. In isolation mode, the stiffness is set to a low level, or even zero if necessary, with a relatively low current, so it can isolate low frequency vibration. With a relatively high current, the stiffness is modified to a high level in maneuver mode so that rapid maneuver can be realized. With these two simple modes, there're only two possible

Corresponding author. E-mail address: [email protected] (H. Wen).

https://doi.org/10.1016/j.actaastro.2019.12.004 Received 5 May 2019; Received in revised form 13 October 2019; Accepted 4 December 2019 Available online 06 December 2019 0094-5765/ © 2019 IAA. Published by Elsevier Ltd. All rights reserved.

Acta Astronautica 168 (2020) 130–137

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current outputs, so the real-time feedback control of the current is no longer needed. Consequently, the complex calculation of control algorithm is avoided either. Different from passive vibration isolators, the proposed isolator has a controllable stiffness so it can adapt to various conditions while the structure is almost as simple as passive isolators. Compared to current noncontact isolators with active control, the proposed isolator is as efficient as them while system complexity is greatly reduced. Besides, compared to the system with variable mechanical stiffness, the stiffness of the proposed isolator can be set down to zero, and its upper limit is only limited by the input current. The remainder of the paper is organized as follows: Section 2 introduces the design process of the isolator and the effects of parameters are discussed. Section 3 shows the controllability of stiffness and the performance of the two working modes by experiments. And in Section 4, some conclusions on the performance of the isolator are given and possible further research is discussed. 2. Design of the isolator Fig. 2. Coordinate system of cylindrical permanent magnet.

This section aims at finding a suitable combination of parameters that can endow the isolator with the best linearity when current is constant, which means that stiffness does not fluctuate too much with displacement and the stiffness can be controlled by regulating the strength of the current in practical use. As shown in Fig. 1, the noncontact isolator consists of square coils and cylindrical magnets. The coils are fixed with SM and the magnets are fixed with PM. In the current design, only four coils are used in each isolator and the total number of turns is 40. More coils can be used if larger stiffness is required in practical use. Ampere force acts on the electrified coils in a magnetic field. Stiffness can be calculated by differentiating the Ampere's force with respect to the displacement. With the size of the coils and the magnets fixed, the stiffness will only be affected by the current and the position of the coil, which is

k=

dF = If (x ) dx

coil [22]. From this point of view, the magnetic field of a cylindrical permanent magnet can be calculated via Biot-Savart Law. The coordinate system of a cylindrical permanent magnet is built as shown in Fig. 2. Providing that the equivalent plane current of the magnet is J, the height of the magnet is h and the diameter of the magnet is r0 , magnetic flux density at point p can be written as

Bz = −

μ0 J 4π

h

∫0 ∫0

2π

⎡ r0 (x − r0 cos θ)cos θ + r0 (y − r0 sin θ)sin θ ⎤ 2 2 2 3/2 ⎢ ⎣ [(x − r0 cos θ) + (y − r0 sin θ ) + (z − z 0) ] ⎥ ⎦

dθdz 0

(2)

Here only z-component of magnetic flux density is given and other components will be ignored because forces generated by them are relatively small and are vertical to the coil plane. The integral in Eq. (2) can't be expressed with fundamental functions because it contains elliptic integral, so the method of curve fitting is used. For example, assuming that J = 1 A/m and the diameter and height of the magnet are both 20 mm, the expression of magnetic flux density in plane z = 30 mm is

(1)

where k is stiffness, F is Ampere's force, x is relative displacement between PM and SM, I is current and f (x) represents the effect of displacement on stiffness. The first step to calculate Ampere's force is analyzing the magnetic field of the magnet. The fictitious charge distribution of a cylindrical permanent magnet is the same as that of a thin coil, so the magnetic field of a permanent magnet is topologically the same as that of a thin

Bz (x ) = −0.7754 exp( −(x /7.98)2 ) − 0.04532 exp( −(x /62.8)2 ) + 0.6929 exp(−(x /15.99)2 ) + 1.645 exp( −(x /9. 772)2 )

(3)

Here in Eq. (3), y is also limited to 0 mm considering the symmetry of the magnetic field generated by cylindrical magnet. The fitting expression is based on Gaussian function. This choice is based on the fact that the real magnetic field distribution is similar to that of a Gaussian function as shown in Fig. 3. If polynomial fitting is used, more number of terms will be needed and accuracy may decrease. With this, the magnetic flux density of cylindrical permanent magnets can be obtained and the next step is to calculate the force. As shown in Fig. 4, the center of the cylindrical magnet is denoted by O and that of the square coil by C. The length of the coil's side is a, the width is d. Initially, the centers of the magnet and coil overlap each other and Ampere's forces applied to the two opposite sides cancel each other out, thus the resultant force of the whole coil is zero. When the coil moves along the x direction for some distance, Ampere's forces applied to the upper side and lower side will differ from each other. With appropriate direction of the current, the resultant force will act as a restoring force. The force applied to the coil is Fig. 1. Structure of the isolator. 131

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Fig. 3. Gaussian distribution of magnetic field. Fig. 6. Stiffness when x = 3 mm and 0 ≤ d ≤ 5 mm.

Fig. 4. Relative position of coil and magnet.

Fig. 7. Stiffness when d = 4 mm, 0 ≤ x ≤ 8 mm and 30 ≤ a≤60 mm.

Fx = Jc

a/2

(a + d)/2

∫−a/2 ∫(a−d)/2

(Bz ( (x + q)2 + p2 ) − Bz ( (x − q)2 + p2 ))dqdp (4)

where Jc represents the current density of the coil and it is calculated by dividing current intensity I by coil width d, which is

Jc =

I d

(5)

With the derivation above, the expression of the force is obtained and the expression of the stiffness can be calculated with Eq. (1), but it is complex and effects of the parameters can be approximately analyzed by numerical simulation in absence of it, so it is not given here. The following part aims to evaluate the parameter effects. Firstly, the impact of the coil's width d is discussed. Assuming that length a = 30 mm, displacement x ranging from 0 to 5 mm and width d ranging from 0 to 5 mm, the change of stiffness k is shown in Fig. 5. To be more intuitive, fixing x = 3 mm and we can see that stiffness k is practically in proportion to width d in Fig. 6. This is an interesting and useful finding, which means that the width of the coil can be used to modify the stiffness without influencing its linearity. Then the impact of the coil's length is discussed. Assuming that width d = 4 mm, displacement x ranging from 0 to 8 mm and

Fig. 5. Stiffness when 0 ≤ x ≤ 5 mm and 0 ≤ d ≤ 5 mm.

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Fig. 10. Internal setup of SM. Table 2 Equipment list. Fig. 8. Detection of the best parameter. Laser sensor Air flotation equipment (including gas cylinder, gas lines and air bearings) Noncontact isolator Industrial computer Exciter Battery USB-RS485 converter Current controller

Table 1 List of predefined parameters. Parameters

Values/mm

Outer diameter of magnet Inner diameter of magnet Height of magnet Closest distance between magnets Distance between center coils Distance between external coils

30 6 20 30 2 1

Number

Quantity

1 2

2 2

3 4 5 6 7 8

2 1 1 2 2 1

Fig. 9. Overall experimental apparatus.

length a ranging from 30 to 60 mm, the change of stiffness is shown in Fig. 7. Changing the view of Fig. 7, the parameter that produces the best linearity can be found in Fig. 8. It is obvious that the length of the coil should be about 44 mm. It should be noted that there are so many parameters in the design of the isolator, such as the radius of the magnet, the distance between the magnets, the distance between layered coils and so on, that it is nearly impossible to consider all of these parameters at the same time. So in this paper, all the parameters, except a, x and d, have been predefined according to actual experimental conditions. These predefined parameters are shown in Table 1. With these specific parameters, Eq. (1) will become

k = kc I

Fig. 11. Free vibration when I = 500 mA.

With f (x ) ≈ k c . So the stiffness can be controlled by regulating the strength of the current.

3. Experimental apparatus and results An experimental prototype of one degree of freedom is built to verify the controllability of the stiffness and the performance of two working modes. In order to simulate the micro-gravity environment in

(6) 133

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Fig. 12. Single-sided amplitude spectrum. Fig. 14. Relationship between k and I.

Fig. 15. Time domain response of PM when excitation frequency is 2 Hz.

Fig. 13. Time-frequency analysis.

displacement of PM and the relative displacement between PM and SM. The absolute displacement of SM is calculated by adding the two sets of measurement data. The output signal format of the sensor is 485 digital signal, and it is transferred to the computer through a USB-RS485 converter. The exciter is fixed to SM and will supply sinusoidal excitation force to SM. The maximum displacement before the two modules would collide with each other is 15 mm, and the range of the sensor is 35 ± 15 mm. The expected relative displacement between PM and SM is within ± 8 mm, because beyond this range, the stiffness will start to decay.

Table 3 Experimental data. I/mA

150

200

300

400

500

600

700

f/Hz m/kg

0.0833 11.84

0.1 11.88

0.12 11.81

0.14 11.73

0.1556 11.65

0.1667 12.44

0.1778 12.35

I/mA

800

900

1000

1100

1200

1300

f/Hz m/kg

0.19 12.27

0.2037 12.24

0.2174 12.19

0.2273 12.15

0.2381 12.11

0.2564 11.75

3.1. Stiffness experiments space, the experiment is carried out on an air-bearing platform. The experimental settings are shown in Figs. 9 and 10, and the equipment overview is shown in Table 2. Both SM and PM are supported by four air bearings so they can float on the granite platform almost without friction. Two isolators are mounted between the two modules. They're electronically connected in series and connected to the current controller. Two laser displacement sensors with a resolution of 6 μm are used to measure the absolute

The stiffness of the isolator with a specific current is indirectly obtained by measuring the free vibration frequency of PM, which is ωn = k /m . In this experiment, SM is fixed and the exciter is not used. The stiffness can also be measured directly by pull test, but the related instruments are not available temporarily. With initial condition x0 = −8 mm and v0 = 0, the free vibration of PM when I = 500 mA is shown in Fig. 11. In a practical experiment, it is 134

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Fig. 16. Frequency domain analysis of experiment data.

Fig. 18. Time domain simulation of PM when excitation frequency is 2 Hz.

Fig. 17. Time-frequency analysis of experiment data. Fig. 19. Relative displacement between PM and SM in maneuver mode.

difficult to achieve pure single-degree-of-freedom motion without physical constraints and environmental disturbances will also influence the experiment, so the vibration in the direction of concern is mixed with vibration of other degrees of freedom, which is vividly shown in the figure. As a consequence, PM may collide with SM in the first several periods, and these data have been neglected. But the frequency component of concern can be distinguished from others easily by spectrum analysis because it should contain the most energy, as shown in Fig. 12, where P1 means the single-sided amplitude spectrum of the data, X and Y correspond to f and P1 respectively and together they show the location of the marked point. Also wavelet transformation has been carried out, as shown in Fig. 13, to illustrate how the frequency components change with time. In the process of the experiment, the mass of gas in the cylinder will decrease because it is consumed to support the modules. The change of mass will influence the natural frequency of the system so the time of each experiment cannot be too long. In order to reduce the effect of the changing mass and obtain enough valid data for further analysis, duration of each experiment is set to be about 2–4 min. In high stiffness

experiment, the experimental time is about 2 min. In low stiffness experiment, the vibration period is longer, so the lasting time is expanded correspondingly to about 4 min. At the beginning and the end of each experiment, the mass of PM is recorded and the data used to calculate the natural frequency is based on the average of these two data. The experimental result is shown in Table 3. The relationship between current and stiffness is shown in Fig. 14. And obviously there is a nearly linear relationship between k and I. There exist some differences between the fitting line and real data because of the effects of neglected damping and probably the nonlinear characteristics of the system. The situation I < 150 mA can't be realized because the force of environmental disturbances will be larger than the output force of the isolator and the situation I > 1300 mA can't be realized either because the maximum stable output of current controller is 1300 mA. It should be noted that due to the current experimental setup, the maximum stiffness of the experimental prototype is approximately 30 N/m. The maximum stiffness can be enlarged easily

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Experiments are carried out to validate the controllability of the stiffness and the performance of two working modes. In practical use, the stiffness of the isolator can be set to a low level or even zero when vibration isolation is needed and it can also be changed to a high level when rapid maneuver control is required, so it is capable of adapting to various situations. At the same time, no real-time feedback is needed any more thus the isolator is relatively more reliable and stable. Further research may focus on how to modify this isolator so that it can work in more degrees of freedom. Although the isolator in this paper is merely used to isolate vibration in one degree of freedom, it can cope with the situation of multiple degrees of freedom in that it has stiffness in both x and y direction, which is clearly shown in Fig. 11. But if this isolator were to be fully used, the problem of coupling should be solved first.

by increasing width and turn number of the coils or by using stronger magnets. 3.2. Working mode experiments In this part, two simple examples are given respectively to validate the performance of isolation mode and maneuver mode. In isolation mode experiment, PM can float freely on the platform while SM is fixed with the exciter. Two laser displacement sensors are used to measure the absolute displacement of PM and SM respectively so that the displacement attenuation can be illustrated. The current used in this experiment is 300 Ma. The preceding experimental result shown in Table 3 indicates that in this case, the natural frequency is about 0.12 Hz, so theoretically the isolator can work when excitation frequency is higher than 0.17 Hz. Due to the limitation of instruments, however, the output force of the exciter is unstable when excitation frequency is lower than 2 Hz, so the excitation frequency is set to 2 Hz in this experiment. The response of PM is shown in Fig. 15. Obviously, the response frequency is not 2 Hz, because its main component is free vibration instead of forced vibration. Frequency domain analysis in Fig. 16 and time-frequency analysis in Fig. 17 vividly prove the situation. Due to the small damping ratio and the limitation of the lasting time of the experiment, the real forced vibration is not obvious during the whole duration of the experiment, which is clearly illustrated in Fig. 17. So a numerical simulation is carried out to validate the isolation performance, which is shown in Fig. 18. The damping ratio is enlarged in numerical simulation, so the free vibration attenuates faster in Fig. 18. The simulation result shows that the amplitude of steady-state vibration is about zero, so it is almost completely submerged in free vibration and hard to be observed in the experiment. The experiment and simulation above can validate the effectiveness of isolation mode in isolating low frequency vibration. In maneuver mode, an experiment is carried out to validate the performance of rapid maneuver of the isolator. In some space missions, the satellite has to adjust its attitude quickly, so experiment should be carried out to validate if PM can follow the quick motion of SM under this circumstance. Compared with isolation mode experiment, the exciter is no longer needed in this experiment and both PM and SM are free to float on the platform and only one laser displacement sensor is used to record the relative displacement between PM and SM. SM is controlled by hand and may suddenly accelerate or decelerate. The current in this mode is set to 1000 mA so that PM can follow SM when it accelerates and avoid collision with SM during its deceleration, as shown in Fig. 19. One can see from Fig. 19 that the velocity profile of SM is not very smooth, due to the limitation of manual operation. The figure shows that when SM accelerates, for example, from 322 to 323 s and from 329 to 330 s, the relative distance between the two modules increases but does not exceed 15 mm, which means they don't separate. When SM decelerates, for example, from 331 to 339 s, the relative displacement reduces but does not exceed −15 mm, which means collision is avoided. Finally, when SM stops gradually, the relative displacement amplitude fluctuates greatly first and then approaches zero. With the preceding analyses, conclusion can be drawn that the proposed isolator is qualified for maneuver mode. Considering that the stiffness of the prototype is quite low due to the limitation of magnets and the turn number of the coils, the practical performance can be greatly improved with the application of better instruments.

Acknowledgements This work was supported by the National Natural Science Foundation of China under Grants 11732006 and 11702146 and the Equipment Pre-research Foundation under Grant 6140210010202, and the Research Fund of State Key Laboratory of Mechanics and Control of Mechanical Structures under Grant NUAA MCMS-0118G01. References [1] Y. Zhang, C. Sheng, Q. Hu, M. Li, Z. Guo, R. Qi, Dynamic analysis and control application of vibration isolation system with magnetic suspension on satellites, Aero. Sci. Technol. 75 (2018) 99–114, https://doi.org/10.1016/j.ast.2017.12.041. [2] M. Li, Y. Zhang, Y. Wang, Q. Hu, R. Qi, The pointing and vibration isolation integrated control method for optical payload, J. Sound Vib. 438 (2019) 441–456, https://doi.org/10.1016/j.jsv.2018.09.038. [3] N. Pedreiro, Spacecraft architecture for disturbance-free payload, J. Guid. Control Dyn. 26 (5) (2003) 794–804, https://doi.org/10.2514/2.5114. [4] H.A. Sodano, D.J. Inman, Non-contact vibration control system employing an active eddy current damper, J. Sound Vib. 305 (4–5) (2007) 596–613, https://doi.org/10. 1016/j.jsv.2007.04.050. [5] W. Robertson, B. Cazzolato, A. Zander, Theoretical analysis of a non-contact spring with inclined permanent magnets for load-independent resonance frequency, J. Sound Vib. 331 (6) (2012) 1331–1341, https://doi.org/10.1016/j.jsv.2011.11.011. [6] Y. Xu, H. Liao, L. Liu, Y. Wang, Modeling and robust H-infinite control of a novel non-contact ultra-quiet Stewart spacecraft, Acta Astronaut. 107 (2015) 274–289, https://doi.org/10.1016/j.actaastro.2014.11.033. [7] M. Guelman, R. Waller, A. Shiryaev, M. Psiaki, Design and testing of magnetic controllers for Satellite stabilization, Acta Astronaut. 56 (1) (2005) 231–239, https://doi.org/10.1016/j.actaastro.2004.09.028. [8] S. Park, S. Min, Design of magnetic actuator with nonlinear ferromagnetic materials using level-set based topology optimization, IEEE Trans. Magn. 46 (2) (2010) 618–621, https://doi.org/10.1109/TMAG.2009.2033336. [9] D.Q. Truong, T.Q. Thanh, K.K. Ahn, Development of a novel linear magnetic actuator with trajectory control based on an online tuning fuzzy PID controller, Int. J. Precis. Eng. Manuf. 13 (8) (2012) 1403–1411, https://doi.org/10.1007/s12541012-0184-2. [10] S. Afshar, M.B. Khamesee, A. Khajepour, Optimal configuration for electromagnets and coils in magnetic actuators, IEEE Trans. Magn. 49 (4) (2013) 1372–1381, https://doi.org/10.1109/TMAG.2012.2232676. [11] X. Lu, F. Chen, Y. Altintas, Magnetic actuator for active damping of boring bars, CIRP Annals 63 (1) (2014) 369–372, https://doi.org/10.1016/j.cirp.2014.03.127. [12] T. Zhu, B. Cazzolato, W.S.P. Robertson, A. Zander, Vibration isolation using six degree-of-freedom quasi-zero stiffness magnetic levitation, J. Sound Vib. 358 (2015) 48–73, https://doi.org/10.1016/j.jsv.2015.07.013. [13] Q. Wu, H. Yue, R. Liu, L. Ding, Z. Deng, Parametric design and multiobjective optimization of maglev actuators for active vibration isolation system, Adv. Mech. Eng. 6 (2015) 215358, https://doi.org/10.1155/2014/215358. [14] H. Ehrpais, J. Kütt, I. Sünter, E. Kulu, A. Slavinskis, M. Noorma, Nanosatellite spinup using magnetic actuators: ESTCube-1 flight results, Acta Astronaut. 128 (2016) 210–216, https://doi.org/10.1016/j.actaastro.2016.07.032. [15] Z. Gong, L. Ding, H. Yue, H. Gao, R. Liu, Z. Deng, Y. Lu, System integration and control design of a maglev platform for space vibration isolation, J. Vib. Control 25 (11) (2019) 1720–1736, https://doi.org/10.1177/1077546319836892. [16] J. Onoda, T. Sanot, K. Kamiyama, Active, passive, and semiactive vibration suppression by stiffness variation, AIAA J. 30 (12) (1992) 2922–2929, https://doi.org/ 10.2514/3.48978. [17] J. Onoda, K. Minesugi, Alternative control logic for type-II variable-stiffness system, AIAA J. 34 (1) (1996) 207–209, https://doi.org/10.2514/3.13049. [18] E. Sonmez, S. Nagarajaiah, Structures with semiactive variable stiffness single/ multiple tuned mass dampers, J. Struct. Eng. 133 (1) (2007) 67–77, https://doi.org/ 10.1061/(ASCE)0733-9445(2007)133:1(67). [19] Y. Liu, H. Matsuhisa, H. Utsuno, Semi-active vibration isolation system with

4. Conclusions In this paper, a noncontact electromagnetic isolator with controllable stiffness is proposed. The effects of system parameters on the stiffness of the isolator are discussed and a combination of parameters which can produce an approximately linear stiffness is found. 136

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systems, J. Sound Vib. 363 (2016) 18–32, https://doi.org/10.1016/j.jsv.2015.10. 024. [22] R. Ravaud, G. Lemarquand, S. Babic, V. Lemarquand, C. Akyel, Cylindrical magnets and coils: fields, forces, and inductances, IEEE Trans. Magn. 46 (9) (2010) 3585–3590, https://doi.org/10.1109/TMAG.2010.2049026.

variable stiffness and damping control, J. Sound Vib. 313 (1–2) (2008) 16–28, https://doi.org/10.1016/j.jsv.2007.11.045. [20] S. Sun, H. Deng, H. Du, W. Li, J. Yang, G. Liu, G. Alici, T. Yan, A compact variable stiffness and damping shock absorber for vehicle suspension, IEEE-ASME T. Mech. 20 (5) (2015) 2621–2629, https://doi.org/10.1109/TMECH.2015.2406319. [21] T. Wu, C. Lan, A wide-range variable stiffness mechanism for semi-active vibration

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