Design and analysis of an adaptive negative resistance shunted electromagnetic isolator

Abstract

The variation of electrical parameters will inevitably influence the performance of electromagnetic shunt damping. To overcome this, this paper presents an adaptive negative resistance shunt to improve the vibration control performance. A novel electromagnetic isolator is designed, modelled and fabricated. The electromagnetic coupling coefficient is obtained according to the current model, and the corresponding parameters are optimized numerically and verified experimentally. An adaptive negative resistance shunt circuit is constructed. The results show that the proposed adaptive negative resistance shunt can adjust resistance to an optimal value and thus improve the vibration isolation performance of the electromagnetic isolator.

Keywords

Shunt damping adaptive negative resistance electromagnetic isolator

1. Introduction

Unexpected vibrations may result in serious damage to a host structure, which can be reduced by vibration isolators. The electromagnetic shunt damping involves an electrical impedance connected acrossing the terminals of electromagnetic transducers. In decade, many researchers have studied this method extensively. The main benefits of electromagnetic transducers are smaller voltages, larger strokes [1]. The most applicated is the resonant shunt [2–5]. The negative impedance can cancel the internal impedance of a electromagnetic transducer, which may result in the increase of the current flowing in the external circuit, and thereby increasing the damping force to improve vibration suppression performance [6–8]. This method has been applicated as a kind of selfsensing technique to control vibrations of a space antenna [9]. In the past few years, the adaptive resonant shunt has been proposed to compensate the frequency shift so as to improve the vibration control performance [10]. We found that the variation of the parameters of the coil will absolutely influence the vibration control performance of the electromagnetic transducer [11]. Hence, this paper proposes an adaptive negative resistance electromagnetic shunt damping to overcome such kind of problem.

An electromagnetic isolator is designed and the corresponding electromagnetic coupling coefficient is obtained with the current model and verified experimentally. After that, the vibration isolation of a single degree of freedom system is analyzed using the proposed adaptive negative resistance shunt.

2. Electromagnetic coupling model of the isolator

Figure 1 shows the prototype of the electromagnetic isolator. According to Faraday’s law, the electromotive force (EMF) is $\begin{eqnarray}\displaystyle \mathbf{V }_{e}=\int \boldsymbol{v}\times \mathbf{B}\cdot dl & & \displaystyle\end{eqnarray}$ (1) where B is the magnetic flux density, v is the velocity of coil cutting the magnetic induction line, and l is the length of the coil.

Fig. 1.

Prototype of the electromagnetic isolator.

2.1. Modeling of a ring permanent magnet

Figure 2 is the amperian current model of the ring permanent magnet. Assume that the magnet is uniformly magnetized along the radial direction M = M r, the volume current density becomes zero, i.e. J _m = ∇ × M = 0, and the equivalent surface current density of the permanent magnet is $\begin{eqnarray}\displaystyle \mathbf{K}_{\text{m}}=\mathbf{M}\times \mathbf{e}_{\text{n}} & & \displaystyle\end{eqnarray}$ (2) where ∇ denotes the gradient operator, r is the unit radial vector, e _n is the outward unit normal of the top or bottom surface of the permanent magnet. The ring magnet is uniformly magnetized along the radial direction, so that the equivalent current distributes only on the top and bottom surfaces of the magnet, and the surface current density is $\begin{eqnarray}\displaystyle k_{m}=B_{r}/{\mu}_{0} & & \displaystyle\end{eqnarray}$ (3) where B _r is the residual magnetic flux density of the magnet, 𝜇₀ is the permeability of the vacuum, |M| can be calculated by B _r∕𝜇₀. According to the Biot-Savart’s law, the magnetic flux density at point P(θ_P, r _p, h _p) produced by k _m rdrdθ at points Q _L(θ, r, 0) and Q _H(θ, r, H) can be written as $\begin{eqnarray}\displaystyle d\mathbf{B}_{\text{P}}=\frac{{\mu}_{0}}{4{\pi}}\left(\frac{\mathbf{K}_{\text{m}}\times \mathbf{Q}_{\text{L}}\mathbf{P}}{|\mathbf{Q}_{\text{L}}\mathbf{P}|^{3}}-\frac{\mathbf{K}_{\text{m}}\times \mathbf{Q}_{\text{H}}\mathbf{P}}{|\mathbf{Q}_{\text{H}}\mathbf{P}|^{3}}\right)rdrd{\theta}. & & \displaystyle\end{eqnarray}$ (4) Integrating Eq. (3), the magnetic flux density of the top and bottom surfaces at point P(θ_P, r _p, h _p) is expressed as $\begin{eqnarray}\displaystyle \mathbf{B}_{\text{P}} & = & \displaystyle \frac{{\mu}_{0}}{4{\pi}}\int _{R_{1}}^{R_{2}}\int _{0}^{2{\pi}}\left(\frac{\mathbf{K}_{\text{m}}\times \mathbf{Q}_{\text{L}}\mathbf{P}}{|\mathbf{Q}_{\text{L}}\mathbf{P}|^{3}}-\frac{\mathbf{K}_{\text{m}}\times \mathbf{Q}_{\text{H}}\mathbf{P}}{|\mathbf{Q}_{\text{H}}\mathbf{P}|^{3}}\right)rdrd{\theta}\nonumber\\ \displaystyle & = & \displaystyle \frac{{\mu}_{0}}{4{\pi}}\int _{R_{1}}^{R_{2}}\int _{0}^{2{\pi}}\left(\frac{\mathbf{K}_{\text{m}}\times (\mathbf{Q}_{\text{L}}\mathbf{P}_{\text{L}}+\mathbf{P}_{\text{L}}\mathbf{P})}{|\mathbf{Q}_{\text{L}}\mathbf{P}|^{3}}-\frac{\mathbf{K}_{\text{m}}\times (\mathbf{Q}_{\text{H}}\mathbf{P}_{\text{H}}+\mathbf{P}_{\text{H}}\mathbf{P})}{|\mathbf{Q}_{\text{H}}\mathbf{P}|^{3}}\right)rdrd{\theta}.\end{eqnarray}$ (5) The magnetic flux density can also be expressed as $\begin{eqnarray}\displaystyle \mathbf{B}_{\text{P}}=\mathbf{B}_{\text{P}{\theta}}+\mathbf{B}_{\Pr }+\mathbf{B}_{\text{Ph}} & & \displaystyle\end{eqnarray}$ (6) where $\begin{eqnarray}\displaystyle \begin{array}{@{}lcl@{}}\displaystyle \mathbf{B}_{\text{Pr}} & = & \displaystyle \frac{{\mu}_{0}}{4{\pi}}\int _{R_{1}}^{R_{2}}\int _{0}^{2{\pi}}\left(\frac{\mathbf{K}_{\text{m}}\times \mathbf{P}_{\text{L}}\mathbf{P}}{|\mathbf{Q}_{\text{L}}\mathbf{P}|^{3}}-\frac{\mathbf{K}_{\text{m}}\times \mathbf{P}_{\text{H}}\mathbf{P}}{|\mathbf{Q}_{\text{H}}\mathbf{P}|^{3}}\right)r\cos ({\theta}_{\text{p}}-{\theta})drd{\theta}\\ \mathbf{B}_{\text{P}{\theta}} & = & 0\\ \displaystyle \mathbf{B}_{\text{Ph}} & = & \displaystyle \frac{{\mu}_{0}}{4{\pi}}\int _{R_{1}}^{R_{2}}\int _{0}^{2{\pi}}\left(\frac{\mathbf{K}_{\text{m}}\times \mathbf{Q}_{\text{L}}\mathbf{P}_{\text{L}}}{|\mathbf{Q}_{\text{L}}\mathbf{P}|^{3}}-\frac{\mathbf{K}_{\text{m}}\times \mathbf{Q}_{\text{H}}\mathbf{P}_{\text{H}}}{|\mathbf{Q}_{\text{H}}\mathbf{P}|^{3}}\right)rdrd{\theta}.\end{array} & & \displaystyle\end{eqnarray}$ (7) According to Fig. 1, the coil only cuts the radial magnetic induction line of the magnet, therefore, the EMF is only dependent on the radial component of the magnetic flux density B _Pr. When there is current flowing in the coil, only the axial component of the ampere force can be generated to prevent the movement of the coil. Thus, it can be concluded that B _Pr is the effective magnetic flux density. |B _Pr| can be simplified as $\begin{eqnarray}\displaystyle B_{\text{Pr}}(\text{r}_{\text{P}},\text{h}_{\text{P}})=\frac{B_{r}}{4{\pi}}\int _{R_{1}}^{R_{2}}\int _{0}^{2{\pi}}\left(\frac{h_{P}}{d_{L}^{3}}+\frac{H-h_{P}}{d_{H}^{3}}\right)r\cos {\theta}drd{\theta} & & \displaystyle\end{eqnarray}$ (8) where R ₁ and R ₂ are inner and outer radius of the ring permanent magnet, θ_P = 0, |P _L P| = h _P, |P _H P| = H − h _P, d _L = |Q _L P| $∼=∼\sqrt{h_{P}^{2}+r_{P}^{2}+r^{2}-2rr_{P}\cos ({\theta})}$ , d _H = |Q _H P| $∼=∼\sqrt{(H-h_{P})^{2}+r_{P}^{2}+r^{2}-2rr_{P}\cos ({\theta})}$ .

Fig. 2.

Amperian current model of the ring permanent magnet.

Figure 3 shows the comparison of the radial component of the magnetic flux density |B _Pr| between the analytical method and finite element method (FEM). The magnetic flux density obtained by FEM shows good coincidence with that obtained by analytical method. Therefore, the geometrical dimensions of the magnet and the coil in this paper are shown in Table 1.

Fig. 3.

Comparison of the magnetic flux density B _Pr between the analytical and FEM methods.

Fig. 4.

Schematic of the electromagnetic coupling coefficient test and the EMF of the coil.

Table 1

The geometric parameters of the ring permanent magnet and the coil

Property	Value
Inner radius of the permanent magnet, R ₁	16 mm
Outer radius of the permanent magnet, R ₂	26 mm
Height of the permanent magnet, H	20 mm
Residual magnetic flux density, B _r	1.35 T
Equivalent radius of the coil, r _P	13 mm
Height of the coil, h	10 mm
Turns of the coil, N	300 turns
The radial component of the magnetic flux density, B _Pr	0.21 T

2.2. Analysis of the electro-mechanic-magnetic coupling coefficient

In the previous work [7], we have obtained expressions of the electromechanical coupling coefficient C _e and the electromagnetic coupling coefficient C _m, which can be expressed as $\begin{eqnarray}\displaystyle C_{e}=C_{m}=2{\pi}r_{P}NB_{\Pr }. & & \displaystyle\end{eqnarray}$ (9) Substituting the data of Table 1 into Eq. (9) which obtains the theoretical value of C_e is 5.14 N/A.

2.3. Test of the electromagnetic coupling coefficient

The experiment is setup to obtain the electromagnetic coupling of the isolator. Figure 4 shows the schematic of the electromagnetic coupling coefficient experimental test.

According to Eqs (1) and (9), the value of the electro-mechanic-magnetic coupling coefficient is 4.49 N/A, which is less than the theoretical value. Other experimental results are listed in Table 2.

Table 2
The parameters of the system

Property Value

Mass, M 0.447 kg

Damping, C 1.2 N⋅s/m

Stiffness, K 1.53 × 10⁴ N/m

Nature frequency, ω_n 29.4 Hz (184.7 rad/s)

The electromagnetic coupling coefficient, C _e, C _m 4.49 N/A

The coil resistance, R _e 26.6 Ω

The coli inductance, L _e 2.78 mH

Property	Value
Mass, M	0.447 kg
Damping, C	1.2 N⋅s/m
Stiffness, K	1.53 × 10⁴ N/m
Nature frequency, ω_n	29.4 Hz (184.7 rad/s)
The electromagnetic coupling coefficient, C _e, C _m	4.49 N/A
The coil resistance, R _e	26.6 Ω
The coli inductance, L _e	2.78 mH

3. Modeling of the adaptive negative resistance shunt

Figure 5 shows the model of the adaptive negative resistance shunted electromagnetic isolator, where R _e and L _e are the inherent resistance and inductance of the coil, respectively. The negative resistance can be adapted online. The electrical equation of is $\begin{eqnarray}\displaystyle L_{e}{\dot{I}}(t)+R_{e}I(t)+ZI(t)=V_{e} & & \displaystyle\end{eqnarray}$ (10) where V _e is the EMF which can be expressed as $C_{e}[{\dot{x}}(t)-{\dot{y}}(t)]$ , I is the shunt current.

Fig. 5.

Model of the adaptive negative resistance shunted electromagnetic isolator.

The equation of motion is $\begin{eqnarray}\displaystyle M\ddot{x}(t)+C{\dot{x}}(t)+Kx(t)+F_{e}(t)=C{\dot{y}}(t)+Ky(t) & & \displaystyle\end{eqnarray}$ (11) where M, C and K are the mass, the damping and stiffness, respectively. F _e(t) is the electromagnetic shunt damping force that can be expressed as C _e I. Z is the shunt impedance represented by Z = R _s + (jω)L _s, where R _s and L _s are the resistance and inductance of the shunt. This paper only considers the negative resistance, so that L _s = 0. Substituting Eq. (10) into Eq. (11) and driving the Laplace transform, the following relationship can be obtained $\begin{eqnarray}\displaystyle s^{2}MX(s)+sC[X(s)-Y(s)]+K[X(s)-Y(s)]+\frac{sC_{e}^{2}[X(s)-Y(s)]}{sLe+(R_{e}+R_{s})}=0. & & \displaystyle\end{eqnarray}$ (12) According to Eq. (12), the transfer function relating the applied base velocity w(s) and isolated mass velocity v(s) is $\begin{eqnarray}\displaystyle T(s)=G_{wv}(s)=\frac{\left(C+\frac{C_{e}^{2}}{sL_{e}+(R_{e}+R_{s})}\right)s+K}{s^{2}M+\left(C+\frac{C_{e}^{2}}{sL_{e}+(R_{e}+R_{s})}\right)s+K} & & \displaystyle\end{eqnarray}$ (13) where v(s) = sX(s), w(s) = sY(s), and T(s) is the transmissibility of the system.

According to Kirchhoff’s voltage law, the relationship between V _e(s) and V _s(s) is $\begin{eqnarray}\displaystyle V_{e}(s)=\left(\frac{sL_{e}+R}{R_{s}}\right)V_{s}(s) & & \displaystyle\end{eqnarray}$ (14) where V _s(s) = Z(s)I(s) is the voltage acrossing the terminals of the shunt impedance, R = R _e + R _s. It can be found that vibration control can be enhanced when the voltage acrossing the terminals of the shunt impedance increases. In the view of energy conservation, the induced EMF will be very small when the structural vibration is controlled. Therefore, Eq. (14) can be used as a performance function for the adaptive negative resistance control system. Figure 6 shows the schematic diagram of the adaptive negative resistance shunt damping system. A new closed loop is employed to construct a new V _e signal. Then we can minimize this new V _e to find the optimal R _s of the shunt. The red dotted line part is the control loop of the adaptive negative resistance shunted electromagnetic isolator. The negative resistance of the shunt controller is then updated using a gradient search algorithm to minimize the performance function.

Fig. 6.

Schematic diagram of the adaptive negative resistance shunt damping system.

In this paper, the same adaptation technique using the EMF is applied to the electromagnetic isolator. We choose the performance function as E[V _e(t)²] i.e., the objective is to minimize the RMS of EMF. The EMF V _e(t) can be expressed as $\begin{eqnarray}\displaystyle E[V_{e}(t)^{2}]=\int _{-\infty }^{+\infty }V_{e}(t)^{2}dt=\frac{1}{2{\pi}}\int _{-\infty }^{+\infty }S_{V_{e}}({\omega})d{\omega}. & & \displaystyle\end{eqnarray}$ (15) The optimal negative resistance shunt controller is then obtained $\begin{eqnarray}\displaystyle R_{s}^{\ast }=\text{arg}∼∼\min _{R_{s}>R_{e}}E[V_{e}(t)^{2}]. & & \displaystyle\end{eqnarray}$ (16) Using the discrete time equivalent of E[V _e(t)²], we can approximate the performance function as $\begin{eqnarray}\displaystyle E[V_{e}(t)^{2}]=\frac{1}{N}\mathop{\sum }_{i=kN}^{(k+1)N-1}V_{e}(i{T}_{s})^{2}. & & \displaystyle\end{eqnarray}$ (17) The update of negative resistance is realized using a gradient search algorithm to minimize V _k $\begin{eqnarray}\displaystyle R_{s_{k+1}}=R_{s_{k}}-{\alpha}\frac{V_{k}}{d_{k}} & & \displaystyle\end{eqnarray}$ (18) where V _k = E[V _e(t)²], $d_{k}=\frac{V_{k}∼-∼V_{k-1}}{R_{s_{k}}-∼R_{s_{k-1}}}$ , 𝛼 = 0.05. Figure 7 shows the schematic of the gradient algorithm.

Fig. 7.

Schematic diagram of the gradient algorithm.

4. Vibration control performance analysis

First, an initial negative resistance value for the shunt is set. When the system reaches up to the steady-state, then taking into the parameters change of the system into consideration. In this paper, we increase the mass of the system so that the natural frequency of the system decrease which influences the vibration control performance of the shunt. The results show that the system can be retuned. Figure 8 shows the frequency response of the system without the shunt and with the shunt cases under a white noise excitation shown in Fig. 9. The result demonstrates that the amplitude can be reduced by 22.32 dB. Figure 10 is the iteration of the negative resistance and the voltage, and the Fig. 11 is the time history response of the isolator.

Fig. 8.

Frequency response of the isolator.

Fig. 9.

Power spectrum of the input excitation.

Fig. 10.

Iteration of (a) the negative resistance and (b) the voltage.

Fig. 11.

Time history response of the isolator with the adaptive negative resistance shunt.

5. Conclusions

This paper proposes a novel adaptive negative resistance shunted electromagnetic isolator. The electromagnetic coupling is modelled theoretically and obtained experimentally. An adaptive negative resistance shunt circuit is constructed and the corresponding isolation system is established. The analysis is carried out and the result shows that the vibration can be reduced by 22.32 dB. Furthermore, the proposed performance function can provide performance robustness under a variable operating condition.

Footnotes

Acknowledgements

This work is national natural science foundation of China under grant no. 11602223, Open Projects of State Key Laboratory for Strength and Vibration of Mechanical Structures (SV2016-KF-14), and the Young Researchers Foundation of Zhejiang Provincial Top Key Academic Discipline of Mechanical Engineering of Zhejiang Sci-tech University under grant no. ZSTUME02B04.

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Design and analysis of an adaptive negative resistance shunted electromagnetic isolator

Abstract

Keywords

1. Introduction

2. Electromagnetic coupling model of the isolator

Table 2 The parameters of the system Property Value Mass, M 0.447 kg Damping, C 1.2 N⋅s/m Stiffness, K 1.53 × 104 N/m Nature frequency, ω n 29.4 Hz (184.7 rad/s) The electromagnetic coupling coefficient, C e , C m 4.49 N/A The coil resistance, R e 26.6 Ω The coli inductance, L e 2.78 mH

Footnotes

Acknowledgements

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