Nonlinear dynamic and electromagnetic interference coupled-field analysis in a semi-active suspension system with magneto-rheological damper

Abstract

A groundhook controlled semi-active suspension system equipped with magneto-rheological damper (MRD) is considered. Both the hysteresis of MRD and control discontinuity, the nonlinearity will greatly affect system stability of the vehicle. In this paper, the performance and nonlinear dynamic of semi-active suspension adopting modified groundhook is evaluated. The influence of frequency of harmonic excitation and parameter of modified groundhook controller on the nonlinear dynamics of the semi-active suspension system is investigated. More importantly, nonlinear dynamic analysis is carried out to study the semi-active suspension system, with the methods of bifurcation diagram, the Lyapunov exponent spectrum, time history, phase plane and power spectrum in detail for the first time. Furtherly, electromagnetic interference (EMI) problems induced by chaotic motion are analyzed and discussed according to EN 55022 for another first time. It is indicated that both of periodic, quasi-periodic and chaotic motions exist. The system undergoes a complex nonlinear dynamical evolution with the excitation change. The coupling influence between chaos and EMI problems cannot be ignored in the further design of the semi-active controller. This work laid a theoretical foundation for EMI reduction and nonlinear control of the intelligent MR semi-active suspension.

Keywords

Magneto-rheological suspension modified groundhook nonlinear dynamic conducted electromagnetic interference

1 Introduction

Magneto-rheological damper (MRD) is considered to be the most promising way to realize semi-active suspension for road vehicle, for its advantage of convenient controllable and fast response [1,2]. Despite a great amount of researches have been conducted concentrating on the calculation modelling and semi-active control strategy, inherent hysteresis as a typical nonlinearity may limit its further practical for commercialization. Because it may cause chaos, and MRD is no exception [3,4].

Recently, more and more researches tend to systematically reveal nonlinear dynamics of the system with hysteresis, such as MR suspension system. Amit Shukla studied the effect of various system and control parameters on the two-degree-of-freedom system with MRD numerically, to reveal parametric bifurcation stability behavior. The results indicated that system could undergo loss of stability via Hopf bifurcation and exhibit limited cycle oscillations [3]. Yang investigated a nonlinear suspension system with hysteretic characteristics. Under harmonic excitation, the dynamic path from quasi-periodic to chaos was verified by deriving Melnikov method [4]. Litak et al. found a possible chaotic motion in a nonlinear vehicle suspension system which was subjected to multi-frequency excitation from a road surface. Then he further investigated global homoclinic bifurcation and transition to chaos in the case of a quarter-car model excited kinematically by a road surface profile [5,6]. Naik used the Melnikov criterion to study the intersection of stable and unstable manifolds and transition to chaos in the system with MRD. The condition for chaotic vibration was found by a bifurcation diagram and the Lyapunov exponents (LE). He thus pointed out the limitation of MRD that inherent hysteresis may cause chaos [7]. Wu built a 4 degree-of-freedom (4-DOF) semi-active suspension model with Sigmoid model of MRD. Its nonlinear dynamics were analyzed that the system would experience complex vibration status with the change of road excitation amplitude, such as periodic motion, and chaotic motion [8]. Siewe applied multiple scales method to analyze local bifurcation in the quarter-car system with periodically excited road profile. A variety of nonlinear behaviors were observed, such as resonance and anti-resonance phenomena, saddle-node bifurcation [9]. In our previous study, dynamic analysis of the MR suspension system was completely performed, based on observation of dynamic trajectories, bifurcation diagrams, and the power spectrum density (PSD) diagram, and thus revealed the effects of frequency and amplitude of the excitation. The main road to chaos was periodic doubling bifurcation and Hopf bifurcation. The periodic-window existed frequently along with intermittent chaos, due to saddle-node bifurcation [10–12]. For the confirmation of the existence of chaos in this kind of model by experiment, more significantly, the experiment was conducted for detecting the chaotic motion in MRD based vibration isolation system. The results verified the theoretical analysis, although the dynamical evolution observed in the experiment was simpler than that observed in the simulation [11].

The previous research rarely analyzed the MR suspension together with semi-active controller. Nevertheless, discontinuous controller especially based on on-off control strategy, such as skyhook, groundhook, bang-bang control etc., may also cause nonlinear dynamic behaviors. Such on-off controller is widely used for semi-active controlling, due to easy implementation. The uncertain nonlinear behaviors caused by hysteresis combing with discontinuous controlling have to be systematic studied. It has significance for optimization of semi-active controller, especially for determining parameters to avoid nonlinear motion such as chaos. In addition, as a typical electronic control system, the semi-active MR suspension may cause the electromagnetic compatibility (EMC) problems, which have been realized and studied by researchers at the level of the electronic circuit. Moreover, it is obvious that the uncertain feedback signal of the nonlinear system motion tends to cause the drive current disordered. Such disordered current will inevitably generate electromagnetic interference noise. To our knowledge, there are no relevant studies in which nonlinear dynamic induced electromagnetic interference (EMI) problems has been evaluated. That is the purpose and values why we conducted this work.

In this paper, nonlinear dynamic and electromagnetic interference coupled-field analysis is presented. A modified groundhook strategy controlled semi-active suspension is systematically analyzed. Suspension performance under different controller parameters is compared. With special parameters, nonlinear characteristics under harmonic excitation are obtained by means of two parameters plots. The detail evolution from periodic to chaotic motion at both varying frequency and varying amplitude is then revealed. Furtherly, EMI noise induced by the drive current of MRD is calculated, along with the nonlinear dynamic evolution. The paper is organized as follows. In Section 2, dynamical model of MR semi-active suspension system is built based on Bouc-wen calculation model for MRD. In Section 3, a modified groundhook-based semi-active control policy in the drive current form is proposed, and its control effect under different control parameters is accessed. In Section 4, the nonlinear dynamic under varying frequency and amplitude of harmonic excitation is systematically analyzed. Then in Section 5, according to GB 9254B_QP EMI characteristics are evaluated by complied with EN 55022. Finally, conclusions of the research and future work are explained in Section 6.

2 Dynamical model of MR semi-active suspension system

Figure 1 is dynamical model of the 2-DOF semi-active suspension system with MRD. Wherein, m _s and m _u denote the sprung and unsprung masses. k _s, i _d and F _d are the suspension stiffness, DC drive current and yielded damping force of MRD. k _t and c _t denote equivalent stiffness and damping coefficient, and x _i, x _s and x _u respectively are the road excitation, vertical vibration displacements of the sprung and unsprung masses. The dynamic model is formulated as $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}m_{\text{s}}\ddot{x}_{\text{s}}=-k_{\text{s}}(x_{\text{s}}-x_{\text{u}})-F_{\text{d}},\\ m_{\text{u}}\ddot{x}_{\text{u}}=k_{\text{s}}(x_{\text{s}}-x_{\text{u}})+F_{\text{d}}-k_{\text{t}}(x_{u}-x_{i})-c_{t}({\dot{x}}_{u}-{\dot{x}}_{i}).\end{array}\right. & & \displaystyle\end{eqnarray}$ (1)

The output F _d of MRD is described by the modified Bouc-wen model proposed in [13] as followed, wherein the hysteretic characteristics was decoupled from current modulation separation. $\begin{eqnarray}\displaystyle F_{\text{d}}=c(i_{\text{d}})F_{\text{h}}(x_{r},v_{\text{r}}),\quad 0\leq i_{d}\leq I_{m} & & \displaystyle\end{eqnarray}$ (2) $\begin{eqnarray}\displaystyle c(i_{\text{d}})=1+\frac{k_{2}}{1+\exp (-a_{0}(i_{\text{d}}+I_{0}))}-\frac{k_{2}}{1+\exp (-a_{0}I_{0})}, & & \displaystyle\end{eqnarray}$ (3) $\begin{eqnarray}\displaystyle F_{\text{h}}(x_{r},v_{\text{r}})=c_{1}{\dot{y}}+k_{1}(x_{r}-x_{0}), & & \displaystyle\end{eqnarray}$ (4) $\begin{eqnarray}\displaystyle {\dot{y}}=\frac{1}{c_{0}+c_{1}}[{\alpha}z+c_{0}{\dot{x}}_{\text{r}}+k_{0}(x_{\text{r}}-y)], & & \displaystyle\end{eqnarray}$ (5) $\begin{eqnarray}\displaystyle {\dot{z}}=-{\gamma}\left|{\dot{x}}_{\text{r}}-{\dot{y}}\right|z\left|z\right|^{n-1}-{\beta}\left({\dot{x}}_{\text{r}}-{\dot{y}}\right)\left|z\right|^{n}+A({\dot{x}}_{\text{r}}-{\dot{y}}), & & \displaystyle\end{eqnarray}$ (6) where i _d denotes DC drive current. x _r is the piston travel of the MRD, which equal to x _s − x _u. The model totally has thirteen parameters of k ₀, k ₁, k ₂, a ₀, I ₀, 𝛼, 𝛽, 𝛾, c ₀, c ₁, n, A, x ₀. The detailed definitions can be found in [13].

A commercially MRD, which has a maximum allowable input DC current 0.5 A at 12 V, is employed. The parameters of modified model are identified as k ₀ = 185.954, k ₁ = 2750.7, k ₂ = 9.991, a ₀ = 7.601, I ₀ = 0.0683, 𝛼 = 19,970, 𝛽 = 179,900, 𝛾 = 8901.5, c ₀ = 1367.4, c ₁ = 6204.1, n = 2, A = 24.518, x ₀ = −0.004. The characteristics of hysteresis and excitation dependency are described properly, as showed in Fig. 2.

3 Semi-active controller based on modified groundhook damping scheme

The groundhook control law suggests that the magnitude of mass acceleration could be reduced by reducing the damping force when the unsprung mass velocity ( ${\dot{x}}_{\text{u}}$ ) is in the same phase with the damper relative velocity across the suspension spring. The groundhook damping algorithm is thus expressed as followed. $\begin{eqnarray}\displaystyle F_{d}=\left\{\begin{array}{@{}l@{}}c_{\text{ground}}{\dot{x}}_{s},\quad \text{if }{\dot{x}}_{u}(-{\dot{x}}_{s}+{\dot{x}}_{u})\geq 0.\\ 0,\quad \text{if }{\dot{x}}_{u}(-{\dot{x}}_{s}+{\dot{x}}_{u})<0.\end{array}\right. & & \displaystyle\end{eqnarray}$ (7) Where c _ground is a large damping coefficient.

The groundhook damping scheme has a superior potential in improving the multi-objective suspension performance of vehicle. A modified groundhook-based semi-active control policy in the drive current form is formulated as $\begin{eqnarray}\displaystyle i_{d}=\left\{\begin{array}{@{}l@{}}k_{\text{ground}}\left|{\dot{x}}_{s}\right|^{2},\quad {\dot{x}}_{u}(-{\dot{x}}_{s}+{\dot{x}}_{u})\geq 0.\\ 0,\quad {\dot{x}}_{u}(-{\dot{x}}_{s}+{\dot{x}}_{u})<0.\end{array}\right. & & \displaystyle\end{eqnarray}$ (8)

The frequency domain responses are utilized to evaluate comprehensive suspension performances of ride comfort and handling safety, including resonance suppression, vibration isolation, limited suspension dynamic travel, road holding, etc. The frequency domain response comparisons of groundhook-based MR suspension systems is presented in Fig. 3, under the harmonic excitation with amplitudes varying at road vehicle dominant frequency band [8,11] as followed. $\begin{eqnarray}\displaystyle x_{\text{i}}=\left\{\begin{array}{@{}l@{}}a_{\text{m}}\sin (2{\pi}ft),\quad f\leq f_{\text{T}}.\\ a_{\text{m}}(f_{\text{T}}/f)\sin (2{\pi}ft),\quad f>f_{\text{T}}.\end{array}\right. & & \displaystyle\end{eqnarray}$ (9)

Where the frequency band f and the cut-off frequency f _T are chosen as 0–20 Hz and 2.1 Hz, respectively, and the system evaluation indexes are chosen as the sprung mass displacement acceleration transmissibility T _as, unsprung mass displacement acceleration transmissibility T _au, travel displacement transmissibility T _dr and dynamic load coefficient (DLC) [13].

Suspension performances comparison of semi-active of different k _ground is presented. As showed in Fig. 3, the first resonance of T _as decreases obviously with the increase of groundhook coefficient, while in the range of 0.5–1 Hz larger k _ground has larger T _as. In high frequency band, magnitude of T _au, T _dr and DLC of different k _ground show little change especially after unsprung mass resonance frequency 9.8 Hz, while has significant change at the first resonance. For T _au, larger k _ground also has larger T _au. Note that for k _ground = 50 and 100, the curve turns to be more fluctuated, which indicate that the system response may change. The same fluctuation exists in 0.5–1.2 Hz of T _dr. Moreover, in the low frequency band of 0.5–1 Hz, magnitudes of T _au, T _dr and DLC show an obvious increase with k _ground getting larger, respectively, which scarify part of unsprung performance. In a word, the modified groundhook controller resolves the contradictory control requirement on suspension damping with a compromise scheme. At the same time, control coefficient k _ground may influence the system characteristics.

4 Nonlinear dynamic analysis of skyhook-based semi-active control MR suspension system

4.1 Stability analysis

In the above section, k _ground is set as 50. The stability of the resonant oscillatory state motion can be determined by investigating the nature of the stationary oscillatory state solutions. By setting x ₁ = x _s, $x_{2}={\dot{x}}_{\text{s}}$ , x ₃ = x _u, $x_{4}={\dot{x}}_{\text{u}}$ , x ₅ = y, x ₆ = z, the state space is derived in (10), which is divided into eight sections from (4) and (8). $\begin{eqnarray}\displaystyle \begin{array}{@{}l@{}}{\dot{x}}_{1}=x_{2}\\ {\dot{x}}_{2}=a_{21}x_{1}+a_{22}x_{2}+a_{23}x_{3}+a_{24}x_{4}+a_{25}x_{5}+a_{26}x_{6}\\ {\dot{x}}_{3}=x_{4}\\ {\dot{x}}_{4}=a_{41}x_{1}+a_{42}x_{2}+a_{43}x_{3}+a_{44}x_{4}+a_{45}x_{5}+a_{46}x_{6}\\ {\dot{x}}_{5}=a_{51}x_{1}+a_{52}x_{2}+a_{53}x_{3}+a_{54}x_{4}+a_{55}x_{5}+a_{56}x_{6}\\ {\dot{x}}_{6}=(A+Bx_{6}^{2})(a_{61}x_{1}+a_{62}x_{2}+a_{63}x_{3}+a_{64}x_{4}+a_{65}x_{5}+a_{66}x_{6}),\end{array} & & \displaystyle\end{eqnarray}$ (10) where the no dimensional coefficients are listed in the Appendix. The Jacobian matrix at fixed point X ₀ = (0, 0, 0, 0, 0, 0) is as followed. $\begin{eqnarray}\displaystyle J=\left[\begin{array}{@{}cccccc@{}}0-{\lambda} & 1 & 0 & 0 & 0 & 0\\ a_{21} & a_{22}-{\lambda} & a_{23} & a_{24} & a_{25} & a_{26}\\ 0 & 0 & 0-{\lambda} & 1 & 0 & 0\\ a_{41} & a_{42} & a_{43} & a_{44}-{\lambda} & a_{45} & a_{46}\\ a_{51} & a_{52} & a_{53} & a_{54} & a_{55}-{\lambda} & a_{56}\\ Aa_{61} & Aa_{62} & Aa_{63} & Aa_{64} & Aa_{65} & Aa_{66}-{\lambda}\end{array}\right]. & & \displaystyle\end{eqnarray}$ (11)

For ${\dot{x}}_{\text{u}}(-{\dot{x}}_{\text{s}}+{\dot{x}}_{\text{u}})\geq 0$ , the Jacbian matrix of the system is calculated as 𝜆₁ = −1.2547 + 3.2171i, 𝜆₂ = −1.2547 −3.2171i, 𝜆₃ = −0.0089 + 0.2156i, 𝜆₄ = −0.0089 −0.2156i, 𝜆₅ = 0. 0073, 𝜆₆ = 0.

For ${\dot{x}}_{\text{u}}(-{\dot{x}}_{\text{s}}+{\dot{x}}_{\text{u}})<0$ , the Jacbian matrix of the system is calculated as 𝜆₁ = −1.2534 + 1.1617i, 𝜆₂ = −1.2534 −1.1617i, 𝜆₃ = −0.0119 + 0.3103i, 𝜆₄ = −0.0119 −0.3103i, 𝜆₅ = 0.0057, 𝜆₆ = 0.

Note, 𝜆_1,2 and 𝜆_3,4 are two couples of conjugate complex roots, while 𝜆₅ is positive and 𝜆₆ is zero under two conditions. Therefore, the system is instable at X ₀.

4.2 Nonlinear response and bifurcation under single-frequency harmonic road excitation

Single-frequency harmonic excitation is commonly used for analysis of non-autonomous dynamical systems. For the above parameter values of the system and control parameter k _ground = 10, bifurcation diagrams are drawn by direct numerical integration of the original equations of motion, to investigate more in depth the evolution of system response. The time step is fixed by dividing the excitation period in 4096 points. The integration is carried out over 1000 cycles of the excitation, by considering those obtained at the previous excitation frequency as initial conditions. The last 50 points of the Poincare map are recorded for the bifurcation diagram. The common range of the frequency for dynamic analysis of the suspension system is from 1 Hz to 20 Hz. Here, the range of the frequency is extended to 30 Hz and 50 Hz, to fully present the evolution. The two parameters space plots are calculated to investigate the effects of the influence of external excitation on chaos globally. Note that Wolf method is used to calculate the largest LE (LLE) in this paper [14]. For each value of the varied parameter, the same initial conditions are used. The time history, phase plane and the power spectrum density (PSD) of the system response are got by utilizing common method.

(1) Two parameters space plots

In the range of A _mp ∈ [0.00, 0.10] m and f ∈ [1.5, 10] Hz, the two parameters space plot is drawn, according to the following equation. The steps for A _mp and f are set as 0.0005 and 0.01 respectively. As showed in Fig. 3, the black region indicates chaotic motions, estimated based on the LLE. In the chaotic region, all the LLEs are positive. $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}LLE_{\text{(Amp,f)i}}>0,\quad \text{Chaotic region}\\ LLE_{\text{(Amp,f)i}}<=0,\quad \text{Non-chaotic region}.\end{array}\right. & & \displaystyle \nonumber\end{eqnarray}$

Obviously, the chaotic motion principally distribute in three regions, in which the low-frequency is narrow. In addition, the chaotic and non-chaotic motions alternately appear in mid-frequency with larger amplitude, which indicates the complicated dynamics behaviors.

(2) Dynamic evolution for frequency varying

In Fig. 5, the bifurcation diagram is drawn in the frequency band of 1–25 Hz with a small amplitude of 0.01 m. Starting from left, period-1 motion continued for f < 7.16 Hz. Subsequently, the period-1 motion experiences a grazing bifurcation, then the quasi-period motion exists. Soon after that, the system turns into the chaotic motion. With f increased to the range of 8.2–8.5 Hz, which determine the handling safety, periodic windows of different periodicity are embedded in the non-periodic regime. The motion of the system undergoes chaos to period-7, and chaos again. Furthermore, the system exhibits a scenario of loss of stability to quasi-periodicity, phase-locking, and torus breakdown. Finally, the system regresses to stable period-1 motion. Corresponding to two parameters space plot, the bifurcation diagram system transmits to the nonlinear region at about 7.16 Hz and escapes from the chaos at about 24.8 Hz. In the chaotic region, the largest LE is varying alternatively positive and negative. It indicates that there are periodic windows in the chaotic region.

The detailed responses are presented through time history, phase portraits and frequency spectrum. In Fig. 6(a), the phase trajectory exhibits a connected two scrolls, and the time history is irregular. In addition, the PSD curve of x _s is continuous, which means abundant frequency components. The existence of chaos is verified. In Fig. 6(b), when f increases to 8.3 Hz, in the periodic window, the time history is periodic. The phase trajectory turns to a closed loop. The PSD curve is discrete. It is showed that the system escapes from chaos. With f further increased, the system tends to the considered limited cycle for f = 10.3 Hz. As showed in Fig. 6(c-1), the time history turns to be periodic from the initial instability, spending 6 s about 600 cycles. A convergent closed loop exists in the phase plane. Moreover, the PSD curve keeps discrete, despite that frequency components are more than that of f = 8.3 Hz.

(3) Dynamic evolution for amplitude varying

The dynamic characteristics of the system for amplitude varying are further analyzed. In Fig. 7, the bifurcation diagrams for the range of A _mp are shown. To fully present the dynamical evolution, the range is also extended to as large as 0.18 m, which seldom exists in practical. Compared with the frequency varying, the motion turns to be more complex. The stable period-1 motion just stays for a short time. Thereafter, until A _mp increases to about 0.045 m, the system gets into the nonlinear region, including quasi-period motion and chaotic motion, through a series of different types of bifurcation. The system suffers the period-doubling bifurcation in a short time near 0.03 m. Soon after that, a grazing bifurcation occurs. Then the system undergoes a short chaotic motion followed another period-4. Through a series of saddle-node bifurcation, chaotic and periodic motions alternate in the range of 0.045–0.17 m. The system undergoes gradually period-5, period-4, period-3 and period-6 motion, and end up in a chaotic region. With amplitude getting larger, the system kept in chaotic region, until simulation breakdown. Combining with Fig. 4, the two parameters plot is also well consistent with bifurcation analysis. Obviously, the motion of the system starts with the periodic motion, through the period-doubling route and the quasi-periodic route to chaotic region intermittently, corresponding to positive largest LE.

The motion state of the specific amplitude is represented through time history, phase portrait and PSD. As showed in Fig. 8, for A _mp = 0.01 m, the closed cycle in the phase plane and discrete PSD show that it is periodic, now that the harmonic of response is serious from time history. As A _mp is increased to 0.05 m, as showed in Fig. 8(b), the time history and power spectra show that variation of the displacement becomes random-like and subharmonics of the modulation frequency in the PSD are noted, which can be considered as a sign of a transition to a chaotic response. With the increasing excitation amplitude, the region of continuous frequency expands. Therefore, chaos appears as a result of destruction of the torus now. As the excitation amplitude is increased further, vibration system displays complicated frequency components including the frequency multiplication. In addition, the system returns to periodic window intermittently. As showed in Fig. 8(c), limit-cycle exists with obvious fluctuation of the amplitude.

5 Current resulting conducted electromagnetic interference noise

Quality of drive current determines directly the suspension performance under semi-active control. It also plays an important role in the safety reliability, because high-frequency harmonic of the current may cause the electromagnetic interference (EMI), which may do harm to other car electric control unit. Especially, the drive current will be disordered when the system is in chaotic motion. The disordered current carries the high-frequency harmonic much higher than the frequency of the input excitation. One of the influences of the impulse current waveform is high heat radiation of the driver module. Complied with EN 55022, the conducted EMI noise of the MR suspension system is derived as followed [15]. $\begin{eqnarray}\displaystyle U=20\lg (5I)+140, & & \displaystyle\end{eqnarray}$ (12) where U denotes the conducted EMI noise of MRD generated under groundhook controlling, dB𝜇V. I is the drive current output by semi-active controller, A.

In Fig. 9, the time history of the current and the calculated conducted EMI noise under critical motion corresponding to Fig. 6 and Fig. 8 are shown. For f = 8 Hz, A _mp = 0.05 m, the system is in chaotic motion. The drive current is not unexpectedly disordered. The conducted EMI noise exceeds the limit in the range between 0.1 to 6 MHz. The possibility of coupling connection between chaotic motion and EMI noise is confirmed. For f = 8.3 Hz, A _mp = 0.05 m, the current looks still disordered, despite the system is periodic now. And the harmonic contents of the signal are mainly in the low frequency domain of about 1 Hz–30 Hz. In range of 0.1–16 MHz, the current meets the EMI requirement. For f = 10.3 Hz, A _mp = 0.05 m, the time history of current also indicates limited cycle. The EMI noise is not exceed. For high frequency of 10 Hz, EMI noise is not exceeded when system is in the periodic motion and the limited cycle like before, as showed in Fig. 9(d) and (f). While for A _mp = 0.05 m as showed in Fig. 9(e), three frequent points exceed the requirement, namely 0.85 MHz, 1.2 MHz and 1.96 MHz. And in the range of 2.7–4 MHz, the EMI noise is close to the GB 9254B_QP. Similarly, the drive current for the MRD when the system falls in the chaotic motion is observed.

6 Conclusion

This paper has assessed the nonlinearity and EMI noise of the groundhook controlled MR semi-active suspension system. Various system responses under single frequency excitation are analyzed, for varying parameters of excitation. Global nonlinear dynamics of the semi-active suspension system are gained, by means of two parameters phase plots, with the range of frequency and amplitude of the excitation. In addition, the different bifurcation patterns between three of multi-periodic, quasi-periodic and chaotic motions have been revealed. For high frequency, the change of amplitude may cause dynamic behavioral complexity. In specific frequency, the alternate change of system response between stable and chaotic state turns to be more frequent. Nonlinear dynamic and electromagnetic interference coupled-field analysis is initially launched. According to EN 55022, the EMI analysis of the drive current of MRD at periodic, quasi-periodic especially chaotic motion is conducted. The results indicate that the chaotic motion is most likely to cause the excess EMI noise.

The analysis may serve as a reference for the MR suspension system and the smart materials with hysteresis characteristics. In our future work, chaotic control, EMI modification and experiments in practice will be conducted.

Footnotes

Acknowledgements

The authors would like to thank the anonymous reviewers for their valuable suggestions that made possible the improvement of this paper. This work was supported by the National Science Foundation of China (Grant Nos. 51475246, 51075215), the National Science Foundation for Young Scientists of Jiangsu Province, China (Grant No. BK20171039), the National Science Foundation for Post-doctoral Scientists of China (Grant No. 2017M611855), Natural Science Foundation of the Higher Education Institutions of Jiangsu Province, China (Grant No.17KJB470010), and the Research Innovation Program for College Graduates of Jiangsu Province (KYCX17_1080).

Appendix

$\begin{eqnarray}\displaystyle a_{21} & = & \displaystyle -\frac{k_{\text{s}}}{m_{\text{s}}{\omega}^{2}}-\frac{c(i_{\text{d}})}{m_{\text{s}}{\omega}}\frac{k_{0}c_{1}}{c_{0}+c_{1}}-\frac{c(i_{\text{d}})k_{1}}{m_{\text{s}}{\omega}},a_{22}=-\frac{c(i_{\text{d}})}{m_{\text{s}}{\omega}}\frac{c_{0}c_{1}}{c_{0}+c_{1}}\nonumber\\ \displaystyle a_{23} & = & \displaystyle \frac{k_{\text{s}}}{m_{\text{s}}{\omega}^{2}}+\frac{c(i_{\text{d}})}{m_{\text{s}}{\omega}}\frac{k_{0}c_{1}}{c_{0}+c_{1}}+\frac{c(i_{\text{d}})k_{1}}{m_{\text{s}}{\omega}},a_{24}=\frac{c(i_{\text{d}})}{m_{\text{s}}{\omega}}\frac{k_{0}c_{1}}{c_{0}+c_{1}},\nonumber\\ \displaystyle a_{25} & = & \displaystyle \frac{c(i_{\text{d}})}{m_{\text{s}}{\omega}}\frac{k_{0}c_{1}}{c_{0}+c_{1}};a_{26}=-\frac{c(i_{\text{d}})}{m_{\text{s}}{\omega}}\frac{{\alpha}c_{1}}{c_{0}+c_{1}};\nonumber\\ \displaystyle a_{41} & = & \displaystyle \frac{k_{\text{s}}}{m_{\text{u}}{\omega}^{2}}+\frac{c(i_{\text{d}})}{m_{\text{u}}{\omega}}\frac{k_{0}c_{1}}{c_{0}+c_{1}}+\frac{c(i_{\text{d}})k_{1}}{m_{\text{u}}{\omega}},a_{42}=\frac{c(i_{\text{d}})}{m_{\text{u}}{\omega}}\frac{c_{0}c_{1}}{c_{0}+c_{1}}\nonumber\\ \displaystyle a_{43} & = & \displaystyle -\frac{k_{\text{s}}}{m_{\text{u}}{\omega}^{2}}-\frac{c(i_{\text{d}})}{m_{\text{u}}{\omega}}\frac{k_{0}c_{1}}{c_{0}+c_{1}}-\frac{c(i_{\text{d}})k_{1}}{m_{\text{u}}{\omega}}-\frac{k_{\text{t}}}{m_{\text{u}}{\omega}^{2}},\nonumber\\ \displaystyle a_{44} & = & \displaystyle -\left(\frac{c(i_{\text{d}})}{m_{\text{s}}{\omega}}\frac{k_{0}c_{1}}{c_{0}+c_{1}}+\frac{c_{\text{t}}}{m_{\text{u}}{\omega}}\right)a_{45}=-\frac{c(i_{\text{d}})}{m_{\text{u}}{\omega}}\frac{k_{0}c_{1}}{c_{0}+c_{1}},a_{46}=\frac{c(i_{\text{d}})}{m_{\text{u}}{\omega}}\frac{{\alpha}c_{1}}{c_{0}+c_{1}};\nonumber\\ \displaystyle a_{51} & = & \displaystyle \frac{k_{0}}{c_{0}+c_{1}},a_{52}=\frac{c_{0}}{c_{0}+c_{1}},a_{53}=-\frac{k_{0}}{c_{0}+c_{1}},a_{54}=-\frac{c_{0}}{c_{0}+c_{1}},a_{55}=-\frac{1}{c_{0}+c_{1}},a_{56}=\frac{{\alpha}}{c_{0}+c_{1}};\nonumber\\ \displaystyle a_{61} & = & \displaystyle -\frac{k_{0}}{c_{0}+c_{1}},a_{62}=\frac{c_{1}}{c_{0}+c_{1}},a_{63}=\frac{k_{0}}{c_{0}+c_{1}},a_{64}=-\frac{c_{1}}{c_{0}+c_{1}},a_{65}=\frac{k_{0}}{c_{0}+c_{1}},a_{66}=-\frac{{\alpha}}{c_{0}+c_{1}};\nonumber\end{eqnarray}$

For x ∈ D ₀ ∪ D ₁, B = −𝛽−𝛾, while x ∈ D ₂ ∪ D ₃, B = −𝛽 + 𝛾. c (i _d) is calculated by combing (3) and (8). $\begin{eqnarray}\displaystyle \begin{array}{@{}l@{}}D_{0}=\big\{(x_{1},x_{2},x_{3},x_{4},x_{5},x_{6})|{\alpha}x_{6}\\ \quad +c_{0}(x_{2}-x_{4})+k_{0}(x_{1}-x_{3}-x_{5})\\ \quad \lt 0\cap x_{6}\lt 0\big\}\end{array},\quad \begin{array}{@{}l@{}}D_{1}=\big\{(x_{1},x_{2},x_{3},x_{4},x_{5},x_{6})|{\alpha}x_{6}\\ \quad +c_{0}(x_{2}-x_{4})+k_{0}(x_{1}-x_{3}-x_{5})\\ \quad >0\cap x_{6}>0\big\}\end{array}, & & \displaystyle \nonumber\\ \displaystyle \begin{array}{@{}l@{}}D_{2}=\big\{(x_{1},x_{2},x_{3},x_{4},x_{5},x_{6})|{\alpha}x_{6}\\ \quad +c_{0}(x_{2}-x_{4})+k_{0}(x_{1}-x_{3}-x_{5})\\ \quad >0\cap x_{6}\lt 0\big\}\end{array},\quad \begin{array}{@{}l@{}}D_{3}=\big\{(x_{1},x_{2},x_{3},x_{4},x_{5},x_{6})|{\alpha}x_{6}\\ \quad +c_{0}(x_{2}-x_{4})+k_{0}(x_{1}-x_{3}-x_{5})\\ \quad \lt 0\cap x_{6}>0\big\}\end{array}. & & \displaystyle \nonumber\end{eqnarray}$

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