Recent advances in dynamics and control of hysteretic nonlinear systems

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Chaos, Solitons and Fractals 40 (2009) 1808–1822 www.elsevier.com/locate/chaos

Recent advances in dynamics and control of hysteretic nonlinear systems Shaopu Yang, Yongjun Shen Shijiazhuang Railway Institute, 050043, China Accepted 17 September 2007

Abstract This paper presents a review of recent advances in dynamics and control of hysteretic nonlinear systems, including high-speed vehicle, Magneto-rheological (MR) damper and mechanical system for data processing and fault diagnosis. Ó 2007 Elsevier Ltd. All rights reserved.

1. Introduction Dynamics and control of hysteretic nonlinear system is one of the leading research fields in nonlinear science. The related research fields include high-speed vehicle dynamics and vibration control via Magneto-rheological (MR) damper, as these systems have hysteretic elements. This paper will review the main advances in all the aforementioned research fields. In the first section of this paper, the bifurcation and chaos in nonlinear system, especially the system with hysteretic nonlinearity, subjected to single and/or multi-frequency excitations are reviewed. In the second section, the investigation on dynamics and control of high-speed vehicle including high-speed railway vehicles, railway locomotive pantographs and automobile system is summarized. In the third section, systems with Magneto-rheological (MR) damper are reviewed and analyzed, including vibration isolation system and MR clutch system. In the fourth section, the techniques for data processing and fault diagnosis of mechanical systems, such as blind source separation (BSS), fractional Fourier transform (FrFT), fractional wavelet transform (FrWT) and singular value decomposition (SVD), are reviewed. In the end, the authors put forward some possible important research directions in the future.

2. Bifurcation and chaos in nonlinear systems Bifurcation and chaos in nonlinear systems are quite hot recently. As a particular and important case, the nonlinear system with hysteretic nonlinearity in mechanical engineering has a particular importance.

E-mail address: [email protected] (S. Yang) 0960-0779/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.09.064

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2.1. Bifurcations and singularities of nonlinear hysteretic systems Hysteretic nonlinearity is one of the intrinsic properties with many engineering systems, such as the wire rope vibration absorbers, elastic–plastic structures, systems with dry-friction component, MR or Electro-rheological (ER) dampers, etc. [1,2]. Hysteretic nonlinearity is caused by the difference between the upload and unload routes as shown in Fig. 1, which is the typical hysteretic loop of Davidenkov’s model demonstrated by n o g

r ¼ E u  ½ðu0  uÞn  2n1 un0  ; ð1Þ n where r, u, E, g, and n is the normal stress, strain, elastic modulus, hysteretic coefficient and hysteretic exponent, respectively. The main features of the hysteretic nonlinearity are their non-smoothness and multiple-valued functions, which might result in complicated dynamical behavior. Due to this non-smooth and multi-valued property, the usual methods for nonlinear system analysis, such as the center manifold theory and normal form theory, would be no longer valid in the investigation for this kind of systems. We would like to review the investigation of hysteretic nonlinear system subjected to single/multiple excitations as follows. 2.1.1. Hysteretic nonlinear system subjected to single-frequency excitation When a system contains one or more hysteretic components, it becomes a hysteretic nonlinear system and its response to excitations will be more complicated than that of a normal nonlinear system. A new method based on the averaging method is proposed in this study [1], and is used to analyze the hysteretic nonlinear system, through which some important results are obtained. A resonance field is firstly presented in this method [3], and verified via an analog computer [4]. This method has been extended to study the complicated dynamical behavior of the nonlinear system. It is verified that the averaged equation is an Arnold normal form of the original nonlinear system [5]. In light of the averaging methods extended to the parameterized system, 3-order truncated normal form of Hopf bifurcation and its coefficients are calculated. This method may simplify the normal form calculation greatly and get the same results as the exact solution. It is shown that the high order terms higher than e3 do not affect the coefficients of the 3-order truncated normal form of Hopf bifurcation. It has been found that the hysteresis has a significant and important effect on the dynamical behavior [1,6–8], such as enlarging the resonance range, changing the bending degree of the amplitude–frequency curves, making the resonance more intensive, etc. In our work [7], this new method is used to investigate the dynamical behavior of the parametrically excited hysteretic nonlinear system. Firstly the stability and local bifurcation in this system is analyzed, and the results are compared with those by the singularity theory. Furthermore, the Hopf bifurcation of the first-order approximated solution is investigated, and some important formulae are derived so as to determine the bifurcation type and stability easily. In addition, other bifurcations of the degenerated vector fields, including homo-clinic orbit bifurcation, hetero-clinic orbit bifurcation, and cusp singular point bifurcation, are also researched, and the engineering application is summarized in [1]. Moreover, the hysteretic nonlinear system with self-excitation is also investigated by this new method [8]. The steady-state bifurcation solution is obtained, and the intrinsic barrier in center manifold method and normal form method that could not deal with the non-smooth system is overcome in the new method. And a new concept, i.e.,

Fig. 1. The hysteretic loop of Davidenkov’s model.

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transition limit point, is proposed in [8]. Based on the concept of transition limit point, the topological structure is classified by singularity theory [1,9,10], and the various bifurcation diagrams in parameter space are obtained and verified numerically. Ding [11] has also researched a similar system with hysteretic nonlinearity, and found that the steady state periodic response existed only over a limited excitation frequency range that lost its stability outside the frequency range through Hopf bifurcation and then the system undergoes quasi-periodic motion. In other work [12] the bifurcation and singularity of a parametrically excited system with Devidenkov’s hysteretic nonlinearity is researched by this method. The ranges in system parameter space are obtained, where the stationary motions will take place. For the different hysteretic exponent n, the universal unfolding and the transition sets in system parameter space are obtained so as to classify the bifurcation modes. At last the dependence of the universal unfolding parameters on the system parameters are investigated and some important conclusions, such as the motions in the ranges corresponding to system parameters, the relation between the unfolding parameters and the system parameters, are also obtained. In a recent work [13], the authors investigated the steady state of non-sticking motion for a single-degree-of-freedom system with dry friction subjected to harmonic excitation. Based on Den Hartog’s and H.K. Hong’s work, the unnecessary hypotheses are dropped. The expression of maximum velocity of the non-sticking motion is presented, which is the product of the magnification factor, the natural cycle frequency and the static displacement of the amplitude of excitation force. Numerical simulations validate the theoretical results of maximum velocity. Some discussions on the magnification factor are presented. The phase lags of maximum displacement and velocity are also discussed. 2.1.2. Bifurcations of hysteretic nonlinear system subjected to multi-frequency excitations When a hysteretic nonlinear system subjected to multi-frequency excitations, some more complicated and different dynamical behavior will take place. It was shown that some typical nonlinear phenomena, such as sub-harmonic, superharmonic, combination, sub-combination resonance would occur in the system. The series of work [14–17] show that the amplitude–frequency curve equation is the universal unfolding of a codimension two normal form, i.e. pitchfork bifurcation. All the transit sets are obtained via the singularity theory, and the topological structure of the bifurcation curves in different parameter space are classified. Due to the engineering background, these results are very useful to the engineering structure designing or revising. The engineers could either select the system parameters in the desired range so as to meet the corresponding demand, or change some system parameters to avoid the undesirable dynamical mode. This method could also be used to research nonlinear multi-degree-of-freedom system. In the work [18,19], the authors investigate a two-degree-of-freedom system under multi-frequency excitations, which could model the dynamical behavior of vehicle suspension system to the road excitation, or the locomotive pantographs system to the disturbance of the catenary, etc. The approximate solution is obtained firstly, and the analysis of stability is also fulfilled. Then the singularity theory is applied to classify all the dynamical modes. At last some typical results are verified through numerical simulation. Zhang and Yao [20] researched the Shilnikov type multi-pulse orbits and chaotic dynamics of visco-elastic moving belt subjected to parametric and external excitations, and the global bifurcation is investigated based on normal form theory. Jing and Wang [21] applied numerical method to Duffing oscillator with twofrequency excitation, and found a lot of complicated nonlinear behaviors, such as cascades of period-doubling, reverse period doubling bifurcations, quasi-periodic orbit, period windows, bubble from period-one to period-two onset of chaos, hopping behavior of chaos, transient chaos, chaotic attractors and strange non-chaotic attractor. 2.2. Bifurcations of a multiple-frequency excited spur gear system As a generalization, the new method has been extended to other nonlinear system, such as Duffing’s oscillator, beam with several rotating machines and spur gear pair system. Researches on the primary resonance and non-primary resonance [22,23] of Duffing’s oscillator to double-frequency excitations have revealed some phenomena different from those in the single-frequency excited Duffing’s system [24–26], and the results have been verified by experimental test on analog computer. Furthermore, a beam supporting three rotating machines simultaneously is simplified as a weakly nonlinear SDOF system with cubic nonlinearity, namely Duffing’s oscillator to three-frequency excitation, and is researched analytically and numerically [27]. At first the uniformly valid, approximate solution of the governing equation for various combination resonance are obtained. The analytical and numerical solutions are in virtually perfect agreement with all case considered. The present results demonstrate that the actual response of a structure could lead to a fatigue life that is much shorter than what is predicted by linear analysis. Hence, the conventional structural engineering practice of considering the structure to be safe from resonant response when none of the frequencies of the excitation matches the natural frequency is shown to be fraught with danger.

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There are many other engineering examples subjected to multi-frequency excitation, one of which is spur gear pair. Ozguven [28] has comprehensively reviewed the mathematical models and dynamical phenomena in gear dynamics, where the focus is mainly on linear case. With the development of nonlinear dynamics theory, the nonlinear characteristics, such as different kinds of periodic solutions, bifurcations, crisis or chaos in gear dynamics, have become the most interesting research topics [29]. Up to now three main kinds of methods, the so-called analytical, numerical and experimental method, have been adopted to analyze the nonlinear dynamics of the gear system. Kahraman [30–32] researched nonlinear dynamics of a spur gear pair by harmonic balance method, where the backlash was represented by truncated series expansion. Theodossiades [33,34] applied the piecewise-linear technique and multi-scale method to the gear system, where the second-order approximate solutions were obtained. The primary disadvantage of these two analytical methods may be that it is difficult to obtain the solutions with high precision. Due to the complexity of gear system the numerical method [35,36] is frequently used to obtain the response and compare it with the results by analytical method, where the parametric continuation technique [35] is interesting and very useful to find higher order subharmonic resonance. The experimental method is also important and some typical results have also been found [37]. From the analytical methods one can find that they have some defects, such as lower computation precision and computation speed, so that it is necessary to develop a new and more efficient method. Incremental harmonic balance method (IHBM) is an efficient method for nonlinear system [38], especially with piece-wise linearity or nonlinearity. This method has been used to research the nonlinear system with piece-wise stiffness [39] or piece-wise viscous damping [40] efficiently. According to the authors’ knowledge, the nonlinear system with piece-wise linearity coupled with parametric excitation and to multi-frequency excitation has not been investigated systemically. The mathematical model of spur gear pair after dimensionless transform is " # L L X X d2 x dx þ 2el þ 1 þ el cosðlXtÞ f ðxÞ ¼ f0  ðlXÞ2 fl cosðlXtÞ: ð2Þ 2 dt dt l¼1 l¼1 Based on the IHBM, the authors have studied Eq. (2) detailedly. At first, they researched Eq. (2) without external excitation that originated from the fluctuation of inputted moment, established the general form of periodic solution based on IHBM, and analyzed the effect of some important system parameters including the damping ratio and the amplitude of the inputted moment on the amplitude–frequency response [41]. Then they studied Eq. (2) with multi-frequency external excitation and single-frequency parametric excitation [42], especially obtained some results to control the single- and double-side impact by adjusting the system parameters. In the recent work [43], the model with multi-frequency external and multi-frequency parametric excitations is studied, where the general form of the periodic solution with arbitrary precision is established based on IHBM and step function. From the computation results one can find the flock of higher order super-harmonic phenomenon observed by Kahraman [37] in experiment that has been omitted by the other researchers [30–34]. And the different damping ratio and amplitude of excitation can be used to control the impact phenomenon, where the single- and double-side impact would disappear gradually if increasing these two system parameters.

3. High-speed railway vehicle dynamics In the suspension system of a railway locomotive or vehicle, dampers are ordinary components, which exhibit hysteretic behavior due to the viscous fluid. As the application of the proposed method, the dynamics and control of highspeed vehicles are studied. 3.1. Dynamics and control of high-speed railway vehicles 3.1.1. Dynamics of high-speed railway vehicles High-speed railway locomotive and vehicle will lead different dynamical behavior that does not take place in lower speed. There are many preliminary works [44–47] in this field. At first we analytically investigate the dynamical behavior of simple but important wheel-set model [48]. The nonlinear critical speed is obtained via bifurcation theory, which gives a new description for the well-known phenomenon, namely, hunting motion. The nonlinearity taken into account is nonlinear damping force in a longitudinal yaw damper, and the Hopf bifurcation could be sub-critical or super-critical, depending on the system parameters. It is found that soft nonlinear damping would lead to a sub-critical Hopf bifurcation, which means that even below the linear critical speed, an unstable limit cycle may occur and leads the wheel-set to unstable hunting, while the hard nonlinear damping would introduce a super-critical Hopf bifurcation.

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It is also found that increasing damping would decrease the amplitude of the hunting vibration and increase the linear critical speed. Then the Hopf bifurcation and hunting behavior of a rail wheel-set with nonlinear primary yaw dampers and wheelrail contact force is researched [49]. This study is intended to complement earlier studies by Hans [50], where they investigated the nonlinearities stemming from creep force saturation and nonlinear wheel-rail profile. The results indicate that the nonlinearities in the primary suspension and flange contact contribute significantly to the hunting behavior. Both the critical speed and the nature of bifurcation are affected by the nonlinear elements. Further, the results show that in some cases, the critical hunting speed from the nonlinear analysis is less than the critical speed from a linear analysis. This indicates that a linear analysis could not predict the actual hunting speed. After that the effect of yaw damping nonlinearities on the hunting behavior of rail vehicles is studied, using a model for a two-axle truck that includes the dynamics of the wheel-sets and the truck frame [51]. Similar to the work [49], in this paper the earlier work by Hans is also extended to this model. The simulation results show that yaw damping could have a mixed effect on the hunting critical speed, which means the yaw damping could make the hunting critical speed higher or lower. And in some ranges, increasing damping may lower the critical speed, unlike the results commonly obtained from a linear model. The further work presents the effect of system parameters on hunting of a rail vehicle with nonlinear yaw dampers and wheel-rail interface [52], where the rail vehicle is represented by a two-axle bogie that includes the dynamical influence of the wheel-sets and the truck frame. The numerical results show that yaw damping could have a mixed effect on the hunting critical speed, similar to that in railway vehicle [51]. Flange contact nonlinearities could also have a significant effect on the hunting behavior. Large lateral stiffness of the rail could increase lateral force to vertical force (L/V) ratio during hunting. Increasing the gauge clearance, however, would have an opposite effect. The effects of a variety of other parameters, such as the primary suspension yaw and lateral stiffness, primary suspension lateral damping, wheelset mass, and truck frame mass, are summarized in a table. Based on the fourth-order Runge–Kutta method, the authors researched the sub-critical Hopf bifurcation of a railway bogie, where the Cooperrider’s simple mode1 is adopted and the lateral stability of bogie is investigated under diverse nonlinear factors [53]. The nonlinear forces considered in the bogie system include creep-force between the rail and wheel, longitudinal damping and the flange contact force. It is found that different nonlinear models of creep-force between the rail and wheel would result in the change of linear and nonlinear critical speed simultaneously when the frictional coefficient keeps constant. On the other hand, the linear critical speed may change very little while the nonlinear one would change more violently when the frictional coefficient varies. These results can provide a reference for bogie optimal design. Furthermore, the authors present a new method for analyzing the lateral stability of locomotive, that is to say, the critical speed can be calculated by using the complex-mode matrix perturbation theory [54]. The primary feature of this method is that the repeated calculation of the matrix’s eigen-values can be avoided so that the calculating time would be reduced significantly. The numerical results indicate that this method is not only highly efficient, but also has the same accuracy as the common algorithmic method based on the decomposition of the matrix. In recent work, a whole locomotive model with seventeen degree-of-freedom was investigated, where the frontal and back bogies are all included [1]. Based on the presented method, it is found that in some range of system parameters the suitable hysteretic damper has better performance than the linear damper, which could increase the critical speed remarkably. And the Hopf bifurcation could not be sub-critical but be super-critical in most cases, which is a favorable result to increase the critical speed. The effects of all the primary system parameters on the dynamical response have been obtained, which presents the theoretical basis for the design of high-speed locomotive and can meet the need of security and stability of modern high-speed locomotive. 3.1.2. Nonlinear control of high-speed railway vehicle dynamics Another important direction about high-speed locomotive is its nonlinear dynamics control. For a simple wheel-set model an output-perturbation uncoupled method is educed based on the differential geometry theory of nonlinear systems, which is used to study and obtain the control result for the Hopf bifurcation of locomotive hunting [55]. The nonlinear model is transformed to a simple linear system by a proper coordinate transform, namely, exact linearization based on nonlinear feedback control, and then the optimal control scheme is applied to the wheel-set model so that the system could maintain asymptotically stable equilibrium state no matter how the system parameters change. At last the effectiveness is verified through the simulation of an actual wheel-set model. Furthermore, the semi-active control of high-speed train suspension system with MR damper is investigated [56], where the model of MR damper is established by experiment data obtained in an INSTRON machine. A scaled half train model is set up including the MR damper and a semi-active control strategy is adopted to control the vibration of suspension system. By the center manifold theorem, the stability of the nonlinear systems with MR damper is also

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studied, which is very useful to the high-speed rail vehicle. The simulation result shows that the vibration of suspension system may be well controlled by the semi-active controller. And a bifurcation control method to stabilize the trivial steady state via the frequency response and to eliminate the jump phenomena in the force response is presented by using state-feedback control [57]. Because the bifurcation of the system is characterized by its modulation equations, the order of the feedback gain is determined at first so that the feedback modifies the modulation equations. Theoretical analysis of the modified modulation equations shows that the unstable region of the trivial steady state can be shifted and the nonlinear characteristic can be changed by means of the bifurcation control with the above feedback. The shift of the unstable region permits the stabilization of the trivial steady state in the frequency response, and the elimination of the jump in the force response can be fulfilled due to the change of the nonlinear characteristic. At last this method is verified by numerical simulation on a locomotive model. In recent work, a revised semi-active control strategy based on MR damper is presented and applied to a locomotive model [58]. The simplified dynamical equations for hunting motion of locomotive are firstly established. Then the dynamical behavior of the semi-active suspension of the locomotive with MR damper is investigated, where the MR model is obtained based on experimental results. In order to compare the performance of the semi-active suspension with MR damper, the passive control suspension, the semi-active on–off control suspension and the active suspension are also discussed. The simulation results show that the semi-active suspension of the locomotive with MR damper could lower the vibration of locomotive almost as effectively as the active control. Based on the control strategy presented here, not only the angular acceleration of the locomotive body can be lowered, which reduces the possibility of hunting motion, but also the yaw angles of the frontal and back bogies would be lowered, which improves the security of operation. 3.2. Dynamics and control of railway locomotive pantographs Pantograph is a device in the locomotive to induce electricity from catenaries to the locomotive. When the locomotive runs at a high-speed, the pantograph would vibrate, and the touch force between pantograph and catenary would fluctuate, which might lead to the undesirable damage on the pantograph and catenaries. In this subsection, we would review the dynamical designing method, nonlinear dynamics and control strategy for the pantograph system. In the work [59], the authors investigated the optimization design of the installing position of the viscous damper on the frame of the SS7 pantograph. And the vibration reduction system is designed for the frame of the SS7 pantograph. After that the more complicated model for the coupled pantograph-catenary system is established based on the generalized coordinate [60]. The effects of all the system parameters, such as the mass, the damping and the stiffness of the panhead, and those of the pantograph frame, on the dynamical behaviors have been investigated. And the optimal values of these parameters are also obtained by numerical method. In the work [61], three kinds of stiffness curves for the pantograph system are fitted by using the nonlinear least square method and a unified fitting formula is established. Based on this result, a 3-DOF model for the pantographcatenary system is presented, where the varying stiffness is denoted as time-varying parameters. Based on the optimal control strategy, the contact force under different train speeds is researched corresponding to different weighting coefficients. From these results it can be found that the contact force can be controlled in the designated range so as to guarantee the desirable contact. In the further work [62], the catenary stiffness is fitted by nonlinear least square method and a nonlinear dynamical model of pantograph-catenary system is established. Then the out-perturbation uncoupled method is educed from differential geometry theory, which is the first time used in the high-speed pantograph-catenary system. The nonlinear model is transformed into a simple linear system by a proper coordinate transform and the optimal control strategy is applied. At last the simulation results show that the controller is very effective. In the recent work [63], the power supply of the high-speed train is analyzed according to the rigorous requirement for high driving velocity, safety and comfort. It is considered that the stable current collection is the prerequisite condition for the smooth running of the train. From Mathieu equation that models the pantograph-catenary system, the stable regions and periodic solution of the dynamics system are obtained. By adopting the actual parameters of the pantograph-catenary system, the different stable current collection curves are calculated. Further study on the system stability including the largest Lyapunov exponents of an actual pantograph-catenary system is fulfilled, and the optimum and most disadvantageous velocities of the train are gained, which would be helpful to the study of the stable current collection for high-speed rail vehicle and would provide a better reference for the design of new types of pantograph. In other work [64], the authors studied the nonlinear dynamics of a pantograph system, where a square velocity damping force is used to describe the nonlinear damping force of the hydraulic vibration damper, and the catenary sys-

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tem is modeled by a spring with time-varying stiffness. The necessary parameter condition for the Hopf bifurcation existing in the pantograph system is obtained. And some complicated motions, such as periodic-doubling bifurcation, quasi-periodical bifurcation and chaos are revealed in the system. Those results could offer the theoretical reference for the design of high-peed pantograph system. 3.3. Dynamics and control of automobile Similar to the railway vehicle, a road vehicle also has a suspension system, which contains dampers and springs. In this subsection, we review the investigation of road vehicle dynamics, which include the vibration control strategy, the optimization design, the bifurcation and chaos, as well as the resonance of the system under the excitation of the road surface roughness. A detailed review of this field can be found in [65], where the preliminary works [66–68] about application of MR damper in automobile suspension system are included. In the work [69], a new model of the road surface roughness is presented, where the very high frequency and very low frequency is disregarded because the response of the vehicle is insensitive to these excitations. Based on the Shannon theorem, inverse Fourier transform, and suitable double-side constraint, the new method modeling the road surface roughness is established by a series of processing of the power spectrum density of known road roughness. The simulation results show the correctness of this new method. In the same work [69], a new design rule for automobile suspension system is put forwarded based on the optimal control theory. This method is illustrated through a model of two-degree-of-freedom vehicle system. Those results of simulation show that this optimizing algorithm could improve the performance of the suspension in comparison with that of the traditional passive suspension, whenever the vehicle running on the sine surface or the random surface stated in this paper. Moreover, it is easy to reform the existing suspension system via the concept put forward in this paper so that it is especially feasible and efficient to the existing suspension system that needs reform. Finally, the possibility of applying this method to other engineering fields is pointed out. In the work [70] an improved semi-active automobile suspension with MR damper is investigated to the deterministic and random excitation. In order to compare the performance, the passive, semi-active on–off, and active control suspensions are also discussed. The results show this improved control method is very good and makes passenger more comfortable, while it is little harmful to security and stability. And this control method is feasible in automobile engineering due to the fine characteristics of the MR damper. In the recent work [71], the authors research the nonlinear dynamics of a suspension system under semi-active control. At first the first-order approximate solution to the main resonance is obtained by the new method in [1]. And then the bifurcation modes are classified in the parametric space composed of the amplitude of excitation and the modulation factor based on the singularity theory. It is shown that the response is the universal unfolding of the winged cusp bifurcation. And the method presented here can be used to analyze the dynamic response of this kind of semi-active control system and obtain all the bifurcation curves. This provides an efficient method that can be used to analyze and/or control the dynamic behavior of this kind of semi-active nonlinear system. On the other hand, the chaos motion is also interesting to some researchers. In the work [72], the condition of occurrence of chaos in a SDOF suspension system is obtained by the Melnikov method. Then the largest Lyapunov exponent and the different Poincare maps are used to verify these results. The further work [73] presents the investigation of possible chaotic motion in a vehicle suspension system with hysteretic nonlinearity, which is subjected to the multi-frequency excitation from road surface. The Melnikov function is used to derive the critical condition for the chaotic motion, and then the effect of nonlinear damping force on the chaotic field is investigated. The path from quasi-periodic to chaotic motion is found via Poincare map, the largest Lyapunov exponent and power spectrum. In the recent work [74] the authors study the chaotic motion in a SDOF vehicle suspension system with hysteretic nonlinearity, which is subjected to the road stochastic excitation. The random Melnikov process is derived firstly, then the necessary critical condition in the sense of mean square is obtained and the effect of noise on the onset of chaos is studied. The result obtained by Melnikov method is also verified via numerical methods, including the largest Lyapunov exponent and Poincare map. It is found that the chaotic motion does not exist when the vehicle runs normally, while when the vehicle comes across protruding slope chaos may occur.

4. Vibration control via MR damper MR fluid is a kind of smart material, whose rheological characteristics would change under the applied magnet field. It is an ideal material for vibration control, whose damping characteristics exhibit strong hysteretic behavior.

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4.1. MR damper modeling In order to investigate the dynamical behavior of the MR system, the first and most important thing is to model the MR device. Three models are proposed in our investigation, which include revised Bingham model, mechanical-electrical coupled model, and revised bi-viscous hysteretic model. Based on the proposed models, the dynamical behaviors of the MR system are researched, which is useful to its application in engineering. Bingham model is often used in the modeling of MR damper, but there is no hysteresis loop in it. In the work [75], the authors propose a new three-parameters bilinear hysteresis model based on the well-known Bingham model. The proposed model may be expressed with simple formulae, which would be used in the analysis of system dynamical behavior. In a whole cycle the revised Bingham model can be proposed as follows: * (

F ¼ C 1 y 0 þ F y0 Signðy 0  V 0 Þ

ð3Þ

where y 0 is the relative velocity between piston and cylinder, C1 is the viscous damping of the MR damper, Fy is the yield force of the MR damper, and V0 is the zero-force velocity. It can be found that the revised Bingham model has a simpler form so as to make dynamical analysis easier. And the analysis of the vibration control system based on this model would be completed in the next subsection. The mechanical–electrical coupled model is researched in another work [76], where the damper and control device are considered as one system. In order to establish the relationship between the damping force and other factors, the physical mechanism of the hysteretic behaviors for Bingham fluid and the dynamic mechanism of excitation control and magnetization of MR fluids (MRF) should be investigated. In the proposed scheme, a phenomenological model is macroscopically established based on relevant testing measures and modeling theories. In this model the rate-dependent hysteretic relation of the damping forces vs. velocity and electromagnetic hysteresis is well demonstrated. At last a simulation model is set up to compare the tested behaviors and the simulated results. According to the comparisons between the measured damping force and the predicted one, the simulations are coincident with the experimental results, which proves the validity of the proposed model. In another work [77], the authors present a new MR model with smooth and concise form that could interpret the biviscous and hysteretic behaviors of the MR damper very well. The bi-viscous and hysteretic behaviors can be characterized by two parameters, and all the parameters in the model have definite physical meanings. Compared with the phenomenological Bouc-Wen model and the bi-viscous hysteresis model, this model makes it convenient to study the effects of the bi-viscous and hysteretic behaviors on the performance of a system with MR damper. The proposed model for the MR damper is shown as      V0 V0 F md ¼ A1 tan h A3 x_ þ x þ A2 x_ þ x ; ð4Þ X0 X0 and some comparisons of the presented model with the Bingham model and experiment results in some cases show that this presented model has enough precision. Besides the above three MR model, other models applied in different cases are also studied. In the work [78], a modified Bouc-Wen model is used to figure the hysteretic character of MR damper. With the optimization method, the parameters of this model are determined. It shows that the model can describe rather accurately the dynamic characteristics of MR damper under fluctuating current. 4.2. Vibration isolation via MR damper MR device is an ideal device for vibration isolation because its coil current is easy to be controlled. Some control methods are investigated for vibration isolation systems. Due to the complexity of MR model it is rather difficult to analytically obtain the steady response of the vibration control and isolation systems. Hence it becomes necessary to adopt the numerical method to research the dynamical response of these systems, especially for multiple degree-of-freedom systems. In order to illustrate the dynamical behavior of MR damper, the bifurcations in a nonlinear system with MR damper are discussed [79]. The normal form and universal unfolding of the double zero eigen-value for this system are achieved for the first time. And the complicated dynamic behaviors of the nonlinear system are also analyzed. By the theoretical analysis, it is shown that the designed parameters have an important effect on the stability of this system. The range of selected parameters for the condition of stability are achieved based on the condition of bifurcation parameters, bifurcation curve, bifurcation set and phase portraits. At last the numerical simulation is fulfilled and the complicated dynamical behaviors are shown, and proved to be correspondent with the theoretic analysis.

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Because the dynamical behavior of a sky-hook damping system with MR damper is open till now, it is important to study how the MR damper affect the dynamical behavior of the system. In the work [80], the primary resonance of a sky-hook damping system with MR damper is investigated, and the steady state primary resonance is obtained via the presented new method [1]. The condition for the primary resonance to take place is obtained which gives a direct hint to quench the resonance. The influence of the system parameters on the primary resonance is investigated and at last the validity of the theoretical results is established through numerical simulations. As the application of the MR damper, a vibration isolation system with MR damper is investigated to obtain its steady motion [77]. From the view of energy conservation, the work done by the excitation force is equal to the one done by the damping force in a cycle when the isolation system attains the steady motion. After the cockamamie mathematical processing and based on the first kind and the second kind complete Elliptic integrals, the amplitude–frequency equation is obtained. And the theoretical results are compared with the numerical results, which shows the validity and high precision of the theoretical results. In another work [81], the authors research a SDOF vibration isolation system with MR damper, where the model is selected as the simple form obtained in the work [77]. Based on the assumption that the steady state motion of the system is periodic and symmetric, and by connecting the two sub-intervals together analytically to construct an entire cycle of the periodic motion, the closed analytical solution can be obtained. From the solution, the hysteretic critical displacement of the steady state motion of the isolation system, the hysteretic critical velocity, the maximum amplitude, and their time lags can be computed. In a special case, the solution can be reduced to the classic results of Den Hartog on the forced vibration with coulomb friction [82]. The results of this study are helpful to understand the characteristics of MR dampers so as to provide the effective damping for the purpose of vibration isolation or vibration control of the suppression system. 4.3. MR clutch As another application of MR fluid, a MR clutch for vehicle radiator as well as the control kit are designed and tested. Investigation shows that the MR clutch does a good job for engineering temperature control [83,84]. At first we research the principle of the torque transmissibility of MR fluid, namely, disc mode’s MR fluid clutch. The flow behavior of MR fluid in the two paralleled discs and the micro cylinder shaped flow units are given in [83]. After the complicated deduction, the transmitted torque of MR fluid in disc clutch is obtained. The key parameters of MR fluid fan clutch are designed and fabricated according to the requirement of the model 1308D5-0500-A silicon oil fan clutch. When its maximal input rotation is 3000 rpm, the output rotation is 2900 ± 30 rpm under the output torque 10 N m. Because of the limited inter-space between engine cool fan and water tank, the appropriate structure model must be considered when designing clutch, where the disc mode is chosen. In the work [84], a simplified model of MR fluid fan clutch is designed, which has two primary work areas, namely, the front face and the back face of the disc, respectively. The experimental setup to investigate the torque transmissibility characteristics of MR fan clutch is established, where an excitation timing motor is employed to drive the clutch. The torque and rotation sensors are used to measure the parameters of the clutch. The rotational speed of input axis (active disc) can be adjusted by tuning the excitation timing motor. Keeping the input rotation constant, the output torque of the clutch and rotation of fan are respectively measured under different excitation currents to the coil, and the mechanical characteristics and timing range of the clutch can be determined therein. From the experimental data it can be concluded that the smaller the gap size is, the smaller the magnetism saturation current. Smaller gap could usually reduce energy consumption, which is preferred in design. But an over-small gap would result in difficulties in manufacturing and high cost. Moreover, constrained by the fluidity of MR fluid, the gap should generally not be less than 0.5 mm. Compared with the shearing rate, the change of magnetic fields would have a tremendous influence on the speed-regulating characteristic of a fan clutch, and the output torque of fan clutch can satisfy the demand of engine cooling fan.

5. Fault diagnosis and signal processing for rotating machinery In modern mechanical systems and other engineering fields the rotating machinery are widely adopted due to their technical advantages. In order to design rotating machinery with good performance, or maintain the desirable condition at operation, the technique of fault diagnosis or signal processing should be considered. The authors have summarized some topics in this field, including non-Gauss’s, non-white and nonlinear signal processing technology in gear and bearing system [85,86]. In this subsection, we would review some new or developing signal processing technique we have researched for some years, including the fractional Fourier transform (FrFT), the fractional wavelet transform (FrWT),

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blind source separation (BSS), an improved feature extraction technique based on higher order statistic (HOS), and singular value decomposition (SVD). 5.1. Fractional Fourier transform FrFT was firstly introduced by Namias [87] to solve the partial differential equation in quantum mechanics. After that Mcbride [88] made a rigorous study mathematically and more operational calculus were developed. Since 1990 the FrFT has received considerable interest, finding many applications in the optical systems and optical signal processing, time-variant filtering and multiplexing, pattern recognition, and study of time-frequency distribution, which have been collected in Ozaktas’s monograph [89]. As a generalization of the ordinary Fourier transform, the FrFT is likely to be applied in every area in which the ordinal Fourier transform and related concepts are used. When the FrFT is applied in any field the problem how the FrFT can be computed fast and accurately is met. Due to the fact that only a small fraction of functions can be computed analytically, the numerical methods become important, which has been gathered in [89]. The decomposed method is proposed by Ozaktas [90] where only O(Nlog N) multiplications are needed, while other numerical methods may all need O(N2) multiplications. The disadvantage of this decomposed method is that double interpolation should be carried out before computation. In the work [91], two kinds of interpolation methods, namely, the method based on polynomial or convolution, especially the spline interpolation method and the Shannon interpolation method, are researched, and the focus is on the precision when applying them to numerical computation of FrFT. We first compare them by the decomposition and recovery of signals, and then apply them in masking the distorting term of the mixed signal composed of the Gaussian signal and the chirp signal. The results show that Shannon interpolation method has a better performance and is preferable to other interpolation methods, which makes the fast computation of FrFT useful for application in many fields. 5.2. Fractional wavelet transform As the generalization of the traditional integral transform, the fractional integral transforms have been used in many fields, and the FrFT may be the earliest and most interesting one in signal processing and optical engineering. Among these fractional integral transforms, the fractional wavelet transform (FrWT) is another promising or powerful tool for signal processing [92,93]. And now FrWT has been preliminarily used in analysis and synthesis of signals, image processing, etc. According to the authors’ knowledge, there was no work on the application of the FrWT in detection of abrupt information. In the work [94], one of the FrWT, named fractional spline wavelet transform (FrSWT) is used to extract the abrupt information from strong noise background. The authors review the construction and some properties of FrSWT at first, including the establishment and the basic properties of the fractional B-Spline functions, the symmetric fractional B-spline functions that form the basis of FrSWT. Then the computation process and approximation characteristic of the FrSWT are discussed, and especially, the truncation error is obtained in the second section. In the last section, the FrSWT is applied to detect the abrupt information of the simulated signal composed of weak abrupt information and strong noise, and the abrupt fault information measured from a gear system in our laboratory. The excellent performance shows the feasibility of applying the FrSWT in detection of abrupt information. 5.3. Blind source separation Blind source separation (BSS) is a typical tool to recover unobserved signals from several observed mixtures, which are usually the output of a set of sensors and combinations of the source signals. This approach is very useful because it is difficult to model the transfer from the sources to the sensors. And the lack of prior knowledge about mixture can be compensated by a static assumption of independence between the sources. This approach originates from array processing and now is applied in many fields, such as the processing of communication signals, biomedical signals, mechanical or acoustical signals, etc. And more application can be found in the monographs [95] or the review papers [96]. When monitoring or diagnosing the condition of mechanical devices, it is important to obtain the vibration signals of the concerned units, e.g. gear and bearing. But in many cases the obtained signals consist of some other useless signals, such as environment noise, or the signals from other unconcerned gears or bearings. In the work [97] some existing methods are analyzed and the general framework of BSS based on joint diagonalization (JD) is derived from these methods. Based on the necessary assumption and analysis, the general framework can be summarized. The first step of this general framework is to whiten the measured signals, which can be achieved

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by applying to them a whitening matrix, so that the whitened vector would be mutually uncorrelated. For convenience, the whitened vector can be assumed to have unit variance, which can be ensured by adjusting the corresponding columns of the whitening matrix. The second step is applying a certain linear transform to the whitened vector after the whitening matrix is obtained, where the linear transform is selected so that the mutually uncorrelated signals would still be uncorrelated after transformation. By applying a series of requested linear transform to the whitened vector, a series of second-order-moment matrixes would be obtained. Then the third step is to solve the unitary matrix. When the JD technique is applied to the set of second-order-moment matrices, the approximate unitary matrix can be obtained. Accordingly the estimates of the source signals and the mixing matrix would be obtained respectively. Furthermore, the authors present a new property of FrFT concerning correlation, namely, the mutually uncorrelated signals would still be uncorrelated after FrFT. Based on the discrete definition of FrFT, this new property can be verified easily. When substituting FrFT for the requested linear transform, one could obtain a new method of BSS. And this new method is an improvement compared to some existing methods [98,99]. For example, some method is based on covariance matrix so that it is not able to process non-stationary signals, but this new method could deal with non-stationary signals due to the relation between FrFT and Wigner distribution, wavelet transform. These existing methods [98,99] may have some other disadvantages, such as slow computation speed, difficult selection of t  f points, etc., but the fast computation method for FrFT [91] could improve the computation speed of this method. As an example, some non-stationary signals are used to illustrate this new method shown in [97]. It can be concluded that this new method has more advantages than the existing technology, so that it may be widely used for fault diagnosis in mechanical engineering. 5.4. An improved feature extraction technique based on higher order statistic In recent years, higher-order statistics (HOS) has received increasing interest in signal processing [100]. It has also been used in many other fields, such as communication, sonar, biomedicine, speech signal, image processing, etc. As we all know, there are some shortcomings of second-order statistics used in signal processing, such as inability to analyze the phase relation between different harmonic signals, and invalidity for nonlinear and non-Gaussian signals. On the other hand, application of HOS in signal processing could avoid these problems, due to its abilities including suppressing additive colored noise, identifying non-minimum phase systems, extracting information due to deviations from Gaussianity, detecting and characterizing nonlinear properties in signals or systems, analyzing the cyclo-stationary time series, and separating unknown sources or BSS, etc. In the work [101] a new method of pattern recognition is presented based on the envelope analysis of the given signals and the corresponding higher-order cumulant, which is one kind of HOS. The procedure of this method is to construct analytical signals by Hilbert Transform from the given signals, and obtain the envelope signals at first, then compute and compare the higher-order cumulants of these envelope signals. The higher-order cumulants can be used as the characteristic quantity to discriminate these given signals. In order to illustrate the validity of this method, it is used to discriminate the measured signals from 197,726 rolling bearings of freight locomotive in different conditions. The comparisons of the second-order, third-order and fourth-order cumulants of different vibration signals of rolling bearing are researched, which show this new method could discriminate the normal and two fault signals distinctly. 5.5. Singular value decomposition The method to detect the abrupt information from measured signals includes wavelet transform and the aforementioned FrWT. Apart from them, the singular value decomposition (SVD) may also be an important method. In the work [102], the SVD technology is used to eliminate the noise from the measured signals. While in the work [103], the authors presented a new method of detecting abrupt information by eliminating the normal vibration signals so that the noise and abrupt fault information would still be saved. In this method the track matrix of attractor is firstly reconstructed by the measured time series. If the measured signal includes abrupt information, the reconstructed matrix must be nonsingular, so that the singular value k1, k2, . . ., kn would be obtained. If letting the front larger singular value be zero, and applying the inverse transform to the result, one would obtain the track matrix corresponding to the abrupt information and the background noise. Applying the inverse process of reconstruction, one would obtain the abrupt information with the background noise. This is the primary process of the new method. It can be concluded that this method is feasible and can be used to detect signal’s singularity, especially for signal with large noise while wavelet transform may not be used.

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6. Conclusions In this paper the dynamics and control of nonlinear system, especially hysteretic nonlinear system is reviewed, which is an interesting and practical field and has been investigated by the scientists. According to the authors’ knowledge, some possibly important developing directions in future can be summarized as follows. 1. Complicated dynamics of multiple DOF hysteretic nonlinear system to complicated excitation may be an important research field when it is coupled with other nonlinear factors, for many analytical methods, including the new method [1], may be very difficult to deal with the multiple DOF nonlinear system, especially when the DOF is higher than three. 2. High efficient and stable numerical method may be crucial to reveal all the dynamical behaviors of the high-speed vehicle. It is almost impossible to solve this kind of model by analytical method so that newly developed or improved numerical method can be helpful in this field. 3. Detailed research on application of MR damper in the momentous engineering has important meaning from both theory and engineering point of view. Due to the defects of the traditional hydraulic damper, the MR damper has been preliminarily used in some key fields. But these applications are not enough. 4. The inverse problems of hysteretic nonlinear dynamics, including the force identification, fault diagnosis and damage assessment of structures may be another important direction. Besides the works mentioned in this paper, there are some other works in this field. But there may be some problems in applying these technologies in engineering, including computation speed and precision, invalidation in the case of stronger noise, complicated structures, etc.

Acknowledgements The authors are thankful to the support by the National Natural Science Foundation of China (Nos. 50625518, 10472073, 10672107 and 10602038).

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