Non-linear control of friction-induced self-excited vibration

The present paper introduces a new method of controlling friction-driven self-excited vibration. The control law is primarily derived using the Lyapunov's second method. A single degree-of-freedom oscillator on a moving belt represents the primary model of the system. The control action is achieved by modulating the normal load at the frictional interface based on the state of the oscillatory system. The basic mechanism of the control action utilises subcritical Hopf bifurcation of the equilibrium followed by cyclic-fold bifurcation (of limit cycle oscillations) to globally stabilise the equilibrium. The basic mechanism is qualitatively independent of the exact model of friction. Different models of friction, like, algebraic model, LuGre model and switch model with time-dependent static friction are considered to substantiate the above claim. An approximate method for estimating the critical value of the control parameter that ensures global stability of the equilibrium is also proposed.     

CHARACTERIZATION OF THE BEHAVIOUR OF A SIMPLE AEROSERVOELASTIC SYSTEM WITH CONTROL NONLINEARITIES

The characterization of the behaviour of nonlinear aeroelastic systems has become a very important research topic in the Aerospace Industry. However, most work carried to-date has concentrated upon systems containing structural or aerodynamic nonlinearities. The purpose of this paper is to study the stability of a simple aeroservoelastic system with nonlinearities in the control system and power control unit. The work considers both structural and control law nonlinearities and assesses the stability of the system response using bifurcation diagrams. It is shown that simple feedback systems designed to increase the stability of the linearized system also stabilize the nonlinear system, although their effects can be less pronounced. Additionally, a nonlinear control law designed to limit the control surface pitch response was found to increase the flutter speed considerably by forcing the system to undergo limit cycle oscillations instead of fluttering. Finally, friction was found to affect the damping of the system but not its stability, as long as the amplitude of the frictional force is low enough not to cause stoppages in the motion.     

串联式叉型滞后簇发振荡及其动力学机制

簇发振荡是自然界和科学技术中广泛存在的快慢动力学现象,其具有与通常的振荡显著不同的特性.根据不同的动力学机制可将其分为多种模式,例如,点-点型簇发振荡和点-环型簇发振荡等.叉型滞后簇发振荡是由延迟叉型分岔诱发的一类具有简单动力学特性的点-点型簇发振荡.研究以多频参数激励Duffing系统为例,旨在揭示一类与延迟叉型分岔相关的具有复杂动力学特性的簇发振荡,即串联式叉型滞后簇发振荡.考虑了一个参激频率是另一个的整倍数情形,利用频率转换快慢分析法得到了多频参数激励Duffing系统的快子系统和慢变量,分析了快子系统的分岔行为.研究结果表明,快子系统可以产生两个甚至多个叉型分岔点;当慢变量穿越这些叉型分岔点时,形成了两个或多个叉型滞后簇发振荡;这些簇发振荡首尾相接,最终构成了所谓的串联式叉型滞后簇发振荡.此外,分析了参数对串联式叉型滞后簇发振荡的影响.      

Nonlinear Response of a Parametrically Excited System Using Higher-Order Method of Multiple Scales

Two fundamentally different versions of the method of multiple scales (MMS) are currently in use in the study of nonlinear resonance phenomena. While the first version is the widely used reconstitution method, the second version is proposed by Rahman and Burton [1]. Both versions of the second-order MMS are applied to the differential equation obtained for a parametrically excited cantilever beam with a lumped mass at an arbitrary position. The bifurcation and stability of the obtained response show the difference between the two versions. While the Hopf bifurcation phenomena with no jump is found in the case of second-order MMS version I, both jump-up and jump-down phenomena are observed in second-order MMS version II, which closely agree with the experimental findings. The results are compared with those obtained by numerically integrating the original temporal equation.     

Self-Excited Oscillations of Structures by Particle Emission

This paper studies the self-excited oscillations of structures by alphaparticle emission. Alpha particles emitting from a source exert a smallmechanical force on the structure. This force is analogous to theso-called follower force which is known to cause structure flutter andcould be used as a source of mechanical power in MEMS devices. Wedevelop a dynamic model for the structure and study the stability of thestructure. The bifurcation analysis renders information about thepossible steady-states for different values of parameters. Depending onthe parameters, the structure can have stable buckled as well asunbuckled equilibria. Moreover, the model predicts stable limit cycleswhich correspond to stable flutter of the structure. Although it isdifficult to achieve the required mechanical force using alpha particleemission, the concept of generating mechanical oscillations throughejecting streams of gaseous or liquid jets is feasible.     

Control of cross-flow-induced vibrations of square cylinders using linear and nonlinear delayed feedbacks

We investigate the effectiveness of linear and nonlinear time-delay feedback controls to suppress high amplitude oscillations of an elastically mounted square cylinder undergoing galloping oscillations. A representative model that couples the transverse displacement and the aerodynamic force is used. The quasi-steady approximation is used to model the galloping force. A linear analysis is performed to investigate the effect of linear time-delay controls on the onset speed of galloping and natural frequencies. It is demonstrated that a linear time-delay control can be used to delay the onset speed of galloping. The normal form of the Hopf bifurcation is then derived to characterize the type of the instability (supercritical or subcritical) and to determine the effects of the linear and nonlinear time-delay parameters on their outputs near the bifurcation. The results show that the nonlinear time-delay control can be efficiently implemented to significantly reduce the galloping amplitude and suppress any dangerous behavior by converting any subcritical Hopf bifurcation into a supercritical one.     

Infinite dimensional slow modulations in a well known delayed model for cutting tool vibrations

We apply the method of multiple scales (MMS) to a well-known model of regenerative cutting vibrations in the large delay regime. By “large” we mean the delay is much larger than the timescale of typical cutting tool oscillations. The MMS up to second order, recently developed for such systems, is applied here to study tool dynamics in the large delay regime. The second order analysis is found to be much more accurate than the first order analysis. Numerical integration of the MMS slow flow is much faster than for the original equation, yet shows excellent accuracy in that plotted solutions of moderate amplitudes are visually near-indistinguishable. The advantages of the present analysis are that infinite dimensional dynamics is retained in the slow flow, while the more usual center manifold reduction gives a planar phase space; lower-dimensional dynamical features, such as Hopf bifurcations and families of periodic solutions, are also captured by the MMS; the strong sensitivity of the slow modulation dynamics to small changes in parameter values, peculiar to such systems with large delays, is seen clearly; and though certain parameters are treated as small (or, reciprocally, large), the analysis is not restricted to infinitesimal distances from the Hopf bifurcation.     

An analytical study of time-delayed control of friction-induced vibrations in a system with a dynamic friction model

We investigate the control of friction-induced vibrations in a system with a dynamic friction model which accounts for hysteresis in the friction characteristics. Linear time-delayed position feedback applied in a direction normal to the contacting surfaces has been employed for the purpose. Analysis shows that the uncontrolled system loses stability via. a subcritical Hopf bifurcation making it prone to large amplitude vibrations near the stability boundary. Our results show that the controller achieves the dual objective of quenching the vibrations as well as changing the nature of the bifurcation from subcritical to supercritical. Consequently, the controlled system is globally stable in the linearly stable region and yields small amplitude vibrations if the stability boundary is crossed due to changes in operating conditions or system parameters. Criticality curve separating regions on the stability surface corresponding to subcritical and supercritical bifurcations is obtained analytically using the method of multiple scales (MMS). We have also identified a set of control parameters for which the system is stable for lower and higher relative velocities but vibrates for the intermediate ones. However, the bifurcation is always supercritical for these parameters resulting in low amplitude vibrations only.     

Bifurcation analysis of a two-DoF mechanical system subject to digital position control. Part I: theoretical investigation

This paper analyzes the double Neimark–Sacker bifurcation occurring in a two-DoF system, subject to PD digital position control. In the model the control force is considered piecewise constant. Introducing a nonlinearity related to the saturation of the control force, the bifurcations occurring in the system are analyzed. The system is generally losing stability through Neimark–Sacker bifurcations, with relatively simple dynamics. However, the interaction of two different Neimark–Sacker bifurcations steers the system to much more complicated behavior. Our analysis is carried out using the method proposed by Kuznetsov and Meijer. It consists of reducing the dynamics of the nonlinear map to its local center manifold, eliminating the non-internally resonant nonlinear terms and transforming the nonlinear map to an amplitude map, that describes the local dynamics of the system. The analysis of this amplitude map allows us to define regions, in the space of the control gains, with a close interaction of the two bifurcations, which generates unstable quasiperiodic motion on a 3-torus, coexisting with two stable 2-torus quasiperiodic motions. Other regions in the space of the control gains show the coexistence of 2-torus quasiperiodic solutions, one stable and the other unstable. All the results described in this work are analytical and obtained in closed form, numerical simulations illustrate and confirm the analytical results.     

Criticality of bifurcation in the tuned axial–torsional rotary drilling model

We study the bifurcation characteristics of a lumped-parameter model of rotary drilling with 1:1 internal resonance between the axial and the torsional modes which leads to the largest stability thresholds. For this special case, the two-degree-of-freedom model for the drill-string reduces to an effectively single-degree-of-freedom system facilitating further analysis. The regenerative effect of the cutting action due to the axial vibrations is incorporated through a delayed term in the cutting force with the delay depending on the torsional oscillations. This state dependency of the delay introduces nonlinearity in the current model. Steady drilling loses stability via a Hopf bifurcation, and the nature of the bifurcation is determined by an analytical study using the method of multiple scales. We find that both subcritical and supercritical Hopf bifurcations are present in this system depending on the choice of operating parameters. Hence, the nonlinearity due to the state-dependent delay term could both be stabilizing or destabilizing in nature, and the self-interruption nonlinearity is essential to capture the global behavior. Numerical bifurcation analysis of a global axial–torsional model of rotary drilling further confirms the analytical results from the method of multiple scales. Further exploration of the rotary drilling dynamics unravels more complex phenomena including grazing bifurcations and possibly chaotic solutions.     

多维磁浮柔性转子控制系统分岔与控制器设计

讨论了多维悬浮柔性转子控制系统局部及全局分岔问题,首先建立了该复杂系统动力学模型,应用中心流形和求规范形综合方法,得到此系统非半简双零特征值问题的规范形及其普适开折,并进一步讨论了此控制系统的分岔 行为（余维二分岔）及稳定性;给出了为实现稳定控制,控制器参数、转子系统结构参数的相互关系及稳定控制域,即给出分岔 参数条件、分岔曲线、转迁集,最后,给出此柔性转子控制系统的数值仿真结果。     

Multiple scales and normal forms in a ring of delay coupled oscillators with application to chaotic Hindmarsh–Rose neurons

We study the appearance and stability of spatiotemporal periodic patterns like phase-locked oscillations, mirror-reflecting waves, standing waves, in-phase or antiphase oscillations, and coexistence of multiple patterns, in a ring of bidirectionally delay coupled oscillators. Hopf bifurcation, Hopf–Hopf bifurcation, and the equivariant Hopf bifurcation are studied in the viewpoint of normal forms obtained by using the method of multiple scales which is a kind of perturbation technique, thus a clear bifurcation scenario is depicted. We find time delay significantly affects the dynamics and induces rich spatiotemporal patterns. With the help of the unfolding system near Hopf–Hopf bifurcation, it is confirmed in some regions two kinds of stable oscillations may coexist. These phenomena are shown for the delay coupled limit cycle oscillators as well as for the delay coupled chaotic Hindmarsh–Rose neurons.     

Numerical stability analysis and control of umbilical–ROV systems in one-degree-of-freedom taut–slack condition

In this paper, the stability of an umbilical–ROV system under nonlinear oscillations in heave motion is analyzed using numerical methods for the uncontrolled and controlled cases comparatively. Mainly the appearance of the so-called taut–slack phenomenon on the umbilical cable produced by interactions of monochromatic waves and an operated ROV is specially focused. Nonlinear elements were considered as nonlinear drag damping, bilinear restoring force and saturation of the actuators. Free-of-taut/slack stability regions are investigated in a space of physical bifurcation parameters involving a set of both operation and design parameters. They indicate a wide diversity in qualitative behaviors, both in the periodicity and possible routes to chaos from the stability regions to outside. For detection of periodicity of the nonlinear oscillations inside and outside the stability regions, a method based on Cauchy series is developed. The first part of the results is dedicated to the stability of the uncontrolled dynamics. These results suggest the design of a control system that is able to counteract hefty hauls of the cable during the sinking/lifting operation under perturbation. A combination of a force and cinematic controller based on nonlinear model–reference control is proposed. Through a comparative study of the stability regions for uncontrolled and controlled dynamics, it is shown that the control system can extend considerably these regions without appearance of the taut–slack phenomenon despite the presence of wave perturbations. The limits between the taut and taut–slack zones are defined by the wave steepness and the available energy of the actuators.     

Two-parameter bifurcation analysis of limit cycles of a simplified railway wheelset model

The effect of the nonlinear terms on bifurcation behaviors of limit cycles of a simplified railway wheelset model is investigated. At first, the stable equilibrium state loses its stability via a Hopf bifurcation. The bifurcation curve is divided into a supercritical branch and a subcritical one by a generalized Hopf point, which plays a key role in determining the occurrence of flange contact and derailment of high-speed railway vehicles, and the occurrence of this critical situation is an important decision-making criteria for design parameters. Secondly, bifurcations of limit cycles are discussed by comparing the bifurcation behavior of cycles for two different nonlinear parameters. Unlike local Hopf bifurcation analysis based on a single bifurcation parameter in most papers, global bifurcation analysis of limit cycles based on two bifurcation parameters is investigated, simultaneously. It is shown that changing nonlinear parameter terms can affect bifurcation types of cycles and division of parameter domains. In particular, near the branch points of cycles, two symmetrical limit cycles are created by a pitchfork bifurcation and then two symmetrical cycles both undergo a period-doubling bifurcation to form two stable period-two cycles. Around the resonant points, period orbits can make several turns, whose number of turns corresponds to the ratio of resonance. Thirdly, near the Neimark–Sacker bifurcation of cycles, a stable torus is created by a supercritical Neimark–Sacker bifurcation, which shows that the orbit of the model exhibits modulated oscillations with two frequencies near the limit cycle. These results demonstrate that nonlinear parameter terms can produce very complex global bifurcation phenomena and make obvious effects on possible hunting motions even though a simple railway wheelset model is concerned.

     

Fractional-order PD control at Hopf bifurcations in a fractional-order congestion control system

In this paper, we address the problem of the bifurcation control of a delayed fractional-order dual model of congestion control algorithms. A fractional-order proportional–derivative (PD) feedback controller is designed to control the bifurcation generated by the delayed fractional-order congestion control model. By choosing the communication delay as the bifurcation parameter, the issues of the stability and bifurcations for the controlled fractional-order model are studied. Applying the stability theorem of fractional-order systems, we obtain some conditions for the stability of the equilibrium and the Hopf bifurcation. Additionally, the critical value of time delay is figured out, where a Hopf bifurcation occurs and a family of oscillations bifurcate from the equilibrium. It is also shown that the onset of the bifurcation can be postponed or advanced by selecting proper control parameters in the fractional-order PD controller. Finally, numerical simulations are given to validate the main results and the effectiveness of the control strategy.

     

Delay-induced bifurcations in collocated position control of an elastic arm

The interference of the elasticity of a single robotic arm and the unavoidable time delay of its position control is analysed from nonlinear vibrations viewpoint. The simplified mechanical model of two blocks and a connecting spring considers the first vibration mode of the arm, while the collocated proportional-derivative (PD) control uses the state of the first block only and actuates also there. It is assumed that the relevant nonlinearity is the saturation of the delayed control force. The linear stability analysis proves that stabilizable and non-stabilizable parameter regions follow each other periodically even for large spring stiffnesses and for tiny time delays. Hopf bifurcation calculation is carried out after an infinite-dimensional centre manifold reduction, and closed-form algebraic expressions are given for the amplitudes of the emerging oscillations. These results support the experimental tuning of the control gains since the parameters of the arising and often unexpected self-excited vibrations can serve as a guide for this practical procedure.

     

反馈时滞对van der Pol振子张弛振荡的影响

研究反馈控制环节时滞对van derPol振子张弛振荡的影响. 首先, 通过稳定性切换分析, 得到了系统的慢变流形的稳定性和分岔点分布图, 结果表明, 当时滞大于某临界值时, 系统慢变流形的结构发生本质的变化.其次, 基于几何奇异摄动理论, 分析了慢变流形附近解轨线的形状, 发现时滞反馈会引起张弛振荡中的慢速运动过程中存在微幅振荡, 其中微幅振荡来自于内部层引起的振荡和Hopf分岔产生的振荡两个方面; 同时, 时滞对张弛振荡的周期也具有显著的影响. 实例分析表明理论分析结果与数值结果相吻合.      

一类混沌系统中的簇发振荡及其延迟叉形分岔行为

由于多时间尺度问题在实际工程系统中广泛存在,关于其复杂动力学行为及其产生机制的研究已成为当前国内外的热点课题之一.簇发振荡是多时间尺度系统复杂动力学行为的典型代表,而分岔延迟又是簇发振荡中的常见现象.本文为探讨非线性系统中分岔延迟所引发的簇发振荡的分岔机制,在一个三维混沌系统中引入参数激励,当激励频率远小于系统的固有频率时,系统产生了两时间尺度簇发振荡.将整个激励项看做慢变参数,激励系统转化为广义自治系统也即快子系统,分析快子系统平衡点的稳定性以及分岔条件,并运用快慢分析法和转换相图揭示了簇发振荡的动力学机理.文中考察了4组参数条件下系统的动力学行为,研究发现当慢变激励项周期性地通过分岔点时,系统产生了明显的超临界叉形分岔延迟行为,随着参数激励振幅的增大,分岔延迟的时间也逐渐延长,当这种延迟的动态行为终止于不同的参数区域时,导致系统轨线围绕不同稳定吸引子(平衡点,极限环)运动,从而得到了不同的簇发振荡行为.      

The Saddle-Node of Nearly Homogeneous Wave Trains in Reaction–Diffusion Systems

We study the saddle-node bifurcation of a spatially homogeneous oscillation in a reaction-diffusion system posed on the real line. Beyond the stability of the primary homogeneous oscillations created in the bifurcation, we investigate existence and stability of wave trains with large wavelength that accompany the homogeneous oscillation. We find two different scenarios of possible bifurcation diagrams which we refer to as elliptic and hyperbolic. In both cases, we find all bifurcating wave trains and determine their stability on the unbounded real line. We confirm that the accompanying wave trains undergo a saddle-node bifurcation parallel to the saddle-node of the homogeneous oscillation, and we also show that the wave trains necessarily undergo sideband instabilities prior to the saddle-node.     

Bifurcations and catastrophes in aerodynamic characteristics

The results of an experimental investigation of the longitudinal stability of a model maneuvering aircraft are presented. The results for a wide angle-of-attack range are obtained in a wind tunnel flow on an aerodynamic setup of free oscillations with a single degree of freedom. It is shown that the static aerodynamic dependences of the normal force and pitch moment coefficients on the angle of attack include catastrophic transitions from one steady state into another. The salient features of these transitions are established. It is experimentally found that the loss of the longitudinal stability of the model aircraft in a flow with variation in the deflection angles of stabilizers is softly realized via the Hopf bifurcation. At high angles of attack the flow regimes are found to exist in which steady motion represents a strange attractor.