一种直流力矩电机系统的滞滑摩擦补偿方法

陀螺漂移测试转台直流力矩电机系统中存在的非线性滞滑摩擦，使转台在PID控制下存在滞滑极限环。为提高转台定位精度，应用具有滞滑(stick-slip)摩擦的直流力矩电机系统模型，推导了一种补偿方法，对含有滞滑摩擦的PID控制转台直流力矩电机伺服系统进行滞滑摩擦补偿。在采用PID控制的转台电机系统定位工作状态下，这种滞滑补偿方法可以减小滞滑极限环的幅值。仿真结果证明了该补偿方法的有效性。     

ASYMPTOTIC ANALYSIS OF A CLASS OF NONLINEAR OSCILLATION EQUATION IN ELECTRICAL ENGINEERING

ＡＳＹＭＰＴＯＴＩＣＡＮＡＬＹＳＩＳＯＦＡＣＬＡＳＳＯＦＮＯＮＬＩＮＥＡＲＯＳＣＩＬＬＡＴＩＯＮＥＱＵＡＴＩＯＮＩＮＥＬＥＣＴＲＩＣＡＬＥＮＧＩＮＥＥＲＩＮＧＣｈｅｎｇＹｏｕ－ｌｉａｎｇ（程友良）（ＤｅｐａｒｔｍｅｎｔｏｆＦｕｎｄａｍｅｎｔａｌＣｏｕ...     

谐和与随机噪声联合作用下非线性系统的响应

研究了Duffing振子在谐和与随噪声联合激励下的响应和稳应性问题。用谐波平均法分析了系统在确定性谐和激励和随机激励联合作用下的响应,用随机平均法讨论了随机扰动项对系统晌应的影响。在一定条件下,系统具有两个均方响应值和跳跃现象。数值模拟表明本文提出的方法是有效的。     

Robust Limit Cycle Suppression for Control Systems with Parametric Uncertainty and Nonlinearity

A systematic limit cycle suppression method is presented to predict and suppress the sustained limit cycles persisting in a class of uncertain control systems containing inherent nonlinearity. The aim is to determine the feasible controller parameter sets in the parameter plane for the entire uncertain system. A family of constant limit cycle amplitude loci is plotted. Successful application to pitch orientation control system with parametric uncertainties and nonlinearity is demonstrated. System output response is further investigated.     

Non-linear oscillator limit cycle analysis using a time transformation approach

A method is presented for the analysis of limit cycle behavior of autonomous non-linear oscillators characterized by second order ordinary differential equations containing a small parameter. The method differs from the classical perturbation methods in that the dependent variable is not expanded in a power series in the small parameter. Rather, a new independent variable is sought such that in its domain the motion is simple harmonic. Use of this time transformation technique to generate limit cycle phase portrait, amplitude and period is presented. We show results of the application of the method to the van der Pol oscillator, to an oscillator with quadratic damping, and to a modified van der Pol oscillator which is statically unstable in the limit of small motion.     

Separatrices and limit cycles of strongly nonlinear oscillators by the perturbation-incremental method

The perturbation-incremental method is applied to determine the separatrices and limit cycles of strongly nonlinear oscillators. Conditions are derived under which a limit cycle is created or destroyed. The latter case may give rise to a homoclinic orbit or a pair of heteroclinic orbits. The limit cycles and the separatrices can be calculated to any desired degree of accuracy. Stability and bifurcations of limit cycles will also be discussed.     

Analysis and synthesis of oscillator systems described by perturbed single well Duffing equations

The paper presents an analysis as well as a synthesis of oscillator systems described by single well Duffing equations under polynomial perturbations of fourth degree. It is proved that such a system can have a unique hyperbolic limit cycle. An analytical condition has been obtained for the arising of a limit cycle and an equation giving the parameters of this limit cycle. There has been proposed a method for the synthesis of oscillator systems of the considered type, having preliminarily assigned properties. The synthesis consists of an appropriate choice of the perturbation coefficients in such a way that the oscillator equation should have a preliminary assigned limit cycle. Both the analysis and the synthesis are performed with the help of the Melnikov function.     

One-to-Two Internal Resonance in Two-Degrees-of-Freedom Nonlinear System with Narrow-Band Excitations

Response of two-degrees-of-freedom nonlinearsystem to narrow-band random parametric excitation isinvestigated. The method of multiple scales is used todetermine the equations of modulation of amplitude andphase. The effect of detunings and amplitude areanalyzed. Theoretical analyses and numerical simulationsshow that the nontrivial steady-state solution may changeform a limit cycle to a diffused limit cycle as theintensity of the random excitation increase. Under someconditions, the system may have two steady-statesolutions.     

Resonant response of a non-linear vibro-impact system to combined deterministic harmonic and random excitations

The resonant resonance response of a single-degree-of-freedom non-linear vibro-impact oscillator, with cubic non-linearity items, to combined deterministic harmonic and random excitations is investigated. The method of multiple scales is used to derive the equations of modulation of amplitude and phase. The effects of damping, detuning, and intensity of random excitations are analyzed by means of perturbation and stochastic averaging method. The theoretical analyses verified by numerical simulations show that when the intensity of the random excitation increases, the non-trivial steady-state solution may change from a limit cycle to a diffused limit cycle. Under certain conditions, impact system may have two steady-state responses. One is a non-impact response, and the other is either an impact one or a non-impact one.     

带外挂二元翼极限环颤振的高次线化分析

本文针对在外挂上带有初偏间隙型非线性刚度的二元翼带外挂系统的极限环颤振,进行KBM法二次渐近解等效线化分析.结果表明,在某些情况下,用二次渐近解等效线化分析极限环颤振,其结果与数值积分结果基本一致,而一次渐近解等效线化分析则得不出满意的结果.这就是说,二次解等效线化比一次解等效线化更为可靠.     

On the design of an electromagnetic aeroelastic energy harvester from nonlinear flutter

A new energy harvester by coupling the electromagnetic induction and the pitch vibration of a rigid wing is built up in this paper. It is aimed: (1) to harvest energy from the pitch limit cycle oscillation (LCO) of the wing due to the preloaded free-play nonlinearity; (2) to introduce a theoretical analysis scheme based on the equivalent linearized method into the design of this harvester. With the equivalent linearized method, the domains of the single stable LCO and double stable LCOs are respectively obtained. Combining the analytical and numerical solutions, the single stable LCO along with the stable limit cycle amplitude greater than its corresponding unstable one is recognized as the better mode for harvesting, since the larger limit cycle domain is induced and the more energy are yielded. Based on such chosen mode, analyses of varying parameters are conducted with respect to the plunge stiffness, pitch stiffness, distance of elastic axis from center of gravity, distance of geometric center from elastic axis, load resistance and magnetic flux density. Meanwhile, three indicators are applied to reveal their effects on the harvesting performances: (1) the size of limit cycle domain, (2) the onset velocity of LCO, and (3) the energy output.     

Analysis and synthesis of oscillator systems described by a perturbed double-well Duffing equation

This paper presents an investigation of limit cycles in oscillator systems described by a perturbed double-well Duffing equation. The analysis of limit cycles is made by the Melnikov theory. Expressing the solutions of the unperturbed Duffing equation by Jacobi elliptic functions allows us to calculate explicitly the Melnikov function, whereupon the final result is a function involving the complete elliptic integrals. The Melnikov function is analyzed with the aid of the Picard–Fuchs and Riccati equations. It has been proved that the considered oscillator system can have two small hyperbolic limit cycles located symmetrically with respect to the y-axis, or one large hyperbolic limit cycle, or two large hyperbolic limit cycles, or one large limit cycle of multiplicity 2. Moreover, we have obtained the conditions under which each of these limit cycles arises. The present work gives the conditions for the arising of limit cycles around the homoclinic trajectory. In this connection, an alternative approach is proposed for obtaining a series expansion of the Melnikov function near the homoclinic trajectory. This approach uses the series expansion of the complete elliptic integrals as the elliptic modulus tends to 1. It is shown that a jumping phenomenon may occur between limit cycles in the analyzed oscillator system. The conditions for the occurrence of this jumping phenomenon are given. A method for the synthesis of an oscillator system with a preliminary assigned limit cycle is also presented in the article. The obtained analytical results are illustrated and confirmed by numerical simulations.     

Limit cycles and homoclinic orbits and their bifurcation of Bogdanov-Takens system

A quantitative analysis of limit cycles and homoclinic orbits, and the bifurcation curve for the Bogdanov-Takens system are discussed. The parameter incremental method for approximate analytical-expressions of these problems is given. These analytical-expressions of the limit cycle and homoclinic orbit are shown as the generalized harmonic functions by employing a time transformation. Curves of the parameters and the stability characteristic exponent of the limit cycle versus amplitude are drawn. Some of the limit cycles and homoclinic orbits phase portraits are plotted. The relationship curves of parameters μ and A with amplitude a and the bifurcation diagrams about the parameter are also given. The numerical accuracy of the calculation results is good.     

Visco-elastic systems under both deterministic harmonic and random excitation

IntroductionForlinearviscoelasticsystemsunderbothadditiveandmultiplicativebroad_bandexcitationexcitations,Ariaratnam[1]studiedthestochasticstabilityofthesystembyusingthemethodofstochasticaveragingprocedure .Itwasshownthatthevisco_elasticforcecontributedtowarddamping ,hence ,stabilityofthesystem .However,thestiffnesseffectofthevisco_elasticcomponentwasnotfullyaccountedfor.FurthermoreAriaratnam[2 ]studiedthestochasticstabilityofthesystembutthemodelislinear.Inthetheoryofnonlinearrandomvibration…     

Dynamic bifurcations analysis of a micro rotating shaft considering non-classical theory and internal damping

In this study, the dynamic bifurcation of a viscoelastic micro rotating shaft is investigated. The non-classical theory (the modified couple stress theory) and the Kelvin Voigt model are used for modeling the viscoelastic micro shaft. The transverse equations of motion are derived using the variational approach. The reduced order model of the system is obtained by the Galerkin method. Using the Routh–Hurwitz criteria the stability regions of the system are extracted in which the effect of the length scale parameter is significant. Using the center manifold theory and the normal form method the double zero eigenvalue bifurcation is analyzed. The results show that the internal and external damping coefficients, the rotational speed and the material length scale parameter influence the critical speed, amplitude, and phase of a non-trivial solution, and radius of limit cycle (periodic solution). Also, it is seen that by increasing the dimensionless length scale parameter (material length scale per radius of the shaft) the radius of the limit cycle is decreased, whereas the critical rotational speed and the rate of the phase are increased. However, the radius of the limit cycle concerning the classical theory is higher than that of regarding the modified couple stress theory. Furthermore, with an increase of the external damping coefficient the radius of the limit cycle is linearly decreased; however, the critical speed of the system is increased. Additionally, by decreasing length scale parameter the results of the modified couple stress theory approach the classical theory ones.     

Computation of phase response curves via a direct method adapted to infinitesimal perturbations

A new numerical algorithm for computation of phase response curves of stable limit cycle oscillators is proposed. The idea of the algorithm originates from a direct method that is based on computation of the oscillator response to short finite pulses delivered at different phases of oscillations. Here we adapt the direct method to the case of infinitesimal perturbations and compare our algorithm with the standard algorithm based on the backward integration of the adjoint equations. In contrast to the standard algorithm, our algorithm does not require any backward integration and it is easier to program since a necessity of numerical interpolation for the Jacobian matrix is avoided. In addition, we demonstrate by examples that our algorithm is faster than the standard algorithm and this advantage is especially notable for weakly stable limit cycle oscillators.     

Limit Cycle Oscillation and Orbital Stability in Aeroelastic Systems with Torsional Nonlinearity

The paper treats the question of the existence of limit cycleoscillations of prototypical aeroelastic wing sections with structuralnonlinearity using the describing function method. The chosen dynamicmodel describes the nonlinear plunge and pitch motion of a wing. Themodel includes an asymmetric structural nonlinearity in the pitchdegree-of-freedom. The dual-input describing functions of thenonlinearity are derived for the limit cycle analysis. Analyticalexpressions for the average value, and the amplitude and frequency ofoscillation of pitch and plunge responses are obtained. Based on ananalytical approach as well as the Nyquist criterion, stability of thelimit cycles is examined. Numerical results are presented for a set ofvalues of the flow velocities and the locations of the elastic axiswhich show that the predicted limit cycle oscillation amplitude andfrequency as well as the mean value are quite close to the actualvalues. Furthermore, for the chosen model with linear aerodynamics, itis seen that the amplitude of the pitch limit cycle oscillation does notalways increase with the flow velocity for certain elastic axislocations.     

Non-linear oscillations of a third-order differential equation

An investigation is conducted into the behavior of the solutions of a third-order non-linear differential equation which is characterized by a non-linearity depending solely upon the Euclidean norm of the associated phase space. The non-linearity represents a central restoring force, which has important applications in modern control theory. For small non-linearities, the existence of a limit cycle is established by a fixed point technique, the approach to the limit cycle is approximated by averaging methods, and the periodic solution is harmonically represented by perturbation. Computer solutions of the differential equation are provided in order to reinforce the analysis. Some related differential equations are discussed including one in which the periodic solution is explicitly prescribed.     

Steady-State Analysis for a Class of Sliding Mode Controlled Systems Using Describing Function Method

In this paper, the analysis of the steady-state response of the slidingmode control system is presented. The nonlinearity of the switching termin the control law is approximately characterized by using itsequivalent describing function. The parasitic dynamics is modeled as afirst-order lag transfer function, and a possible transport delay isconsidered. Subsequently, a frequency domain method is used for theprediction of limit cycles. The stability-equation method together withthe parameter plane method is proposed to predict graphically limitcycles in the system coefficient plane. Four common types of switchingfunctions are investigated. This analysis further provides an approachof switching control gain selection for suppressing the limit cycle inthe sliding mode.