Solar sail chaotic pitch dynamics and its control in Earth orbits

The attitude dynamics and control for solar sail orbiting a celestial body (e.g., the Earth) are critical for the space missions. In the paper, the pitch dynamics is addressed by considering the torques by the center-of-mass and center-of-pressure offset, the gravity gradient, the internal damping and the control vane. The chaotic pitch motion is analytically detected for the sailcraft in the circular and elliptical orbits with small eccentricities using the Melnikov’s method. The validity of the Melnikov method is numerically verified by checking the Poincare surface of section and the power spectral density. The stability criterion method with some improvements is utilized to stabilize the chaotic pitch motion onto the reference unstable periodic motion embedded in the chaotic attractor. The reference unstable periodic motion is obtained based on the calculation of the close return pairs. The small control input torques and the stabilization effects are presented, and the advantages of the modified stabilization method are clarified based on the numerical simulations.     

Noise-induced chaos in the elastic forced oscillators with real-power damping force

The chaotic behavior of the elastic forced oscillators with real-power exponents of damping and restoring force terms under bounded noise is investigated. By using random Melnikov method, a mean square criterion is used to detect the necessary conditions for chaotic motion of this stochastic system. The results show that the threshold of bounded noise amplitude for the onset of chaos in the system increases as the intensity of the random frequency increases, and decrease as the real-power exponent of damping term increase. The threshold of bounded noise amplitude for the onset of chaos is determined by the numerical calculation via the largest Lyapunov exponents. The effects of bounded noise and real-power exponent of damping term on bifurcation and Poincaré map are also investigated. Our results may provide a valuable guidance for understanding the effect of bounded noise on a class of generalized double well system.     

Melnikov analysis for a ship with a general roll-damping model

In the framework of a general roll-damping model, we study the influence of different damping models on the nonlinear roll dynamics of ships through a detailed Melnikov analysis. We introduce the concept of the Melnikov equivalent damping and use phase-plane concepts to obtain simple expressions for what we call the Melnikov damping coefficients. We also study the sensitivity of these coefficients to parameter variations. As an application, we consider the equivalence of the linear-plus-cubic and linear-plus-quadratic damping models, and we derive a condition under which the two models yields the same Melnikov predictions. The free- and forced-oscillation behaviors of the models satisfying this condition are also compared.     

Global bifurcations and multi-pulse chaotic dynamics of rectangular thin plate with one-to-one internal resonance

Global bifurcations and multi-pulse chaotic dynamics for a simply supported rectangular thin plate are studied by the extended Melnikov method.The rectangular thin plate is subject to transversal and in-plane excitation.A two-degree-of-freedom nonlinear nonautonomous system governing equations of motion for the rectangular thin plate is derived by the von Karman type equation and the Galerkin approach.A one-toone internal resonance is considered.An averaged equation is obtained with a multi-scale method.After transforming the averaged equation into a standard form,the extended Melnikov method is used to show the existence of multi-pulse chaotic dynamics,which can be used to explain the mechanism of modal interactions of thin plates.A method for calculating the Melnikov function is given without an explicit analytical expression of homoclinic orbits.Furthermore,restrictions on the damping,excitation,and detuning parameters are obtained,under which the multi-pulse chaotic dynamics is expected.The results of numerical simulations are also given to indicate the existence of small amplitude multi-pulse chaotic responses for the rectangular thin plate.     

Bifurcation response and Melnikov chaos in the dynamic of a Bose–Einstein condensate loaded into a moving optical lattice

We investigate global bifurcation of a Bose–Einstein condensate with both repulsive two-body interaction between atoms and attractive three-body interaction loaded into a traveling optical lattice. Slow-flow equations of the traveling wave function are the first to derive and the reduced amplitude equation is obtained. The Melnikov method is applied on the reduced parametrically driven system and the Melnikov function is subsequently established. Effects of different physical parameters on the global bifurcation are studied analytically and numerically, and different chaotic regions of the parameter space are found. The results suggest that optical intensity may help to enhance chaos while the strength of the effective three-body interaction, the velocity of the optical lattice, and the damping coefficients annihilate or reduce chaotic behavior of the steady-state traveling wave solution of the particle number density of a Bose–Einstein condensate.     

Chaotic thresholds for the piecewise linear discontinuous system with multiple well potentials

In this paper we investigate the bifurcations and the chaos of a piecewise linear discontinuous (PWLD) system based upon a rig-coupled SD oscillator, which can be smooth or discontinuous (SD) depending on the value of a system parameter, proposed in [18], showing the equilibrium bifurcations and the transitions between single, double and triple well dynamics for smooth regions. All solutions of the perturbed PWLD system, including equilibria, periodic orbits and homoclinic-like and heteroclinic-like orbits, are obtained and also the chaotic solutions are given analytically for this system. This allows us to employ the Melnikov method to detect the chaotic criterion analytically from the breaking of the homoclinic-like and heteroclinic-like orbits in the presence of viscous damping and an external harmonic driving force. The results presented here in this paper show the complicated dynamics for PWLD system of the subharmonic solutions, chaotic solutions and the coexistence of multiple solutions for the single well system, double well system and the triple well dynamics.     

Analytical study of nonlinear synchronous full annular rub motion of flexible rotor–stator system and its dynamic stability

The nonlinear synchronous full annular rub motion of a flexible rotor induced by the mass unbalance and the contact-rub force with rigid and flexible stator is studied analytically. The nonlinear property is due to the dry friction force between stator and rotor. The exact solutions of the synchronous full annular rub motion and its run speed regions are obtained. The stability of the synchronous full annular rub motion is discussed analytically. The stability criterion and the stability regions of the synchronous full annular rub motion are obtained. A simplified approximate criterion formula for dynamic stability is also derived under the conditions of large impact stiffness, small damping and small friction. The simplified criterion formula can be used conveniently in engineering and matches the real situations of industry.     

有约束阻尼的非保守力学系统的稳定性

利用Ｌｙａｐｕｎｏｖ直接法，研究了有势力、陀螺力、Ｒａｙｌｅｉｇｈ阻尼和约束阻尼同时作用的非线性非保守力学系统的稳定性．假设陀螺力依赖某参数ｈ，得到系统渐近稳定的两个定理．     

Landau Damping in Sobolev Spaces for the Vlasov-HMF Model

We consider the Vlasov-HMF (Hamiltonian Mean-Field) model. We consider solutions starting in a small Sobolev neighborhood of a spatially homogeneous state satisfying a linearized stability criterion (Penrose criterion). We prove that these solutions exhibit a scattering behavior to a modified state, which implies a nonlinear Landau damping effect with polynomial rate of damping.     

A method to calculate period doubling bifurcation

Based on physical meaning of Melnikov function, we establish a method to calculate period doubling bifurcation and discuss this kind of bifurcation of soft spring Duffing system and find that the result is analogous to subharmonic bifurcation, that is, period doubling bifurcation will appear if damping is small and amplitude of excitation is big. Thi coincides with facts of physics.     

一类非自治动力系统超次谐周期解的不存在性

在非线性动力系统的研究中， Melnikov函数被广泛地用来作为微扰哈密顿系统是否发生次谐或超次谐分岔乃至混沌的判 据. 但是在大多数情况下，经典的Melnikov方法往往只给出存在次谐周期解的结论. 产生 该结果的原因被归之为在经典的Melnikov方法中只采取了一阶近似，因而高阶Melnikov方 法被发展用来判断超次谐周期解的存在性. 本文对一类非自治微分动力系统进行了研究，证 明了在这样一类系统中如果存在周期解则只可能是次谐周期解，超次谐周期解不可能存在， 并进一步证明了在一类平面问题中所定义的旋转(R)型超次谐周期解同样不可能存在.作为 该结论的一个应用，文中考察了几个典型的算例，结果表明现有的二阶Melnikov方法判断 平面扰动系统是否存在超次谐周期解的结论是不恰当的，并提供了一个简单的几何上的解释.     

Stability and Hamiltonian Hopf bifurcation for a nonlinear symmetric bladed rotor

The modal interaction which leads to Hamiltonian Hopf bifurcation is studied for a nonlinear rotating bladed-disk system. The model, which is discussed in the paper, is a Jeffcott rotor carrying a number of planar blades which bend in the plane of the motion. The rigid rotating disk is supported on nonlinear bearings. It is supposed that this dynamical system is a Hamiltonian system which is perturbed by small dissipative and nonlinear forces. Krein’s theorem is employed for obtaining a stability criterion. The nonlinear eigenvalue equations on the stability boundary are turned into ordinary differential equations (ODEs) by differentiating them over the rotating speed. By solving these ODEs, the eigenmodes and the eigenvalues on the stability boundary are obtained. The bifurcation analysis is performed by applying multiple scales method around the boundary. The rotor nonlinear behavior and damping effects are studied for different conditions on the rotating speed and nonlinearity type by the bifurcation equation. It is shown that the damping distribution between the blades and bearings may shift the unstable mode. Depending on the nonlinearity type, subcritical and supercritical Hopf bifurcation are possible.     

Poynting oscillations of a rigid disk supported by a neo-Hookean shaft

Simultaneous axial and torsional oscillations of a rigid disk attached to an elastomeric shaft are investigated. Five cases are solved exactly. The uncoupled, small amplitude axial and torsional oscillations of the disk are investigated for neo-Hookean and Mooney-Rivlin shafts with static stretch. The finite torsional vibration of the load superimposed on a static stretch of the shaft is studied for the Mooney-Rivlin model. Solutions for both small and finite amplitude, uniaxial vibrations of the body superimposed on a pretwisted neo-Hookean shaft with static stretch are derived. Simple bounds on the period for the finite motion are provided; and various universal frequency relations for neo-Hookean and Mooney-Rivlin materials are identified.Finally, the main problem of finite, uniaxial vibrations accompanied by a small twisting motion is studied for the neo-Hookean model. The exact periodic solution for the axial response is obtained; and the coupled, small torsional motion is then determined by Hill's equation. A stability criterion for the Mathieu-Hill equation is used to obtain stability maps in a physical parameter space. Geometrical conditions sufficient for universal stability of the motion are read from this graph. Instability of the torsional oscillation, the beating phenomenon and exchange of energies, and the relation of the stability diagram to amplitude bounds on the uncoupled, linearized motion sufficient to assure universal stability predicted for small amplitude vibrations, are discussed and described graphically with the aid of a numerical model. It is shown that an unstable configuration may be stabilized by increasing the diameter of the disk.     

Mode coupling instability in friction-induced vibrations and its dependency on system parameters including damping

Friction-induced vibrations due to coupling modes can cause severe damage and are recognized as one of the most serious problems in industry. In order to avoid these problems, engineers must find a design to reduce or to eliminate mode coupling instabilities in braking systems. Though many researchers have studied the problem of friction-induced vibrations with experimental, analytical and numerical approaches, the effects of system parameters, and more particularly damping, on changes in stable-unstable regions and limit cycle amplitudes are not yet fully understood.The goal of this study is to propose a simple non-linear two-degree-of-freedom system with friction in order to examine the effects of damping on mode coupling instability. By determining eigenvalues of the linearized system and by obtaining the analytical expressions of the Routh–Hurwitz criterion, we will study the stability of the mechanical system's static solution and the evolution of the Hopf bifurcation point as functions of the structural damping and system parameters. It will be demonstrated that the effects of damping on mode coupling instability must be taken into account to avoid design errors. The results indicate that there exists, in some cases, an optimal structural damping ratio between the stable and unstable modes which decreases the unstable region. We also compare the evolution of the limit cycle amplitudes with structural damping and demonstrate that the stable or unstable dynamic behaviour of the coupled modes are completely dependent on structural damping.     

Fourth body gravitation effect on the resonance orbit characteristics of the restricted three-body problem

In this paper, the gravitational effect of a fourth body on the resonance orbit defined in the restricted three-body problem (RTBP) is considered. In this regard, Resonance Hamiltonian of the RTBP and the Hamiltonian associated with the fourth gravitational body that perturbs the resonance orbit are computed. The Melnikov approach is utilized as a mean for the detection of chaos in resonance orbit under the influence of the fourth gravitation body. In addition, the numerical simulation of RTBP and bicircular four-body model, time–frequency analysis (TFA), and fast Lyapunov indicator (FLI) are performed to verify the results of the Melnikov approach. The results indicate that for the (2:1) resonance orbit, the Melnikov integral computed over outer loop of separatrix does not cross the zero line, and consequently chaos is unexpected. On the other hand, the Melnikov integral computed over the inner sepratrix loop crosses the zero line indicating a potential for chaos. Similarly, it is shown that inclusion of the fourth body gravitation leads the (3:1) as well as the (4:1) resonance orbits to chaos. Additionally, simulation results indicate that for some initial conditions on the separatrix, the fourth body effect bounds the amplitude of the resonance orbits while diffusing its corresponding trajectory in the bounded phase space. TFA and the FLI verify similar results.     

Chaotic pitch motion of an asymmetric non-rigid spacecraft with viscous drag in circular orbit

We study the pitch motion dynamics of an asymmetric spacecraft in circular orbit under the influence of a gravity gradient torque. The spacecraft is perturbed by a small aerodynamic drag torque proportional to the angular velocity of the body about its mass center. We also suppose that one of the moments of inertia of the spacecraft is a periodic function of time. Under both perturbations, we show that the system exhibits a transient chaotic behavior by means of the Melnikov method. This method gives us an analytical criterion for heteroclinic chaos in terms of the system parameters which is numerically contrasted. We also show that some periodic orbits survive for perturbation small enough.     

Heteroclinic bifurcations in completely liquid-filled spacecraft with flexible appendage

In this paper, the chaotic dynamics in an attitude transition maneuver of a rigid body with a completely liquid-filled cavity in going from minor axis to major axis spin under the influence of viscous damping and a small flexible appendage constrained to undergo only torsional vibration is investigated. The focus in this paper is on the way in which the dynamics of the liquid and flexible appendage vibration are coupled. The equations of motion are derived and then transformed into a form suitable for the application of Melnikov's method. Melnikov's integral is used to predict the transversal intersections of the stable and unstable manifolds for the perturbed system. An analytical criterion for chaotic motion is derived in terms of the system parameters. This criterion is evaluated for its significance to the design of spacecraft. The dependence of the onset of chaos on quantities such as body shape and magnitude of damping values, fuel fraction and frequency of flexible appendage vibration are investigated.     

Heteroclinic motion and energy transfer in coupled oscillator with nonlinear magnetic coupling

In this paper, we consider two coupled oscillators exhibiting both transient chaos and energy transfer from mechanical to electrical oscillators. Melnikov method is applied to these oscillators with linear damping and strongly nonlinear coupling terms in order to study the possibility of existence of chaos and transversal heteroclinic orbits and their control in a dynamical system. The energy transfer is studied using a qualitative measure of the system which can be obtained by computing the energy dissipated in it. At last, the numerical simulation is carried out for this system.     

Non-linear parametric vibration and stability of axially moving visco-elastic Rayleigh beams

An axially moving visco-elastic Rayleigh beam with cubic non-linearity is considered, and the governing partial-differential equation of motion for large amplitude vibration is derived through geometrical, constitutive, and dynamical relations. By directly applying the method of multiple scales to the governing equations of motion, and considering the solvability condition, the linear and non-linear frequencies and mode shapes of the system are analytically formulated. In the presence of damping terms, it can be seen that the amplitude is exponentially time-dependent, and as a result, the non-linear natural frequencies of the system will be time-dependent. For the resonance case, through considering the solvability condition and Routh–Hurwitz criterion, the stability conditions are developed analytically. Eventually, the effects of system parameters on the vibrational behavior, stability and bifurcation points of the system are investigated through parametric studies.     

Nonlinear dynamics in the flexible shaft rotating–lifting system of silicon crystal puller using Czochralski method

Silicon crystal puller (SCP) is key equipment in silicon wafer manufacture, which is, in turn, the base material for the most currently used integrated circuit chips. With the development of the techniques, the demand for longer mono-silicon crystal rod with larger diameter is continuously increasing in order to reduce the manufacture time and the price of the wafer. This demand calls for larger SCP with an increasing height, though it causes serious swing phenomenon of the crystal seed. The strong swing of the seed increases the possibility of defects in the mono-silicon rod and the risk of mono-silicon growth failure. The main aim of this paper is to analyze the nonlinear dynamics in flexible shaft rotating–lifting (FSRL) system of the SCP. A mathematical model for the swing motion of the FSRL system is derived. The influence of relevant parameters, such as system damping, excitation amplitude, and rotation speed, on the stability and the responses of the system is analyzed. The stability of the equilibrium, bifurcation, and chaotic motion is demonstrated, which have been observed in practical situations. Melnikov method is used to derive the possible parameter region which leads to chaotic motion. Three routes to chaos are identified in the FSRL system, including period doubling, symmetry-breaking bifurcation, and crisis. The work in this paper analyzes and explains the complex dynamics in FSRL system of the SCP, which will be helpful for the designers in the designing process in order to avoid the swing phenomenon in the SCP.