Fixed Final Time Optimal Control via Simple Cell Mapping

A cell mapping strategy based on Bellman'sPrinciple of optimality (BP) is proposed to solve the fixed final time optimalcontrol of strongly nonlinear systems with state and control bounds. Thefixed final time problem is transformed into a fixed final time-fixed finalstate optimal control problem by reversing the time. This transformationallows to solve the problem in the framework of the BP. Backward searchingalgorithms within the cell mapping context are used to obtain the solution ofthe new problem. This approach reduces considerably the computational effortrequired for the original problem when it is solved by a forward searchingapproach. The vibration control problem of damped and undamped variablestiffness oscillators with bounded stiffness tuning range is studied todemonstrate the effectiveness of the approach. A numerical study on theconvergence of the method is also presented.     

Log-likelihood maximization and response surface in reliability assessment

Nonlinear dynamics problems can generally be solved only in a numerical way. This prevents from a direct application of standard reliability methods. A technique which makes use of iterated response-surface analytical approximations of the system performance function was therefore proposed in view of reliability assessment. The limitation of this technique was of working in a standard normalized space, so that appropriate space transformations are preliminarly required.This paper shows how this response-surface iterative scheme can also be used in the original space of the random variables, provided a maximum log-likelihood constrained optimization problem is solved. Moreover, asymptotic theory also provides a better estimate of the probability of failure of the dynamical system against any assigned limit state.     

A Wavelet-Balance Method to Investigate the Vibrations of Nonlinear Dynamical Systems

The scope of this paper is to introduce a new wavelet-balanceprocedure allowing to give a genuine time-scale representation ofvibrations of nonlinear dynamical systems by adopting a waveletmultiresolution approach. In a former paper, a wavelet-Galerkinoriented procedure was developed to analyze vibrations of lineartime-periodic systems. The topic is here to extend the process tothe nonlinear case using a perturbation technique. The underlyingidea consists in successively balancing the linearized equationsof motion into wavelet spaces with increasing resolution scales.Here we demonstrate the wavelet-balance procedure may accuratelyexhibit both transient and stationary vibrations of any nonlinearproblem in general, whatever smooth nonlinearity shape or externalforcing may be. In addition, wavelets inherit of fairly goodtime-frequency localization properties that are likely to permitthe investigation of strong nonlinear problems. Numericalexperiments achieved on a well known Duffing oscillator involvinga cubic nonlinearity then illustrate the procedure. Simulationsattest the relevance of the method by comparison with eitherpurely numerical results obtained with a Runge–Kutta integrationscheme or with an analytical study based on the multiple scalesmethod. We demonstrate that this semi-analytical semi-numericalperturbation method permits to capture stable limit cycles of theDuffing oscillator and its related amplitude spectrum response orstill responses to pulse-like excitations. Finally, key propertiesof the method are discussed and future prospective works areoutlined.     

Duffing振子强迫振动的混沌特性仿真分析

根据Ｄｕｆｆｉｎｇ振子强迫振动系统的数学模型，编制该动力系统的计算机仿真软件．通过对Ｄｕｆｆｉｎｇ方程实例仿真分析，探讨该类方程出现混沌的各参数之间的关系，可全面分析混沌动力系统特性．     

Bifurcation Analysis of Parametrically Excited Rayleigh–Liénard Oscillators

A parametrically excited Rayleigh–Liénard oscillator is investigatedby an asymptotic perturbation method based on Fourier expansion and timerescaling. Two coupled equations for the amplitude and the phase ofsolutions are derived and the stability of steady-state periodic solutionsas well as parametric excitation-response and frequency-response curvesare determined. Comparison with the parametrically excited Liénardoscillator is performed and analytic approximate solutions are checkedusing numerical integration. Dulac's criterion, thePoincaré–Bendixson theorem, and energy considerations are used in order to study the existence and characteristics of limit cycles of the twocoupled equations. A limit cycle corresponds to a modulated motion forthe Rayleigh–Liénard oscillator. Modulated motion can be also obtainedfor very low values of the parametric excitation, and in this case, anapproximate analytic solution is easily constructed. If the parametricexcitation is increased, an infinite-period bifurcation is observed because the modulation period lengthens and becomes infinite, while themodulation amplitude remains finite and suddenly the attractor settlesdown into a periodic motion. Floquet's theory is used to evaluatethe stability of the periodic solutions, and in certain cases,symmetry-breaking bifurcations are predicted. Numerical simulationsconfirm this scenario and detect chaos and unbounded motions in theinstability regions of the periodic solutions.     

Heat transfer augmentation and pumping power in double-pipe heat exchangers

One of the criteria for evaluating the performance of a heat exchanger with extended surfaces is the pumping power required for a specified heat duty. The results of an experimental project to relate the pumping power to heat transfer augmentation in a double-pipe heat exchanger are reported. The inner, electrically heated pipe was provided with external, rectangular, axial extended surfaces with interruptions. Heat transfer augmentation and friction factors were determined for different configurations with air as the fluid. Starting with continuous fins, cuts were introduced in the fins to give four ratios of the finssegment length to the gap between the segments, and finally all the fins were removed, which resulted in smooth pipes. Five different mass flow rates in two different inner pipes were employed. Lengths, surface areas, and pumping powers for finned pipes are compared with those for smooth pipes. The average heat transfer coefficient increases with an increase in the frequency of the interruptions. For equal heat transfer rates a significant reduction in the lengths can be achieved by interrupted fins. In many cases the reduction in the length is also accompanied by a reduction in the pumping power.     

A decoupled modal form Duffing equation for an orthotropic laminate in a clamped boundary condition with thermal diffusion

The non-linear thermo-elasticity response of an orthotropic laminate in a clamped boundary condition is analyzed by means of the method of weighted residuals for its behaviors subject to thermal and mechanical loading. The equation of motion for the laminate's deflection is obtained in the form of a decoupled modal form Duffing equation, without Berger's approximation. The thermal field, with both the in-plane and transverse temperature variations, is incorporated in both a steady-state and transient state. The formulation indicates that a transverse thermal field with a temperature rise produces a load onto the laminate, while the in-plane temperature variation affects the system frequency and stability. We demonstrate the numerical results of the modal response of an isotropic laminate for the thermal buckling and vibration.     

A nonlinear oscillatory system subjected to driving forces of elliptic type

The aim of this paper is to show how Jacobi elliptic functions in combination with the averaging and the harmonic balance methods can be applied to obtain the approximate solution of two coupled, ordinary differential equations having a spring with cubic nonlinearity and subjected to driving forces of elliptic type. By an appropriate choice of the system parameter values, it is possible to show that our derived solution represents the exact steady-state solution of the undamped Duffing equation with driving force of elliptic type. At the end of this work, we also demonstrate the validity of our derived solution by comparing the amplitude–time response curves with those of the numerical integration solutions.     

Effects of Multi Global Bifurcations on Basin Organization,Catastrophes and Final Outcomes in a Driven Nonlinear Oscillator at the 2T-Subharmonic Resonance

The behavior of the escape driven oscillator at the 2T-periodic subharmonic resonance is considered, and the mechanism of generating different fractal patterns of the basins of attraction of coexisting attractors, as well as its effects on the unpredictable asymptotic system behaviors, are the main points of interest. The analysis is based on the numerical study of the sudden qualitative changes of the structure of basin-phase portraits, the changes implied by multi global bifurcations. Attention is focused on two qualitatively different regions of control space: the region prior to the subcritical flip bifurcation, where all three attractors (2T-periodic, T-periodic and the attractor at infinity) coexist, and the region after the bifurcation, where only two attractors (2T-periodic and the attractor at infinity) coexist. In particular, the concept of the global (homoclinic and heteroclinic) bifurcations is extended to the latter region, where the arising flip saddle (instead of the direct saddle) is involved in the events. The possible forms of unpredictable outcomes, which arise in both regions of control parameters, are pointed out.     

Analysis of Hopf and Takens–Bogdanov Bifurcations in a Modified van der Pol–Duffing Oscillator

We analyze a modified van der Pol–Duffing electronic circuit, modeled by a tridimensional autonomous system of differential equations with Z₂-symmetry. Linear codimension-one and two bifurcations of equilibria give rise to several dynamical behaviours, including periodic, homoclinic and heteroclinic orbits. The local analysis provides, in first approximation, the different bifurcation sets. These local results are used as a guide to apply the adequate numerical methods to obtain a global understanding of the bifurcation sets. The study of the normal form of the Hopf bifurcation shows the presence of cusps of saddle-node bifurcations of periodic orbits. The existence of a codimension-four Hopf bifurcation is also pointed out. In the case of the Takens–Bogdanov bifurcation, several degenerate situations of codimension-three are analyzed in both homoclinic and heteroclinic cases. The existence of a Hopf–Shil'nikov singularity is also shown.