Nonlinear normal modes in multi-mode models of an inertially coupled elastic structure

Nonlinear normal modes for elastic structures have been studied extensively in the literature. Most studies have been limited to small nonlinear motions and to structures with geometric nonlinearities. This work investigates the nonlinear normal modes in elastic structures that contain essential inertial nonlinearities. For such structures, based on the works of Crespo da Silva and Meirovitch, a general methodology is developed for obtaining multi-degree-of-freedom discretized models for structures in planar motion. The motion of each substructure is represented by a finite number of substructure admissible functions in a way that the geometric compatibility conditions are automatically assured. The multi degree-of-freedom reduced-order models capture the essential dynamics of the system and also retain explicit dependence on important physical parameters such that parametric studies can be conducted. The specific structure considered is a 3-beam elastic structure with a tip mass. Internal resonance conditions between different linear modes of the structure are identified. For the case of 1:2 internal resonance between two global modes of the structure, a two-mode nonlinear model is then developed and nonlinear normal modes for the structure are studied by the method of multiple time scales as well as by a numerical shooting technique. Bifurcations in the nonlinear normal modes are shown to arise as a function of the internal mistuning that represents variations in the tip mass in the structure. The results of the two techniques are also compared.     

Nonlinear and nonsmooth dynamics in a DC-DC Buck converter: Two experimental set-ups

It is well known that many nonlinear phenomena such as bifurcations and chaotic behavior occur in DC–DC converters mainly due to the switching action among all the different topologies of the circuit. Such behavior has been described with detail numerically, and also mathematical reasoning has been provided. In this paper we focuss on the experimental side of a DC–DC Buck converter controlled with two different strategies: classical Pulse Width Modulation (PWM) with a ramp and a more recently described Zero Average Dynamics (ZAD). We show some nonsmooth events and we explain with detail the experimental set-ups. In one of them, we use a FPGA card to obtain on-line results. In the other we use Virtual Instrumentation to generate an experimental two-dimensional bifurcation diagram, which will be compared to the numerical data. After the data acquisition of the system state variables some elaborated post-processing must be made. This is done through LabView. Although the main application of these results is centered in avoiding non-periodic or high-amplitude periodic behavior, they can also be applied to reducing the generated electro-magnetic interference and to the information transmission.Partially funded by SICONOS.     

A degenerate Hopf–saddle-node bifurcation analysis in a family of electronic circuits

This paper presents a study of a three-parameter unfolding of a degenerate case in the Hopf--saddle-node singularity. This analysis shows that this nonlinear degeneracy is a source of interesting bifurcations of periodic orbits as well as global bifurcations of equilibria. The results achieved are applied to the study of a simple autonomous electronic circuit, which has just only one nonlinearity. The numerical results include the analysis of interesting resonance behaviors.     

On the dynamics of European two-axle railway freight wagons

The standardized design of European two-axle freight wagons has been used since the 1890s. The basic construction is a carbody suspended on leaf springs by a system of links working as pendulums. The internationally approved suspension system for these vehicles is the UIC standard suspension. The design is advantageous because it is simple, cheap, and robust. Despite the simple design, its influence on the dynamic performance of the vehicle is highly intricate due to dry friction damping and motion delimiters causing impacts. This paper deals with the dynamic behavior of European two-axle freight wagons analyzed through the use of a multibody model developed specifically for this purpose. The modeling of the vehicle and the integration of the dynamic equations of motion are far from trivial due to the complicated nature of the interacting forces. The model exhibits interesting bifurcation patterns emphasizing the complexity of the dynamic behavior of these wagons.     

Discontinuity-induced bifurcations of equilibria in piecewise-smooth and impacting dynamical systems

A rich variety of dynamical scenarios has been shown to occur when a fixed point of a non-smooth map undergoes a border-collision. This paper concerns a closely related class of discontinuity-induced bifurcations, those involving equilibria of n-dimensional piecewise-smooth flows. Specifically, transitions are studied which occur when a boundary equilibrium, that is one lying within a discontinuity manifold, is perturbed. It is shown that such equilibria can either persist under parameter variations or can disappear giving rise to different bifurcation scenarios. Conditions to classify among the possible simplest scenarios are given for piecewise-smooth continuous, Filippov and impacting systems. Also, we investigate the possible birth of other attractors (e.g. limit cycles) at a boundary-equilibrium bifurcation.     

The fate of self-aligned rolls in gravity modulated magnetoconvection

We explore the modifications brought about by gravity modulation in self-aligned two-dimensional (2D) stationary rolls and higher-order convective instabilities with horizontal magnetic field. The Chandrasekhar number plays no role in determining critical values of the forcing parameters for convection to begin. The onset is in the form of 2D rolls with periodically varying intensity. The external magnetic field is solely responsible for aligning the rolls while the induced magnetic field comes into play after three-dimensional convection sets in. Gravity modulation destabilizes the system and results in chaotic flow much earlier above onset.     

Bifurcation of attractor in the falling film problem

Film flows down a vertical plane, the so-called falling films, are covered by waves whose lengths are longer than a threshold value corresponding to neutral stability of the waveless flow. On the other hand, experiments also demonstrate existence of the upper threshold wavelength in unstable interval. This phenomenon is explained through careful analysis of dynamical system modelling the wave dynamics in falling films.     

A Mathematical Model of Intermittent Androgen Suppression for Prostate Cancer

For several decades, androgen suppression has been the principal modality for treatment of advanced prostate cancer. Although the androgen deprivation is initially effective, most patients experience a relapse within several years due to the proliferation of so-called androgen-independent tumor cells. Bruchovsky et al. suggested in animal models that intermittent androgen suppression (IAS) can prolong the time to relapse when compared with continuous androgen suppression (CAS). Therefore, IAS has been expected to enhance clinical efficacy in conjunction with reduction in adverse effects and improvement in quality of life of patients during off-treatment periods. This paper presents a mathematical model that describes the growth of a prostate tumor under IAS therapy based on monitoring of the serum prostate-specific antigen (PSA). By treating the cancer tumor as a mixed assembly of androgen-dependent and androgen-independent cells, we investigate the difference between CAS and IAS with respect to factors affecting an androgen-independent relapse. Numerical and bifurcation analyses show how the tumor growth and the relapse time are influenced by the net growth rate of the androgen-independent cells, a protocol of the IAS therapy, and the mutation rate from androgen-dependent cells to androgen-independent ones.

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The set of periodic scalar differential equations with cubic nonlinearities

We study the structure induced by the number of periodic solutions on the set of differential equations x^′=f(t,x) where f∈C³(R²) is T-periodic in t, fx³(t,x)<0 for every (t,x)∈R², and f(t,x)→?∞ as x→∞, uniformly on t. We find that the set of differential equations with a singular periodic solution is a codimension-one submanifold, which divides the space into two components: equations with one periodic solution and equations with three periodic solutions. Moreover, the set of differential equations with exactly one periodic singular solution and no other periodic solution is a codimension-two submanifold.