IUTAM Symposium on Nonlinear Dynamics for Advanced Technologies and Engineering Design

We present a scheme of analysis for predicting the approach to a fold in the noisy time series of a slowly evolving system. It provides estimates of the evolution rate of the control parameter, the variation of the stability coefficient, the path itself, and the level of noise in the time series. Finally, it gives probability estimates of the future time at which the system will jump to a remote (dangerous) steady state. We apply the technique, first, to the output of a

computer simulation

from a model for sudden cooling of the Earth. Second, we use the algorithms to give probabilistic tipping estimates for the end of the most recent glaciation of the Earth using actual

geological data

from the Vostok ice core.

The paper discusses several issues related to the numerical computation of the stable manifold of saddle-like periodic cycles in piecewise smooth dynamical systems. Results are presented for a particular stick–slip system. In the second part of the paper the same mechanical model is used to briefly describe the interaction between fold and adding-sliding bifurcations.

Two formulations of nonlinear normal vibration modes (NNMs) in conservative and near conservative systems are considered. Construction of the NNMs and their applications in some mechanical problems are presented. Namely, the nonlinear vibro-absorption problem, the cylindrical shell nonlinear dynamics and the vehicle suspension nonlinear dynamics are analyzed.

After recalling features of solitons in the Toda (more precisely an adapted Morse-Toda) lattice a succint discussion is provided about the stability of such solitons when the lattice is heated up to physiological temperatures for values of parameters typical of bio-macro-molecules. Then the discussion is focused on the soliton trapping of an added

excess

(originally free) electron thus creating the solectron electric carrier. Results are presented for 1d- and 2d-anharmonic crystal lattices.

In this paper we study the dynamics of a large ring of unidirectionally coupled autonomous Duffing oscillators. We paid our attention to the role of unstable periodic solutions for the appearance of spatio-temporal structures and the Eckhaus effect. We provide an explanation for the fast transition to chaos showing that the parameter interval, where the transition from a stable periodic state to chaos occurs, scales like the inverse square of the number of oscillators in the ring.

In order to monitor the dynamics of a distributed or high-dimensional dynamical system, an infinite- or high-dimensional Hilbert space is required as this class of systems may show various and complex dynamics. For example, different operational regimes can be observed depending on the current external loads. However, very often, the relevant dynamics of each of the observed operational regimes take place in a low-dimensional active subspace which is spanned by a low number of active modes. For changes in operational conditions, the active modes and therefore the low-dimensional subspace will be subjected to change as well. As Karhunen–Loève–Transform (KLT) is always applied to a history of measurements, good convergence and fast detection of changes in system dynamics conflict when choosing the length of the time interval. We present an algorithm based on KLT which uses an adaptive sliding time window. It can be employed for real-time tracking of the active subspace. The data from the real-time subspace tracking can be used to categorize different operational conditions and thus monitor the system in real time. We show that with currently available technology, an efficient implementation of the subspace tracking with sampling frequencies of 1,000 Hz is possible.

In this paper we investigate the dynamics of a pendulum subject to an elliptical pattern of excitation. The physical model is motivated by the development of sea wave energy extraction systems which exploit the rotating solutions of pendulum systems to drive generation. We formulate the homoclinic Melnikov function for the system and then demonstrate bounds on the set of parameters which can support homoclinic bifurcation. As the homoclinic bifurcation is a precursor to escape and the formation of rotating solutions in the evolution of the system under increasing forcing, these estimates provide bounds on the parameter space outwith which stable rotating solutions are not observed.

Curious dynamic behaviour that beams may experience is discussed in this article. In the model of the beams a simple type of viscous damping, an elasto-plastic stress–strain relation and Von Kármán type of non-linearity are assumed. The continuum system is discretized following a finite element,

p

-version approach, and the equations of motion are numerically solved in the time domain. The analyses focuses on the qualitative differences that appear in the oscillations of damped beams when plasticity is present and on a buckling like phenomenon that plasticity may induce, leading to a dynamic behaviour difficult to anticipate.

Locomotion of a mechanical system consisting of two rigid bodies, a main body and a tail, connected by a cylindrical joint, is considered. The system moves in a resistive fluid and is controlled by periodic angular oscillations of the tail relative to the main body. The resistance force acting upon each body is a quadratic function of its velocity. Under certain assumptions, a nonlinear equation of motion is derived and simplified. The average velocity of locomotion is estimated. This velocity is positive, if the deflection of the tail is performed slower that its retrieval. The optimal time history of oscillations is found that corresponds to the maximal locomotion velocity.

We present a detailed bifurcation structure and associated flow patterns for lowPrandtl-number (P = 0.005, 0.02) Rayleigh-Bénard convection near its onset. We use both direct numerical simulations and a 30-mode low-dimensional model for this study. The main flow patterns observed for this range are 2D straight rolls, stationary squares, asymmetric squares, oscillating asymmetric squares, relaxation oscillations, and chaos. At the onset of convection, low-P convective flows have stationary 2D rolls and associated stationary and oscillatory asymmetric squares. The range of Rayleigh numbers for which the stationary 2D rolls exist decreases rapidly with decreasing Prandtl numbers and vanishes in the zero-P limit giving rise to chaotic solutions at the onset itself. Our results are in qualitative agreement with results reported earlier on this topic.

An experimental and theoretical investigation was conducted into the spatial motion of the self-excited oscillation of a pipe which is built-in at upper end and has an attached mass at the other. A certain characteristic mode of the pipe vibration is self-excited when the axial flow velocity in the pipe exceeds a certain value. For higher flow velocity, the two distinct eigen modes of the pipe can be simultaneously self-excited with two different natural frequencies, which called mixed-mode flutter. Equations governing amplitudes and the phases were derived and used to clarify nonlinear modal interactions numerically the above specific cases. It is theoretically clarified that the planes, on which second modal oscillation and the third modal oscillation are produced, are perpendicular. Furthermore experiments were conducted with the silicon rubber pipe conveying fluid. As a result, typical features of the mixed-mode flutter were confirmed qualitatively by experiments.

An observation that a parameter plot of a harmonically forced wake oscillator, which shows three distinct regions, in qualitative sense corresponds well with results of numerical and experimental investigations has prompted further investigations into dynamical behaviour of a flexible cylinder experiencing vortex-induced vibration. For this purpose, a reduced-order model representing a vertical pipe excited by surrounding fluid flow was built and certain aspects of its dynamics were studied. When resonance curves of a numerical and an analytical solution obtained with the method of multiple scales were compared, it became apparent that the former gives a much wider resonance region as the latter. This difference expectedly narrowed for lower values of the coupling strength parameter, indicating not only that the initial value might have been very large as defined in the method used, but also that because of this the overall dynamics were dominated by the linear part, the structure.

Parametric rolling has become an important design issue in the safe operation of large high-speed container ships. This interest is partly based upon the well-published large amplitude rolling incident of the APL China several years ago and numerous other incidents which have also since come to light. What we have found in our research is that a simple linear model of roll damping does not accurately predict the threshold nature of this phenomenon. However, when we accurately include nonlinear damping we are able to capture the qualitative nature of this phenomenon. Moreover the Mathieu Ince-Strutt stability diagram is limited to linear harmonic excitation. What we have done in this work is to extend the Mathieu Ince-Strutt stability diagram to include non-harmonic excitation and nonlinear damping.

Analysis of ship parametric rolling has generally been restricted to simple analytical models and sophisticated time domain simulations. However, simple analytical models do not capture all the critical dynamics while time-domain simulations are time consuming to implement. Our model captures the essential dynamics of the system without over simplification. This work incorporates important aspects of the system and assesses the significance of including or ignoring these aspects. Many of the previous works on parametric rolling make the assumption of linearized and harmonic behaviour of the time-varying restoring arm or metacentric height. This assumption enables modelling the roll as a Mathieu equation.

This paper aims at showing how global safety can be used to interpret experimental results of a parametrically excited pendulum. The experimental investigation shows that rotations exist in a region smaller than the theoretical one, a discrepancy which has deeper motivations than the sole experimental approximations. By comparing the experimental results with the dynamical integrity profiles we understand that rotations exist only where a measure of dynamical integrity, accounting for both attractor robustness and basin compactness, is enough large to support experimental imperfections leading to changes in initial conditions.

In this work the dynamics and synchronization of the coupled parametric pendulums system is examined with a view to its application for wave energy extraction. The system consisting of two parametric pendulums on a common support has been modeled and its response studied numerically, with a main focus on synchronized rotation. Different methods of controlling the response the pendulums have been introduced and compared. Numerical results have been verified in experimental studies.

A combined semi-analytical and experimental approach is proposed for predictive modelling of the nonlinear dynamic behaviour of microelectromechanical resonators. The approach is demonstrated for a clamped-clamped beam resonator, for which the mechanical, electrical, and thermal domains are relevant. Multiphysics modelling is applied, based on first principles, to derive a reduced-order model of the resonator. A qualitative correspondence between numerical and experimental steady-state responses has been obtained. Depending on the excitation values, both simulations and experiments show hardening and softening nonlinear dynamic behaviour. Since the model captures the observed experimental behaviour, it can be used to optimize the resonator behaviour with respect to nonlinear dynamic effects.

Smart materials have a growing technological importance due to their unique thermomechanical characteristics. Shape memory alloys (SMAs) belong to this class of materials being easy to manufacture, relatively lightweight, and able of producing high forces or displacements with low power consumption. These aspects could be explored in different applications including vibration control. Nevertheless, there is a lack in literature concerning the experimental analysis of SMA dynamical systems. This contribution deals with the experimental analysis of SMA dynamical systems by considering an experimental apparatus composed of low-fiction cars free to move in a rail. A shaker that provides harmonic forcing excites the system. Vibration analysis reveals that SMA elements introduce complex dynamical responses to the system and different thermomechanical loadings are of concern showing the main aspects of SMA dynamical response. Special attention is dedicated to the analysis of vibration reduction that can be achieved by considering different approaches exploiting either temperature changes or vibration absorber techniques.

This paper analyses the static and dynamic buckling behavior of a simplified 2-DOF model of a cable stayed tower with emphasis on the safety of the pre-buckling solutions whose stability must be preserved for a safe design. First, the influence of the inherent symmetries of the model on the buckling loads and the post-buckling paths emerging from the bifurcation point is investigated. Then, a global dynamic analysis is conducted to investigate the degree of safety of the static pre-buckling solution. To understand the behavior of the guyed mast in a dynamic environment, a base excitation is considered and the influence of its direction on the escape stability boundary in the force control space is studied. Finally, the erosion and integrity of the basins of attraction of the stable solutions are investigated. The paper shows how the tools of nonlinear dynamics can help in the understanding of the global safety and integrity of the model, thus leading to a safe structural design.

The paper presents results of stability analysis of milling process. Machining of nickel superalloys Inconel 713C, titanium alloy Ti6Al4V and epoxide-polymer matrix composite reinforced carbon fibers (EPMC) is studied here, classically using stability lobe diagrams (SLD) received by modal analysis and next verified by recurrence quantification analysis (RQA). Finally some measures of recurrence quantification analysis are proposed as a tool for stability examination.

A dynamical integrity analysis is performed for an electrostatic micro-electro-mechanical system (MEMS) device. The analysis starts from the experimental data of dynamic pull-in due to a frequency-sweeping process in a capacitive accelerometer. The loss of dynamical integrity is investigated by curves of constant percentage of integrity factor. We found that these curves follow exactly the experimental data and succeed in interpreting the existence of disturbances. On the other hand, instead, the theoretical curves of disappearance of the attractors represent the limit when disturbances are absent, which never occurs in practice. Also, the obtained behavior chart can serve as a design guideline in order to ensure safety of the device.

We study a problem of passive nonlinear targeted energy transfer between a two degree of freedom suspension bridge model and a single degree of freedom nonlinear energy sink (NES). The system is studied under 1:1:1 nonlinear resonance involved in targeted energy transfer mechanisms. Analytical expansions are performed by mean of complexification methods, multiple scales expansions and exploits also the concept of limiting phase trajectories (LPTs). Several control mechanisms for aeroelastic instability are identified, and analytical calculations bring to efficient parameters for the absorber design. Numerical simulations are performed and good agreement with analytical predictions is observed. It results that the concept of Limiting Phase Trajectories (LPT) allows formulating adequately the problem of intensive energy transfer from a bridge to a nonlinear energy sink.

In this article, the dynamics of base-excited elastic cantilevers with non-linear attractive and repulsive forces in macro-scale and micro-scale systems are studied through experimental and numerical means. The macro-scale set-up consists of a base-excited elastic cantilever with a long-range attractive force and a short-range repulsive force. The attractive force is generated by a combination of two magnets, one located at the cantilever structure’s tip and another on a movable translatory base. The repulsive force is generated through impacts between the cantilever tip and a compliant material that covers the magnet on the translatory stage. This macro-scale experimental system is motivated by micro-scale cantilevers used in tapping mode or dynamic mode atomic force microscopy (AFM). In tapping mode AFM, the micro-cantilever undergoes a long-range van der Waals attractive force and a short-range repulsive force as the cantilever tip approaches the sample. The authors study the macro-scale system and the micro-scale system, when the excitation frequency is away from the first natural frequency. For off-resonance excitations, period-doubling events are observed in these impacting systems. Areduced-order model is developed to numerically study these systems on the basis of a single mode assumption. In the numerical studies, similar nonlinear tip-sample forces are used to model the interaction forces on the cantilever’s tip in both macro-scale and micro-scale systems. In an effort to understand the effects of noise on the dynamics, the responses of the systems are studied when Gaussian white noise is introduced into the base excitation, along with a harmonic component. It is observed that the addition of Gaussian white noise facilitates contact between the tip and the sample, for low levels of a harmonic base excitation.

A method for reducing the size of filter inductance in DC-DC converter circuits based on Filippov’s theory is presented in this paper. In this method, the state transitional matrix of the system over one complete switching cycle, including the state transitional matrices across the converter switching events, is used to stabilize converter operation with a substantially reduced inductor size while maintaining circuit average currents and voltages. An analysis of circuit conduction losses shows that losses are not significantly affected by higher inductor current ripple resulting from the use of the smaller inductor. The switching frequency of the converter will not vary or change compared to a conventionally designed DC-DC converter. The new control/design method is demonstrated using an experimental voltage-mode-controlled buck converter.

A new model of the progression phase of a drifting oscillator is proposed to account more accurately for the penetration of a conical impactor through elasto-plastic solids under a combination of a static and a harmonic excitation. The dynamic response of the semi-infinite elasto-plastic medium subjected to repeated impacts by a rigid impactor with conical contacting surface is considered and a power law force–penetration relationship is adopted to describe the loading and unloading phases of contact. These relationships are then used to develop a physical and mathematical model of this drifting oscillator, where the time histories of the progression through the medium include both the loading and unloading phases. A limited nonlinear dynamic analysis of the system was performed and it confirms that the maximum progressive motion of the oscillator occurs when the system exhibits periodic motion.

A recently proposed two degrees-of-freedom model for axial and torsional vibrations of drill string elsewhere is here revisited. The model involves state-dependent time delay, with discontinuous cutting and friction force nonlinearities. The original model is, here, enhanced by introducing axial and torsional damping, and axial stiffness. Stability analysis, which is relatively recent for systems with state-dependent delays, is conducted on the enhanced model. For representative parameter values, it is concluded from the analysis that the original model (with no damping) had no stable operating regime, while the enhanced model possesses some practically relevant stable operating regime. The steady drilling state corresponding to the enhanced model is, however, still unstable for a major portion of the operating regime leading to stick-slip and bit-bounce situations.

An integrated model for the study of the complex dynamical and geometrical behaviour of a drill-string is developed based on the modified Cosserat rod element method. The model includes general deformation of the rod with flexure, extension, torsion and shear, and allows analysis of the main vibration modes, torsional, axial and lateral, and also their couplings. Shape functions are obtained from the solution of nonlinear partial differential equations of motion in quasi-static sense, containing up to third order nonlinear terms, and the final ordinary differential equations of motion for the elements are derived from Lagrange’s approach. Boundary conditions are taken as applied to a drill-string to account for the motor and the drill-bit. The model offers significant performance advantage over a standard Finite Element approach, which facilitates numerical analysis. Such model is capable of simulating stick-slip oscillations in the drill-bit–rock interface and contact of the drill-string with borehole wall along its length.

An analytical solution for a system exhibiting oscillations of a conductor in magnetic field which is controlled by a discrete waveform is developed by means of multiple scales. The solution provides a guideline to design an effective control strategy so as to guide the system to a desirable attractor. Initial tests were also conducted to investigate the effect of hydrodynamic forces on an inertia excited by this mechanism.

Intrinsic localized mode (ILM) in coupled oscillators is one of nonlinear modes, which show the temporally periodic and spatially localized characteristics. Since the theoretical discovery of the mode, the existence of ILM has been confirmed in many system including biological molecular chains, 1D material structure, and MEMS. In the paper, we are going to discuss control of ILM in a coupled cantilever array.

The paper presents the deployment and retrieval control of a tethered satellite system from both theoretical and experimental points of view. The ideas of online optimization and receding horizon control enable one to design a feedback controller for the tethered satellite system. The presented feedback control law is not in any closed-form, but can accounts for the nonlinearity in the system model and the mission-related constraints. Furthermore, the paper outlines a technically and economically feasible solution to verify the efficacy of the controller via a ground-based experiment. The two key issues concerning the design of the ground-based experiment are the principle of dynamic similarity and the idea of using thrusts for simulating the gravity gradient field and the Coriolis forces. Finally, the paper presents the experimental verification of proposed control scheme for the deployment and retrieval processes of the satellite simulator.

This paper deals with the application of chaos control methods in order to perform bifurcation control of a parametric pendulum-shaker system. The extended time-delayed feedback control method is employed to maintain stable rotational solution of the system avoiding bifurcation to chaos. The considered pendulum system is chosen due to its potential application for extracting energy from sea waves. This alternative concept of energy harvesting is based on exploiting the rotational unbounded solution of the pendulum dynamics. The bifurcation control proposed allows the system to keep the desired rotational solution over extended parameter range avoiding undesirable changes in system dynamics.

A parametric pendulum inherently demonstrates a conversion from external vibration to rotational motion and this property is applicable to energy scavenging from vibration of external source in nature. The periodic rotation of the parametric pendulum has a benefit to convert the mechanical energy to electric energy through conventional electric machines. On the other hand, the onset of the periodic rotation depends on the initial condition. We propose a control method for starting up the parametric pendulum to the periodic rotations based on an external force input with time delay. The feasibility of proposed method is verified numerically and experimentally. The results show that there exists a certain range of control gain to achieve the control from any initial condition. This paper advocates that the proposed method is suitable for crossing over a separatrix which governs the dynamics.

This paper deals with a (MEMS) Gyroscope nonlinear dynamical system, modeled with a proof mass constrained to move in a plane with two resonant modes, which are nominally orthogonal. We present some modifications to the governing equations of the considered system, taking into account the nonlinear interactions between the parts of the systems. We also develop a linear optimal control design, for reducing the oscillatory movement of the nonlinear system to a stable point.

This paper is devoted to the adaptive control of worm-systems, which are inspired by biological ideas. We introduce a certain type of mathematical models of finite DOF worm-like locomotion systems: modeled as a chain of

k

interconnected (linked) point masses in a common straight line (a discrete straight worm). We assume that these systems contact the ground via (1) spikes and then (2) stiction combined with Coulomb sliding friction (modification of a Karnopp friction model). In general, one cannot expect to have complete information about a sophisticated mechanical or biological system, only structural properties (known type of actuator with unknown parameters) are known. Additionally, in a rough terrain, unknown or changing friction coefficients lead to uncertain systems, too. The consideration of uncertain systems leads to the use of adaptive control. Gaits from the kinematical theory (preferred motion patterns to achieve movement) can be tracked by means of adaptive controllers (λ-trackers). Simulations are aimed at the justification of theoretical results.