Time-periodic Systems Research Articles

The rate of convergence of new Lax–Oleinik type operators for time-periodic positive definite Lagrangian systems

Assume that the Aubry set of the time-periodic positive definite Lagrangian L consists of one hyperbolic 1-periodic orbit. We provide an upper bound estimate of the rate of convergence of the family of new Lax–Oleinik type operators associated with L introduced by the authors in Wang and Yan (2012 Commun. Math. Phys. 309 663–91). In addition, we construct an example where the Aubry set of a time-independent positive definite Lagrangian system consists of one hyperbolic periodic orbit and the rate of convergence of the Lax–Oleinik semigroup cannot be better than O(1/t) as t → +∞.

Output-only Modal Analysis using Continuous-Scan Laser Doppler Vibrometry and application to a 20 kW wind turbine

Continuous-Scan Laser Doppler Vibrometry (CSLDV) is a technique where the measurement point continuously sweeps over a structure while measuring, capturing both spatial and temporal information. The continuous-scan approach can greatly accelerate measurements, allowing one to capture spatially detailed mode shapes in the same amount of time that conventional methods require to measure the response at a single point. The method is especially beneficial when testing large structures, such as wind turbines, that have low natural frequencies and hence may require very long time records at each measurement point. Several CSLDV methods have been presented that use sinusoidal excitation or impulse excitation, but CSLDV has not previously been employed with an unmeasured, broadband random input. This work extends CSLDV to that class of input, developing an Output-only Modal Analysis method (OMA-CSLDV). A recently developed algorithm for linear time-periodic system identification, which makes use of harmonic power spectra and the harmonic transfer function concept developed by Wereley [17], is used in conjunction with CSLDV measurements. One key consideration, the choice of the scan frequency, is explored. The proposed method is validated on a randomly excited free-free beam, where one-dimensional mode shapes are captured by scanning the laser along the length of the beam. The first seven natural frequencies and mode shapes are extracted from the harmonic power spectrum of the vibrometer signal and show good agreement with the analytically-derived modes of the beam. The method is then applied to identify the mode shapes of a parked 20kW wind turbine using a ground based laser and with only a light breeze providing excitation.

Reduction of Self-Excited, Time-Periodic Systems Using Proper Orthogonal Decomposition

AbstractIn almost all fields of mechanical engineering, the mathematical models of dynamical systems become more and more computationally expensive. A repeated evaluation as it is needed in the field of model based design, for example, or an evaluation in real-time, necessary for model based control applications, is challenging. Therefore, the necessity of reduced models, reflecting the principal system dynamics is growing continuously.The reduction of dynamical systems with time-periodic coefficients, termed as parametrically excited systems, subjected to self-excitation is addressed in this work. For certain frequencies of the time-periodic coefficients, referred to as parametric antiresonance frequencies, vibration suppression is achieved, as it is known from the literature. Reduced models are set up using the method of Proper Orthogonal Decomposition, and the stability of the original as well as the reduced model is investigated using the Floquet-Theory. From an analysis of the Floquet-stability parameter in the vicinity of the parametric antiresonance frequency, it is concluded that the reduced model predicts the stability properties of the original model in a satisfactory manner.

Fredholmness and smooth dependence for linear time-periodic hyperbolic systems

This paper concerns n × n linear one-dimensional hyperbolic systems of the type ∂ t u j + a j ( x ) ∂ x u j + ∑ k = 1 n b j k ( x ) u k = f j ( x , t ) , j = 1 , … , n , with periodicity conditions in time and reflection boundary conditions in space. We state conditions on the data a j and b j k and the reflection coefficients such that the system is Fredholm solvable. Moreover, we state conditions on the data such that for any right-hand side there exists exactly one solution, that the solution survives under small perturbations of the data, and that the corresponding data-to-solution map is smooth with respect to appropriate function space norms. In particular, those conditions imply that no small denominator effects occur. We show that perturbations of the coefficients a j lead to essentially different results than perturbations of the coefficients b j k , in general. Our results cover cases of non-strictly hyperbolic systems as well as systems with discontinuous coefficients a j and b j k , but they are new even in the case of strict hyperbolicity and of smooth coefficients.

A fiber-modified adenoviral vector interacts with immunoevasion molecules of the B7 family at the surface of murine leukemia cells derived from dormant tumors

Tumor cells can escape the immune system by overexpressing molecules of the B7 family, e.g. B7-H1 (PD-L1 or CD86), which suppresses the anti-tumor T-cell responses through binding to the PD-1 receptor, and similarly for B7.1 (CD80), through binding to CTLA-4. Moreover, direct interactions between B7-H1 and B7.1 molecules are also likely to participate in the immunoevasion mechanism. In this study, we used a mouse model of tumor dormancy, DA1-3b leukemia cells. We previously showed that a minor population of DA1-3b cells persists in equilibrium with the immune system for long periods of time, and that the levels of surface expression of B7-H1 and B7.1 molecules correlates with the dormancy time. We found that leukemia cells DA1-3b/d365 cells, which derived from long-term dormant tumors and overexpressed B7-H1 and B7.1 molecules, were highly permissive to Ad5FB4, a human adenovirus serotype 5 (Ad5) vector pseudotyped with chimeric human-bovine fibers. Both B7-H1 and B7.1 were required for Ad5FB4-cell binding and entry, since (i) siRNA silencing of one or the other B7 gene transcript resulted in a net decrease in the cell binding and Ad5FB4-mediated transduction of DA1-3b/d365; and (ii) plasmid-directed expression of B7.1 and B7-H1 proteins conferred to Ad5FB4-refractory human cells a full permissiveness to this vector. Binding data and flow cytometry analysis suggested that B7.1 and B7-H1 molecules played different roles in Ad5FB4-mediated transduction of DA1-3b/d365, with B7.1 involved in cell attachment of Ad5FB4, and B7-H1 in Ad5FB4 internalization. BRET analysis showed that B7.1 and B7-H1 formed heterodimeric complexes at the cell surface, and that Ad5FB4 penton, the viral capsomere carrying the fiber projection, could negatively interfere with the formation of B7.1/B7-H1 heterodimers, or modify their conformation. As interactors of B7-H1/B7.1 molecules, Ad5FB4 particles and/or their penton capsomeres represent potential therapeutic agents targeting cancer cells that had developed immunoevasion mechanisms.

Stability and robustness analysis of a linear time-periodic system subjected to random perturbations

In this work, new methods of guaranteeing the stability of linear time periodic dynamical systems with stochastic perturbations are presented. In the approaches presented here, the Lyapunov–Floquet (L–F) transformation is applied first so that the linear time-periodic part of the equations becomes time-invariant. For the linear time periodic system with stochastic perturbations, a stability theorem and related corollary have been suggested using the results previously obtained by Infante. This technique is not only applicable to systems with stochastic parameters but also to systems with deterministic variation in parameters. Some illustrative examples are presented to show the practical applications. These methods can be used to investigate the degree of robustness and design controllers for systems with time periodic coefficients subjected to random perturbations.

Analytic Bounds for Instability Regions in Periodic Systems With Delay via Meissner’s Equation

A method for obtaining analytic bounds for period doubling and cyclic fold instability regions in linear time-periodic systems with piecewise constant coefficients and time delay is suggested. The method is based on the use of transition matrices for Meissner’s equation corresponding to the desired type of instability. Analytic expressions for the disconnected regions of fold and flip instability for two- and three-segment coefficients including both complex and real eigenvalues in Meissner’s equation are obtained. The proposed method when applied to the example of two-segment interrupted turning with complex eigenvalues in each segment yields the same results as those obtained recently for the boundaries of the flip regions (Szalai and Stepan, 2006, “Lobes and Lenses in the Stability Chart of Interrupted Turning,” J Comput. Nonlinear Dyn., 1, pp. 205–211). Next, the period-doubling instability regions for a particular delay differential equation related to the damped Meissner’s equation and the fold instabilities for a model of delayed position feedback control are analytically obtained. Finally, we extend the method to a single degree-of-freedom milling model with a three-piecewise-constant-segment approximation to the true specific cutting force in which lower bounds for and horizontal locations of the regions of flip instability are obtained. The analytic results are verified through numerical stability charts obtained using the temporal finite element method. Conditions for the existence of islands of instability are also obtained.

Delay-Integral-Quadratic Constraints and Stability Multipliers for Systems With MIMO Nonlinearities

The formula for stability multipliers for systems with MIMO nonlinearities is derived using the method of delay-integral-quadratic constraints. The results, applicable to both time-invariant and time-periodic systems, are then used to derive a solution for a special case of a nonlinear regulation problem and a criterion for stability of forced oscillations. A linear time-periodic system is considered to illustrate the difference between the delay-integral-quadratic constraints that yield stability criteria in multiplier form and those that do not.

Pseudo analytical solution to time periodic stiffness systems

An analytical form of state transition matrix for a system of equations with time periodic stiffness is derived in order to solve the free response and also allow for the determination of system stability and bifurcation. A pseudo-closed form complete solution for parametrically excited systems subjected to inhomogeneous generalized forcing is developed, based on the Fourier expansion of periodic matrices and the substitution of matrix exponential terms via Lagrange—Sylvester theorem. A Mathieu type of equation with large amplitude is presented to demonstrate the method of formulating state transition matrix and Floquet multipliers. A two-degree-of-freedom system with irregular time periodic stiffness characterized by spiral bevel gear mesh vibration is presented to find forced response in stability and instability. The obtained results are presented and discussed.

State Dependent Regenerative Effect in Milling Processes

The governing equation of milling processes is generalized with the help of accurate chip thickness derivation resulting in a state dependent delay model. This model is valid for large amplitude machine tool vibrations and uses accurate nonlinear screen functions describing the entrance and exit positions of the cutting edges of the milling tool relative to the workpiece. The periodic motions of this nonlinear system are calculated by a shooting method. The stability calculation is based on the linearization around these periodic solutions by means of the semidiscretization method applied for the corresponding time-periodic delay system. Predictor-corrector method is developed to continue the periodic solutions as the bifurcation parameter, that is, the axial immersion is varied. It is observed that the influence of the state dependent delay on linear stability can be significant close to resonance where large amplitude forced vibrations occur. The existence of an unusual fold bifurcation is shown where a kind of hysteresis phenomenon appears between two different stable periodic motions.

Reduced order modeling of nonlinear time periodic systems subjected to external periodic excitations

In this work, new methodologies for order reduction of nonlinear systems with periodic coefficients subjected to external periodic excitations are presented. The periodicity of the linear terms is assumed to be non-commensurate with the periodicity of forcing vector. The dynamical equations of motion are transformed using the Lyapunov–Floquet (L–F) transformation such that the linear parts of the resulting equations become time-invariant while the forcing and nonlinearity takes the form of quasiperiodic functions. The techniques proposed here construct a reduced order equivalent system by expressing the non-dominant states as time-varying functions of the dominant (master) states. This reduced order model preserves stability properties and is easier to analyze, simulate and control since it consists of relatively small number of states in comparison with the large scale system. Specifically, two methods are discussed to obtain the reduced order model. First approach is a straightforward application of linear method similar to the ‘Guyan reduction’. The second novel technique proposed here extends the concept of ‘ invariant manifolds’ for the forced problem to construct the fundamental solution. Order reduction approach based on this extended invariant manifold technique yields unique ‘ reducibility conditions’. If these ‘ reducibility conditions’ are satisfied only then an accurate order reduction via extended invariant manifold approach is possible. This approach not only yields accurate reduced order models using the fundamental solution but also explains the consequences of various ‘ primary’ and ‘ secondary resonances’ present in the system. One can also recover ‘ resonance conditions’ associated with the fundamental solution which could be obtained via perturbation techniques by assuming weak parametric excitation. This technique is capable of handling systems with strong parametric excitations subjected to periodic and quasi-periodic forcing. It is anticipated that these order reduction techniques will provide a useful tool in the analysis and control system design of large-scale parametrically excited nonlinear systems subjected to external periodic excitations.

UNIFORM DISSIPATIVENESS, ROBUST SYNCHRONIZATION AND LOCATION OF THE ATTRACTOR OF PARAMETRIZED NONAUTONOMOUS DISCRETE SYSTEMS

In this series of papers, we study issues related to the synchronization of two coupled chaotic discrete systems arising from secured communication. The first part deals with uniform dissipativeness with respect to parameter variation via the Liapunov direct method. We obtain uniform estimates of the global attractor for a general discrete nonautonomous system, that yields a uniform invariance principle in the autonomous case. The Liapunov function is allowed to have positive derivative along solutions of the system inside a bounded set, and this reduces substantially the difficulty of constructing a Liapunov function for a given system. In particular, we develop an approach that incorporates the classical Lagrange multiplier into the Liapunov function method to naturally extend those Liapunov functions from continuous dynamical system to their discretizations, so that the corresponding uniform dispativeness results are valid when the step size of the discretization is small. Applications to the discretized Lorenz system and the discretization of a time-periodic chaotic system are given to illustrate the general results. We also show how to obtain uniform estimation of attractors for parametrized linear stable systems with nonlinear perturbation.

Model reduction of nonlinear systems with external periodic excitations via construction of invariant manifolds

A methodology for determining reduced order models of periodically excited nonlinear systems with constant as well as periodic coefficients is presented. The approach is based on the construction of an invariant manifold such that the projected dynamics is governed by a fewer number of ordinary differential equations. Due to the existence of external and parametric periodic excitations, however, the geometry of the manifold varies with time. As a result, the manifold is constructed in terms of temporal and dominant state variables. The governing partial differential equation (PDE) for the manifold is nonlinear and contains time-varying coefficients. An approximate technique to find solution of this PDE using a multivariable Taylor–Fourier series is suggested. It is shown that, in certain cases, it is possible to obtain various reducibility conditions in a closed form. The case of time-periodic systems is handled through the use of Lyapunov–Floquet (L–F) transformation. Application of the L–F transformation produces a dynamically equivalent system in which the linear part of the system is time-invariant; however, the nonlinear terms get multiplied by a truncated Fourier series containing multiple parametric excitation frequencies. This warrants some structural changes in the proposed manifold, but the solution procedure remains the same. Two examples; namely, a 2-dof mass–spring–damper system and an inverted pendulum with periodic loads, are used to illustrate applications of the technique for systems with constant and periodic coefficients, respectively. Results show that the dynamics of these 2-dof systems can be accurately approximated by equivalent 1-dof systems using the proposed methodology.

Output-only modal analysis of linear time-periodic systems with application to wind turbine simulation data

Many important systems, such as wind turbines, helicopters and turbomachinery, must be modeled with linear time-periodic equations of motion to correctly predict resonance phenomena. Time periodic effects in wind turbines might arise due to blade-to-blade manufacturing variations, stratification in the velocity of the wind with height and changes in the aerodynamics of the blades as they pass the tower. These effects may cause parametric resonance or other unexpected phenomena, so it is important to properly characterize them so that these machines can be designed to achieve high reliability, safety, and to produce economical power. This work presents a system identification methodology that can be used to identify models for linear, periodically time-varying systems when the input forces are unmeasured, broadband and random. The methodology is demonstrated for the well-known Mathieu oscillator and then used to interrogate simulated measurements from a rotating wind turbine. The measurements were simulated for a 5MW turbine modeled in the HAWC2 simulation code, which includes both structural dynamic and aerodynamic effects. This simulated system identification provides insights into the test and measurement requirements and the potential pitfalls, and simulated experiments such as this may be useful to obtain a set of time-periodic equations of motion from a numerical model, since a closed form model is not readily available by other means due to the way in which the aeroelastic effects are treated in the simulation code.

Lyapunov stability for conservative systems with lower degrees of freedom

It is a central theme to study the Lyapunov stability of periodic solutions of nonlinear differential equations or systems. For dissipative systems, the Lyapunov direct method is an important tool to study the stability. However, this method is not applicable to conservative systems such as Lagrangian equations and Hamiltonian systems. In the last decade, a method that is now known as the 'third order approximation' has been developed by Ortega, and has been applied to particular types of conservative systems including time periodic scalar Lagrangian equations (Ortega, J. Differential Equations, 128(1996), 491-518). This method is based on Moser's twist theorem, a prototype of the KAM theory. Latter, the twist coefficients were re-explained by Zhang in 2003 through the unique positive periodic solutions of the Ermakov-Pinney equation that is associated to the first order approximation (Zhang, J. London Math. Soc., 67(2003), 137-148). After that, Zhang and his collaborators have obtained some important twist criteria and applied the results to some interesting examples of time periodic scalar Lagrangian equations and planar Hamiltonian systems. In this survey, we will introduce the fundamental ideas in these works and will review recent progresses in this field, including applications to examples such as swing, the (relativistic) pendulum and singular equations. Some unsolved problems will be imposed for future study.

Experimental study on dynamic characteristics of linear parametrically excited system

An electromagnetic device, acting like a spring with alternating stiffness, has been designed to parametrically excite the cantilever beam transversely in the laboratory. It showed good potential for the experimental investigation of linear parametric system. Therefore, experiments for the natural frequency, the response (both free and forced) spectra and frequency response characteristics of a cantilever beam under electromagnetic excitation, are, respectively, conducted to study the dynamic characteristics of linear parametrically excited system. The vibrational model of the cantilever beam system is established using the assumed mode method. Theoretical results, related to the natural frequency, response spectra and external resonant condition, are presented to verify the experimental results. It is shown that the dynamic characteristics of linear time-periodic system do have some typical features, which differ distinctly from that of the linear time-invariant system.

Cramer–Rao Bound Development for Linear Time Periodic Systems

System identification techniques are often used to determine the parameters required to define a model of a linear time invariant (LTI) system. The Cramer–Rao bound can be used to validate those parameters in order to ensure that the system model is an accurate representation of the system. Unfortunately, the Cramer–Rao bound is only valid for LTI systems and is not valid for linear time periodic (LTP) systems such as a helicopter rotor in forward flight. This paper describes an extension of the Cramer–Rao bound to LTP systems and demonstrates the methodology for a simple LTP system.

Application of teledentistry in oral medicine in a Community Dental Service, N. Ireland

Currently, patients with oral medicine conditions from all areas of Northern Ireland are referred by dentists and doctors to a small number of specialist services: predominantly, the Regional Oral Medicine Consultant at the School of Dentistry, Belfast. On receipt of the referral the consultant makes an assessment of the urgency of the case and the patient is placed on a waiting list. Until the recent implementation of waiting list initiatives (Elective Access Protocol, Department of Health, N. Ireland, 2006), patients remained on the waiting list for long periods of time. Analysis of these patient profiles highlights that many need both multiple treatment and review appointments of their chronic conditions, and consequently remain in the hospital system for significant periods of time. This increases the waiting time for these services. The idea of using teledentistry to triage referrals, and its potential as a tool to support locally based treatment, poses an alternative approach to the management of oral medicine referrals. It may be of particular interest to practitioners in rural locations where distance from the regional centre is significant. In 2005, to test this theory, a prototype teledentistry system was set up as part of a service improvement scheme by the Community Dental Service of the Homefirst Legacy Trust (now Northern Trust) in partnership with the Oral Medicine Department at the School of Dentistry, Royal Group of Hospitals Legacy Trust (now Belfast Trust). This paper describes the feasibility study.

Generalized synchronization in linearly coupled time periodic systems

We consider the synchronization of a network of linearly coupled and not necessarily identical oscillators. We present an approach to the existence of the synchronization manifold which is based on some results developed by R. Smith for the study of periodic solutions of ODEs. Our framework allows the study of a large class of systems and does not assume that they are small perturbations of linear systems. Moreover, it provides a practical way to compute estimations on the parameters of the system for which generalized synchronization occurs. Additionally, we give a new proof of the main result of R. Smith on invariant manifolds using Wazewski's principle. Several examples of application are presented.

Statistical mechanics of Floquet systems with regular and chaotic states

We investigate the asymptotic state of time-periodic quantum systems with regular and chaotic Floquet states weakly coupled to a heat bath. The asymptotic occupation probabilities of these two types of states follow fundamentally different distributions. Among regular states the probability decreases from the state in the center of a regular island to the outermost state by orders of magnitude, while chaotic states have almost equal probabilities. We derive an analytical expression for the occupations of regular states of kicked systems, which depends on the winding numbers of the regular tori and the parameters temperature and driving frequency. For a constant winding number within a regular island it simplifies to Boltzmann-like weights exp(-β{eff}E{m}^{reg}) , similar to time-independent systems. For this we introduce the regular energies E{m}^{reg} of the quantizing tori and an effective winding-number-dependent temperature 1/β{eff} , different from the actual bath temperature. Furthermore, the occupations of other typical Floquet states in a mixed phase space are studied, i.e., regular states on nonlinear resonances, beach states, and hierarchical states, giving rise to distinct features in the occupation distribution. Avoided crossings involving a regular state lead to drastic consequences for the entire set of occupations. We introduce a simplified rate model whose analytical solutions describe these occupation changes quite accurately.