Numerical Continuation Research Articles

Nonlinear dynamics and bifurcation analysis of journal bearings based on second order stiffness and damping coefficients

Investigating dynamics and stability of rotors supported on journal bearings is a crucial step in the design of an efficient and reliable rotating machine. In the current work, a model for flexible rotor supported on two symmetric journal bearings is investigated. The nonlinear bearing forces are evaluated by either using direct solution of Reynolds equation or analyzing Reynolds equation to obtain linear and nonlinear bearing stiffness and damping coefficients using time dependent second order perturbation method. These coefficients are obtained for different operating conditions and bearing parameters such as length to diameter ratio, groove angle or applied groove pressure. The present results are validated with the previous literature and a perturbation analysis is used to investigate the validity range of the bearing linear and nonlinear coefficients. A novel technique based on polynomial fitting is used to present the bearing coefficient as a function of the bearing parameters. This enables the investigation of the dynamics of flexible rotor model using numerical continuation technique. Also, the effect of the bearing design parameters such as groove angle, length to diameter ratio and static pressure on the system stability is investigated. • Nonlinear journal bearing coefficients are investigated. • Time dependent perturbation method is used to obtain bearing coefficients. • Parametric analysis to the journal bearing coefficients is introduced. • Continuation analysis to the dynamics of rotor supported on Journal bearing is introduced.

Complex dynamics of a Kaldor model of business cycle with discrete-time

This paper studies the dynamical behavior of a Kaldor model of business cycle with discrete-time analytically and numerically. The conditions and the critical coefficients for the flip (period-doubling), Neimark-Sacker, and strong resonances are computed analytically. By using the critical coefficients, the bifurcation scenarios are determined for each of the deleted bifurcation points. Bifurcation curves of fixed points and cycles with periods up to sixteen by changing one and two parameters along with all codim-1 and codim-2 bifurcations on the corresponding curves are computed using the numerical continuation method. Numerical analysis confirms our analytical results and reveals more complex dynamical behaviors.

Stability of cycling behaviour near a heteroclinic network model of Rock–Paper–Scissors–Lizard–Spock

The well-known game of Rock–Paper–Scissors can be used as a simple model of competition between three species. When modelled in continuous time using differential equations, the resulting system contains a heteroclinic cycle between the three equilibrium solutions representing the existence of only a single species. The game can be extended in a symmetric fashion by the addition of two further strategies (‘Lizard’ and ‘Spock’): now each strategy is dominant over two of the remaining four strategies, and is dominated by the remaining two. The differential equation model contains a set of coupled heteroclinic cycles forming a heteroclinic network. In this paper we carefully consider the dynamics near this heteroclinic network. We develop a technique to use a previously defined definition of stability (known as fragmentary asymptotic stability) in numerical continuation software. We are able to identify regions of parameter space in which arbitrarily long periodic sequences of visits are made to the neighbourhoods of the equilibria, which form a complicated pattern in parameter space.

Bubbling, Bistable Limit Cycles and Quasi-Periodic Oscillations in Queues with Delayed Information

We consider a model describing the length of two queues that incorporates customer choice behavior based on delayed queue length information. The symmetric case, where the values of the time-delay parameter in each queue are the same, was recently studied. It was shown that under some conditions, the stable equilibrium solution becomes unstable as the common time delay passes a threshold value. This one-time stability switch occurs only at a symmetry-breaking Hopf bifurcation where a family of stable asynchronous limit-cycle solutions arise. In this paper, we examine the non-symmetric case, wherein the values of the time-delay parameter in each queue are different. We show that, in contrast to the symmetric case, the non-symmetric case allows bubbling, multiple stability switches and coexistence of distinct families of stable limit cycles. An investigation of the dynamical behavior of the non-symmetric system in a neighborhood of a double-Hopf bifurcation using numerical continuation explains the occurrence of the bistable limit cycles. Quasi-periodic oscillations were also observed due to the presence of torus bifurcations near the double-Hopf bifurcation. These identifications of the underlying mechanisms that cause unwanted oscillations in the system give a better understanding of the effects of providing delayed information and consequently help in better management of queues.

Periodic Motion and Frequency Energy Plots of Dynamical Systems Coupled With Piecewise Nonlinear Energy Sink

Abstract The frequency-energy plots (FEPs) of two-degree-of-freedom linear structures attached to a piecewise nonlinear energy sink (PNES) are generated here and thoroughly investigated. This study provides the FEP analysis of such systems for further understanding to nonlinear targeted energy transfer (TET) by the PNES. The attached PNES to the considered linear dynamical systems incorporates a symmetrical clearance zone of zero stiffness content where the boundaries of the zone are coupled with the linear structure by linear stiffness elements. In addition, linear viscous damping is selected to be continuous during the PNES mass oscillation. The underlying nonlinear dynamical behavior of the considered structure-PNES systems is investigated by generating the fundamental backbone curves of the FEP and the bifurcated subharmonic resonance branches using numerical continuation methods. Accordingly, interesting dynamical behavior of the nonlinear normal modes (NNMs) of the structure-PNES system on different backbones and subharmonic resonance branches has been observed. In addition, the imposed wavelet transform frequency spectra on the FEPs have revealed that the TET takes place in multiple resonance captures where it is dominated by the nonlinear action of the PNES.

Effect of damping nonlinearity on the dynamics and performance of a quasi-zero-stiffness vibration isolator

• Damping nonlinearity of the QZS isolator originates from horizontal and rotational dampers. • Effects of sub- and superharmonic resonances on the response is considered. • Performance metrics are defined and the effects of linear and nonlinear damping compared. • Geometric nonlinear damping is beneficial for removal of isolated periodic branches. • Nonlinear damping can greatly enhance vibration isolation for both base and force excitations. Quasi-zero-stiffness (QZS) vibration isolators that take advantage of stiffness nonlinearity are commonly studied with linear damping elements. This paper concerns the effects of damping nonlinearity on their dynamics and performance. Geometric nonlinear damping originating from horizontal and rotational dampers is considered. Analytical expressions for displacement and force transmissibility and stability of the isolator are derived using the harmonic balance method and, especially for large-amplitude excitation, numerical continuation. The beneficial effect of nonlinear damping on the removal or minimization of unwanted isolated subharmonic branches is shown. Metrics are defined to assess the performance of the QZS isolator. Results indicate that, for both small and large-amplitude base excitation, increasing nonlinear damping can effectively suppress peak transmissibility; however, as base excitation level increases, larger nonlinear damping increases transmissibility at intermediate to high frequencies. On the other hand, results reveal that for all levels of force excitation, force transmissibility at higher frequencies is not dependent on the variations of nonlinear damping and is just a function of the linear damping. Whatever the input level, the peak transmissibility of the forced QZS isolator can be reduced by increasing nonlinear damping.

Nonlinear dynamics of bistable composite cantilever shells: An experimental and modelling study

The nonlinear dynamics of cantilever bistable shells with asymmetric stable configurations is investigated. The possibility of maximizing the kinetic energy associated with snap-through motion offered by the considered cantilevered shells is pursued to enhance the sought-after energy harvesting capabilities. Through an experimental campaign under harmonic forcing the resonance scenarios around the stable configurations are analysed. The resulting nonlinear behaviour observed for high amplitude excitation is adopted to guide a reduced order model derivation. By combining the experimental response with two double-well oscillators derived from FE simulations, a double-well quintic oscillator capturing the observed softening behaviour is proposed. It involves a parameter identification phase in which a nonlinear damping model is introduced. A numerical continuation approach is adopted to discuss the local periodic solutions around each stable configuration in terms of resonance curves, bifurcation diagrams and basins of attraction. The global dynamics involving regular and chaotic dynamic regimes as well as snap-through motion is eventually described.

Nonlinear dynamic analysis and numerical continuation of periodic orbits in high-index differential–algebraic equation systems

Nonlinear dynamic analysis and numerical continuation of periodic orbits in high-index differential–algebraic equation systems

Near-onset dynamics in natural doubly diffusive convection

Doubly diffusive convection is considered in a vertical slot where horizontal temperature and solutal variations provide competing effects to the fluid density while allowing the existence of a conduction state. In this configuration, the linear stability of the conductive state is known, but the convection patterns arising from the primary instability have only been studied for specific parameter values. We have extended this by determining the nature of the primary bifurcation for all values of the Lewis and Prandtl numbers using a weakly nonlinear analysis. The resulting convection branches are extended using numerical continuation and we find large-amplitude steady convection states can coexist with the stable conduction state for sub- and supercritical primary bifurcations. The stability of the convection states is investigated and attracting travelling waves and periodic orbits are identified using time stepping when these steady states are unstable.

Bifurcations of Limit Cycles in Rotating Shafts Mounted On Partial Arc and Lemon Bore Journal Bearings in Elastic Pedestals

Abstract The paper investigates the bifurcations encountered in a simple rotor dynamic system interacting with nonlinear impedance forces, generated by the supporting journal bearings of realistic profile geometry. Bearing configurations of finite arc length and of finite width, as implemented in standard design of turbomachinery have been selected, namely the cylindrical partial arc and the elliptical (lemon) bore profile. The way in which the key design parameters influence the stability of elastic or rigid Jeffcott rotor is discussed. In the scope of this study, the following bearing design parameters are considered: arc length, length to diameter ratio, geometric preload and offset, and properties of the supporting pedestal by codimension-two studies. The bearing model is coupled to a six degree of freedom shaft-disc-pedestal model with nonlinear forces calculated from the journal kinematics, bearing design and operating conditions by numerical evaluation of the Reynolds equation for laminar, isothermal flow on a two dimensional mesh. An autonomous system of differential equations is implemented. Stability of fixed points and of limit cycles for this system is evaluated applying numerical continuation. The results confirm that minor variations in journal bearing design and pedestal properties have the potential to render substantial changes in the quality of stability and the bifurcation set of the rotor dynamic system. Specific bearing profiles render significant increment of instability threshold speed while at the same time supercritical Hopf bifurcations can be shifted to subcritical with resulting instability envelopes to be generated at speeds lower than the threshold speed.

The spatial Hill four-body problem I—An exploration of basic invariant sets

In this work we perform a first study of basic invariant sets of the spatial Hill’s four-body problem, where we have used both analytical and numerical approaches. This system depends on a mass parameter μ in such a way that the classical Hill’s problem is recovered when μ=0. Regarding the numerical work, we perform a numerical continuation, for the Jacobi constant C and several values of the mass parameter μ by applying a classical predictor–corrector method, together with a high-order Taylor method considering variable step and order and automatic differentiation techniques, to specific boundary value problems related with the reversing symmetries of the system. The solution of these boundary value problems defines initial conditions of symmetric periodic orbits. Some of the results were obtained departing from periodic orbits within Hill’s three-body problem. The numerical explorations reveal that a second distant disturbing body has a relevant effect on the stability of the orbits and bifurcations among these families. We have also found some new families of periodic orbits that do not exist in the classical Hill’s three-body problem; these families have some desirable properties from a practical point of view.

Lane-keeping control of automated vehicles with feedback delay: Nonlinear analysis and laboratory experiments

The feedback control of automated lane-keeping of passenger cars is analyzed in this paper. The calculations are based on the single-track vehicle model with the consideration of steering system dynamics. A linear feedback controller is used to control the lateral dynamics, while taking into account feedback delay and the effects of the lower-level steering controller. Subcritical Hopf bifurcations are detected along the linear stability limits and the emerging branches of periodic orbits are followed using numerical continuation. It is shown that low amplitude unstable limit cycles exist around the stable equilibrium of straight-line motion for certain parameter ranges. Based on the limit cycle amplitudes, safe and unsafe zones of stabilizing control gains are specified. The unsafe control gain zones within the linearly stable domain are also identified under laboratory conditions using small-scale experiments on a conveyor belt. The theoretical and the practical results show good accordance with respect to the domain of attraction of the straight-line motion. • Linear and nonlinear analyses of a single-track vehicle model with linear feedback control for lane-keeping. • Subcritical Hopf bifurcations and low amplitude unstable limit cycles are detected around the stable equilibrium. • Safe and unsafe zones of stabilizing control gains are specified, based on limit cycle amplitudes. • The safe and unsafe control gain zones are also identified in a series of small-scale laboratory experiments on a conveyor belt.

A numerical approach for detecting switch-like bistability in mass action chemical reaction networks with conservation laws

BackgroundTheoretical analysis of signaling pathways can provide a substantial amount of insight into their function. One particular area of research considers signaling pathways capable of assuming two or more stable states given the same amount of signaling ligand. This phenomenon of bistability can give rise to switch-like behavior, a mechanism that governs cellular decision making. Investigation of whether or not a signaling pathway can confer bistability and switch-like behavior, without knowledge of specific kinetic rate constant values, is a mathematically challenging problem. Recently a technique based on optimization has been introduced, which is capable of finding example parameter values that confer switch-like behavior for a given pathway. Although this approach has made it possible to analyze moderately sized pathways, it is limited to reaction networks that presume a uniterminal structure. It is this limited structure we address by developing a general technique that applies to any mass action reaction network with conservation laws.ResultsIn this paper we developed a generalized method for detecting switch-like bistable behavior in any mass action reaction network with conservation laws. The method involves (1) construction of a constrained optimization problem using the determinant of the Jacobian of the underlying rate equations, (2) minimization of the objective function to search for conditions resulting in a zero eigenvalue, (3) computation of a confidence level that describes if the global minimum has been found and (4) evaluation of optimization values, using either numerical continuation or directly simulating the ODE system, to verify that a bistability region exists. The generalized method has been tested on three motifs known to be capable of bistability.ConclusionsWe have developed a variation of an optimization-based method for the discovery of bistability, which is not limited to uniterminal chemical reaction networks. Successful completion of the method provides an S-shaped bifurcation diagram, which indicates that the network acts as a bistable switch for the given optimization parameters.

Generalized Regularization of Constrained Optimal Control Problems

Constraints in optimal control problems introduce challenges with traditional indirect methods. Bang-bang/singular solutions with discontinuous or indefinite control laws add further difficulty in numerical solution. Recent efforts in control regularization strategies have sought to overcome these limitations. Regularization generates a smoothed constraint transformation of a multiphase Hamiltonian boundary value problem to a single-phase unconstrained problem. This work develops a new approach to regularization using orthogonal error-control saturation functions. The method is developed for problems in bang-bang/singular form. The method is then applied to problems of general Hamiltonian structure using system extension and differential control. Applications in state constraint regularization are discussed. A key feature of the new approach is to eliminate ambiguity of the control law derived from the first-order necessary conditions of optimality. Results show desirable stability and convergence in numerical continuation. The method is applied to classical problems in optimal control, as well as problems of interest in aerospace mission design.

К ОБОСНОВАНИЮ ОДНОЗНАЧНОСТИ ПРОДОЛЖЕНИЯ РЕШЕНИЯ ЗАДАЧ О ДЕФОРМИРОВАНИИ МЯГКИХ ОБОЛОЧЕК МЕТОДОМ ДИФФЕРЕНЦИРОВАНИЯ ПО ПАРАМЕТРУ

Criterion of hyperelastic soft shell deforming nonlinear problem numerical solution continuation uniqueness assessment is suggested in the work. The criterion can be used when carrying out calculations. It is based on investigation of properties of Jacobi matrix of linear algebraic equations system which is formed when using parameter differentiation method. This method allows reducing nonlinear boundary value problem solution to a couple of quasilinear boundary value and nonlinear initial problems and applying initial parameters method of solving linear boundary value problems. For assessment of solution continuation uniqueness in each point of integration interval one needs controlling magnitudes of resolving differential equation system right-hand sides vector components, as well as calculating determinant and rank of Jacobi matrix of algebraic equations system formed as a result of initial parameters method using, and expanded Jacobi matrix rank calculation. For testing suggested criterion the problem of neohookean material hemisphere static inflation by uniformly applied pressure is considered. Solution of this problem as certain values of numerical algorithm parameters leads to various calculation difficulties – calculation stability loss, big errors of calculation results, non-uniqueness of solution which reason requires additional investigations. It is shown that in the points, where mentioned difficulties are met, determinateness conditions for the function of right-hand sides of differential equation system formulated within the framework of the suggested criterion are broken.

Thermodynamic Bethe Ansatz past turning points: the (elliptic) sinh-Gordon model

We analyze the Thermodynamic Bethe Ansatz (TBA) for various integrable S-matrices in the context of generalized T overline{mathrm{T}} deformations. We focus on the sinh-Gordon model and its elliptic deformation in both its fermionic and bosonic realizations. We confirm that the determining factor for a turning point in the TBA, interpreted as a finite Hagedorn temperature, is the difference between the number of bound states and resonances in the theory. Implementing the numerical pseudo-arclength continuation method, we are able to follow the solutions to the TBA equations past the turning point all the way to the ultraviolet regime. We find that for any number k of resonances the pair of complex conjugate solutions below the turning point is such that the effective central charge is minimized. As k → ∞ the UV effective central charge goes to zero as in the elliptic sinh-Gordon model. Finally we uncover a new family of UV complete integrable theories defined by the bosonic counterparts of the S-matrices describing the Φ1,3 integrable deformation of non-unitary minimal models mathcal{M} 2,2n+3.

Pressure-driven wrinkling of soft inner-lined tubes

A simple equation modelling an inextensible elastic lining of an inner-lined tube subject to an imposed pressure difference is derived from a consideration of the idealised elastic properties of the lining and the pressure and soft-substrate forces. Two cases are considered in detail, one with prominent wrinkling and a second one in which wrinkling is absent and only buckling remains. Bifurcation diagrams are computed via numerical continuation for both cases. Wrinkling, buckling, folding, and mixed-mode solutions are found and organised according to system-response measures including tension, in-plane compression, maximum curvature and energy. Approximate wrinkle solutions are constructed using weakly nonlinear theory, in excellent agreement with numerics. Our approach explains how the wavelength of the wrinkles is selected as a function of the parameters in compressed wrinkling systems and shows how localised folds and mixed-mode states form in secondary bifurcations from wrinkled states. Our model aims to capture the wrinkling response of arterial endothelium to blood pressure changes but applies much more broadly.

Computation of nonreciprocal dynamics in nonlinear materials

<p style='text-indent:20px;'>The reciprocity theorem in elastic materials states that the response of a linear, time-invariant system to an external load remains invariant with respect to interchanging the locations of the input and output. In the presence of nonlinear forces within a material, circumventing the reciprocity invariance requires breaking the mirror symmetry of the medium, thus allowing different wave propagation characteristics in opposite directions along the same transmission path. This work highlights the application of numerical continuation methods for exploring the steady-state nonreciprocal dynamics of nonlinear periodic materials in response to external harmonic drive. Using the archetypal example of coupled oscillators, we apply continuation methods to analyze the influence of nonlinearity and symmetry on the reciprocity invariance. We present symmetry-breaking bifurcations for systems with and without mirror symmetry, and discuss their influence on the nonreciprocal dynamics. Direct computation of the reciprocity bias allows the identification of response regimes in which nonreciprocity manifests itself as a phase shift in the output displacements. Various operating regimes, bifurcations and manifestations of nonreciprocity are identified and discussed throughout the work.</p>

A trajectory design and optimization framework for transfers from the Earth to the Earth-Moon triangular L4 point

• A synthetic computation procedure for transfer families using tangential velocity increments is presented. • A fast optimization method for direct transfer is constructed based on primer vector theory. • The necessary optimality conditions for a transfer utilizing powered lunar gravity assist under a minimum swing-by height constraint are deriver. • An optimization procedure for the optimal transfer utilizing powered lunar gravity assist is presented. The triangular libration points in the Earth-Moon system are considered to be potential candidates for future space missions because of their unique positions and dynamics. This paper presents a comprehensive transfer trajectory design and optimization framework for transfers from the Earth to L 4 in the Earth-Moon system, including direct transfer and transfer utilizing powered lunar gravity assist. In this framework, first, a synthetic procedure combining an initial guess, a differential correction algorithm and numerical continuation for generating tangential transfer families is illustrated. Subsequently, the optimization models for optimal transfer trajectories with the minimum velocity increment requirement are constructed. For direct transfer, the optimal direct transfer trajectory is obtained using primer vector theory. For the optimal transfer utilizing powered lunar gravity assist under a minimum swing-by height constraint, the necessary optimality conditions are found through optimal control theory, and an optimization procedure that allows for numerical convergence to the optimal solution is constructed. Numerical results indicate that the proposed transfer trajectory design and optimization framework has sufficient effectiveness to provide various transfer trajectories associated with different insertion points and fuel-optimal transfers.

Parameter identification of a physical model of brass instruments by constrained continuation

Numerical continuation using the Asymptotic Numerical Method (ANM), together with the Harmonic Balance Method (HBM), makes it possible to follow the periodic solutions of non-linear dynamical systems such as physical models of wind instruments. This has been recently applied to practical problems such as the categorization of musical instruments from the calculated bifurcation diagrams [V. Fréour et al. Journal of the Acoustical Society of America 148 (2020) https://doi.org/10.1121/10.0001603]. Nevertheless, one problem often encountered concerns the uncertainty on some parameters of the model (reed parameters in particular), the values of which are set almost arbitrarily because they are too difficult to measure experimentally. In this work we propose a novel approach where constraints, defined from experimental measurements, are added to the system. This operation allows uncertain parameters of the model to be relaxed and the continuation of the periodic solution with constraints to be performed. It is thus possible to quantify the variations of the relaxed parameters along the solution branch. The application of this technique to a physical model of a trumpet is presented in this paper, with constraints derived from experimental measurements on a trumpet player.