A piece wise linear suspension bridge model: nonlinear dynamics and orbit continuation

A fully nonlinear model of suspension bridges parameterized by one single space coordinate is proposed to describe overall three-dimensional motions. The nonlinear equations of motion are obtained via a direct total Lagrangian formulation and the kinematics, for the deck-girder and the suspension cables, feature the finite displacements of the associated base lines and the flexural and torsional rotations of the deck cross-sections assumed rigid in their own planes. The strain-displacement relationships for the generalized strain parameters, the elongations in the cables, the deck elongation, and the three curvatures, retain the full geometric nonlinearities. The proposed nonlinear model with its full extensional-flexural-torsional coupling is employed to study the torsional divergence caused by the static part of the wind-induced forces. Two suspension bridges are considered as case studies: the Runyang bridge (main span 1,490 m) and the Hu Men bridge (main span 888 m) in China. The evaluation of the onset of the static instability and the post-critical behavior takes into account the prestressed condition of the bridge subject to dead loads. The dynamic bifurcation that occurs at the onset of flutter is also studied accounting for the prestressed equilibrium state about which the equations of motion are obtained via an updated Lagrangian formulation. Such a bifurcation is investigated in the context of the parametric nonlinear model considering the model parameters of the Runyang Suspension Bridge together with its aeroelastic derivatives. The calculated critical wind speeds for the onset of the static and dynamic bifurcations are compared with the results obtained via linear analysis and the main differences are highlighted. Parametric sensitivity studies are carried out to assess the influence of the design parameters on the instabilities associated with the bridge aeroelastic response.

A model is developed for the seif-organisation of zones of enzymatic activity along a liver capillary (hepatic sinusoid) lined with cells of two types, which contain different enzymes and compete for sites on the wall of the sinusoid. An effectively non-local interaction between the cells arises from local consumption of oxygen from blood flowing throug1 the sinusoid, which gives rise to gradients of oxygen concentration in turn influencing rates of division and of death of the two cell-types. The process is modelled by a pair of coupled non-linear integro-differential equations for the cell-densities as functions of time and position along the sinusoid. Existence of a unique, bounded, non-negative solution of the equations is proved, for prescribed initial values. The equations admit infinitely many stationary solutions, but it is shown that all except one are unstable, for any given set of the model parameters. The remaining solution is shown to be asymptotically stable against a large class of perturbations. For certain ranges of the model parameters, the asymptotically stable stationaxy solution has a zonal structure, with cells of one type located entirely upstream of cells of the other type, and with jump discontinuities in the cell densities at a certain distance along the sinusoid. Such sinusoidal zones can account for zones of enzymatic activity observed in the intact liver. Exceptional cases are found for singular choices of model parameters, such that stationary cell-densities cannot be asymptotically stable individually, but together form an asymptotically stable set. Certain mathematical questions are left open, notably the behaviour of large deviations from stationary solutions, and the global stability of such solutions. Possible generalisations of the model are described.

We consider nonlinearly coupled string-beam equations modelling time-periodic oscillations in suspension bridges. We prove the existence of a unique solution under suitable assumptions on certain parameters of the bridge.

Near-fault and far-fault ground motion effects on cable-supported bridges

Most of the tragedies on any bridge structure have been the cause of high‐density crowd behavior as a response to trampling as well as the crushing scenario. Therefore, it is most important to monitor such unforeseen situations by sensing the load imposed on the bridge structures. This scenario may arise where crowd movement is huge on these types of bridges. Similarly, the fiber Bragg grating (FBG) is a promising technology for structural health monitoring applications. In this work, an Internet of Things based FBG optical sensing scheme is proposed to monitor real‐time strain distribution throughout the bridge structures and localization of load imposed on the structure from a central control room. A suspension bridge model is designed by referring to a real bridge scenario and these FBG sensors are deployed to validate the proposed machine learning models. In this article, the performances of two machine learning strategies are discussed for the accurate estimation of load and its position by acquiring high sensitive FBG sensors signals at a very high data rate. The algorithms include K‐nearest neighbor (KNN) and random forest (RF); which are applied on each sensing data source, and then validated using a prototype suspension bridge model integrated with three FBG sensors (1532 nm, 1538 nm, and1541 nm) on a single optical fiber cable.

<p>A central buckle is generally set in the middle of long-span suspension bridge to improve the structure mechanical performance. With a large number and sophisticated composition of steel plates, the mechanical behaviour of central buckle region is obscure. In this paper, a shell-beam hybrid finite element model of a 428m main span self-anchored suspension bridge was established, and the mechanical characteristics of central buckle region were analysed. The result shows that the normal stress of steel plates in central buckle region is mainly dependent on by dead loads, the effect caused by live loads only accounts for approximately 15% of that caused by dead loads; axial force of central buckle is mainly transmitted to the diaphragms and webs of the girder.</p>

Rotorcraft and their associated flight dynamics are highly non-linear systems, especially near extremities of the flight envelope. A global picture of the flight dynamics can be obtained by incorporating in the equations of motion non-linearities in aerodynamics, control system features and inertial terms. This paper deals with the coupling of these phenomena with the non-linear dynamics of an underslung load system. Dynamical systems theory provides the basis for the continuation and bifurcation method of analysing non-linear systems of ordinary differential equations, and forms the basis of this study. The principal strength of the bifurcation-based approach is to identify and give insight into phenomena that globally affect behaviour of an aircraft model, as the full system of non-linear equations is used in analysis. Implemented alongside appropriate numerical simulation runs, the technique promotes an understanding of the underlying structure governing the behaviour of non-linear dynamical systems. Bifurcation and continuation analysis is carried out here on a 21st order non-linear flight dynamics model representing a utility helicopter. Also, the equations of motion for an underslung load system are formulated and then analyzed in isolation. Thereafter they are appended to the helicopter model and a non-linear analysis of the complete system is performed. The effect of the underslung load dynamics on the nominal helicopter system is revealed by the bifurcation-based studies. The influence of varying parameters of the underslung model on the underlying equilibria of the rotorcraft is also investigated. The results reveal changes in helicopter and underslung load stability once the two systems are coupled. The underslung system reduces the extent of the stable equilibrium solution branches, and causes the stability augmentation system to saturate closer to the straight and level trim initial condition. The parameter dependance of the combined dynamics are presented and highlight areas of instability, which may be missed using conventional methods, and show how the transition between stable and unstable solutions is exactly determined as an underslung parameter is varied. This characterisation of the coupled system in terms of stability and response illustrates the potential for bifurcation-based methods to contribute to such non-linear analysis and synthesis problems.

In this paper, we investigate the existence, uniqueness, and exponential stability of almost periodic mild solutions for the chemotaxis-Navier-Stokes system with a matrix-valued sensitivity on a real hyperbolic space H d ( R ) ( d ⩾ 2 ) \mathbb {H}^d(\mathbb {R})\, (d\geqslant 2) and on the whole-line time-axis R t \mathbb {R}_t . First, we prove the existence and uniqueness of almost periodic mild solutions for the corresponding linear systems by using the dispersive estimates of the scalar and vectorial heat semigroups and the boundedness of the matrix-valued sensitivity. Then, we use the results of linear systems and fixed-point arguments, and we establish the well-posedness of almost periodic mild solutions for the chemotaxis-Navier-Stokes systems. Finally, we employ the Gronwall inequality to prove the exponential stability of such solutions. Our obtained results are also valid for such systems on a bounded domain with smooth boundary in the Euclidean space R d ( d ⩾ 2 ) \mathbb {R}^d\, (d\geqslant 2) .

We consider a simplified Ginzburg–Landau model of superconductivity in a two-dimensional infinite strip domain under the assumption of the periodicity in the infinite direction. This model equation has two physical parameters, λ, h, coming from the Ginzburg–Landau parameter and the strength of an applied magnetic field, respectively. We study the bifurcation of non-trivial solutions in the parameter space (h, λ), in particular through a bifurcation of the existence of a vortex solution, that is, a solution with isolated zeros. We first observe that in the parameter space there is a smooth (bifurcation) curve on which a solution with k-mode in the periodic direction takes place. This bifurcating solution, however, is vortexless. Then analysing the local bifurcation structure around the critical point at which two bifurcation curves for k and m(>k) intersect, we prove the existence of vortex solutions under a generic condition. Moreover, we show that the solutions have vortices lying on a boundary if the parameters belong to a certain curve emanating from the critical point. The stability of such solutions is also discussed.

Dynamics of species in a model with two predators and one prey

ABSTRACTIn this paper, we investigate the existence and stability of almost periodic solutions of impulsive fractional-order differential systems with uncertain parameters. The impulses are realised at fixed moments of time. For the first time, we determine the impact of the uncertainties on the qualitative behaviour of such systems. The main criteria for the existence of almost periodic solutions are proved by employing the fractional Lyapunov method. The global perfect robust uniform-asymptotic stability of such solutions is also considered. We apply our results to uncertain impulsive neural network systems of fractional order.

In a numerical study we investigate the influence of noise on the shape-invariant solitons in nematic liquid crystals. We use the modified Petviashvili’s method for finding eigenvalues and eigenfunctions of the evolution partial differential equations, to determine the shape-invariant solitons in a realistic scalar three-dimensional model that includes the highly nonlocal nature of uniaxial nematic liquid crystals. We check the stability of such solutions by propagating them for long distances, without or within the presence of white noise. Without noise, we find them stable. In the presence of noise (added to the medium), we find them breathing, which renders shape-invariant solitons difficult to observe. After prolonged propagation, the noise leads to the dissipation of solitons.KeywordsNematic Liquid CrystalOptical FieldLiquid Crystal CellSpatial SolitonMolecular ReorientationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We present the notion of monotone solution of mean field games master equations in the case of a continuous state space. We establish the existence, uniqueness and stability of such solutions under standard assumptions. This notion allows us to work with solutions which are merely continuous in the measure argument, in the case of first order master equations. We study several structures of common noises, in particular ones in which common jumps (or aggregate shocks) can happen randomly, and ones in which the correlation of randomness is carried by an additional parameter.

We have obtained the conditions of existence and a scheme for constructing solutions of a weakly nonlinear boundary value problem for a degenerate differential-algebraic system in the noncritical case. The boundary condition is determined by a weakly nonlinear vector functional. The linear part of the problem is a linear boundary value problem for a degenerate differential-algebraic system. Linear differential-algebraic boundary value problems have been studied in monographs by S. Campbell, J.R. Magnus, A.M. Samoilenko and V.P. Yakovets. In the works of A.M. Samoilenko and O.A. Boichuk, using the central canonical form, the necessary and sufficient conditions for the existence of solutions of nonlinear differential-algebraic boundary value problems were obtained. We have obtained necessary and sufficient conditions for the existence of solutions of nonlinear differential-algebraic systems without using the central canonical form, which allows us to study the solvability of differential-algebraic boundary value problems that depend on arbitrary continuous functions. This approach significantly varies the classification of nonlinear differential-algebraic boundary value problems in critical and noncritical cases. Our formulation of the weakly nonlinear differential-algebraic boundary value problem generalises the boundary value problems studied in the works of Yu.O. Mitropolsky, A.M. Samoilenko, and O.A. Boichuk. The case when a differential-algebraic system is not solvable with respect to the derivative is considered, and substitutions of the unknown are proposed. It leads the original system to a nonlinear differential-algebraic system solvable with respect to the derivative. Finally, we present an example of a nonlinear differential-algebraic antiperiodic boundary value problem for a Riccati-type equation, which demonstrates the constructiveness of the obtained necessary and sufficient conditions for the existence of solutions of nonlinear differential-algebraic systems. The obtained results can be transferred to the problems of finding conditions for the existence and schemes for constructing solutions of nonlinear degenerate differential-algebraic boundary value problems in critical cases, as well as to the problems of finding conditions for the stability of such solutions.

Periodic standing waves are considered for a Schrödinger system. The existence of periodic waves of dnoidal-type as well as the stability of such solutions is studied. Two kinds of the stability are considered, namely, the nonlinear and spectral stabilities.