Bifurcation analysis of an arch structure with parametric and forced excitation

The subharmonic bifurcation and universal unfolding problems are discussed for an arch structure with parametric and forced excitation in this paper. The amplitude–frequency curve and some dynamical behavior have been shown for this class of problems by Liu et al. Here, by means of singularity theory, in the case of strict 1:2 internal resonance, the bifurcation behavior of the amplitude with respect to a parameter (which is related to the amplitude of the live load imposed on the arch structures) is studied. The results indicate that it is a high codimensional bifurcation problem with codimension 5, and the universal unfolding is given. From the mechanical background, 20 forms of two parameter unfoldings with some constraints are studied. The transition sets in the parameter plane and the bifurcation diagrams are plotted. The results obtained in this paper present some new dynamic buckling patterns and abundant bifurcation phenomena.     

Bifurcation and stability analysis of an anaerobic digestion model

This paper presents the dynamic behaviour of the anaerobic digestion process, based on a simplified model. The hydraulic, biological and physicochemical processes such as those which underpin anaerobic digestion have more than one stable stationary solution and they compete with each other. Further, the attractive domains of the stable solutions vary with the key parameters. Thus, some initial transient process moving toward one stable solution could suddenly move towards another solution, at which a so-call catastrophe takes places (e.g. washout). The paper systematically analyses the stationary solutions with their associated stability, which provides insight and guidance for anaerobic digestion reactor design, operation and control.     

Bifurcation and buckling analysis of a unilaterally confined self-rotating cantilever beam

A nonlinear dynamic model of a simple non-holonomic system comprising a self-rotating cantilever beam subjected to a unilateral locked or unlocked constraint is established by employing the general Hamilton's Variational Principle. The critical values, at which the trivial equilibrium loses its stability or the unilateral constraint is activated or a saddle-node bifurcation occurs, and the equilibria are investigated by approximately analytical and numerical methods. The results indicate that both the buckled equilibria and the bifurcation mode of the beam are different depending on whether the distance of the clearance of unilateral constraint equals zero or not and whether the unilateral constraint is locked or not. The unidirectional snap-through phenomenon (i.e. catastrophe phenomenon) is destined to occur in the system no matter whether the constraint is lockable or not. The saddle-node bifurcation can occur only on the condition that the unilateral constraint is lockable and its clearance is non-zero. The results obtained by two methods are consistent. The project supported by the National Natural Science Foundation of China (10272002) and the Doctoral Program from the Ministry of Education of China (20020001032) The English text was polished by Yunming Chen.     

Nonlinear analysis of the forced response of a beam with three mode interaction

An analysis is presented for the primary resonance of a clamped-hinged beam, which occurs when the frequency of excitation is near one of the natural frequencies,_n . Three mode interaction (₂ 3₁ and _{3 1} + 2₂) is considered and its influence on the response is studied. The case of two mode interaction (₂ 3₁) is also considered to compare it with the case of three mode interaction. The straight beam experiencing mid-plane stretching is governed by a nonlinear partial differential equation. By using Galerkin's method the governing equation is reduced to a system of nonautonomous ordinary differential equations. The method of multiple scales is applied to solve the system. Steady-state responses and their stability are examined. Results of numerical investigations show that there exists no significant difference between both modal interactions' influences on the responses.     

Bifurcation and stability analysis of a neural network model with distributed delays

In this paper, the dynamics of a generalized two-neuron model with self-connections and distributed delays are investigated, together with the stability of the equilibrium. In particular, the conditions under which the Hopf bifurcation occurs at the equilibrium are obtained for the weak kernel. This means that a family of periodic solutions bifurcates from the equilibrium when the bifurcation parameter exceeds a critical value. Explicit algorithms for determining the stability of the bifurcating periodic solutions and the direction of the Hopf bifurcation are derived by using the theory of normal form and center manifold [20]. Some numerical simulations are given to illustrate the effectiveness of the results found. The obtained results are new and they complement previously known results.This work was supported by the National Natural Science Foundation of China under Grants 60574043 and 60373067, the Natural Science Foundation of Jiangsu Province, China under Grants BK2003053.     

BIFURCATION ANALYSIS OF A MITOTIC MODEL OF FROG EGGS

ＩｎｔｒｏｄｕｃｔｉｏｎＷｉｔｈｔｈｅｒａｐｉｄｄｅｖｅｌｏｐｍｅｎｔｏｆｂｉｏｌｏｇｙｓｃｉｅｎｃｅｓ,ｃｅｌｌｓｉｇｎａｌｔｒａｎｓｄｕｃｔｉｏｎ ,ｃｅｌｌｅｐｏｐｔｏｓｉｓ,ｇｅｎｏｍｅａｎｄｐｏｓｔ＿ｇｅｎｏｍｉｃａｎａｌｙｓｉｓｈａｖｅａｔｔｒａｃｔｅｄｉｎｃｒｅａｓｉｎｇａｔｔｅｎｔｉｏｎ[1- 6 ].Ｔｈｅｃｅｌｌｄｉｖｉｓｉｏｎｃｙｃｌｅｉｓｔｈｅｓｅｑｕｅｎｃｅｏｆｅｖｅｎｔｓｂｙｗｈｉｃｈａｇｒｏｗｉｎｇｃｅｌｌｄｕｐｌｉｃａｔｅｓａｌｌｉｔｓｃｏｍｐｏｎｅｎｔｓａｎｄｔｈｅｎｄｉｖ…     

Examining a modified algorithm of smoothed particle hydrodynamics for a high velocity perforation of an aluminum beam

The wide spreading of utilizing of smoothed particle hydrodynamics (SPH) for numerical studies of the complex and high rate deformations of continuums, led the current study to gain a more reliable simulation by employing a modified compressible smoothed particle hydrodynamics (MCSPH) algorithm which could be a more accurate and stable technique in high tension regions, in despite of incompressible standard SPH. The main feature of the modified compressible SPH algorithm relies on a three steps solution procedure to calculate the pressure gradient, the deviatoric stress tensor, and the body forces separately. This algorithm is free of any artificial viscosity in its formulations, as well as welcoming to compressible effects which permits the pressure shock waves in high rate plastic deformation. To examine the accuracy of the algorithm, a benchmark problem of colliding rubber cylinders was simulated first and then a high velocity perforation process of an aluminum beam struck by a rigid projectile was simulated in various projectile speeds, and the failure response of the beam in each case was accompanied by crack propagation process. The prominent capability of the utilized MCSPH can be more illustrated when it was used in simulation of thickness crack propagation a tiny crack paths and defragmentation which can be encountered as a not easy numerical case study. The adequate assurance has been more fortified when the results were compared to those reported from a Finite Element method study.     

Method of model analysis for flexible head impacting with elastic plane

ＩｎｔｒｏｄｕｃｔｉｏｎＲｅｓｅａｒｃｈｅｓｓｈｏｗ :70ｔｏ 80ｐｅｒｃｅｎｔｏｆｔｈｅｖｅｈｉｃｌｅｓｔｈａｔｃａｕｓｅｔｈｅａｃｃｉｄｅｎｔｓａｒｅｓａｌｏｏｎｃａｒｓ[1].Ｉｎｔｈｅ 70ｐｅｒｃｅｎｔｏｆｔｈｅａｃｃｉｄｅｎｔｓ,ｔｈｅｈｕｍａｎｂｏｄｙｗｉｌｌｃｏｌｌｉｄｅｗｉｔｈｔｈｅｆｒｏｎｔｏｆｔｈｅｃａｒａｎｄｔｈｅｈｅａｄａｎｄｌｅｇｗｉｌｌｂｅｉｎｊｕｒｅｄ .Ｍｅａｎｔｉｍｅ ,ｔｈｅｈｏｏｄａｎｄｗｉｎｄｓｈｉｅｌｄｗｉｌｌｃａｕｓｅｔｈｅｉｎｊｕｒｙｏｆｔｈｅｈｅａｄａｎｄｔｈ…     

Bifurcation analysis near a double eigenvalue of a model chemical reaction

In this paper we analyze the steady-state bifurcations from the trivial solution of the reaction-diffusion equations associated to a model chemical reaction, the so-called Brusselator. The present analysis concentrates on the case when the first bifurcation is from a double eigenvalue. The dependence of the bifurcation diagrams on various parameters and perturbations is analyzed. The results of reference [2] are invoked to show that further complications in the model would not lead to new behavior.     

Bifurcation analysis of an aeroelastic system with concentrated nonlinearities

Analytical and numerical analyses of the nonlinear response of a three-degree-of-freedom nonlinear aeroelastic system are performed. Particularly, the effects of concentrated structural nonlinearities on the different motions are determined. The concentrated nonlinearities are introduced in the pitch, plunge, and flap springs by adding cubic stiffness in each of them. Quasi-steady approximation and the Duhamel formulation are used to model the aerodynamic loads. Using the quasi-steady approach, we derive the normal form of the Hopf bifurcation associated with the system??s instability. Using the nonlinear form, three configurations including supercritical and subcritical aeroelastic systems are defined and analyzed numerically. The characteristics of these different configurations in terms of stability and motions are evaluated. The usefulness of the two aerodynamic formulations in the prediction of the different motions beyond the bifurcation is discussed.     

Three-dimensional simulation of two viscoelastic droplets impacting onto a rigid plate using smoothed particle hydrodynamics

In this article, a smoothed particle hydrodynamics method is developed to simulate the dynamic process of the impact of two viscoelastic droplets onto a rigid plate. The Oldroyd-B fluid is considered as the rheological model to describe the viscoelastic characteristics. An artificial stress is added into the momentum equation to remove the tensile instability. The solution of the problem of two successive impacts of droplets are demonstrated to be in good agreement with the literature data. The problem of two droplets impacting simultaneously onto a rigid plate is investigated.     

Bifurcation and path-following continuation analysis of periodic orbits of an extended Fermi oscillator model

In this research, we offer eigenvalue analysis and path following continuation to describe the impact, stick, and non-stick between the particle and boundaries to understand the nonlinear dynamics of an extended Fermi oscillator. The principles of discontinuous dynamical systems will be utilized to explain the moving process in such an extended Fermi oscillator. The motion complexity and stick mechanism of such an oscillator are demonstrated using periodic and chaotic motions. The major parameters are the frequency, amplitude in periodic excitation force, and the gap between the top and bottom boundary. We employ path-following analysis to illustrate the bifurcations that lead to solution destabilization. We present the evolution of the period solutions of the extended Fermi oscillator as the parameter varies. From the viewpoint of eigenvalue analysis, the essence of period-doubling, saddle-node, and Torus bifurcation is revealed. Numerical continuation methods are used to do a complete one- and two-parameter bifurcation analysis of the extended Fermi oscillator. The presence of codimension-one bifurcations of limit cycles, such as saddle-node, period-doubling, and Torus bifurcations, is shown in this work. Bifurcations cause all solutions to lose stability, according to our findings. The acquired results provide a better understanding of the extended Fermi oscillator mechanism and demonstrate that we may control the system dynamics by modifying the parameters.

     

Bifurcation analysis in a time-delay model for prey–predator growth with stage-structure

A time-delay model for prey–predator growth with stage-structure is considered. At first, we investigate the stability and Hopf bifurcations by analyzing the distribution of the roots of associated characteristic equation. Then, an explicit formula for determining the stability and the direction of periodic solutions bifurcating from Hopf bifurcations is derived, using the normal form theory and center manifold argument. Finally, some numerical simulations are carried out for supporting the analytic results.     

Bifurcation and evolution of a forced and damped Duffing system in two-parameter plane

Nonlinear Dynamics - A general method to calculate multi-parameter bifurcation diagram in the parameter space is designed based on top Lyapunov exponent and Floquet multiplier to study the effect...     

Nonlinear forced vibration analysis of composite beam considering internal damping

Nonlinear Dynamics - In a structure made of composite material, dynamic behavior must be analyzed in consideration of internal damping that occurs due to the damping property of the matrix...     

Free and forced nonlinear oscillations of a two-layer composite beam with interface slip

Free and forced flexural nonlinear vibrations of a two-layer beam are investigated. Each beam is assumed to have Euler?CBernoulli kinematics and free-free boundary conditions. The interface allows only nonlinear elastic slip between adjacent sides of the beams, so that the transversal displacement is unique. Free vibrations are considered first by the multiple time scale method, which allows to determine the amplitude dependent nonlinear natural frequencies of the system. It is shown that the nonlinear coefficient of the backbone curve is positive, so that hardening/softening behavior of the interface generates hardening/softening behavior of the whole structure. The modifications of the linear normal modes for moderate excitation amplitudes have been computed. Forced and damped nonlinear oscillations are then considered by the same mathematical method, and the nonlinear frequency response curves are obtained.     

Asymptotic analysis on weakly forced vibration of axially moving viscoelastic beam constituted by standard linear solid model

The weakly forced vibration of an axially moving viscoelastic beam is investigated.The viscoelastic material of the beam is constituted by the standard linear solid model with the material time derivative involved.The nonlinear equations governing the transverse vibration are derived from the dynamical,constitutive,and geometrical relations.The method of multiple scales is used to determine the steady-state response.The modulation equation is derived from the solvability condition of eliminating secular terms.Closed-form expressions of the amplitude and existence condition of nontrivial steady-state response are derived from the modulation equation.The stability of nontrivial steady-state response is examined via the Routh-Hurwitz criterion.