Global analysis of secondary bifurcation of an elastic bar

In a three dimensional framework of finite deformation configurations, this paper investigates the secondary bifurcation of a uniform, isotropic and linearly elastic bar under compression in a large range of parameters. The governing differential equations and finite dimensional equations of this problem are discussed. It is found that, for a bar with two ends hinged, usually many secondary bifurcation points appear on the primary branches which correspond to the maximum bending stiffness. Results are shown on parameter charts. Secondary modes and branches are also calculated with numerical methods. The project supported in part by the National Natural Science Foundation of China     

Impact bifurcation of semi-infinite elastic thin bars

The dynamic buckling problem of elastic bars subjected to axial impact has been investigated by many authors in different ways. In this paper the problem, in which the elastic bars are assumed to be ideally straight, is reformulated in connection with the bifurcation due to the stress wave propagation. The example of a semi-infinite elastic bar is used for illustration.     

Stability and bifurcation in an electromechanical system with time delays

The effect of time delays occurring in a proportional-integral-derivative feedback controller on the linear stability of a simple electromechanical system is investigated by analyzing the characteristic transcendental equation. It is found that the trivial fixed point of the system can lose its stability through Hopf bifurcations when the time delay crosses certain critical values. Codimension two bifurcations, which result from non-resonant and resonant Hopf–Hopf bifurcation interactions, are also found to exist in the system.     

A modified averaging scheme with application to the secondary Hopf bifurcation of a delayed van der Pol oscillator

In this paper, a modified averaging scheme is presented for a class of time-delayed vibration systems with slow variables. The new scheme is a combination of the averaging techniques proposed by Hale and by Lehman and Weibel, respectively. The averaged equation obtained from the modified scheme is simple enough but it retains the required information for the local nonlinear dynamics around an equilibrium. As an application of the present method, the delay value for which a secondary Hopf bifurcation occurs is successfully located for a delayed van der Pol oscillator.     

On the sensitivity of non-generic bifurcation of non-linear normal modes

It is shown that a non-generic bifurcation of non-linear normal modes may occur if the ratio of linear natural frequencies is near r-to-one, r=1,3,5,… . Non-generic bifurcations are explicitly obtained in the systems having certain symmetry, as observed frequently in literatures. It is found that there are two kinds of non-generic bifurcations, super-critical and sub-critical. The normal mode generated by the former kind is extended to large amplitude, but that by the latter kind is limited to small amplitude which depends on the difference between two linear natural frequencies and disappears when two frequencies are equal. Since a non-generic bifurcation is not generic, it is expected generically that if a system having a non-generic bifurcation is perturbed then the non-generic bifurcation disappears, and generic bifurcation appears in the perturbed system. Examples are given to verify the change in bifurcations and to obtain the stability behavior of normal modes. It is found that if a system having a super-critical non-generic bifurcation is perturbed, then two new normal modes are generated, one is stable, but the other unstable, implying a saddle-node bifurcation. If the system having a sub-critical non-generic bifurcation is perturbed, then no new normal mode is generated, but there is an interval of instability on a normal mode, implying two saddle-node bifurcations on the mode. Application of this study is discussed.     

Symmetry, cusp bifurcation and chaos of an impact oscillator between two rigid sides

Both the symmetric period n-2 motion and asymmetric one of a one-degree- of-freedom impact oscillator are considered.The theory of bifurcations of the fixed point is applied to such model,and it is proved that the symmetric periodic motion has only pitchfork bifurcation by the analysis of the symmetry of the Poincarémap.The numerical simulation shows that one symmetric periodic orbit could bifurcate into two antisymmet- ric ones via pitchfork bifurcation.While the control parameter changes continuously, the two antisymmetric periodic orbits will give birth to two synchronous antisymmetric period-doubling sequences,and bring about two antisymmetric chaotic attractors subse- quently.If the symmetric system is transformed into asymmetric one,bifurcations of the asymmetric period n-2 motion can be described by a two-parameter unfolding of cusp, and the pitchfork changes into one unbifurcated branch and one fold branch.     

Stability and Hopf bifurcation of a delayed ratio-dependent predator-prey system

Since the ratio-dependent theory reflects the fact that predators must share and compete for food, it is suitable for describing the relationship between predators and their preys and has recently become a very important theory put forward by biologists. In order to investigate the dynamical relationship between predators and their preys, a so-called Michaelis-Menten ratio-dependent predator-prey model is studied in this paper with gestation time delays of predators and preys taken into consideration. The stability of the positive equilibrium is investigated by the Nyquist criteria, and the existence of the local Hopf bifurcation is analyzed by employing the theory of Hopf bifurcation. By means of the center manifold and the normal form theories, explicit formulae are derived to determine the stability, direction and other properties of bifurcating periodic solutions. The above theoretical results are validated by numerical simulations with the help of dynamical software WinPP. The results show that if both the gestation delays are small enough, their sizes will keep stable in the long run, but if the gestation delays of predators are big enough, their sizes will periodically fluc-tuate in the long term. In order to reveal the effects of time delays on the ratio-dependent predator-prey model, a ratiodependent predator-prey model without time delays is considered. By Hurwitz criteria, the local stability of positive equilibrium of this model is investigated. The conditions under which the positive equilibrium is locally asymptotically stable are obtained. By comparing the results with those of the model with time delays, it shows that the dynamical behaviors of ratio-dependent predator-prey model with time delays are more complicated. Under the same conditions, namely, with the same parameters, the stability of positive equilibrium of ratio-dependent predator-prey model would change due to the introduction of gestation time delays for predators and preys. Moreover, with the variation of time delays, the positive equilibrium of the ratio-dependent predator-prey model subjects to Hopf bifurcation.     

Non-linear dynamic behaviour of multi-folding microstructure systems based on origami skill

This paper presents the theoretical basis for both static and dynamic numerical approaches to the elastic stability and elasto-plastic stability of a folding multi-layered truss. Both analyses are based on bifurcation theory and include geometrical non-linearity. The dynamic analysis includes an allowance for contact between nodes. Comparisons are made between elastic behaviour and elasto-plastic behaviour obtained from both numerical dynamic methods in which bifurcations are demonstrated as a theoretically elastic analysis. Our analysis reveals that several folding behaviours in the pantographic truss are identified during the elastic and elasto-plastic instabilities, where the folding behaviour of the truss is shown to be a function of the initial geometry and velocity of the dynamic loading. The authors suggest that understanding this behaviour will be very useful for developing light-weight structures subject to dynamic loading based on static bifurcation analysis and dynamic analysis.     

Elastoplastic microscopic bifurcation and post-bifurcation behavior of periodic cellular solids

In this study, a general framework is developed to analyze microscopic bifurcation and post-bifurcation behavior of elastoplastic, periodic cellular solids. The framework is built on the basis of a two-scale theory, called a homogenization theory, of the updated Lagrangian type. We thus derive the eigenmode problem of microscopic bifurcation and the orthogonality to be satisfied by the eigenmodes. The orthogonality allows the macroscopic increments to be independent of the eigenmodes, resulting in a simple procedure of the elastoplastic post-bifurcation analysis based on the notion of comparison solids. By use of this framework, then, bifurcation and post-bifurcation analysis are performed for cell aggregates of an elastoplastic honeycomb subject to in-plane compression. Thus, demonstrating a basic, long-wave eigenmode of microscopic bifurcation under uniaxial compression, it is shown that the eigenmode has the longitudinal component dominant to the transverse component and consequently causes microscopic buckling to localize in a cell row perpendicular to the loading axis. It is further shown that under equi-biaxial compression, the flower-like buckling mode having occurred in a macroscopically stable state changes into an asymmetric, long-wave mode due to the sextuple bifurcation in a macroscopically unstable state, leading to the localization of microscopic buckling in deltaic areas.     

Simulation and interpretation of diffuse mode bifurcation of elastoplastic solids

The diffuse mode bifurcation of elastoplastic solids at finite strain is investigated. The multiplicative decomposition of deformation gradient and the hyperelasto-plastic constitutive relationship are adapted to the numerical bifurcation analysis of the elastoplastic solids. First, bifurcation analyses of rectangular plane strain specimens subjected to uniaxial compression are conducted. The onset of the diffuse mode bifurcations from a homogeneous state is detected; moreover, the post-bifurcation states for these modes are traced to arrive at localization to narrow band zones, which look like shear bands. The occurrence of diffuse mode bifurcation, followed by localization, is advanced as a possible mechanism to create complex deformation and localization patterns, such as shear bands. These computational diffuse modes and localization zones are shown to be in good agreement with the associated experimental ones observed for sand specimens to ensure the validity of this mechanism. Next, the degradation of horizontal sway stiffness of a rectangular specimen due to plane strain uniaxial compression is pointed out as a cause of the bifurcation of the first antisymmetric diffuse mode, which triggers the tilting of the specimen. Last, circular and punching failures of a footing on a foundation are simulated.     

Stability boundaries of two-parameter non-linear elastic structures

The load-bearing capacity of structures can be influenced by variations in parameters, such as initial geometric defects, multi-parameter loadings, material specifications and temperature. This paper aims to introduce a new formulation to trace the stability boundaries of two-parameter elastic structures. The proposed procedure can find a set of critical points, both limit and bifurcation ones, via a modified Newton’s method. In the authors’ formulation, the residual force is set to zero, and a critically constraint is satisfied simultaneously. Numerical examples presented in this paper demonstrate the efficiency of the suggested method.     

Surface instability of an elastic half space with material properties varying with depth

If a body with a stiffer surface layer is loaded in compression, a surface wrinkling instability may be developed. A bifurcation analysis is presented for determining the critical load for the onset of wrinkling and the associated wavelength for materials in which the elastic modulus is an arbitrary function of depth. The analysis leads to an eigenvalue problem involving a pair of linear ordinary differential equations with variable coefficients which are discretized and solved using the finite element method.The method is validated by comparison with classical results for a uniform layer on a dissimilar substrate. Results are then given for materials with exponential and error-function gradation of elastic modulus and for a homogeneous body with thermoelastically induced compressive stresses.     

Non-linear dynamic responses of elastic two-story structures with partially filled liquid tanks

This paper deals with the non-linear vibrations of an elastic two-story structure with two liquid tanks installed under horizontal harmonic excitation. The influence of the configuration of the two rectangular tanks on the response of the structure is investigated. In the theoretical analysis, Galerkin's method is applied to derive the equations of motion for the structure and the modal equations for sloshing, while considering the non-linear liquid forces. Then, van der Pol's method is used to determine the frequency response curves. Three cases are investigated: In the first case two tanks are installed, one on the top and one on the second story of the structure, in the second case one tank is installed on top, and in the third case two tanks are installed on top. The theoretical results of the first case are compared with those of the second and third cases. In the numerical calculations, it is found that Hopf bifurcations occur near the tuning frequency and then amplitude modulated motion appears in both the first and third cases. It is thus concluded that multiple tanks yield less effectiveness in suppressing the vibrations of the structure. The experimental data confirm the validity of the theoretical results for the first and third cases.     

Microscopic symmetric bifurcation condition of cellular solids based on a homogenization theory of finite deformation

In this paper, we establish a homogenization framework to analyze the microscopic symmetric bifurcation buckling of cellular solids subjected to macroscopically uniform compression. To this end, describing the principle of virtual work for infinite periodic materials in the updated Lagrangian form, we build a homogenization theory of finite deformation, which satisfies the principle of material objectivity. Then, we state a postulate that at the onset of microscopic symmetric bifurcation, microscopic velocity becomes spontaneous, yet changing the sign of such spontaneous velocity has no influence on the variation in macroscopic states. By applying this postulate to the homogenization theory, we derive the conditions to be satisfied at the onset of microscopic symmetric bifurcation. The resulting conditions are verified by analyzing numerically the in-plane biaxial buckling of an elastic hexagonal honeycomb. It is thus shown that three kinds of experimentally observed buckling modes of honeycombs i.e., uniaxial, biaxial and flower-like modes, are attained and classified as microscopic symmetric bifurcation. It is also shown that the multiplicity of bifurcation gives rise to the complex cell-patterns in the biaxial and flower-like modes.     

Numerical analysis of bifurcation buckling for rotationally periodic structures under rotationally periodic loads

By considering the characteristics of deformation of rotationally periodic structures under rotationally periodic loads, the periodic structure is divided into some identical substructures in this study. The degrees-of-freedom (DOFs) of joint nodes between the neighboring substructures are classified as master and slave ones. The stress and strain conditions of the whole structure are obtained by solving the elastic static equations for only one substructure by introducing the displacement constraints between master and slave DOFs. The complex constraint method is used to get the bifurcation buckling load and mode for the whole rotationally periodic structure by solving the eigenvalue problem for only one substructure without introducing any additional approximation. The finite element (FE) formulation of shell element of relative degrees of freedom (SERDF) in the buckling analysis is derived. Different measures of tackling internal degrees of freedom for different kinds of buckling problems and different stages of numerical analysis are presented. Some numerical examples are given to illustrate the high efficiency and validity of this method.     

The global bifurcation and chaos for the coupling between longitudinal and transverse vibration of a thin elastic plate in large overall motion

This paper presents the global bifurcation and chaotic behavior for the coupling of longitudinal and transverse vibration of a thin elastic plate in large overall motion. First the parametric equations of the homoclinic orbits of such a system is obtained. Then, by using the Melnikov method and digital computer simulation. the behavior of bifurcation and chaos of this vibration system is investigated in the cases of different resonances. The obvious difference between the transverse vibration and the coupling of transverse and longitudinal vibration is also shown.The project supported by the National Natural Science Foundation of China.     

Stability of plates made of a particulate composite with nonlinear elastic matrix and damaged inclusions

The bifurcation instability problem for rectangular plates made of particulate composites with nonlinear elastic matrix and damaged inclusions is formulated and solved __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 7, pp. 79–89, July 2007.     

Stability of cylindrical shells made of a particulate composite with nonlinear elastic matrix and damaged inclusions

The bifurcation instability problem for cylindrical shells made of particulate composites with nonlinear elastic matrix and damaged inclusions is formulated and solved __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 8, pp. 80–91, August 2007.     

An experimental study of bifurcation,chaos, and dimensionality in a system forced through a bifurcation parameter

An experimental study of a system that is parametrically excited through a bifurcation parameter is presented. The system consits of a lightly-damped, flexible beam which is buckled and unbuckled magnetically: it is parametrically excited by driving an electromagnet with a low-frequency sine wave. For voltage amplitudes in excess of the static bifurcation value, the beam slowly switches between the one-and two-well configurations. Experimental static and dynamic bifurcation results are presented. Static bifurcatons for the system are shown to involve a butterfly catastrophe. The dynamic bifurcation diagram, obtained with an automated data acquisition system, shows several period-doubling sequences, jump phenomena, and a chaotic region. Poincaré sections of a chaotic steady-state are obtained for various values of the driving phase, and the correlation dimension of the chaotic attractor is estimated over a large scaling region. Singular system analysis is used to demonstrate the effect of delay time on the noise level in delay-reconstructions, and to provide an independent check on the dimension estimate by directly estimating the number of independent coordinates from time series data. The correlation dimension is also estimated using the delay-reconstructed data and shown to be in good agrement with the value obtained from the Poincaré sections. The bifurcation and dimension results are used together with physical sonsiderations to derive the general form of a single-degree-of-freedom model for the experimental system.