Oscillations of a?beam with a?time-varying mass

This paper analyzes dynamical behavior of a simply supported Euler?CBernoulli beam with a time-varying mass on its surface. Though the system under consideration is linear, it exhibits dynamics similar to a nonlinear system behavior including internal resonances. The asymptotical solutions for the beam displacement has been found by combining the classical Galerkin method with the averaging method for equations in Banach spaces. The resonance conditions have been derived. It has been proposed a method for finding a number of possible resonances.Effect of the beam parameters on its dynamical behavior is investigated as well.     

Can a monkey with a computer create art?

A computer can be programmed to search through the solution of millions of equations to find a few hundred whose graphical display is aesthetically pleasing to humans. This paper describes some methods for performing such an exhaustive search, criteria for automatically judging aesthetic appeal, and examples of the results.     

Global dynamics and integrity of a two-dof model of?a?parametrically excited cylindrical shell

In this paper the global dynamics and topological integrity of the basins of attraction of a parametrically excited cylindrical shell are investigated through a two-degree-of-freedom reduced order model. This model, as shown in previous authors?? works, is capable of describing qualitatively the complex nonlinear static and dynamic buckling behavior of the shell. The discretized model is obtained by employing Donnell shallow shell theory and the Galerkin method. The shell is subjected to an axial static pre-loading and then to a harmonic axial load. When the static load is between the buckling load and the minimum post-critical load, a three potential well is obtained. Under these circumstances the shell may exhibit pre- and post-buckling solutions confined to each of the potential wells as well as large cross-well motions. The aim of the paper is to analyze in a systematic way the bifurcation sequences arising from each of the three stable static solutions, obtaining in this way the parametric instability and escape boundaries. The global dynamics of the system is analyzed through the evolution of the various basins of attraction in the four-dimensional phase space. The concepts of safe basin and integrity measures quantifying its magnitude are used to obtain the erosion profile of the various solutions. A detailed parametric analysis shows how the basins of the various solutions interfere with each other and how this influences the integrity measures. Special attention is dedicated to the topological integrity of the various solutions confined to the pre-buckling well. This allows one to evaluate the safety and dynamic integrity of the mechanical system. Two characteristic cases, one associated with a sub-critical parametric bifurcation and another with a super-critical parametric bifurcation, are considered in the analysis.     

Can a single-wall carbon nanotube be modeled as a thin shell?

Single-wall carbon nanotubes (SWCNT) have been frequently modeled as thin shells, but the shell thickness and Young's modulus reported in literatures display large scattering. The order of error to approximate SWCNTs as thin shells is studied in this paper via an atomistic-based finite-deformation shell theory, which avoids the shell thickness and Young's modulus, but links the tension and bending rigidities directly to the interatomic potential. The ratio of atomic spacing (Δ≈0.14 nm) to the radius of SWCNT, Δ/R, which ranges from zero (for graphene) to 40% [for a small (5,5) armchair SWCNT (R=0.35 nm)], is used to estimate the order of error. For the order of error O[(Δ/R)³], SWCNTs cannot be represented by a conventional thin shell because their constitutive relation involves the coupling between tension and curvature and between bending and strain. For the order of error O[(Δ/R)²], the tension and bending (shear and torsion) rigidities of SWCNTs can be represented by an elastic orthotropic thin shell, but the thickness and elastic modulus cannot. Only for the order of error O(Δ/R), a universal constant shell thickness can be defined and SWCNTs can be modeled as an elastic isotropic thin shell.     

Forced convective flow and heat transfer over a?porous plate in a?Darcy-Forchheimer porous medium in presence of?radiation

A mathematical model is presented for analyzing the boundary layer forced convective flow and heat transfer of an incompressible fluid past a plate embedded in a Darcy-Forchheimer porous medium. Thermal radiation term is considered in the energy equation. The similarity solutions for the problem are obtained and the reduced nonlinear ordinary differential equations are solved numerically. It is noticed that the boundary layer decreases with an increase in the value of inertial parameter and in this case the temperature profile is found to decrease smoothly within the boundary layer. In case of porous plate, fluid velocity increases whereas non-dimensional temperature decreases for increasing values of suction parameter. The rate of heat transfer increases with the increasing values of Prandtl number. The effect of thermal radiation on temperature field is also analyzed.     

Is the suppression of short waves by a swell a three-dimensional effect?

We consider the phenomenon of suppression of short waves by a long wave, observed by Mitsuyasu in 1966. The recently proposed [1] essentially 3-D explanation of this phenomenon is reviewed and compared with more traditional 2-D explanations. Several physical implications of this 3-D explanation are suggested and the experimental verification is discussed.     

When is a symmetric pin-jointed framework isostatic?

Maxwell’s rule from 1864 gives a necessary condition for a framework to be isostatic in 2D or in 3D. Given a framework with point group symmetry, group representation theory is exploited to provide further necessary conditions. This paper shows how, for an isostatic framework, these conditions imply very simply stated restrictions on the numbers of those structural components that are unshifted by the symmetry operations of the framework. In particular, it turns out that an isostatic framework in 2D can belong to one of only six point groups. Some conjectures and initial results are presented that would give sufficient conditions (in both 2D and 3D) for a framework that is realized generically for a given symmetry group to be an isostatic framework.     

Controlling Hopf�CHopf interaction bifurcations of?a?two-degree-of-freedom self-excited system with?dry?friction

The feedback control problem of designing Hopf?CHopf interaction bifurcations into a dry friction system at a pre-specified parameter point is addressed. A new bifurcation criterion without using eigenvalues is established to preferably determine the control gains. Numerical simulation shows that the torus solution of Hopf?CHopf interaction bifurcation can be created in the friction system at a desired parameter location.     

Generalized Homoclinic Solutions of a Coupled Schr?dinger System Under a Small Perturbation

This paper is devoted to study a coupled Schr?dinger system with a small perturbation $$\begin{array}{ll}u_{xx} - u + u^{3} + \beta uv^{2} + \epsilon f( \epsilon, u, u_{x}, v, v_{x}) = 0 \quad {\rm in} \, {\bf R}, \\v_{xx} + v - v^{3} + \beta u^{2}v + \epsilon g( \epsilon, u, u_{x}, v, v_{x}) = 0 \quad {\rm in} \, {\bf R} \end{array}$$ where β is a constant and ε is a small parameter. We first show that this system has a periodic solution and its dominant system has a homoclinic solution exponentially approaching zero. Then we apply the fixed point theorem and the perturbation method to prove that this homoclinic solution deforms to a homoclinic solution exponentially approaching the obtained periodic solution (called generalized homoclinic solution) for the whole system. Our methods can be used to other four dimensional dynamical systems like the Schr?dinger-KdV system.     

Nonlinear dynamic analysis and sliding mode control for?a?gyroscope system

This paper employs differential transformation (DT) method to analyze and control the dynamic behavior of a gyroscope system. The analytical results reveal a complex dynamic behavior comprising periodic, subharmonic, quasiperiodic, and chaotic responses of the center of gravity. Furthermore, the results reveal the changes which take place in the dynamic behavior of the gyroscope system as the external force is increased. The current analytical results by DT method are found to be in good agreement with those of Runge?CKutta (RK) method. In order to suppress the chaotic behavior in gyroscope system, the sliding mode controller (SMC) is used and guaranteed the stability of the system from chaotic motion to periodic motion. Numerical simulations are shown to verify the results. The proposed DT method and controlling scheme provide an effective means of gaining insights into the nonlinear dynamics and controlling of gyroscope systems.     

Can one measure the temperature of a curve?

The entropy of a plane curve is defined in terms of the number of intersection points with a random line. The Gibbs distribution which maximizes the entropy enables one to define the temperature of the curve. At 0 temperature, the curve reduces to a straight segment. At high temperature, the curve is somewhat chaotic and behaves like a perfect gas. We attempt to show that thermodynamic formalism can be used for the study of plane curves. The curves we discuss have finite length, unlike Mandelbrot's fractal curves [1], yet we feel our approach to the mathematics is not far from his.     

Stability analysis and modeling for the three-dimensionalDarcy-Forchheimer stagnation point nano?uid ?owtowards a moving surface?

正Yuming CHU1,2, M. I. KHAN3,, M. I. U. REHMAN4, S. KADRY5,S. QAYYUM4, M. WAQAS6     

Nonlinear analysis of chatter vibration in a cylindrical transverse grinding process with two time delays using?a?nonlinear time transformation method

A nonlinear dynamic system of cylindrical transverse grinding process is studied in this paper. The system consists of a grinding wheel and a workpiece, which are connected to the base by spring-damper elements, interacting with nonlinear normal forces. This two DOF model includes two time delays originated from the regenerative effects of the workpiece and the grinding wheel. Bifurcation points are located using a numerical algorithm by which we can find all the eigenvalues in a given rectangular region on the complex plane for the delayed differential equations. Supercritical bifurcation has been found for some sets of system parameter values. The amplitudes of the limit cycles are predicted using a nonlinear time transformation method, which is similar to the harmonic balance approach in that a periodic solution is approximated by a Fourier series. However, the main difference is that a nonlinear time ? is introduced in the Fourier series rather than the physical time t. The analytical solutions of stable limit cycles up to the third harmonics are compared with numerical simulations for the retarded system. It is shown that the proposed method gives accurate approximate solutions.     

Analysis of a Belyakov homoclinic connection with ?2-symmetry

We study the existence, in a two-parameter plane, of double- and triple-pulse homoclinic orbits in a ?₂-symmetric three-dimensional system, in the vicinity of a Belyakov point (a?point where the involved equilibrium in the homoclinic connection changes from saddle to saddle-focus) in the Shil??nikov zone. The first-order computation of these global connections allows us to describe their position and organization in the parameter plane. The analytical results are successfully applied in the study of such degeneration in Chua??s equation.     

Is research publication a catastrophic phenomenon among medical faculty?

Studies seeking to predict publication rates among faculty have found contradictory results. The purpose of this study was to determine whether short- or long-term research publications among family medicine faculty were better accounted for using cusp catastrophe modeling (CCM) rather than linear modeling. This secondary analysis of annual research publications used data collected from family medicine faculty in a university department. To predict the number of research publications, two service variables -- national service and administrative responsibility -- were used. There were three bifurcation variables: Scholarly Activity, Professional Status, and 'proportion of studies asprincipal investigator'. Research publications at two and five years were modeled using CCM as well as two linear models. Based upon the amount of variance explained, while linear models accounted for more variance in publications at -year intervals, CCM was superior at explaining publications for all three bifurcation variables at -year intervals. Entering all of the bifurcation variables into the models found that CCM explained more of the -year publication variance with Scholarly Activity and national service as significant predictors. In conclusion, short-term career planning needs to consider its irregular cusp behavior and to minimize the possible impact of bifurcation factors.     

A robust method on estimation of Lyapunov exponents from?a?noisy time series

Lyapunov exponents can indicate the asymptotic behaviors of nonlinear systems, and thus can be used for stability analysis. However, it is notoriously difficult to estimate these exponents reliably from experimental data due to the measurement error (noise). In this paper, a novel method for estimating Lyapunov exponents from a time series in the presence of additive noise corruption is presented. The method combines the ideas of averaging the noisy data to form new neighbors and of nonlinear mapping to determine neighborhood mapping matrices. Two case studies of balancing control of a bipedal robot and the Lorenz systems are presented to demonstrate the efficacy of the proposed method. The bipedal robot system has two negative Lyapunov exponents while the Lorenz system has one positive, zero, and negative exponents, respectively. It is shown that, as compared with the existing methods, our proposed one is more robust to the ratio of signal to noise, and is particularly effective in estimating negative Lyapunov exponents. We believe that the work can contribute significantly to the stability analysis of nonlinear systems using a noisy time series.     

Decentralized adaptive output feedback fuzzy controller for?a?class of large-scale nonlinear systems

In the previous work of Huang et al., a coordinated decentralized hybrid adaptive output feedback fuzzy control scheme of large-scale nonlinear systems is obtained predicated upon this prerequisite assumption that the local controllers can share the a priori information about their individual reference models. In this note, we concentrate in the absence of the coordination assumption on developing a classical decentralized combined indirect and direct adaptive fuzzy controller for a class of uncertain large-scale nonlinear systems. The output feedback and adaptation mechanisms proposed for each subsystem hinges just upon its individual output, regardless of any other output reference. Neither the famous strictly positive real (SPR) condition nor a high-gain observer (HGO) is required to realize the overall output feedback algorithm. The tracking errors of the closed-loop large-scale system are shown to converge to tunable neighborhoods of the origin. Simulation results on correlated inverted pendulums verify the validity of the decentralized controller modification.     

Multiple scales analysis of wave�Cwave interactions in?a?cubically nonlinear monoatomic chain

The interaction of waves in nonlinear media is of practical interest in the design of acoustic devices such as waveguides and filters. This investigation of the monoatomic mass?Cspring chain with a cubic nonlinearity demonstrates that the interaction of two waves results in different amplitude and frequency dependent dispersion branches for each wave, as opposed to a single amplitude-dependent branch when only a single wave is present. A theoretical development utilizing multiple time scales results in a set of evolution equations which are validated by numerical simulation. For the specific case where the wavenumber and frequency ratios are both close to 1:3 as in the long wavelength limit, the evolution equations suggest that small amplitude and frequency modulations may be present. Predictable dispersion behavior for weakly nonlinear materials provides additional latitude in tunable metamaterial design. The general results developed herein may be extended to three or more wave?Cwave interaction problems.     

Threshold dynamics for a HFMD epidemic model with?periodic transmission rate

In this paper, a periodic epidemic model is proposed in order to simulate the dynamics of HFMD transmission. We consider the effects of quarantine in the children population. We obtain a threshold value which determines the extinction and uniform persistence of the disease. Our results show that the disease-free equilibrium is globally asymptotically stable if the threshold value is less than unity. Otherwise, the system has a positive periodic solution and the disease persists. Numerical simulations show that quarantine has a positive impact on the spread of disease, i.e., quarantine is beneficial to the intervention and control of the disease outbreak in the children population.