Non-linearly parametric resonances of an axially moving viscoelastic sandwich beam with time-dependent velocity

Non-linearly parametric resonances of an axially moving viscoelastic sandwich beam are investigated in this paper. The beam is moving with a time-dependent velocity, namely a harmonically varied velocity about the mean velocity. The partial differential equation is discretized into nonlinear ordinary differential equations via the method of Galerkin truncation and then the steady-state response is obtained using the method of multiple scales, an approximate analytical method. The tuning equations are obtained by eliminating secular terms and the amplitude of the vibration is derived from the tuning equations expressed in polar form, and two bifurcation points are obtained as well. Additionally, the stability conditions of trivial and nontrivial solutions are analyzed using the Routh–Hurwitz criterion. Eventually, the effects of various parameters such as the thickness of core layer, mean velocity, initial tension, and the amplitude of axially moving velocity on amplitude–frequency response curves and unstable regions are investigated.     

Nonlinear study of a rotor–AMB system under simultaneous primary-internal resonance

A rotor–active magnetic bearing (AMB) system subjected to a periodically time-varying stiffness with quadratic and cubic non-linearities under multi-parametric excitations is studied and solved. The method of multiple scales is applied to analyze the response of two modes of a rotor–AMB system with multi-parametric excitations and time-varying stiffness near the simultaneous primary and internal resonance. The stability of the steady state solution for that resonance is determined and studied using Rung–Kutta method of fourth order. It is shown that the system exhibits many typical non-linear behaviors including multiple-valued solutions, jump phenomenon, hardening and softening non-linearities and chaos in the second mode of the system. The effects of the different parameters on the steady state solutions are investigated and discussed also. A comparison to published work is reported.     

轴向变速运动弦线的非线性振动的稳态响应及其稳定性

研究具有几何非线性的轴向运动弦线的稳态横向振动及其稳定性．轴向运动速度为常平均速度与小简谐涨落的叠加．应用Hamilton原理导出了描述弦线横向振动的非线性偏微分方程．直接应用于多尺度方法求解该方程．建立了避免出现长期项的可解性条件．得到了近倍频共振时非平凡稳态响应及其存在条件．给出数值例子说明了平均轴向速度、轴向速度涨落的幅值和频率的影响．应用Liapunov线性化稳定性理论,导出倍频参数共振时平凡解和非平凡解的不稳定条件．给出数值算例说明相关参数对不稳定条件的影响．     

The onset of resonance processes in the Couette–Taylor problem close to the point of codimension-2 bifurcation

The bifurcations on passing around the point of intersection of two neutral curves (points of codimension-2 bifurcation) are considered in the Couette–Taylor problem of the fluid motion between rotating cylinders. The secondary modes in a small neighbourhood of a point of codimension-2 bifurcation are studied using a system of non-linear amplitude equations in a central manifold. The steady-state solutions of the amplitude systems, to which secondary periodic modes of the travelling-wave type, non-linear mixtures of travelling waves and unsteady two-, three- and four-frequency quasiperiodic solutions of the system of Navier–Stokes equations correspond, are analysed. A numerical analysis of the conditions for the existence and stability of irrotationally symmetric steady-state fluid flows between unidirectionally rotating cylinders is carried out.     

The effect of the slip condition on flows of an Oldroyd 6-constant fluid

The steady flows of a non-Newtonian fluid are considered when the slippage between the plate and the fluid is valid. The constitutive equations of the fluid are modeled by those for an Oldroyd 6-constant fluid. They give rise to non-linear boundary value problems. The analytical solutions are obtained using powerful, easy-to-use analytic technique for non-linear problems, the homotopy analysis method. It is shown that solutions exist for all values of non-Newtonian parameters. The solutions valid for no-slip condition for all values of non-Newtonian parameters can be derived as the special cases of the present analysis. Finally, graphs are plotted and critical assessment is made for the cases of slip and no-slip conditions.     

Non-linear forced vibration analysis of nanobeams subjected to moving concentrated load resting on a viscoelastic foundation considering thermal and surface effects

Analytical solution for the steady-state response of an Euler–Bernoulli nanobeam subjected to moving concentrated load and resting on a viscoelastic foundation with surface effects consideration in a thermal environment is investigated in this article. At first, based on the Eringen's nonlocal theory, the governing equations of motion are derived using the Hamilton's principle. Then, in order to solve the equation, Galerkin method is applied to discretize the governing nonlinear partial differential equation to a nonlinear ordinary differential equation; solution is obtained employing the perturbation technique (multiple scales method). Results indicate that by increasing of various parameters such as foundation damping, linear stiffness, residual surface stress and the temperature change, the jump phenomenon is postponed and with increasing the amplitude of the moving force and the nonlocal parameter, the jump phenomenon occurs earlier and its frequency and the peak value of amplitude of vibration increases. In addition, it is seen that the non-linear stiffness and the critical velocity of the moving load are important factors in studying nanobeams subjected to moving concentrated load. Presence of the non-linear stiffness of Winkler foundation resulting nanobeam tends to instability and so, the jump phenomenon occurs. But, presence of the linear stiffness will lead to stability of the nanobeam. In the next sections of the paper, frequency responses of the nanobeam made of temperature-dependent material properties under multi-frequency excitations are investigated.     

Heat transfer in unsteady axisymmetric second grade fluid

This work looks at the heat transfer effects on the flow of a second grade fluid over a radially stretching sheet. The axisymmetric flow of a second grade fluid is induced due to linear stretching of a sheet. Mathematical analysis has been carried out for two heating processes, namely (i) with prescribed surface temperature (PST case) and (ii) prescribed surface heat flux (PHF case). The modelled non-linear partial differential equations in two dependent variables are reduced into a partial differential equation with one dependent variable. The resulting non-linear partial differential equations are solved analytically using homotopy analysis method (HAM). The series solutions are developed and the convergence is properly discussed. The series solutions and graphs of velocity and temperature are constructed. Particular attention is given to the variations of emerging parameters such as second grade parameter, Prandtl and Eckert numbers.     

同伦分析法求解非线性多孔收缩表面上黏性磁流体的流动

研究在非线性多孔收缩表面上黏性磁流体(MHD)的流动.先用相似变换简化其控制方程,然后用同伦分析法(HAM)求解该简化问题.用图表的形式对问题的相关参数进行讨论,发现在有磁流体时,收缩解存在.同时得到,在不同参数下f″(0)的解是收敛的.     

Positive Steady-state Solutions of a Non-linear Reaction–Diffusion System

The steady-state problem of the non-linear reaction-diffusion system u_t−u_xx=u(av−b) , v_t−v_xx=cu is considered. The existence of positive steady-solutions is established by using a fixed point theorem in ordered Banach space. The uniqueness of ordered positive steady-state solutions and an application to the associated reaction-diffusion system are presented. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Analytical expressions for the concentrations of substrate,oxygen and mediator in an amperometric enzyme electrode

The theoretical model of Martens and Hall [N. Martens, E.A.H. Hall, Model for an immobilized oxidase enzyme electrode in the presence of two oxidants, Anal. Chem. 66 (1994) 2763–2770] in an immobilized oxidase enzyme electrodes is discussed. This model contains a non-linear term related to enzyme reaction system. In this paper, we obtain approximate analytical solutions for the non-linear equations under steady-state condition by using the homotopy perturbation method (HPM) and homotopy analysis method (HAM). Simple and approximate polynomial expressions for the concentration of substrate, oxygen, reduced mediator and current were obtained in terms of Thiele moduli and the normalized surface concentrations of species. Furthermore, in this work the numerical simulation of the problem is also reported using Scilab/Matlab program. An agreement between analytical and numerical results is noted.     

Properties and exact solutions of the equations of motion of shallow water in a spinning paraboloid

A transformation is found and, using this, the non-linear system of equations describing the spatial oscillations of a thin layer of liquid in a spinning circular parabolic basin is reduced to the conventional equations of the model of shallow water over a level fixed bottom. This transformation is obtained by analyzing the properties of the symmetry of the equations of motion of spinning shallow water. The existence of non-trivial symmetries in the case of the model considered enabled group multiplication of the solutions to be carried out. Using the known steady-state rotationally symmetric solution, a class of time-periodic solutions is obtained that describes the non-linear oscillations of the liquid in a circular paraboloid with closed or quasiclosed (ergodic) trajectories of the motion of the liquid particles.     

Positive solutions and dynamics of a finite difference reaction–diffusion system

We investigate the dynamics and methods of computation for some nonlinear finite difference systems that are the discretized equations of a time-dependent and a steady-state reaction–diffusion problem. The formulation of the discrete equations for the time-dependent problem is based on the implicit method for parabolic equations, and the computational algorithm is based on the method of monotone iterations using upper and lower solutions as the initial iterations. The monotone iterative method yields improved upper and lower bounds of the solution in each iteration, and the sequence of iterations converges monotonically to a solution for both the time-dependent and the steady-state problems. An important consequence of this method is that it leads to a bifurcation point that determines the dynamic behavior of the time-dependent problem in relation to the corresponding steady-state problem. This bifurcation point also determines whether the steady-state problem has one or two non-negative solutions, and is explicitly given in terms of the physical parameters of the system and the type of boundary conditions. Numerical results are presented for both the time-dependent and the steady-state problems under various boundary conditions, including a test problem with known analytical solution. These numerical results exhibit the predicted dynamic behavior of the time-dependent solution given by the theoretical analysis. Also discussed are the numerical stability of the computational algorithm and the convergence of the finite difference solution to the corresponding continuous solution of the reaction–diffusion problem. © 1993 John Wiley & Sons, Inc.     

一类具有抑制剂的非均匀恒化器模型的共存态

研究了一类具有抑制剂和Beddington-DeAngelis功能反应项的非均匀恒化器模型.根据单调动力系统理论得到了正平衡解的存在性.利用度理论、分歧理论以及摄动理论,分析了抑制剂对系统正平衡解及渐近行为的影响.结果表明当体现抑制作用的参数μ充分大时,此模型或者没有正解,并且一个半平凡晌非负解是全局吸引的;或者模型的所有正解均由一个极限问题决定.     

Solutions faibles H2 pour un modèle de fluide non newtonien

This Note is devoted to the study a non-linear system modelling a flow of a non-Newtonian fluid, namely aqueous polymer solutions. In the present work, we prove a theorem on the existence of weak solutions for the steady-state problem in a bounded or exterior plane domain. A first difficulty is due to the fact that we search these solutions in the natural space given by the energy inequalities. The second difficulty stems from the fact that the non-linear term involves a third-order derivative, whereas its elliptic term is only a Laplace operator. To cite this article: C. Amrouche, E.-H. Ouazar, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Single-mode control and chaos of cantilever beam under primary and principal parametric excitations

A non-linear control law is proposed to suppress the vibrations of the first mode of a cantilever beam when subjected to primary and principal parametric excitations. The dynamics of the beam are modeled with a second-order non-linear ordinary-differential equation. The model accounts for viscous damping air drag, and inertia and geometric non-linearities. A control law based on quantic velocity feedback is proposed. The method of multiple scales method is used to derive two-first ordinary differential equations that govern the evolution of the amplitude and phase of the response. These equations are used to determine the steady state responses and their stability. Amplitude and phase modulation equations as well as external force–response and frequency–response curves are obtained. Numerical simulations confirm this scenario and detect chaos and unbounded motions in the instability regions of the periodic solutions.     

New exact solutions corresponding to the second problem of Stokes for second grade fluids

New exact solutions for the velocity field corresponding to the second problem of Stokes, for second grade fluids, have been established by the Laplace transform method. These solutions, presented as a sum of the steady-state and transient solutions, are in accordance with the previous solutions obtained by a different technique. The required time to reach the steady state is determined by graphical illustrations. This time decreases if the frequency of the velocity increases. The effects of the material parameters on the decay of the transients are also investigated by graphs.     

1:2 and 1:3 internal resonance active absorber for non-linear vibrating system

Vibration and dynamic chaos should be controlled in either structures or machines. An active vibration absorber for suppressing the vibration of the non-linear plant when subjected to external and parametric excitations is studied in the presence of one-to-two and one-to-three internal resonance. The main attention is focused on the study of the active control and stability of two systems, which can be used to reduce vibrations due to rotor blade flapping motion. The method of multiple scale perturbation technique is applied to determine four first-order non-linear ordinary differential equations that govern the modulation of the amplitudes and phases in the presence of internal resonance of the two systems with quadratic and cubic order of control. These equations are used to determine the steady state solutions and their stability. The stability study of non-linear periodic solution for two cases (1:2 and 1:3 internal resonance) and the stability of the obtained numerical solution are investigated using frequency, force-response curves and phase-plane method. Also, effects of some parameters on the steady state solution of the vibrating system are investigated and reported in this paper. Variation of some parameters leads to the bending of the frequency, force-response curves and hence to the jump phenomenon occurrence. The reported results are compared to the available published work.     

Unsteady convective boundary layer flow of a viscous fluid at a vertical surface with variable fluid properties

In this paper we present numerical solutions to the unsteady convective boundary layer flow of a viscous fluid at a vertical stretching surface with variable transport properties and thermal radiation. Both assisting and opposing buoyant flow situations are considered. Using a similarity transformation, the governing time-dependent partial differential equations are first transformed into coupled, non-linear ordinary differential equations with variable coefficients. Numerical solutions to these equations subject to appropriate boundary conditions are obtained by a second order finite difference scheme known as the Keller-Box method. The numerical results thus obtained are analyzed for the effects of the pertinent parameters namely, the unsteady parameter, the free convection parameter, the suction/injection parameter, the Prandtl number, the thermal conductivity parameter and the thermal radiation parameter on the flow and heat transfer characteristics. It is worth mentioning that the momentum and thermal boundary layer thicknesses decrease with an increase in the unsteady parameter.     

The method of multiple scales for forced oscillators with some real-power nonlinearities in the stiffness and damping force

The steady-state response of forced damped nonlinear oscillators is considered, the restoring force of which has a non-negative real power-form nonlinear term and the linear term of which can be negative, zero or positive. The damping term is also assumed in a power form, thus covering polynomial and non-polynomial damping. The method of multiple scales with a new expansion parameter is presented in order to cover the cases when the nonlinearity is not necessarily small. Amplitude-frequency equations and approximate solutions for the steady-state response at the frequency of excitation are obtained and compared with numerical results, showing good agreement.