VIBRATION ANALYSIS OF A ROD WITH COMPLEX BOUNDARY CONDITIONS

This paper deals with the forced longitudinal vibration ora rod carrying a concentrated mass and supported by a spring at one end. The vibration of the rod is excited by the motion of the support point at the other end Since the boundary conditions of the problem are complex and it is necessary to consider the damping, we determine only the steady state periodic solution. First the linear system is analysed, then the material nonlinearity is considered and the approximate analytic solution of nonlinear partial differential equation with nonlinear boundary conditions is obtained by the perturbation method.     

Soret and Dufour effects on mixed convection unsteady MHD boundary layer flow over stretching sheet in porous medium with chemically reactive species

This paper studies the thermal-diffusion and diffusion thermo-effects in the hydro-magnetic unsteady flow by a mixed convection boundary layer past an imperme- able vertical stretching sheet in a porous medium in the presence of chemical reaction. The velocity of t~he stretching surface, the surface temperature, and the concentration are assumed to vary linearly with the distance along the surface. The governing partial differential equations are transformed into self-similar unsteady equations using similarity transformations .and solved numerically by the Runge-Kutta fourth order scheme in as- sociation with the shooting method for the whole transient domain from the initial state to the final steady state flow. Numerical results for the velocity, the temperature, the concentration, the skin friction, and the Nusselt and Sherwood numbers are shown graph- ically for various flow parameters. The results reveal that there is a smooth transition of flow from unsteady state to the final steady state. A special case of our results is in good agreement with an earlier published work.     

A convolution-type semi-analytic DQ approach to transient response of rectangular plates

The convolution-type Gurtin variational principle is known as the only variational principle that is, from the mathematics point of view, totally equivalent to the initial value problem system. In this paper, the equation of motion of rectangular thin plates is first transformed to a new governing equation containing initial conditions by using a convolution method. A convolution-type semi-analytical DQ approach, which involves differential quadrature （DQ） approximation in the space domain and an analytical series expansion in the time domain, is proposed to obtain the transient response solution. This approach offers the same advantages as the Gurtin variational principle and, at the same time, is much simpler in calculation. Numerical results show that it is very accurate yet computationally efficient for the dynamic response of plates.     

Bursting-like motion induced by time-varying delay in an internet congestion control model

Time delay is an important parameter in the problem of internet congestion control. According to some researches, time delay is not always constant and can be viewed as a periodic function of time for some cases. In this work, an internet congestion control model is considered to study the time-varying delay induced bursting-like motion, which consists of a rapid oscillation burst and quiescent steady state. Then, for the system with periodic delay of small amplitude and low frequency, the method of multiple scales is employed to obtain the amplitude of the oscillation. Based on the expression of the asymptotic solution, it can be found that the relative length of the steady state increases with amplitude of the variation of time delay and decreases with frequency of the variation of time delay. Finally, an effective method to control the bursting-like motion is proposed by introducing a periodic gain parameter with appropriate amplitude. Theoretical results are in agreement with that from numerical method.     

Effect of thermal diffusion and electrostatic force on evolution of wind-blown sand flow

A theoretical model is suggested to mathematically describe the effect of thermal diffusion from a sand-bed on evolution of a wind-blown sand flow.An upward wind field is engendered by the thermal diffusion and the coupling interaction among the horizontal and upward wind flow,saltating grains,and a kind of electrostatic force exerted on the grains are considered in this theoretical model.The numerical results show that the effect of the thermal diffusion on the evolution process of wind-blown grain flow is quite obvious and very similar to the effect of the electrostatic force on the evolution.Not only the time for the entire system to reach a steady state(called the duration time),the transport rate of grains,the mass-flux profiles and the trajectory of saltating grains are affected by the thermal diffusion and the electrostatic force exerted on saltating grains, but also the wind profiles and the temperature profiles at the steady state are affected by the wind-blown sand flow.     

ON THE STABILITY OF A FUNCTIONALLY GRADED RECTANGULAR MICRO-PLATE SUBJECTED TO HYDROSTATIC AND NONLINEAR ELECTROSTATIC PRESSURES

This article studies the stability of a functionally graded clamped-clamped micro-plate subjected to hydrostatic and electrostatic pressures. Equilibrium positions of the micro-plate are determined and shown in the state control space. To study the stability of the equilibrium positions, the motion trajectories are given for different initial conditions in the phase plane. Effects of the electrostatic and hydrostatic pressure changes on the deflection and stability of the micro-plate for some sample value of k are studied and values of the applied voltage and hydrostatic pressure leading system to unstable conditions by undergoing a saddle node and homoclinic bifurcations are determined.     

Free vibrations of beam-mass-spring systems: analytical analysis with numerical confirmation

Free vibrations of a beam-mass-spring system with different boundary conditions are analyzed both analytically and numerically.In the analytical analysis,the system is divided into three subsystems and the effects of the spring and the point mass are considered as internal boundary conditions between any two neighboring subsystems.The partial differential equations governing the motion of the subsystems and internal boundary conditions are then solved using the method of separation of variables.In the numerical analysis,the whole system is considered as a single system and the effects of the spring and point mass are introduced using the Dirac delta function.The Galerkin method is then employed to discretize the equation of motion and the resulting set of ordinary differential equations are solved via eigenvalue analysis.Analytical and numerical results are shown to be in very good agreement.     

On decentralized stabilization of linear large scale systems with symmetric circulant structure

The decentralized stabilization of continuous and discrete linear large scale systems with symmetric circulant structure was studied. A few sufficient conditions on decentralized stabilization of such systems were proposed. For the continuous systems, by introducing a concept called the magnitude of interconnected structure, a very important property that the decentralized stabilization of such systems is fully determined by the structure of each isolated subsystem that is obtained when the magnitude of interconnected structure of the overall system is given. So the decentralized stabilization of such systems can be got by only appropriately designing or modifying the structure of each isolated subsystem, no matter how complicated the interconnected structure of the overall system is. A algorithm for obtaining decentralized state feedback to stabilize the overall system is given. The discrete systems were also discussed. The results show that there is a great dfference on decentralized stabilization between continuous case and discrete case.     

Optimal feedback gains of a delayed proportional-derivative (PD) control for balancing an inverted pendulum

In the dynamics analysis and synthesis of a con-trolled system, it is important to know for what feedback gains can the controlled system decay to the demanded steady state as fast as possible. This article presents a sys-tematic method for finding the optimal feedback gains by taking the stability of an inverted pendulum system with a delayed proportional-derivative controller as an example. First, the condition for the existence and uniqueness of the stable region in the gain plane is obtained by using the D-subdivision method and the method of stability switch. Then the same procedure is used repeatedly to shrink the stable region by decreasing the real part of the rightmost charac-teristic root. Finally, the optimal feedback gains within the stable region that minimizes the real part of the rightmost root are expressed by an explicit formula. With the optimal feedback gains, the controlled inverted pendulum decays to its trivial equilibrium at the fastest speed when the initial val-ues around the origin are fixed. The main results are checked by numerical simulation.     

ANALYSIS OF THE CHAOTIC MOTION FOR A ROTOR SYSTEM WITH A TRANSVERSE CRACK

The chaotic motion of an elastic shaft-disk rotor system including the geometricnonlinearity of the shaft with a transverse crack is investigated.The critical condition for thesystem to enter chaotic state is given by the Melnikov method.Meanwhile,whether the chaosoccurs is determined by the Poincaré map,phase portrait and time-displacement history diagram.     

HOPF BIFURCATION OF A NONLINEAR RESTRAINED CURVED PIPE CONVEYING FLUID BY DIFFERENTIAL QUADRATURE METHOD

This paper proposes a new method for investigating the Hopf bifurcation of a curved pipe conveying fluid with nonlinear spring support. The nonlinear equation of motion is derived by forces equilibrium on microelement of the system under consideration. The spatial coordinate of the system is discretized by the differential quadrature method and then the dynamic equation is solved by the Newton-Raphson method. The numerical solutions show that the inner fluid velocity of the Hopf bifurcation point of the curved pipe varies with different values of the parameter,nonlinear spring stiffness. Based on this, the cycle and divergent motions are both found to exist at specific fluid flow velocities with a given value of the nonlinear spring stiffness. The results are useful for further study of the nonlinear dynamic mechanism of the curved fluid conveying pipe.     

Nonlinear vibration of slightly curved pipe with conveying pulsating fluid

The nonlinear governing motion equation of slightly curved pipe with conveying pulsating fluid is set up by Hamilton’s principle. The motion equation is discretized into a set of low dimensional system of nonlinear ordinary differential equations by the Galerkin method. Linear analysis of system is performed upon this set of equations. The effect of amplitude of initial deflection and flow velocity on linear dynamic of system is analyzed. Curves of the resonance responses about \(\varOmega \approx {\omega _\mathrm{{1}}}\) and \(\varOmega \approx \mathrm{{2}}{\omega _\mathrm{{1}}}\) are performed by means of the pseudo-arclength continuation technique. The global nonlinear dynamic of system is analyzed by establishing the bifurcation diagrams. The dynamical behaviors are identified by the phase diagram and Poincare maps. The periodic motion, chaotic motion and quasi-periodic motion are found in this system.     

Nonlinear dynamics of a fluid-conveying curved pipe subjected to motion-limiting constraints and a harmonic excitation

This paper investigates the dynamical behaviour of a fluid-conveying curved pipe subjected to motion-limiting constraints and a harmonic excitation. Based on a Newtonian method, the in-plane equation of motion of this curved pipe is derived. Then a set of discrete equations in spatial space obtained by the differential quadrature method (DQM) is solved numerically. Emphasis is placed on the possible dynamical behaviour of the curved pipe conveying fluid. The numerical results show that the pipe without motion-limiting constraints but with a harmonic force behaves as an ordinary linear system. If, however, the pipe is subjected to cubic motion-limiting constraints, nonlinear dynamic phenomena of the system will occur. Calculations of bifurcation diagrams, phase-plane portraits, time responses, power spectrum diagrams, and Poincaré maps of the oscillations clearly demonstrate the existence of chaotic and quasiperiodic motions. Moreover, it is shown that the route to chaos is via a sequence of period-doubling bifurcations.     

输液管的非线性振动、分叉与混沌——现状与展望

围绕输液管的非线性振动、分叉与混沌问题，对近几年来的主要研究工作加以综述并提出预测，其中包括运动方程中非线性项的归纳与讨论、非线性动力分析的一些现代计算方法、定常流速下输液管的分叉与混沌行为、振荡流速下输液管的参数共振以及今后值得进一步研究的某些问题．     

Stability analysis of viscoelastic curved pipes conveying fluid

Based on the Hamilton' s principle for elastic systems of changing mass, a differential equation of motion for viscoelastic curved pipes conveying fluid was derived using variational method, and the complex characteristic equation for the viscoelastic circular pipe conveying fluid was obtained by normalized power series method. The effects of dimensionless delay time on the variation relationship between dimensionless complex frequency of the clamped-clamped viscoelastic circular pipe conveying fluid with the Kelvin-Voigt model and dimensionless flow velocity were analyzed. For greater dimensionless delay time, the behavior of the viscoelastic pipe is that the first, second and third mode does not couple, while the pipe behaves divergent instability in the first and second order mode, then single-mode flutter takes place in the first order mode.     

粘弹性地基上粘弹性输流管道的稳定性分析

从Winkler假设和单轴线性粘弹性本构方程出发，推导了Kelvin-Voigt粘弹性地基上三参量固体模型输流管道的运动微分方程，采用改进的有限差分法，分析了管道和地基的粘弹性参数对输流管道无量纲复频率和无量纲流速之间的变化关系的影响。     

A parametric study on nonlinear flow-induced dynamics of a fluid-conveying cantilevered pipe in post-flutter region from macro to micro scale

A mathematical formulation is proposed to investigate the nonlinear flow-induced dynamic characteristics of a cantilevered pipe conveying fluid from macro to micro scale. The model is developed by using the extended Hamilton's principle in conjunction with the inextensibility condition and laminar and turbulent flow profiles as well as modified couple stress theory. The current model is capable of recovering the classical model of cantilevered pipe conveying fluid by neglecting the couple stress effect. The governing equation of motion is presented in dimensionless form in a convenient and usable manner. To solve the problem at hand, the integro-partial-differential equation of motion is discretized into a set of ordinary differential equations via Galerkin method. Afterward, a Runge–Kutta's finite difference scheme is employed to evaluate the nonlinear dynamic response of the cantilevered pipe conveying fluid. A parametric study is carried out to examine the influences of mass parameter and dimensionless mean flow velocity on the nonlinear dynamic characteristics of the cantilevered pipe conveying fluid in post-flutter region. The role of size-dependency in the nonlinear behavior of pipe is explored by converting the new set of dimensionless parameters into the conventional one. Eventually, some convergence studies are performed to indicate the reliability of present results.     

Nonlinear dynamics of functionally graded pipes conveying hot fluid

For improved stability of fluid-conveying pipes operating under the thermal environment, functionally graded materials (FGMs) are recommended in a few recent studies. Besides this advantage, the nonlinear dynamics of fluid-conveying FG pipes is an important concern for their engineering applications. The present study is carried out in this direction, where the nonlinear dynamics of a vertical FG pipe conveying hot fluid is studied thoroughly. The FG pipe is considered with pinned ends while the internal hot fluid flows with the steady or pulsatile flow velocity. Based on the Euler–Bernoulli beam theory and the plug-flow model, the nonlinear governing equation of motion of the fluid-conveying FG pipe is derived in the form of the nonlinear integro-partial-differential equation that is subsequently reduced as the nonlinear temporal differential equation using Galerkin method. The solutions in the time or frequency domain are obtained by implementing the adaptive Runge–Kutta method or harmonic balance method. First, the divergence characteristics of the FG pipe are investigated and it is found that buckling of the FG pipe arises mainly because of temperature of the internal fluid. Next, the dynamic characteristics of the FG pipe corresponding to its pre- and post-buckled equilibrium states are studied. In the pre-buckled equilibrium state, higher-order parametric resonances are observed in addition to the principal primary and secondary parametric resonances, and thus the usual shape of the parametric instability region deviates. However, in the post-buckled equilibrium state of the FG pipe, its chaotic oscillations may arise through the intermittent transition route, cyclic-fold bifurcation, period-doubling bifurcation and subcritical bifurcation. The overall study reveals complex dynamics of the FG pipe with respect to some system parameters like temperature of fluid, material properties of FGM and fluid flow velocity.     

粘弹性桩的混沌运动分析

研究了在轴向载荷和周期性横向载荷共同作用下非线性粘弹性嵌岩桩的混沌运动情况。假定桩和土体分别满足Leaderman非线性粘弹性和线性粘弹性本构关系，得到的运动方程为非线性偏微分．积分方程；利用Galerkin方法将方程简化为非线性常微分一积分方程，同时利用非线性动力系统中的数值方法，进行了数值计算，得到了不同载荷参数、几何参数、材料参数时粘弹性桩发生周期运动、多周期运动及混沌运动的时程曲线、相图、功率谱、Poincare截面图，同时得到了挠度-载荷、挠度-几何参数、挠度-材料参数等分叉图，考察了各种参数的影响。数值结果表明非线性粘弹性桩在一定的条件下可以通过倍周期分叉的方式进入混沌运动状态，且桩的载荷参数、几何参数、材料参数对其运动状态有较大的影响。     

参-强激励联合作用下输流管的分岔和混沌行为研究

研究输送脉动流的两端固定输流管道在其基础简谐运动激励下的分岔和混沌行为,考虑管道变形的几何非线性和管道材料的非线性因素,推导了系统的非线性运动方程,并应用Galerkin方法对其进行了离散化处理。通过采用数值模拟方法,对系统的运动响应进行仿真,重点探讨了流体平均流速、流速脉动振幅以及基础简谐运动激励振幅对系统动态特性的影响。结果表明,系统在不同的参数下会发生围绕不同平衡点的周期和混沌等运动,并在系统中发现了两条通向混沌运动的途径:倍周期分岔和阵发混沌运动。