The small length scale effect for a non-local cantilever beam: a paradox solved

Non-local continuum mechanics allows one to account for the small length scale effect that becomes significant when dealing with microstructures or nanostructures. This paper presents some simplified non-local elastic beam models, for the bending analyses of small scale rods. Integral-type or gradient non-local models abandon the classical assumption of locality, and admit that stress depends not only on the strain value at that point but also on the strain values of all points on the body. There is a paradox still unresolved at this stage: some bending solutions of integral-based non-local elastic beams have been found to be identical to the classical (local) solution, i.e.?the small scale effect is not present at all. One example is the Euler-Bernoulli cantilever nanobeam model with a point load which has application in microelectromechanical systems and nanoelectromechanical systems as an actuator. In this paper, it will be shown that this paradox may be overcome with a gradient elastic model as well as an integral non-local elastic model that is based on combining the local and the non-local curvatures in the constitutive elastic relation. The latter model comprises the classical gradient model and Eringen's integral model, and its application produces small length scale terms in the non-local elastic cantilever beam solution.     

On the stability of a Timoshenko beam on an elastic foundation under a moving spring-mass system

Summary The stability problem of densely distributed oscillators moving along a Timoshenko beam on an elastic foundation is considered. The forward speed of the moving subsystem is assumed to be constant. The friction at the contact line between the beam and the oscillator set is neglected. A qualitatively new instability region is found. It is pointed out that the critical velocity for some system parameters takes smaller values than the velocity of shear waves or the velocity of longitudinal waves.With 8 Figures     

Vibration of double-walled carbon nanotubes coupled by temperature-dependent medium under a moving nanoparticle with multi physical fields

A numerical model on nonlinear vibration of double-walled carbon nanotubes (DWCNTs) subjected to a moving nanoparticle and multi physical fields is proposed. DWCNTs are considered with the kinematic assumption of Euler–Bernoulli beam theory. The surrounding elastic substrate is simulated as Pasternak foundation, which is assumed to be temperature-dependent. Hamilton's principle, incremental harmonic balanced method, Galerkin, and time integration method with direct iteration are employed to establish the equations of motion of zigzag DWCNTs. The study reveals that for the weak van der Waals forces, DWCNTs have the positive and the negative deflections as if it vibrates under a moving nanoparticle.     

Numerical analysis for dynamic contact between high‐speed wheel and elastic beam with Coulomb friction

For numerical analysis of the dynamic contact between a high‐speed wheel and an elastic beam, the equation of motion of each body is time integrated by a simple ODE solution technique and frictional contact conditions are imposed by the augmented Lagrange multiplier method using the contact errors defined in this work. For the stability of the numerical solution, the velocity and acceleration contact conditions as well as the displacement contact condition are imposed with special consideration for the high‐velocity contact point moving on the deformed beam. Especially, it is shown that the Coriolis and centripetal accelerations of the contact point moving rapidly on the deformed beam play crucial roles for the stability of the solution. It is also shown that, for a wheel rolling on a beam with friction, the acceleration constraint in the tangential direction is important for the stability of the solution. Copyright © 2008 John Wiley & Sons, Ltd.     

A simple finite element for beams on elastic foundations

A simple "routine" beam on elastic foundation finite element using a polynomial displacement function has been developed which yields acceptably accurate deflection, shear and bending moment values for prismatic or non-prismatic beams of elastic material resting on foundations with varying or nonlinear subgrade reactions. Limited extension of the formulation to an "exact" finite element using the exact displacement function of a beam on elastic foundation has also been carried out. The subgrade is represented by a non-homogeneous solid medium to include nonlinear parameters if required. The iterative solution is extended to cases where the beam may uplift because the foundation is a no tension material. The model is also suitable for calculating the elastic deflections, membrane. and bending stress resultants for axisymmetrically loaded variable thickness shells of revolution. A computer program called FEBEF [finite element: beam on elastic foundation] incorporating the routine finite element has been prepared for the solution of beams on elastic foundations and axi symmetrically loaded shells of revolution.     

Stability analysis of a capacitive micro-resonator with embedded pre-strained SMA wires

This paper presents a study on the effects of the SMA wires’ characteristics on tuning the stability of a capacitive micro-resonator. In the proposed model, pre-strained SMA wires have been embedded in a double clamped resonant microbeam which is actuated electrostatically. The governing equations of the system have been introduced and then an eigen-value problem has been adopted to investigate stability. Galerkin-based numerical methods have been used to solve the governing equation of motion for obtaining the motion trajectories of the beam. The effects of the number of SMA wires, their diameter, pre-strain and temperature on the pull-in instability and frequency response of the resonator have been shown. Critical values of recovery stress and SMA temperature for avoiding instability, with and without applying DC voltage have been obtained. The results have shown that by changing each of the SMA parameters, one can reach a needed magnitude of recovery stress as well as transmitted longitudinal force to the host beam, and consequently control and tune the stability and resonance frequency of the micro-resonator.

     

梁结构刚度动态非概率可靠性分析

 为了定量分析在疲劳载荷作用下梁在不同寿命期内刚度的可靠性,建立梁结构物理性能退化的精确公式就十分重要.依据疲劳载荷造成的累积损伤对材料极限应力的影响,基于材料剩余强度模型,利用材料强度与弹性模量之间的关系,推导出结构弹性模量的退化表达式,并在此基础上,提出梁弹性模量退化系数的递推表达式,推导出圆截面梁剩余抗弯刚度的表达式.在对结构可靠性分析时,概率可靠性模型和模糊可靠性模型对于原始数据信息要求较高.为了充分利用结构的不确定性信息弥补原始数据的不足,将梁的初始弹性模量及所受的疲劳载荷等看作区间变量,利用区间模型建立基于刚度退化的梁刚度动态非概率可靠性模型.最后,结合工程实例的计算表明了该方法对梁的刚度退化分析及其刚度动态可靠性分析是可行、有效和合理的.     

Excitation of Surface Plasma Waves in Microwave Radiation Sources by Electron Beam with Allowance for Thermal Velocity Spread

Based on the hydrodynamic model, in the linear approximation, the problem of excitation by an electron beam of surface waves in electrodynamic systems of plasma relativistic microwave electronics has been considered with consideration of electron velocity spread. Complex instability increments have been determined for complex parameters of the beam-plasma system. Two instability regimes differing in the dynamics of beam plasma oscillations during instability development have been observed: Compton (singleparticle) and Raman (collective) instabilities. The role of thermal effects in an electron beam and its density in the formation of a particular mode of beam instability has been analyzed.     

Pulsating fluid induced dynamic instability of visco-double-walled carbon nano-tubes based on sinusoidal strain gradient theory using DQM and Bolotin method

This research deals with the dynamic instability analysis of double-walled carbon nanotubes (DWCNTs) conveying pulsating fluid under 2D magnetic fields based on the sinusoidal shear deformation beam theory (SSDBT). In order to present a realistic model, the material properties of DWCNTs are assumed viscoelastic using Kelvin–Voigt model. Considering the strain gradient theory for small scale effects, a new formulation of the SSDBT is developed through the Gurtin–Murdoch elasticity theory in which the effects of surface stress are incorporated. The surrounding elastic medium is described by a visco-Pasternak foundation model, which accounts for normal, transverse shear and damping loads. The van der Waals interactions between the adjacent walls of the nanotubes are taken into account. The size dependent motion equations and corresponding boundary conditions are derived based on the Hamilton’s principle. The differential quadrature method in conjunction with Bolotin method is applied for obtaining the dynamic instability region. The detailed parametric study is conducted, focusing on the combined effects of the nonlocal parameter, magnetic field, visco-Pasternak foundation, Knudsen number, surface stress and fluid velocity on the dynamic instability of DWCNTs. The results depict that the surface stress effects on the dynamic instability of visco-DWCNTs are very significant. Numerical results of the present study are compared with available exact solutions in the literature. The results presented in this paper would be helpful in design and manufacturing of nano/micro mechanical systems in advanced biomechanics applications with magnetic field as a parametric controller.     

Instability and vibration of a vehicle moving on curved beams with different boundary conditions

The instability and vibration of a concentrated mass moving along the curved beam is investigated. In most literature, the moving mass model is approximated by the moving load model. The semi-analytical method for this moving mass problem is presented here. Comparison between the two models is made. The mechanism of instability of the vehicle separating from a curved bridge is studied. Moreover, the effects of several parameters and boundary conditions on the vibration and instability are investigated.     

A viscoelastic analysis of ground-movement due to an advancing coal face

Summary While and elastic analysis of subsidence problems gives results in accord with practical data for essentially static situations, it fails when applied to the problem of a moving coal face. This failure appears to be due to a time-dependent phenomenon associated with the material in which the excavation is moving. This time effect may be thought of as a lag (that is, the ground does not immediately realize the full extent of the mining disturbance). It is assumed therefore that the ground behaves as though it were a viscoelastic material, exhibiting both initial and delayed elastic states, and it is shown that by a suitable choice of parameters the main features of the practical results can be reproduced. It is also shown that for the model used here the difficulties are due to the large number of parameters involved rather than to any theoretical consideration.     

Stochastic analysis of the critical velocity of an axially moving cracked elastic plate

In this study, a probabilistic analysis of the critical velocity for an axially moving cracked elastic and isotropic plate is presented. Axially moving materials are commonly used in modelling of manufacturing processes, like paper making and plastic forming. In such systems, the most serious threats to runnability are instability and material fracture, and finding the critical value of velocity is essential for efficiency. In this paper, a formula for the critical velocity is derived under constraints for the probabilities of instability and fracture. The significance of randomness in different model parameters is investigated for parameter ranges typical of paper material and paper machines. The results suggest that the most significant factors are variations in the crack length and tension magnitude.     

钢筋混凝土梯形截面桁粱桥侧倾稳定性分析

根据李国豪教授的桁梁桥挠曲扭转理论，针对钢筋混凝土梯形截面桁梁桥的侧倾稳定性进行了研究。基于结构体系的变分原理，本文导出了梯形截面桁梁桥侧倾失稳微分方程及相应的边界条件，通过采用伽辽金法得到了相应的侧倾失稳临界荷载ｑ（ｃｒ），文末给出了一个实际工程算例。     

Stability conditions for non-conservative dynamical systems

The present paper treats dynamic instability problems of non-conservative elastic systems. Starting from general equations of motion, the equations of the perturbed motion are derived. The boundedness of the perturbed motions is studied and sufficient conditions for instability and a necessary condition for stability are deduced. These conditions may determine the instability of non-conservative systems and they are expressed in terms of the properties of generalized tangent damping and stiffness matrices of the systems. Thus, they can easily be incorporated with finite element computations of arbitrary structures.     

弹性地基上受切向力作用梁稳定性研究

用广义微分求积法(GDQR)分析了弹性地基上复杂弹性支承条件下受切向力作用梁的稳定性问题。基于弹性支承梁的运动微分方程及边界条件,采用GDQR进行离散化,获得由动力方程组及边界条件合成的特征值矩阵方程。通过对相应特征值方程的具体分析,讨论了弹性地基模量、剪切系数、复杂边界条件对临界载荷的影响,研究了一端固定约束、另一端弹性约束梁弹性失稳区域随弹性地基模量和支承弹簧刚度变化的情况,得到了一些有益的结论。结果表明：GDQR能很好地解决此类系统的稳定性问题。     

On reliability of systems with moving material subjected to fracture and instability

The reliability of systems with moving cracked elastic and isotropic material is considered. The material is modeled as a moving plate which continually has a crack on the edge. The plate is subjected to homogeneous tension acting in the traveling direction and the tension varies temporally around a constant value, the set tension. The tension and the length of the crack are modeled by an Ornstein–Uhlenbeck process and an exponential Ornstein–Uhlenbeck process, respectively. Failure is regarded as the state at which the plate becomes unstable or fractures (or both) and a lower bound for the reliability of the system is derived. Considering reliability of the system leads to first passage time problems and, in solving them, a known explicit result for the first passage time of an Ornstein–Uhlenbeck process to a constant boundary is exploited. A change in the set tension has opposite effects on the probabilities of instability and fracture, and a safe range of set tension is studied. Numerical examples are computed for material and machine parameters typical of paper and printing presses. The results suggest that tension variations may significantly affect the pressroom runnability.     

A parameter to improve stability for a family of dissipative integration methods

There is a family of integration methods which has unconditional stability for linear elastic and stiffness softening-type systems; however, it becomes conditionally stable for stiffness hardening-type systems. Consequently, its applications are inconvenient or limited due to the conditional stability in stiffness hardening-type systems. This drawback can be overcome by introducing a free parameter into its formulation. The numerical properties of this family method are almost unaffected by this free parameter except that the stability property is improved. Thus, the method’s unconditional stability is successfully extended into stiffness hardening-type systems in addition to linear elastic and stiffness softening-type systems.     

Vibration analysis and identification of breathing cracks in beams subjected to single or multiple moving mass using online sequential extreme learning machine

In this paper, a novel approach was presented to vibration analysis and identification of breathing cracks in Timoshenko beam under single or multiple moving mass. Dynamic strain energies (DSEs) and translational accelerations in beam structures under moving mass were used as forward problem and application of an emergent learning algorithm called the online sequential extreme learning machine algorithm as inverse problem to predict crack depths and locations. To demonstrate the potential of the proposed vibration analysis over existing ones, two validation studies have been done. To evaluate the proposed method to identify breathing cracks, two examples, namely, clamped–clamped beam and two span continuous beams have been studied. Also, the effect of the discrepancy in stiffness between the finite-element model and the actual tested dynamic system has been investigated. Another examination has been performed in which moving mass with different speeds are utilized. Also, the effect of multi mass passing through the beam has been studied. The obtained results indicated that the proposed method could identify the breathing cracks existence and severity in the beam under moving mass using DSE and accelerations, which may be noisy or noise free.